From 70d4d3155e71c4f95e3fdebc97c9cfc1492e93f1 Mon Sep 17 00:00:00 2001 From: Parker Ewen Date: Mon, 8 Jul 2024 22:05:17 -0400 Subject: [PATCH] Updated scripts Integrated simplified B matrix Removed config.py All scripts updated 6D example script added Refactored grid container Refactored code to make initialization faster Refactored marginalize function in scripts Fixed bug in 6D example Updated scripts Updated README Assets added Update README.md Update README.md --- .gitignore | 5 + README.md | 133 ++++---- assets/2D_example.gif | Bin 0 -> 630584 bytes assets/3D_example.gif | Bin 0 -> 672653 bytes assets/system_diagram.pdf | Bin 0 -> 136432 bytes pyproject.toml | 25 -- scripts/1D_Lp_error.py | 253 ++++++++++++++ scripts/1D_example.py | 104 ++++++ scripts/2D_example.py | 216 ++++++++++++ scripts/3D_example.py | 283 ++++++++++++++++ scripts/4D_example.py | 251 ++++++++++++++ scripts/6D_example.py | 403 +++++++++++++++++++++++ scripts/{test.py => fit_sparse_grid.py} | 25 +- scripts/monte_carlo_2D.py | 65 ++++ scripts/monte_carlo_4D.py | 194 +++++++++++ scripts/monte_carlo_6D.py | 271 +++++++++++++++ setup.py | 15 +- {src/sparselib => sparselib}/__init__.py | 1 + sparselib/gridContainer.py | 37 +++ sparselib/specgal.py | 112 +++++++ sparselib/spgrid.py | 224 +++++++++++++ src/sparselib/gridContainer.py | 9 - src/sparselib/spgrid.py | 227 ------------- 23 files changed, 2527 insertions(+), 326 deletions(-) create mode 100644 .gitignore create mode 100644 assets/2D_example.gif create mode 100644 assets/3D_example.gif create mode 100644 assets/system_diagram.pdf delete mode 100644 pyproject.toml create mode 100644 scripts/1D_Lp_error.py create mode 100644 scripts/1D_example.py create mode 100644 scripts/2D_example.py create mode 100644 scripts/3D_example.py create mode 100644 scripts/4D_example.py create mode 100644 scripts/6D_example.py rename scripts/{test.py => fit_sparse_grid.py} (50%) create mode 100644 scripts/monte_carlo_2D.py create mode 100644 scripts/monte_carlo_4D.py create mode 100644 scripts/monte_carlo_6D.py rename {src/sparselib => sparselib}/__init__.py (65%) create mode 100644 sparselib/gridContainer.py create mode 100644 sparselib/specgal.py create mode 100644 sparselib/spgrid.py delete mode 100644 src/sparselib/gridContainer.py delete mode 100644 src/sparselib/spgrid.py diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..335f81e --- /dev/null +++ b/.gitignore @@ -0,0 +1,5 @@ +*.npy +sparselib.egg-info/ +build/ +*.png +debug/ \ No newline at end of file diff --git a/README.md b/README.md index 011565b..71db253 100644 --- a/README.md +++ b/README.md @@ -1,61 +1,72 @@ -Funcs: - - Variables: - - dynamics: list(function) - - uncertainty: function - -Domain: - - Variables - - max_level: (int) - - domain: (Dx2 array) - - dim: int - -SparseGrid: - - Methods: - - init: - - Inputs: domain, max level - - Outputs: N/A - - fit: - - Inputs: func - - Outputs: N/A - - eval: - - Inputs: coords (NxD array) - - Outputs: vals (N array) - -GridContainer: - - Methods: - - fit: - - Inputs: Funcs class - - Output: N/A - -SolverParams: - - max_level - - dimension - - domain - - dynamics function(s) - - uncertainty function - - funcs: [uncertainty, dynamics] - -SpectralDiscretization: - - Methods: - - init: - - Inputs: SolverParams - - Outputs: N/A - - solve: - - Inputs: time (float) - coords (NxD array) - - Output: vals (NxD array) - - -solverParams --> SpectralDiscretization -SpectralDiscretization(): - 1) Constructs GridContainer - 2) Calls gridContainer.fit() -SpectralDiscretization.solve(t, xs): - 1) Runs propagation script - - - -FIXES: - - Fix issue with basis_der always multiplying 1j * k[d] for each - dim even though der is not over each dim - - Similar fix for basis function \ No newline at end of file +# Uncertainty Propagation Using Half-Densities Via Sparse Spectral Discretization on Compact Manifolds +Parker Ewen, Lucas Lymburner, Henry Jacobs, Ram Vasudevan
+| [Webpage](https://roahmlab.github.io/sparselib_page/) | [Full Paper](https://repo-sam.inria.fr/fungraph/3d-gaussian-splatting/3d_gaussian_splatting_high.pdf)
+

+ +

+ +This repository contains the *sparselib* implementation associated with the paper "Uncertainty Propagation Using Half-Densities Via Sparse Spectral Discretization on Compact Manifolds". + +## Setup +This package can be installed using the pip package installer: + +```shell +git clone https://github.com/ParkerEwen5441/sparselib.git +cd sparselib +pip install . +``` + +## Overview + +The codebase has 4 main components: +- A grid object used for computing the hyperbolic Fourier coefficients +- A grid container for storing all grid objects used to approximate the uncertainty and vector fields +- A spectral scheme for computing solutions to the linear advection equation + +Any of these components may be modified to suit the user's particular implementation. +Algorithms used in the component implementations are constructive so as to alleviate the curse-of-dimensionality. + + +## Running + +To run the an example script for the 1D example seen in the paper, simply use + +```shell +cd scripts +python 1D_example.py +``` + +Additionally, other scripts are available for the 2D, 3D, and 6D cases. +To run the 2D and 3D cases, there is an additional dependency to plot the uncertainty over the torus: + +```shell +pip install s3dlib==1.0.0 +``` + +## Abstract +Uncertainty propagation is a critical component in various applications such as stochastic optimal control, optimal transport, probabilistic inference, and filtering. +This paper frames uncertainty propagation of a dynamical system with compact support thru the lens of advecting a probability density through a vector field via the transport and continuum equations. +These equations exhibit a number of conservation laws. +For example, scalar multiplication is conserved by the transport equation, while positivity of probabilities is conserved by the continuum equation. +Certain discretization techniques, such as particle based methods, conserve these properties, but converge slower than spectral discretization methods on smooth data. +Standard spectral discretization methods, on the other hand, do not conserve the invariants of the transport equation and the continuum equation. +This article constructs a novel sparse spectral discretization technique that conserves these important invariants, namely positivity of probability measure, while simultaneously preserving spectral convergence rates. +The performance of this proposed method is illustrated on several numerical experiments and the quantitative convergence rate of the proposed approach is derived. + +If you use this library, please cite the following work. + +
+
+

BibTeX

+
@Article{kerbl3Dgaussians,
+      author       = {Ewen, Parker and Lymburner, Lucas and Jacobs, Henry and Vasudevan, Ram},
+      title        = {Uncertainty Propagation Using Half-Densities Via Sparse Spectral Discretization on Compact Manifolds},
+      journal      = {In Submission},
+      number       = {1},
+      volume       = {1},
+      month        = {September},
+      year         = {2024},
+      url          = {https://roahmlab.github.io/sparselib_page/}
+}
+
+
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zD_N>%6};DJ`nPQfOHJ;6o}ezSdgi)}ywv{ai%;j0du}aJdEYl%w%=#T(^9<7`xk@3 zCO0)$5A?xZ?P=S!g*w+)3~zhG)akmkvrJa!f{bYhn)@%%KAQu$hn;csFl8l_shO22 OT3HDOb8yEiEBzn5mm=c; literal 0 HcmV?d00001 diff --git a/pyproject.toml b/pyproject.toml deleted file mode 100644 index 22a3704..0000000 --- a/pyproject.toml +++ /dev/null @@ -1,25 +0,0 @@ -[build-system] -requires = ["setuptools>=45", "setuptools_scm>=6.2", "wheel"] -build-backend = "setuptools.build_meta" - -[project] -name = "sparselib" -version = "1.0.0" -authors = [ - { name="Parker Ewen", email="parker.ewen5441@gmail.com" }, -] -description = "Sparse spectral discretization for linear advection." -readme = "README.md" -requires-python = ">=3.10" -classifiers = [ - "Programming Language :: Python :: 3", - "License :: OSI Approved :: MIT License", - "Operating System :: OS Independent", -] -dependencies = [ - "argparse", - "pyyaml", - "numpy==1.24.2", - "matplotlib" -] - diff --git a/scripts/1D_Lp_error.py b/scripts/1D_Lp_error.py new file mode 100644 index 0000000..94399aa --- /dev/null +++ b/scripts/1D_Lp_error.py @@ -0,0 +1,253 @@ +import numpy as np +import matplotlib +from tqdm import tqdm +import matplotlib.pyplot as plt + +import sparselib + +def dynamics(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return -np.sin(2 * x) + +def uniform_dense(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + return np.ones_like(x[:,0]) * 1 / (2 * np.pi) + +def uniform_sparse(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + return np.sqrt(np.ones_like(x[:,0]) * 1 / (2 * np.pi)) + +def ground_truth(t, x): + return np.power(np.exp(2*t) * np.power(np.sin(x), 2) + \ + np.exp(-2*t) * np.power(np.cos(x), 2), -1) + +def normalize(ys, N): + return np.squeeze(ys / (2 * np.pi * np.sum(ys) / N)) + + +class SolverParamsDense(): + max_level: int = 6 + dim: int = 1 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [uniform_dense, dynamics] + + +class SolverParamsSparse(): + max_level: int = 6 + dim: int = 1 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [uniform_sparse, dynamics] + + +def Lp_error_vs_time(): + # Initialize solver parameters + paramsDense = SolverParamsDense() + paramsSparse = SolverParamsSparse() + + # Standard Galerkin method + specgalDense = sparselib.SpectralGalerkin(paramsDense) + + # Our sparse method + specgalSparse = sparselib.SpectralGalerkin(paramsSparse) + + # Evaluate results + N = 1000 + xs = np.linspace(paramsDense.domain[0], paramsDense.domain[1], N) + xs = np.expand_dims(xs, axis=1) + + # Compute the propagated uncertainty for our proposed sparse, half-density + # method, a standard Galerkin approach, and the ground-truth distribution. + interpDense = normalize(np.real(specgalDense.container.grids[0].eval(xs)), N) + interpSparse = normalize(np.power(np.real(specgalSparse.container.grids[0].eval(xs)), 2), N) + gt = normalize(ground_truth(0, xs), N) + + L1sparse = [] + L2sparse = [] + Linfsparse = [] + + L1dense = [] + L2dense = [] + Linfdense = [] + + L1sparse.append(np.sum(2 * np.pi * np.abs(interpSparse - gt) / N)) + L2sparse.append(np.sqrt(np.sum(2 * np.pi * np.power(interpSparse - gt, 2)) / N)) + Linfsparse.append(np.max(np.abs(interpSparse - gt))) + + L1dense.append(np.sum(2 * np.pi * np.abs(interpDense - gt) / N)) + L2dense.append(np.sqrt(np.sum(2 * np.pi * np.power(interpDense - gt, 2)) / N)) + Linfdense.append(np.max(np.abs(interpDense - gt))) + + total_time = 1.5 + M = 100 + t = 0 + dt = total_time / M + + print("Computing Lp errors ...") + pbar = tqdm(total=M) + for i in range(M): + t += dt + specgalSparse.solve(dt) + specgalDense.solve(dt) + + interpDense = normalize(np.real(specgalDense.container.grids[0].eval(xs)), N) + interpSparse = normalize(np.power(np.real(specgalSparse.container.grids[0].eval(xs)), 2), N) + gt = normalize(ground_truth(t, xs), N) + + L1sparse.append(np.sum(2 * np.pi * np.abs(interpSparse - gt) / N)) + L2sparse.append(np.sqrt(np.sum(2 * np.pi * np.power(interpSparse - gt, 2)) / N)) + Linfsparse.append(np.max(np.abs(interpSparse - gt))) + + L1dense.append(np.sum(2 * np.pi * np.abs(interpDense - gt) / N)) + L2dense.append(np.sqrt(np.sum(2 * np.pi * np.power(interpDense - gt, 2)) / N)) + Linfdense.append(np.max(np.abs(interpDense - gt))) + + pbar.update(1) + pbar.close() + + L1sparse = np.array(L1sparse) + L2sparse = np.array(L2sparse) + Linfsparse = np.array(Linfsparse) + + L1dense = np.array(L1dense) + L2dense = np.array(L2dense) + Linfdense = np.array(Linfdense) + + ts = np.linspace(0, total_time, M+1) + + matplotlib.rcParams.update({'font.size': 18}) + fig, ax = plt.subplots(figsize=(18, 6)) + ax.plot(ts, L1sparse, color='#05445e', linestyle='-', marker='^', markevery=10, markersize=10) + ax.plot(ts, L2sparse, color='#189ab4', linestyle='-', marker='v', markevery=10, markersize=10) + ax.plot(ts, Linfsparse, color='#75e6da', linestyle='-', marker='*', markevery=10, markersize=10) + ax.plot(ts, L1dense, color='#fc2e20', linestyle='--', marker='^', markevery=10, markersize=10) + ax.plot(ts, L2dense, color='#fd7f20', linestyle='--', marker='v', markevery=10, markersize=10) + ax.plot(ts, Linfdense, color='#fdb750', linestyle='--', marker='*', markevery=10, markersize=10) + ax.set_xlabel("Time [s]") + ax.set_ylabel("$L^p$-norm Error") + ax.set_xlim([0, total_time]) + plt.gca().legend(('$L^1$-norm (Ours)', + '$L^2$-norm (Ours)', + '$L^\\infty$-norm (Ours)', + '$L^1$-norm (Galerkin)', + '$L^2$-norm (Galerkin)', + '$L^\\infty$-norm (Galerkin)')) + fig.tight_layout() + plt.show() + +def Lp_Error_vs_num_bases(): + N = 1000 + xs = np.linspace(0, 2*np.pi, N) + xs = np.expand_dims(xs, axis=1) + + L1sparse = [] + L2sparse = [] + Linfsparse = [] + + L1dense = [] + L2dense = [] + Linfdense = [] + + num_bases_sparse = [] + num_bases_dense = [] + + t = 1.0 + + pbar = tqdm(total=12) + for i in range(1, 12): + # Initialize solver parameters + paramsDense = SolverParamsDense() + paramsSparse = SolverParamsSparse() + + paramsSparse.max_level = int(i) + paramsDense.max_level = int(i) + + # Standard Galerkin method + specgalDense = sparselib.SpectralGalerkin(paramsDense) + specgalDense.solve(t) + + # Our sparse method + specgalSparse = sparselib.SpectralGalerkin(paramsSparse) + specgalSparse.solve(t) + + interpDense = normalize(np.real(specgalDense.container.grids[0].eval(xs)), N) + interpSparse = normalize(np.power(np.real(specgalSparse.container.grids[0].eval(xs)), 2), N) + gt = normalize(ground_truth(t, xs), N) + + L1sparse.append(np.sum(2 * np.pi * np.abs(interpSparse - gt) / N)) + L2sparse.append(np.sqrt(np.sum(2 * np.pi * np.power(interpSparse - gt, 2)) / N)) + Linfsparse.append(np.max(np.abs(interpSparse - gt))) + + L1dense.append(np.sum(2 * np.pi * np.abs(interpDense - gt) / N)) + L2dense.append(np.sqrt(np.sum(2 * np.pi * np.power(interpDense - gt, 2)) / N)) + Linfdense.append(np.max(np.abs(interpDense - gt))) + + num_bases_sparse.append(specgalSparse.container.grids[0].N) + num_bases_dense.append(specgalDense.container.grids[0].N) + + pbar.update(1) + pbar.close() + + L1sparse = np.array(L1sparse) + L2sparse = np.array(L2sparse) + Linfsparse = np.array(Linfsparse) + + L1dense = np.array(L1dense) + L2dense = np.array(L2dense) + Linfdense = np.array(Linfdense) + + num_bases_sparse = np.array(num_bases_sparse) + num_bases_dense = np.array(num_bases_dense) + + matplotlib.rcParams.update({'font.size': 18}) + fig, ax = plt.subplots(figsize=(18, 6)) + ax.plot(num_bases_sparse, L1sparse, color='#05445e', linestyle='-', marker='^', markersize=10) + ax.plot(num_bases_sparse, L2sparse, color='#189ab4', linestyle='-', marker='v', markersize=10) + ax.plot(num_bases_sparse, Linfsparse, color='#75e6da', linestyle='-', marker='*', markersize=10) + ax.plot(num_bases_dense, L1dense, color='#fc2e20', linestyle='--', marker='^', markersize=10) + ax.plot(num_bases_dense, L2dense, color='#fd7f20', linestyle='--', marker='v', markersize=10) + ax.plot(num_bases_dense, Linfdense, color='#fdb750', linestyle='--', marker='*', markersize=10) + ax.set_xlabel("Num. of Basis Functions") + ax.set_ylabel("$L^p$-norm Error") + ax.set_xscale('log') + ax.set_yscale('log') + ax.set_xlim([1, np.max(num_bases_dense)]) + plt.gca().legend(('$L^1$-norm (Ours)', + '$L^2$-norm (Ours)', + '$L^\\infty$-norm (Ours)', + '$L^1$-norm (Galerkin)', + '$L^2$-norm (Galerkin)', + '$L^\\infty$-norm (Galerkin)')) + fig.tight_layout() + plt.show() + + +# Lp_error_vs_time() +Lp_Error_vs_num_bases() \ No newline at end of file diff --git a/scripts/1D_example.py b/scripts/1D_example.py new file mode 100644 index 0000000..affbb4a --- /dev/null +++ b/scripts/1D_example.py @@ -0,0 +1,104 @@ +import matplotlib +import numpy as np +import matplotlib.pyplot as plt + +import sparselib + +def dynamics(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return -np.sin(2 * x) + +def uniform_dense(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + return np.ones_like(x[:,0]) * 1 / (2 * np.pi) + +def uniform_sparse(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + return np.sqrt(np.ones_like(x[:,0]) * 1 / (2 * np.pi)) + +def ground_truth(t, x): + return np.power(np.exp(2*t) * np.power(np.sin(x), 2) + \ + np.exp(-2*t) * np.power(np.cos(x), 2), -1) + +def normalize(ys, N): + return np.squeeze(ys / (2 * np.pi * np.sum(ys) / N)) + + +class SolverParamsDense(): + max_level: int = 6 + dim: int = 1 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [uniform_dense, dynamics] + + +class SolverParamsSparse(): + max_level: int = 6 + dim: int = 1 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [uniform_sparse, dynamics] + + +# Initialize solver parameters +paramsDense = SolverParamsDense() +paramsSparse = SolverParamsSparse() + +# Standard Galerkin method +specgalDense = sparselib.SpectralGalerkin(paramsDense) +specgalDense.solve(1.5) + +# Our sparse method +specgalSparse = sparselib.SpectralGalerkin(paramsSparse) +specgalSparse.solve(1.5) + +# Evaluate results +N = 1000 +xs = np.linspace(paramsDense.domain[0], paramsDense.domain[1], N) +xs = np.expand_dims(xs, axis=1) + +# Compute the propagated uncertainty for our proposed sparse, half-density +# method, a standard Galerkin approach, and the ground-truth distribution. +interpDense = normalize(np.real(specgalDense.container.grids[0].eval(xs)), N) +interpSparse = normalize(np.power(np.real(specgalSparse.container.grids[0].eval(xs)), 2), N) +gt = normalize(ground_truth(1.5, xs), N) + +matplotlib.rcParams.update({'font.size': 18}) +fig, ax = plt.subplots(figsize=(18, 6)) +ax.plot(xs, interpSparse, color='#189ab4') +ax.plot(xs, interpDense, color='#fd7f20') +ax.plot(xs, gt, color='green') +ax.set_xlabel("Domain") +ax.set_ylabel("Probability Density") +ax.set_xlim([0, 2*np.pi]) +ax.set_ylim(np.minimum(0, np.min(interpDense)), np.maximum(1, np.max(gt))) + +fig.tight_layout() +plt.gca().legend(('Half-Densities','Standard Galerkin', 'Ground Truth')) +plt.show() \ No newline at end of file diff --git a/scripts/2D_example.py b/scripts/2D_example.py new file mode 100644 index 0000000..e7487d1 --- /dev/null +++ b/scripts/2D_example.py @@ -0,0 +1,216 @@ +import sparselib +import numpy as np +import matplotlib +import s3dlib.surface as s3d +import matplotlib.pyplot as plt + +from tqdm import tqdm +from scipy import stats +from matplotlib import cm, colorbar +from scipy.integrate import odeint +from scipy.interpolate import griddata +from matplotlib.ticker import LinearLocator + + +def torusFunc(rtz) : + r,t,z = rtz + Z = 0.45 * np.sin(z*np.pi) + R = r + 0.45 * np.cos(z*np.pi) + return R,t,Z + +def dynamics1(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return 1.0 * np.sin(x[:,1]) + \ + 0.2 * np.cos(x[:,0]) + +def dynamics2(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return 0.5 * np.sin(x[:,0]) + \ + 1.0 * np.cos(x[:,1]) + +def gaussian_uncertainty_sparse(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + mu = np.array([np.pi, np.pi]) + cov = np.array([0.2, 0.3]) + + vals = np.ones((x.shape[0])) + for d in range(x.shape[1]): + vals *= 1 / np.sqrt(2 * np.pi * cov[d]) * np.exp(-0.5 * \ + np.power(x[:,d] - mu[d], 2) / cov[d]) + + return np.sqrt(vals) + +def gaussian_uncertainty_dense(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + mu = np.array([np.pi, np.pi]) + cov = np.array([0.2, 0.3]) + + vals = np.ones((x.shape[0])) + for d in range(x.shape[1]): + vals *= 1 / np.sqrt(2 * np.pi * cov[d]) * np.exp(-0.5 * \ + np.power(x[:,d] - mu[d], 2) / cov[d]) + + return vals + +def plot_torus(zq, filename, title): + cm = plt.get_cmap('magma') + vals = cm(zq)[:,:,:] + fig = plt.figure(frameon=False) + plt.axis('off') + plt.imshow(vals, interpolation='bicubic') + plt.savefig('prob.png', bbox_inches='tight', pad_inches=0) + plt.close(fig) + + vmin = np.minimum(0, np.min(zq)) + vmax = np.maximum(1, np.max(zq)) + + torus = s3d.CylindricalSurface(6).map_geom_from_op(torusFunc) + torus.map_color_from_image('prob.png') + + fig = plt.figure(figsize=plt.figaspect(0.75)) + ax = plt.axes(projection='3d') + ax.set(xlim=(-1,1), ylim=(-1,1), zlim=(-1,1) ) + ax.xaxis.set_major_locator(LinearLocator(5)) + ax.yaxis.set_major_locator(LinearLocator(5)) + ax.zaxis.set_major_locator(LinearLocator(5)) + minc = torus.bounds['vlim'][0] + maxc = torus.bounds['vlim'][1] + ax.add_collection3d(torus) + + # Normalizer + # norm = matplotlib.colors.Normalize(vmin=vmin, vmax=vmax) + + # creating ScalarMappable + # sm = plt.cm.ScalarMappable(cmap=cm, norm=norm) + # sm.set_array([]) + + # plt.colorbar(sm) + fig.tight_layout() + plt.axis('off') + # plt.title(title) + plt.savefig(filename, bbox_inches='tight', pad_inches=0) + +def monte_carlo_sample(N): + t = np.arange(0, 1.51, 0.01) + mu = np.array([np.pi, np.pi]) + cov = np.array([0.2, 0.3]) + samples = np.random.normal(loc=mu, scale=cov, size=(N,2)) + prop_samples = np.zeros((N, samples.shape[1])) + + pbar = tqdm(total=N) + for i in range(samples.shape[0]): + vals = odeint(func, samples[i,:], t) + prop_samples[i,:] = vals[-1,:] + pbar.update(1) + pbar.close() + + return prop_samples + +def func(x, t): + return [dynamics1_MC(x), dynamics2_MC(x)] + +def dynamics1_MC(x): + return 1.0 * np.sin(x[1]) + \ + 0.2 * np.cos(x[0]) + +def dynamics2_MC(x): + return 0.5 * np.sin(x[0]) + \ + 1.0 * np.cos(x[1]) + +def kernel_estimate(samples): + """ + Plots the gaussian_kde estimate given samples + """ + kde = stats.gaussian_kde(samples.T) + + nLevel = [np.linspace(0, 2*np.pi, 100),np.linspace(0, 2*np.pi, 100)] + coordinates = np.array(np.meshgrid(*nLevel)).T.reshape(-1, 2) + z = kde(np.transpose(coordinates)) + + dx = 2 * np.pi / 1024 + xq,yq = np.meshgrid(np.arange(0,2*np.pi,dx), + np.arange(0,2*np.pi,dx)) + zq = griddata(coordinates, z, (xq, yq), method='cubic') + return zq + +class SolverParamsSparse(): + max_level: int = 7 + dim: int = 2 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [gaussian_uncertainty_sparse, dynamics1, dynamics2] + + +class SolverParamsDense(): + max_level: int = 7 + dim: int = 2 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [gaussian_uncertainty_dense, dynamics1, dynamics2] + + +# global specgalSparse +paramsSparse = SolverParamsSparse() +specgalSparse = sparselib.SpectralGalerkin(paramsSparse, logging=True) +specgalSparse.solve(t=1.5) + +paramsDense = SolverParamsDense() +specgalDense = sparselib.SpectralGalerkin(paramsDense) +specgalDense.solve(t=1.5) + +M = 100 +nLevel = [np.linspace(paramsSparse.domain[0], paramsSparse.domain[1], M), + np.linspace(paramsSparse.domain[0], paramsSparse.domain[1], M)] +coordinates = np.array(np.meshgrid(*nLevel)).T.reshape(-1, 2) +interpSparse = np.power(np.real(specgalSparse.eval(coordinates, container_id=0)), 2) +interpDense = np.real(specgalDense.eval(coordinates, container_id=0)) + +dx = 2 * np.pi / 1024 +xq, yq = np.meshgrid(np.arange(paramsSparse.domain[0], paramsSparse.domain[1], dx), + np.arange(paramsSparse.domain[0], paramsSparse.domain[1], dx)) + +zqSparse = griddata(coordinates, interpSparse, (xq, yq), method='cubic') +zqDense = griddata(coordinates, interpDense, (xq, yq), method='cubic') + +# Plot spectral methods +plot_torus(zqSparse, 'torusSparseGrid.png', "Sparse Grid Method") +input("WAIT") +plot_torus(zqDense, 'torusDenseGrid.png', "Standard Galerkin Method") + +# Plot Monte Carllo method +particles = monte_carlo_sample(N=5000) +plot_torus(kernel_estimate(particles), 'torusMonteCarlo.png', "Monte Carlo Method") \ No newline at end of file diff --git a/scripts/3D_example.py b/scripts/3D_example.py new file mode 100644 index 0000000..6e720af --- /dev/null +++ b/scripts/3D_example.py @@ -0,0 +1,283 @@ +import sparselib +import numpy as np +import matplotlib +import s3dlib.surface as s3d +import matplotlib.pyplot as plt + +from tqdm import tqdm +from scipy import stats +from matplotlib import cm, colorbar +from scipy.integrate import odeint +from scipy.interpolate import griddata +from matplotlib.ticker import LinearLocator + + +def torusFunc(rtz) : + r,t,z = rtz + Z = 0.45 * np.sin(z*np.pi) + R = r + 0.45 * np.cos(z*np.pi) + return R,t,Z + +def dynamics1(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return 1.0 * np.sin(x[:,2]) + \ + 0.2 * np.cos(x[:,1]) + \ + 0.5 * np.cos(x[:,0]) + +def dynamics2(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return 0.5 * np.sin(x[:,2]) + \ + 1.0 * np.cos(x[:,1]) + \ + 0.5 * np.cos(x[:,1]) + +def dynamics3(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return 1.0 * np.sin(x[:,2]) + \ + 0.5 * np.cos(x[:,1]) + \ + 0.5 * np.cos(x[:,2]) + +def gaussian_uncertainty_sparse(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + mu = np.array([np.pi, np.pi, np.pi]) + cov = np.array([0.2, 0.3, 0.3]) + + vals = np.ones((x.shape[0])) + for d in range(x.shape[1]): + vals *= 1 / np.sqrt(2 * np.pi * cov[d]) * np.exp(-0.5 * \ + np.power(x[:,d] - mu[d], 2) / cov[d]) + + return np.sqrt(vals) + +def gaussian_uncertainty_dense(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + mu = np.array([np.pi, np.pi, np.pi]) + cov = np.array([0.2, 0.3, 0.3]) + + vals = np.ones((x.shape[0])) + for d in range(x.shape[1]): + vals *= 1 / np.sqrt(2 * np.pi * cov[d]) * np.exp(-0.5 * \ + np.power(x[:,d] - mu[d], 2) / cov[d]) + + return vals + +def plot_torus(zq, filename, title): + vmin = np.minimum(0, np.min(zq)) + # vmax = np.maximum(1, np.max(zq)) + vmax = np.max(zq) + + cm = plt.get_cmap('magma') + vals = cm(zq)[:,:,:] + fig = plt.figure(frameon=False) + plt.axis('off') + plt.imshow(zq, cmap=cm, interpolation='bicubic', norm=matplotlib.colors.Normalize(vmin=vmin, vmax=vmax)) + # plt.imshow(vals, interpolation='bicubic') + plt.clim(vmin, vmax) + plt.savefig('prob.png', bbox_inches='tight', pad_inches=0) + plt.close(fig) + + torus = s3d.CylindricalSurface(6).map_geom_from_op(torusFunc) + torus.map_color_from_image('prob.png') + + fig = plt.figure(figsize=plt.figaspect(0.75)) + ax = plt.axes(projection='3d') + ax.set(xlim=(-1,1), ylim=(-1,1), zlim=(-1,1) ) + ax.xaxis.set_major_locator(LinearLocator(5)) + ax.yaxis.set_major_locator(LinearLocator(5)) + ax.zaxis.set_major_locator(LinearLocator(5)) + minc = torus.bounds['vlim'][0] + maxc = torus.bounds['vlim'][1] + ax.add_collection3d(torus) + + # Normalizer + norm = matplotlib.colors.Normalize(vmin=vmin, vmax=vmax) + + # creating ScalarMappable + sm = plt.cm.ScalarMappable(cmap=cm, norm=norm) + sm.set_array([]) + + plt.colorbar(sm) + fig.tight_layout() + plt.axis('off') + plt.title(title) + plt.savefig(filename, bbox_inches='tight', pad_inches=0) + +def monte_carlo_sample(N): + t = np.arange(0, 1.51, 0.01) + mu = np.array([np.pi, np.pi, np.pi]) + cov = np.sqrt(np.array([0.2, 0.3, 0.3])) + samples = np.random.normal(loc=mu, scale=cov, size=(N,3)) + prop_samples = np.zeros((N, samples.shape[1])) + # return samples + + pbar = tqdm(total=N) + for i in range(samples.shape[0]): + vals = odeint(func, samples[i,:], t) + prop_samples[i,:] = vals[-1,:] + pbar.update(1) + pbar.close() + + return prop_samples + +def func(x, t): + return [dynamics1_MC(x), dynamics2_MC(x), dynamics3_MC(x)] + +def dynamics1_MC(x): + return 1.0 * np.sin(x[2]) + \ + 0.2 * np.cos(x[1]) + \ + 0.5 * np.cos(x[0]) + +def dynamics2_MC(x): + return 0.5 * np.sin(x[2]) + \ + 1.0 * np.cos(x[1]) + \ + 0.5 * np.cos(x[1]) + +def dynamics3_MC(x): + return 1.0 * np.sin(x[2]) + \ + 0.5 * np.cos(x[1]) + \ + 0.5 * np.cos(x[2]) + +def kernel_estimate(samples): + """ + Plots the gaussian_kde estimate given samples + """ + kde = stats.gaussian_kde(samples.T) + + nLevel = [np.linspace(0, 2*np.pi, 100),np.linspace(0, 2*np.pi, 100)] + coordinates = np.array(np.meshgrid(*nLevel)).T.reshape(-1, 2) + z = kde(np.transpose(coordinates)) + + dx = 2 * np.pi / 1024 + xq,yq = np.meshgrid(np.arange(0,2*np.pi,dx), + np.arange(0,2*np.pi,dx)) + zq = griddata(coordinates, z, (xq, yq), method='cubic') + + M = xq.shape[0] + zq /= np.sum(zq * (2*np.pi/M)**2) + return zq + +def marginalize(specgal, half=True): + """ + Compute marginal distribution of dims, integrating out + additional dimensions from total probability density function. + + :param dims: The dimensions to compute marginals for + :type dims: list(int) + """ + print("Marginalizing...") + + # Initialize new 1D sparse grid + params1D = SolverParams1D() + spgrid1D = sparselib.SparseGrid(params1D.domain, params1D.max_level, params1D.dim) + spgrid1D.build() + + M = 20 + nLevel = [np.linspace(params1D.domain[0], params1D.domain[1], M), + np.linspace(params1D.domain[0], params1D.domain[1], M)] + coordinates = np.array(np.meshgrid(*nLevel)).T.reshape(-1, 2) + + interp = np.zeros((coordinates.shape[0],)) + for val in spgrid1D.sparseGrid: + extended_coordinates = np.hstack((coordinates, val * np.ones((coordinates.shape[0],1)))) + + if half: + interp += np.power(np.real(specgal.eval(extended_coordinates)), 2) + else: + interp += np.real(specgal.eval(extended_coordinates)) + + return coordinates, interp/spgrid1D.sparseGrid.shape[0] + +class SolverParams1D(): + max_level: int = 7 + dim: int = 1 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [gaussian_uncertainty_sparse, dynamics1] + + +class SolverParamsSparse(): + max_level: int = 7 + dim: int = 3 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [gaussian_uncertainty_sparse, dynamics1, dynamics2, dynamics3] + + +class SolverParamsDense(): + max_level: int = 5 + dim: int = 3 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [gaussian_uncertainty_dense, dynamics1, dynamics2, dynamics3] + + +global specgalSparse +paramsSparse = SolverParamsSparse() +specgalSparse = sparselib.SpectralGalerkin(paramsSparse, logging=True) +specgalSparse.solve(t=1.5) + +# paramsDense = SolverParamsDense() +# specgalDense = sparselib.SpectralGalerkin(paramsDense) +# specgalDense.solve(t=1.5) + +coordinates, valsSparse = marginalize(specgalSparse) +# coordinates, valsDense = marginalize(specgalDense) + +dx = 2 * np.pi / 1024 +xq, yq = np.meshgrid(np.arange(paramsSparse.domain[0], paramsSparse.domain[1], dx), + np.arange(paramsSparse.domain[0], paramsSparse.domain[1], dx)) + +zqSparse = griddata(coordinates, valsSparse, (xq, yq), method='cubic') +# zqDense = griddata(coordinates, valsDense, (xq, yq), method='cubic') + +M = xq.shape[0] +zqSparse /= np.sum(zqSparse * (2*np.pi/M)**2) +# zqDense /= (2**2 * np.pi**2 * np.sum(zqDense) / (M * M)) + +# # Plot spectral methods +plot_torus(zqSparse, 'torusSparseGrid.png', "Sparse Grid Method") +# plot_torus(zqDense, 'torusDenseGrid.png', "Standard Galerkin Method") + +# Plot Monte Carllo method +particles = monte_carlo_sample(N=5000) +plot_torus(kernel_estimate(particles[:,:2]), 'torusMonteCarlo.png', "Monte Carlo Method") \ No newline at end of file diff --git a/scripts/4D_example.py b/scripts/4D_example.py new file mode 100644 index 0000000..34ccea0 --- /dev/null +++ b/scripts/4D_example.py @@ -0,0 +1,251 @@ +import sparselib +import numpy as np +import matplotlib +import matplotlib.pyplot as plt + +from tqdm import tqdm +from scipy import stats +from scipy import interpolate +from matplotlib import cm, colorbar +from scipy.integrate import odeint +from scipy.interpolate import griddata +from scipy.signal import savgol_filter +from matplotlib.ticker import LinearLocator + + +omega = [np.pi, np.pi, np.pi, np.pi] +K = np.array([[1.0, 0.1, 0.1, 0.1], + [0.1, 1.0, 0.1, 0.1], + [0.1, 0.1, 1.0, 0.1], + [0.1, 0.1, 0.1, 1.0]]) + +def normalize(ys, N): + return np.squeeze(ys / (2 * np.pi * np.sum(ys) / N)) + +def marginalize(specgal, target_dim=0, num_points=20): + """ + Compute marginal distribution over the specified target dimension, + integrating out the additional dimensions from the total probability density function. + + :param specgal: The spectral Galerkin object with a method 'eval' that evaluates the PDF. + :type specgal: object + :param target_dim: The dimension to marginalize over (default is the first dimension). + :type target_dim: int + :param num_points: The number of points for interpolation along the target dimension. + :type num_points: int + :returns: Tuple containing the coordinates of the marginal distribution and the interpolated values. + :rtype: tuple(np.ndarray, np.ndarray) + """ + print(f"Marginalizing over dimension {target_dim}...") + + class SolverParams1D(): + max_level: int = specgal.container.grids[0].max_level + dim: int = 1 + domain: np.ndarray = specgal.container.grids[0].domain + funcs: list = [] + + # Determine the number of dimensions in the problem + total_dim = specgal.spgridUncertainty.dim + + # Create interpolation points along the target dimension + params1D = SolverParams1D() # Assuming params1D.domain gives you the range for each dimension + coordinates = np.linspace(params1D.domain[0], params1D.domain[1], num_points) + coordinates = np.expand_dims(coordinates, axis=1) # Shape (num_points, 1) + + # Initialize the result array for the marginal distribution + interp = np.zeros(coordinates.shape[0]) + + # Create the multi-dimensional sparse grid (excluding the target dimension) + spgrid_params = SolverParamsSparse() # Assume this gives parameters for the N-1 dimensional grid + spgridND = sparselib.SparseGrid(spgrid_params.domain, spgrid_params.max_level, total_dim - 1) + spgridND.build() + + # Loop over all points in the N-1 dimensional sparse grid + for point, weight in zip(spgridND.sparseGrid, spgridND.sparseGridWeights): + # for point in spgridND.sparseGrid: + # Create extended coordinates by inserting the current sparse grid point + # into every position except the target dimension + extended_coordinates = np.zeros((coordinates.shape[0], total_dim)) + extended_coordinates[:, target_dim] = coordinates[:, 0] # Set target dimension + extended_coordinates[:, np.arange(total_dim) != target_dim] = point # Set other dimensions + + # Evaluate the PDF at these coordinates and accumulate the results + interp += weight * np.power(np.real(specgal.eval(extended_coordinates)), 2) + + # Normalize the marginal distribution by the number of points in the N-1 dimensional sparse grid + interp /= np.sum(spgridND.sparseGridWeights) + + return coordinates, normalize(interp, coordinates.shape[0]) + +def dynamics1(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[0] + K[0,1] * np.sin(x[:,1] - x[:,0]) \ + + K[0,2] * np.sin(x[:,2] - x[:,0]) \ + + K[0,3] * np.sin(x[:,3] - x[:,0]) + +def dynamics2(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[1] + K[1,0] * np.sin(x[:,0] - x[:,1]) \ + + K[1,2] * np.sin(x[:,2] - x[:,1]) \ + + K[1,3] * np.sin(x[:,3] - x[:,1]) + +def dynamics3(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[2] + K[2,0] * np.sin(x[:,0] - x[:,2]) \ + + K[2,1] * np.sin(x[:,1] - x[:,2]) \ + + K[2,3] * np.sin(x[:,3] - x[:,2]) + +def dynamics4(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[3] + K[3,0] * np.sin(x[:,0] - x[:,3]) \ + + K[3,1] * np.sin(x[:,1] - x[:,3]) \ + + K[3,2] * np.sin(x[:,2] - x[:,3]) + +def gaussian_uncertainty_sparse(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + mu = np.array([np.pi, np.pi, np.pi, np.pi]) + cov = np.array([0.3, 0.3, 0.3, 0.3]) + + vals = np.ones((x.shape[0])) + for d in range(x.shape[1]): + vals *= 1 / np.sqrt(2 * np.pi * cov[d]) * np.exp(-0.5 * \ + np.power(x[:,d] - mu[d], 2) / cov[d]) + + return np.sqrt(vals) + +class SolverParamsSparse(): + max_level: int = 5 + dim: int = 4 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [gaussian_uncertainty_sparse, + dynamics1, dynamics2, dynamics3, + dynamics4] + +paramsSparse = SolverParamsSparse() +specgalSparse = sparselib.SpectralGalerkin(paramsSparse, logging=True) + +# Propoagation time parameters +total_time = 0.5 +M = 10 +t = 0 +dt = total_time / M +ts = np.linspace(0, total_time, M+1) + +# Results +coords, t1 = marginalize(specgalSparse, target_dim=0) +# plt.plot(t1) +# plt.show() +_, t2 = marginalize(specgalSparse, target_dim=1) +_, t3 = marginalize(specgalSparse, target_dim=2) +_, t4 = marginalize(specgalSparse, target_dim=3) + +xs, ys = np.meshgrid(coords, ts) + +print("Computing coordinates wrt time ...") +pbar = tqdm(total=M) +for i in range(M): + t += dt + specgalSparse.solve(dt) + + # Marginalize along each dimension + _, t1s = marginalize(specgalSparse, target_dim=0) + _, t2s = marginalize(specgalSparse, target_dim=1) + _, t3s = marginalize(specgalSparse, target_dim=2) + _, t4s = marginalize(specgalSparse, target_dim=3) + + t1 = np.vstack((t1, t1s)) + t2 = np.vstack((t2, t2s)) + t3 = np.vstack((t3, t3s)) + t4 = np.vstack((t4, t4s)) + pbar.update(1) +pbar.close() + +fig1 = plt.figure() +ax1 = fig1.add_subplot(111, projection='3d') +ax1.set_xlabel('Frequency') +ax1.set_ylabel('Time [s]') +ax1.set_zlabel('Probability Density') +ax1.set_title('Harmonic Oscillation Frequency vs Time of First Mode') + +fig2 = plt.figure() +ax2 = fig2.add_subplot(111, projection='3d') +ax2.set_xlabel('Frequency') +ax2.set_ylabel('Time [s]') +ax2.set_zlabel('Probability Density') +ax2.set_title('Harmonic Oscillation Frequency vs Time of Second Mode') + +fig3 = plt.figure() +ax3 = fig3.add_subplot(111, projection='3d') +ax3.set_xlabel('Frequency') +ax3.set_ylabel('Time [s]') +ax3.set_zlabel('Probability Density') +ax3.set_title('Harmonic Oscillation Frequency vs Time of Third Mode') + +fig4 = plt.figure() +ax4 = fig4.add_subplot(111, projection='3d') +ax4.set_xlabel('Frequency') +ax4.set_ylabel('Time [s]') +ax4.set_zlabel('Probability Density') +ax4.set_title('Harmonic Oscillation Frequency vs Time of Fourth Mode') + +xnew, ynew = np.mgrid[0:2*np.pi:100j, 0:total_time:100j] +tck = interpolate.bisplrep(xs, ys, t1, s=1) +t1 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t2, s=1) +t2 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t3, s=1) +t3 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t4, s=1) +t4 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + + +ax1.plot_surface(xnew, ynew, t1.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax2.plot_surface(xnew, ynew, t2.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax3.plot_surface(xnew, ynew, t3.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax4.plot_surface(xnew, ynew, t4.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +plt.show() \ No newline at end of file diff --git a/scripts/6D_example.py b/scripts/6D_example.py new file mode 100644 index 0000000..2405838 --- /dev/null +++ b/scripts/6D_example.py @@ -0,0 +1,403 @@ +import sparselib +import numpy as np +import matplotlib +import matplotlib.pyplot as plt + +from tqdm import tqdm +from scipy import stats +from scipy import interpolate +from matplotlib import cm, colorbar +from scipy.integrate import odeint +from scipy.interpolate import griddata +from scipy.signal import savgol_filter +from matplotlib.ticker import LinearLocator + + +omega = np.array([np.pi, np.pi, np.pi, np.pi, np.pi, np.pi]) + \ + np.random.normal(scale=0.01, size=(6,)) +K = np.array([[1.0, 0.1, 0.1, 0.1, 0.1, 0.1], + [0.1, 1.0, 0.1, 0.1, 0.1, 0.1], + [0.1, 0.1, 1.0, 0.1, 0.1, 0.1], + [0.1, 0.1, 0.1, 1.0, 0.1, 0.1], + [0.1, 0.1, 0.1, 0.1, 1.0, 0.1], + [0.1, 0.1, 0.1, 0.1, 0.1, 1.0]]) + +def normalize(ys, N): + return np.squeeze(ys / (2 * np.pi * np.sum(ys) / N)) + +def monte_carlo_integration(specgal, coords, target_dim=0, N=1000): + """ + Monte Carlo integration for a 6D function. + + Parameters: + func (callable): The 6D function to integrate. It should take a numpy array of shape (6,) as input. + bounds (list of tuples): A list of 6 tuples, each representing the (min, max) bounds for the corresponding dimension. + N (int): The number of random samples to use for the integration. + + Returns: + float: The estimated integral of the function over the specified bounds. + """ + print(f"Marginalizing over dimension {target_dim}...") + + # Generate random samples within the bounds + random_points = np.random.rand(N, 5) + + # Scale points to the specified bounds + min_bound = 0 + max_bound = 2*np.pi + scaled_points = random_points * max_bound + + M = coords.shape[0] + func_values = np.zeros((N, M)) + + pbar = tqdm(total=M) + for i, val in enumerate(coords): + points = np.insert(scaled_points, target_dim, val, axis=1) + func_values[:,i] = np.power(np.real(specgal.eval(points)), 2) + + pbar.update(1) + + # Evaluate the function at each sampled point + # func_values = np.apply_along_axis(specgal.eval, 1, points) + + pbar.close() + + # Calculate the volume of the 6D space + volume = np.power(2*np.pi, 6) + + # Estimate the integral + integral = volume * np.mean(func_values, axis=0) + + return normalize(integral, M) + +def marginalize(specgal, target_dim=0, num_points=20): + """ + Compute marginal distribution over the specified target dimension, + integrating out the additional dimensions from the total probability density function. + + :param specgal: The spectral Galerkin object with a method 'eval' that evaluates the PDF. + :type specgal: object + :param target_dim: The dimension to marginalize over (default is the first dimension). + :type target_dim: int + :param num_points: The number of points for interpolation along the target dimension. + :type num_points: int + :returns: Tuple containing the coordinates of the marginal distribution and the interpolated values. + :rtype: tuple(np.ndarray, np.ndarray) + """ + print(f"Marginalizing over dimension {target_dim}...") + + class SolverParams1D(): + max_level: int = specgal.container.grids[0].max_level + dim: int = 1 + domain: np.ndarray = specgal.container.grids[0].domain + funcs: list = [] + + # Determine the number of dimensions in the problem + total_dim = specgal.spgridUncertainty.dim + + # Create interpolation points along the target dimension + params1D = SolverParams1D() # Assuming params1D.domain gives you the range for each dimension + coordinates = np.linspace(params1D.domain[0], params1D.domain[1], num_points) + coordinates = np.expand_dims(coordinates, axis=1) # Shape (num_points, 1) + + # Initialize the result array for the marginal distribution + interp = np.zeros(coordinates.shape[0]) + + # Create the multi-dimensional sparse grid (excluding the target dimension) + spgrid_params = SolverParamsSparse() # Assume this gives parameters for the N-1 dimensional grid + spgridND = sparselib.SparseGrid(spgrid_params.domain, spgrid_params.max_level, total_dim - 1) + spgridND.build() + + # Loop over all points in the N-1 dimensional sparse grid + for point, weight in zip(spgridND.sparseGrid, spgridND.sparseGridWeights): + # for point in spgridND.sparseGrid: + # Create extended coordinates by inserting the current sparse grid point + # into every position except the target dimension + extended_coordinates = np.zeros((coordinates.shape[0], total_dim)) + extended_coordinates[:, target_dim] = coordinates[:, 0] # Set target dimension + extended_coordinates[:, np.arange(total_dim) != target_dim] = point # Set other dimensions + + # Evaluate the PDF at these coordinates and accumulate the results + interp += weight * np.power(np.real(specgal.eval(extended_coordinates)), 2) + + # Normalize the marginal distribution by the number of points in the N-1 dimensional sparse grid + interp /= np.sum(spgridND.sparseGridWeights) + + return coordinates, normalize(interp, coordinates.shape[0]) + +def dynamics1(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[0] + K[0,1] * np.sin(x[:,1] - x[:,0]) \ + + K[0,2] * np.sin(x[:,2] - x[:,0]) \ + + K[0,3] * np.sin(x[:,3] - x[:,0]) \ + + K[0,4] * np.sin(x[:,4] - x[:,0]) \ + + K[0,5] * np.sin(x[:,5] - x[:,0]) + +def dynamics2(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[1] + K[1,0] * np.sin(x[:,0] - x[:,1]) \ + + K[1,2] * np.sin(x[:,2] - x[:,1]) \ + + K[1,3] * np.sin(x[:,3] - x[:,1]) \ + + K[1,4] * np.sin(x[:,4] - x[:,1]) \ + + K[1,5] * np.sin(x[:,5] - x[:,1]) + +def dynamics3(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[2] + K[2,0] * np.sin(x[:,0] - x[:,2]) \ + + K[2,1] * np.sin(x[:,1] - x[:,2]) \ + + K[2,3] * np.sin(x[:,3] - x[:,2]) \ + + K[2,4] * np.sin(x[:,4] - x[:,2]) \ + + K[2,5] * np.sin(x[:,5] - x[:,2]) + +def dynamics4(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[3] + K[3,0] * np.sin(x[:,0] - x[:,3]) \ + + K[3,1] * np.sin(x[:,1] - x[:,3]) \ + + K[3,2] * np.sin(x[:,2] - x[:,3]) \ + + K[3,4] * np.sin(x[:,4] - x[:,3]) \ + + K[3,5] * np.sin(x[:,5] - x[:,3]) + +def dynamics5(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[4] + K[4,0] * np.sin(x[:,0] - x[:,4]) \ + + K[4,1] * np.sin(x[:,1] - x[:,4]) \ + + K[4,2] * np.sin(x[:,2] - x[:,4]) \ + + K[4,3] * np.sin(x[:,3] - x[:,4]) \ + + K[4,5] * np.sin(x[:,5] - x[:,4]) + +def dynamics6(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[5] + K[5,0] * np.sin(x[:,0] - x[:,5]) \ + + K[5,1] * np.sin(x[:,1] - x[:,5]) \ + + K[5,2] * np.sin(x[:,2] - x[:,5]) \ + + K[5,3] * np.sin(x[:,3] - x[:,5]) \ + + K[5,4] * np.sin(x[:,4] - x[:,5]) + +def gaussian_uncertainty_sparse(x): + """ + Initial Uniform uncertainty, independent of dimension + + :param x: Coordinates + :type x: np.array + :param mu: Optional mean + :type mu: np.array + + :returns: initial uncertainty at coords + :rtype: np.array + """ + # mu = np.linspace(0, 2*np.pi, 6) + # mu = np.array([0, 0, 0, 0, 0, 0]) + mu = np.array([np.pi, np.pi, np.pi, np.pi, np.pi, np.pi]) + # mu = np.linspace(0, 2*np.pi, 6) + cov = np.array([0.3, 0.3, 0.3, 0.3, 0.3, 0.3]) + + vals = np.ones((x.shape[0])) + for d in range(x.shape[1]): + vals *= 1 / np.sqrt(2 * np.pi * cov[d]) * np.exp(-0.5 * \ + np.power(x[:,d] - mu[d], 2) / cov[d]) + + return np.sqrt(vals) + +class SolverParamsSparse(): + max_level: int = 4 + dim: int = 6 + domain: np.ndarray = np.array([0, 2*np.pi]) + funcs: list = [gaussian_uncertainty_sparse, + dynamics1, dynamics2, dynamics3, + dynamics4, dynamics5, dynamics6] + +paramsSparse = SolverParamsSparse() +specgalSparse = sparselib.SpectralGalerkin(paramsSparse, logging=True) + +# Propoagation time parameters +total_time = 0.5 +M = 10 +t = 0 +dt = total_time / M +ts = np.linspace(0, total_time, M+1) + +# Coordinates for marginalization +x = np.linspace(0, 2*np.pi, 50) + +# Results +t1 = monte_carlo_integration(specgalSparse, x, target_dim=0) +t2 = monte_carlo_integration(specgalSparse, x, target_dim=1) +t3 = monte_carlo_integration(specgalSparse, x, target_dim=2) +t4 = monte_carlo_integration(specgalSparse, x, target_dim=3) +t5 = monte_carlo_integration(specgalSparse, x, target_dim=4) +t6 = monte_carlo_integration(specgalSparse, x, target_dim=5) +# plt.plot(t1) +# plt.show() +# plt.plot(t2) +# plt.show() +# plt.plot(t3) +# plt.show() +# plt.plot(t4) +# plt.show() +# plt.plot(t5) +# plt.show() +# plt.plot(t6) +# plt.show() + +# coords, t1 = marginalize(specgalSparse, target_dim=0) +# _, t2 = marginalize(specgalSparse, target_dim=1) +# _, t3 = marginalize(specgalSparse, target_dim=2) +# _, t4 = marginalize(specgalSparse, target_dim=3) +# _, t5 = marginalize(specgalSparse, target_dim=4) +# _, t6 = marginalize(specgalSparse, target_dim=5) + +xs, ys = np.meshgrid(x, ts) + +print("Computing coordinates wrt time ...") +pbar = tqdm(total=M) +for i in range(M): + t += dt + specgalSparse.solve(dt) + + # Marginalize along each dimension + # _, t1s = marginalize(specgalSparse, target_dim=0) + # _, t2s = marginalize(specgalSparse, target_dim=1) + # _, t3s = marginalize(specgalSparse, target_dim=2) + # _, t4s = marginalize(specgalSparse, target_dim=3) + # _, t5s = marginalize(specgalSparse, target_dim=4) + # _, t6s = marginalize(specgalSparse, target_dim=5) + + t1s = monte_carlo_integration(specgalSparse, x, target_dim=0) + t2s = monte_carlo_integration(specgalSparse, x, target_dim=1) + t3s = monte_carlo_integration(specgalSparse, x, target_dim=2) + t4s = monte_carlo_integration(specgalSparse, x, target_dim=3) + t5s = monte_carlo_integration(specgalSparse, x, target_dim=4) + t6s = monte_carlo_integration(specgalSparse, x, target_dim=5) + + t1 = np.vstack((t1, t1s)) + t2 = np.vstack((t2, t2s)) + t3 = np.vstack((t3, t3s)) + t4 = np.vstack((t4, t4s)) + t5 = np.vstack((t5, t5s)) + t6 = np.vstack((t6, t6s)) + pbar.update(1) +pbar.close() + +fig1 = plt.figure() +ax1 = fig1.add_subplot(111, projection='3d') +ax1.set_xlabel('Frequency') +ax1.set_ylabel('Time [s]') +ax1.set_zlabel('Probability Density') +ax1.set_title('Harmonic Oscillation Frequency vs Time of First Mode') + +fig2 = plt.figure() +ax2 = fig2.add_subplot(111, projection='3d') +ax2.set_xlabel('Frequency') +ax2.set_ylabel('Time [s]') +ax2.set_zlabel('Probability Density') +ax2.set_title('Harmonic Oscillation Frequency vs Time of Second Mode') + +fig3 = plt.figure() +ax3 = fig3.add_subplot(111, projection='3d') +ax3.set_xlabel('Frequency') +ax3.set_ylabel('Time [s]') +ax3.set_zlabel('Probability Density') +ax3.set_title('Harmonic Oscillation Frequency vs Time of Third Mode') + +fig4 = plt.figure() +ax4 = fig4.add_subplot(111, projection='3d') +ax4.set_xlabel('Frequency') +ax4.set_ylabel('Time [s]') +ax4.set_zlabel('Probability Density') +ax4.set_title('Harmonic Oscillation Frequency vs Time of Fourth Mode') + +fig5 = plt.figure() +ax5 = fig5.add_subplot(111, projection='3d') +ax5.set_xlabel('Frequency') +ax5.set_ylabel('Time [s]') +ax5.set_zlabel('Probability Density') +ax5.set_title('Harmonic Oscillation Frequency vs Time of Fifth Mode') + +fig6 = plt.figure() +ax6 = fig6.add_subplot(111, projection='3d') +ax6.set_xlabel('Frequency') +ax6.set_ylabel('Time [s]') +ax6.set_zlabel('Probability Density') +ax6.set_title('Harmonic Oscillation Frequency vs Time of Sixth Mode') + +xnew, ynew = np.mgrid[0:2*np.pi:100j, 0:total_time:100j] +tck = interpolate.bisplrep(xs, ys, t1, s=1) +t1 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t2, s=1) +t2 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t3, s=1) +t3 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t4, s=1) +t4 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t5, s=1) +t5 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t6, s=1) +t6 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +ax1.plot_surface(xnew, ynew, t1.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax2.plot_surface(xnew, ynew, t2.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax3.plot_surface(xnew, ynew, t3.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax4.plot_surface(xnew, ynew, t4.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax5.plot_surface(xnew, ynew, t5.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax6.plot_surface(xnew, ynew, t6.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +plt.show() + +np.savez('t1.npz', x=xnew, y=ynew, t1=t1) +np.savez('t2.npz', x=xnew, y=ynew, t1=t2) +np.savez('t3.npz', x=xnew, y=ynew, t1=t3) +np.savez('t4.npz', x=xnew, y=ynew, t1=t4) +np.savez('t5.npz', x=xnew, y=ynew, t1=t5) +np.savez('t6.npz', x=xnew, y=ynew, t1=t6) \ No newline at end of file diff --git a/scripts/test.py b/scripts/fit_sparse_grid.py similarity index 50% rename from scripts/test.py rename to scripts/fit_sparse_grid.py index 509c540..f4b9db2 100644 --- a/scripts/test.py +++ b/scripts/fit_sparse_grid.py @@ -1,10 +1,11 @@ import sparselib import numpy as np +import matplotlib.pyplot as plt class SolverParams(): max_level: int = 6 dim: int = 1 - domain: np.ndarray = np.array(0, 2*np.pi) + domain: np.ndarray = np.array([0, 2*np.pi]) funcs: list = [] def dynamics(x): @@ -34,7 +35,27 @@ def uniform_uncertainty(x): return np.sqrt(np.ones_like(x[:,0]) * 1 / (2 * np.pi)) +# Initialize all the parameters params = SolverParams() params.funcs = [uniform_uncertainty, dynamics] -container = sparselib.GridContainer(params) \ No newline at end of file +# Create the series of sparse grids used to approximate the +# time-varying uncertainty and time-invariant vector field. +container = sparselib.GridContainer(params) +container.fit(params) + +# Plot the approximation +xs = np.linspace(params.domain[0], params.domain[1], 1000) +xs = np.expand_dims(xs, axis=1) + +# Grids[0] for uncertainty, grids[1] for vector field +interp = np.real(container.grids[0].eval(xs)) + +# Square the half-density if plotting uncertainty +interp = np.power(interp, 2) +plt.ylim(np.minimum(0, np.min(interp)), np.maximum(1, np.max(interp))) + +plt.plot(xs, interp) +plt.xlabel("Domain") +plt.ylabel("Probability Density") +plt.show() diff --git a/scripts/monte_carlo_2D.py b/scripts/monte_carlo_2D.py new file mode 100644 index 0000000..6fa8481 --- /dev/null +++ b/scripts/monte_carlo_2D.py @@ -0,0 +1,65 @@ +import numpy as np +import matplotlib.pyplot as plt + +from tqdm import tqdm +from scipy import stats +from matplotlib import cm +from scipy.integrate import odeint +from scipy.interpolate import griddata + + +def monte_carlo_sample(N): + t = np.arange(0, 1.01, 0.01) + mu = np.array([np.pi, np.pi]) + cov = np.array([0.2, 0.3]) + samples = np.random.normal(loc=mu, scale=cov, size=(N,2)) + + plot_kernel_estimate(samples) + + prop_samples = np.zeros((N, samples.shape[1])) + + pbar = tqdm(total=N) + for i in range(samples.shape[0]): + vals = odeint(func, samples[i,:], t) + prop_samples[i,:] = vals[-1,:] + pbar.update(1) + pbar.close() + + return prop_samples + +def func(x, t): + return [dynamics1(x), dynamics2(x)] + +def dynamics1(x): + return 1.0 * np.sin(x[1]) + \ + 0.2 * np.cos(x[0]) + +def dynamics2(x): + return 0.5 * np.sin(x[0]) + \ + 1.0 * np.cos(x[1]) + +def plot_kernel_estimate(samples): + """ + Plots the gaussian_kde estimate given samples + """ + kde = stats.gaussian_kde(samples.T) + + nLevel = [np.linspace(0, 2*np.pi, 100),np.linspace(0, 2*np.pi, 100)] + coordinates = np.array(np.meshgrid(*nLevel)).T.reshape(-1, 2) + z = kde(np.transpose(coordinates)) + + dx = 2 * np.pi / 1024 + xq,yq = np.meshgrid(np.arange(0,2*np.pi,dx), + np.arange(0,2*np.pi,dx)) + zq = griddata(coordinates, z, (xq, yq), method='cubic') + + inferno = plt.get_cmap('inferno') + vals = inferno(zq)[:,:,:] + fig = plt.figure(frameon=False) + plt.axis('off') + plt.imshow(vals, interpolation='nearest') + plt.show() + + +particles = monte_carlo_sample(N=5000) +plot_kernel_estimate(particles) \ No newline at end of file diff --git a/scripts/monte_carlo_4D.py b/scripts/monte_carlo_4D.py new file mode 100644 index 0000000..5dcf4ca --- /dev/null +++ b/scripts/monte_carlo_4D.py @@ -0,0 +1,194 @@ +import sparselib +import numpy as np +import matplotlib +import s3dlib.surface as s3d +import matplotlib.pyplot as plt + +from tqdm import tqdm +from scipy import stats +from scipy import interpolate +from matplotlib import cm, colorbar +from scipy.integrate import odeint +from scipy.interpolate import griddata +from scipy.signal import savgol_filter +from matplotlib.ticker import LinearLocator + + +omega = [np.pi, np.pi, np.pi, np.pi] +K = np.array([[1.0, 0.1, 0.1, 0.1], + [0.1, 1.0, 0.1, 0.1], + [0.1, 0.1, 1.0, 0.1], + [0.1, 0.1, 0.1, 1.0]]) + +def dynamics1(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[0] + K[0,1] * np.sin(x[1] - x[0]) \ + + K[0,2] * np.sin(x[2] - x[0]) \ + + K[0,3] * np.sin(x[3] - x[0]) + +def dynamics2(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[1] + K[1,0] * np.sin(x[0] - x[1]) \ + + K[1,2] * np.sin(x[2] - x[1]) \ + + K[1,3] * np.sin(x[3] - x[1]) + +def dynamics3(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[2] + K[2,0] * np.sin(x[0] - x[2]) \ + + K[2,1] * np.sin(x[1] - x[2]) \ + + K[2,3] * np.sin(x[3] - x[2]) + +def dynamics4(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[3] + K[3,0] * np.sin(x[0] - x[3]) \ + + K[3,1] * np.sin(x[1] - x[3]) \ + + K[3,2] * np.sin(x[2] - x[3]) + +def func(x, t): + return [dynamics1(x), dynamics2(x), dynamics3(x), + dynamics4(x)] + +def monte_carlo_sample(N): + # mu = np.array([0, 0, 0, 0]) + mu = np.array([np.pi, np.pi, np.pi, np.pi]) + cov = np.sqrt(np.array([0.3, 0.3, 0.3, 0.3])) + cov = np.array([0.9, 0.9, 0.9, 0.9]) + samples = np.random.normal(loc=mu, scale=cov, size=(N,4)) + return samples + +def kernel_estimate(samples): + """ + Plots the gaussian_kde estimate given samples + """ + kde = stats.gaussian_kde(samples.T) + coords = np.linspace(0, 2*np.pi, 100) + z = kde(np.transpose(coords)) + return z + +def propagate(samples, dt): + t = np.arange(0, dt, dt/10) + prop_samples = np.zeros((samples.shape[0], samples.shape[1])) + + for i in range(samples.shape[0]): + vals = odeint(func, samples[i,:], t) + prop_samples[i,:] = vals[-1,:] + + return prop_samples + +# Propoagation time parameters +total_time = 0.5 +M = 10 +t = 0 +dt = total_time / M +ts = np.linspace(0, total_time, M+1) + +# Sample using Monte Carlo +samples = monte_carlo_sample(N=100000) + +# Results +t1 = kernel_estimate(samples[:,0]) +t2 = kernel_estimate(samples[:,1]) +t3 = kernel_estimate(samples[:,2]) +t4 = kernel_estimate(samples[:,3]) + +N = 100 +x = np.linspace(0, 2*np.pi, N) +xs, ys = np.meshgrid(x, ts) + +print("Computing coordinates wrt time ...") +pbar = tqdm(total=M) +for i in range(M): + t += dt + samples = propagate(samples, dt) + + # Marginalize along each dimension + t1s = kernel_estimate(samples[:,0]) + t2s = kernel_estimate(samples[:,1]) + t3s = kernel_estimate(samples[:,2]) + t4s = kernel_estimate(samples[:,3]) + + t1 = np.vstack((t1, t1s)) + t2 = np.vstack((t2, t2s)) + t3 = np.vstack((t3, t3s)) + t4 = np.vstack((t4, t4s)) + pbar.update(1) +pbar.close() + +fig1 = plt.figure() +ax1 = fig1.add_subplot(111, projection='3d') +ax1.set_xlabel('Frequency') +ax1.set_ylabel('Time [s]') +ax1.set_zlabel('Probability Density') +ax1.set_title('Harmonic Oscillation Frequency vs Time of First Mode') + +fig2 = plt.figure() +ax2 = fig2.add_subplot(111, projection='3d') +ax2.set_xlabel('Frequency') +ax2.set_ylabel('Time [s]') +ax2.set_zlabel('Probability Density') +ax2.set_title('Harmonic Oscillation Frequency vs Time of Second Mode') + +fig3 = plt.figure() +ax3 = fig3.add_subplot(111, projection='3d') +ax3.set_xlabel('Frequency') +ax3.set_ylabel('Time [s]') +ax3.set_zlabel('Probability Density') +ax3.set_title('Harmonic Oscillation Frequency vs Time of Third Mode') + +fig4 = plt.figure() +ax4 = fig4.add_subplot(111, projection='3d') +ax4.set_xlabel('Frequency') +ax4.set_ylabel('Time [s]') +ax4.set_zlabel('Probability Density') +ax4.set_title('Harmonic Oscillation Frequency vs Time of Fourth Mode') + +xnew, ynew = np.mgrid[0:2*np.pi:100j, 0:total_time:100j] +tck = interpolate.bisplrep(xs, ys, t1, s=1) +t1 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t2, s=1) +t2 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t3, s=1) +t3 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t4, s=1) +t4 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +ax1.plot_surface(xnew, ynew, t1.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax2.plot_surface(xnew, ynew, t2.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax3.plot_surface(xnew, ynew, t3.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +ax4.plot_surface(xnew, ynew, t4.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +plt.show() \ No newline at end of file diff --git a/scripts/monte_carlo_6D.py b/scripts/monte_carlo_6D.py new file mode 100644 index 0000000..ab4fe62 --- /dev/null +++ b/scripts/monte_carlo_6D.py @@ -0,0 +1,271 @@ +import sparselib +import numpy as np +import matplotlib +import s3dlib.surface as s3d +import matplotlib.pyplot as plt + +from tqdm import tqdm +from scipy import stats +from scipy import interpolate +from matplotlib import cm, colorbar +from scipy.integrate import odeint +from scipy.interpolate import griddata +from scipy.signal import savgol_filter +from matplotlib.ticker import LinearLocator + + +omega = np.array([np.pi, np.pi, np.pi, np.pi, np.pi, np.pi]) + \ + np.random.normal(scale=0.01, size=(6,)) +K = np.array([[1.0, 0.1, 0.1, 0.1, 0.1, 0.1], + [0.1, 1.0, 0.1, 0.1, 0.1, 0.1], + [0.1, 0.1, 1.0, 0.1, 0.1, 0.1], + [0.1, 0.1, 0.1, 1.0, 0.1, 0.1], + [0.1, 0.1, 0.1, 0.1, 1.0, 0.1], + [0.1, 0.1, 0.1, 0.1, 0.1, 1.0]]) + +def normalize(ys, N): + return np.squeeze(ys / (2 * np.pi * np.sum(ys) / N)) + +def dynamics1(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[0] + K[0,1] * np.sin(x[1] - x[0]) \ + + K[0,2] * np.sin(x[2] - x[0]) \ + + K[0,3] * np.sin(x[3] - x[0]) \ + + K[0,4] * np.sin(x[4] - x[0]) \ + + K[0,5] * np.sin(x[5] - x[0]) + +def dynamics2(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[1] + K[1,0] * np.sin(x[0] - x[1]) \ + + K[1,2] * np.sin(x[2] - x[1]) \ + + K[1,3] * np.sin(x[3] - x[1]) \ + + K[1,4] * np.sin(x[4] - x[1]) \ + + K[1,5] * np.sin(x[5] - x[1]) + +def dynamics3(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[2] + K[2,0] * np.sin(x[0] - x[2]) \ + + K[2,1] * np.sin(x[1] - x[2]) \ + + K[2,3] * np.sin(x[3] - x[2]) \ + + K[2,4] * np.sin(x[4] - x[2]) \ + + K[2,5] * np.sin(x[5] - x[2]) + +def dynamics4(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[3] + K[3,0] * np.sin(x[0] - x[3]) \ + + K[3,1] * np.sin(x[1] - x[3]) \ + + K[3,2] * np.sin(x[2] - x[3]) \ + + K[3,4] * np.sin(x[4] - x[3]) \ + + K[3,5] * np.sin(x[5] - x[3]) + +def dynamics5(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[4] + K[4,0] * np.sin(x[0] - x[4]) \ + + K[4,1] * np.sin(x[1] - x[4]) \ + + K[4,2] * np.sin(x[2] - x[4]) \ + + K[4,3] * np.sin(x[3] - x[4]) \ + + K[4,5] * np.sin(x[5] - x[4]) + +def dynamics6(x): + """ + Dynamics for x + + :param x: Coordinates + :type x: np.array + + :returns: vector field at coords + :rtype: np.array + """ + return omega[5] + K[5,0] * np.sin(x[0] - x[5]) \ + + K[5,1] * np.sin(x[1] - x[5]) \ + + K[5,2] * np.sin(x[2] - x[5]) \ + + K[5,3] * np.sin(x[3] - x[5]) \ + + K[5,4] * np.sin(x[4] - x[5]) + +def func(x, t): + return [dynamics1(x), dynamics2(x), dynamics3(x), + dynamics4(x), dynamics5(x), dynamics6(x)] + +def monte_carlo_sample(N): + # mu = np.array([0, 0, 0, 0, 0, 0]) + # mu = np.linspace(0, 2*np.pi, 6) + mu = np.array([np.pi, np.pi, np.pi, np.pi, np.pi, np.pi]) + # cov = np.sqrt(np.array([0.3, 0.3, 0.3, 0.3, 0.3, 0.3])) + cov = np.array([0.9, 0.9, 0.9, 0.9, 0.9, 0.9]) + samples = np.random.normal(loc=mu, scale=cov, size=(N,6)) + return samples % (2 * np.pi) + +def kernel_estimate(samples): + """ + Plots the gaussian_kde estimate given samples + """ + kde = stats.gaussian_kde(samples.T) + coords = np.linspace(0, 2*np.pi, 100) + z = kde(np.transpose(coords)) + return normalize(z, 100) + +def propagate(samples, dt): + t = np.arange(0, dt, dt/10) + prop_samples = np.zeros((samples.shape[0], samples.shape[1])) + + for i in range(samples.shape[0]): + vals = odeint(func, samples[i,:], t) + prop_samples[i,:] = vals[-1,:] + + return prop_samples % (2 * np.pi) + +# Propoagation time parameters +total_time = 0.5 +M = 10 +t = 0 +dt = total_time / M +ts = np.linspace(0, total_time, M+1) + +# Sample using Monte Carlo +samples = monte_carlo_sample(N=100000) + +# Results +t1 = kernel_estimate(samples[:,0]) +t2 = kernel_estimate(samples[:,1]) +t3 = kernel_estimate(samples[:,2]) +t4 = kernel_estimate(samples[:,3]) +t5 = kernel_estimate(samples[:,4]) +t6 = kernel_estimate(samples[:,5]) + +N = 100 +x = np.linspace(0, 2*np.pi, N) +xs, ys = np.meshgrid(x, ts) + +print("Computing coordinates wrt time ...") +pbar = tqdm(total=M) +for i in range(M): + t += dt + samples = propagate(samples, dt) + + # Marginalize along each dimension + t1s = kernel_estimate(samples[:,0]) + t2s = kernel_estimate(samples[:,1]) + t3s = kernel_estimate(samples[:,2]) + t4s = kernel_estimate(samples[:,3]) + t5s = kernel_estimate(samples[:,4]) + t6s = kernel_estimate(samples[:,5]) + + t1 = np.vstack((t1, t1s)) + t2 = np.vstack((t2, t2s)) + t3 = np.vstack((t3, t3s)) + t4 = np.vstack((t4, t4s)) + t5 = np.vstack((t5, t5s)) + t6 = np.vstack((t6, t6s)) + pbar.update(1) +pbar.close() + +fig1 = plt.figure() +ax1 = fig1.add_subplot(111, projection='3d') +ax1.set_xlabel('Frequency') +ax1.set_ylabel('Time [s]') +ax1.set_zlabel('Probability Density') +ax1.set_title('Harmonic Oscillation Frequency vs Time of First Mode') + +fig2 = plt.figure() +ax2 = fig2.add_subplot(111, projection='3d') +ax2.set_xlabel('Frequency') +ax2.set_ylabel('Time [s]') +ax2.set_zlabel('Probability Density') +ax2.set_title('Harmonic Oscillation Frequency vs Time of Second Mode') + +fig3 = plt.figure() +ax3 = fig3.add_subplot(111, projection='3d') +ax3.set_xlabel('Frequency') +ax3.set_ylabel('Time [s]') +ax3.set_zlabel('Probability Density') +ax3.set_title('Harmonic Oscillation Frequency vs Time of Third Mode') + +fig4 = plt.figure() +ax4 = fig4.add_subplot(111, projection='3d') +ax4.set_xlabel('Frequency') +ax4.set_ylabel('Time [s]') +ax4.set_zlabel('Probability Density') +ax4.set_title('Harmonic Oscillation Frequency vs Time of Fourth Mode') + +fig5 = plt.figure() +ax5 = fig5.add_subplot(111, projection='3d') +ax5.set_xlabel('Frequency') +ax5.set_ylabel('Time [s]') +ax5.set_zlabel('Probability Density') +ax5.set_title('Harmonic Oscillation Frequency vs Time of Fifth Mode') + +fig6 = plt.figure() +ax6 = fig6.add_subplot(111, projection='3d') +ax6.set_xlabel('Frequency') +ax6.set_ylabel('Time [s]') +ax6.set_zlabel('Probability Density') +ax6.set_title('Harmonic Oscillation Frequency vs Time of Sixth Mode') + +xnew, ynew = np.mgrid[0:2*np.pi:100j, 0:total_time:100j] +tck = interpolate.bisplrep(xs, ys, t1, s=1) +t1 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t2, s=1) +t2 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t3, s=1) +t3 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t4, s=1) +t4 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t5, s=1) +t5 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +tck = interpolate.bisplrep(xs, ys, t6, s=1) +t6 = interpolate.bisplev(xnew[:,0], ynew[0,:], tck) + +# ax1.plot_surface(xnew, ynew, t1.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +# ax2.plot_surface(xnew, ynew, t2.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +# ax3.plot_surface(xnew, ynew, t3.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +# ax4.plot_surface(xnew, ynew, t4.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +# ax5.plot_surface(xnew, ynew, t5.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +# ax6.plot_surface(xnew, ynew, t6.clip(min=0), rstride=1, cstride=1, alpha=None, antialiased=True) +# plt.show() + +np.savez('t1_monte_carlo.npz', x=xnew, y=ynew, t1=t1) diff --git a/setup.py b/setup.py index 22ed8c2..b9314cd 100644 --- a/setup.py +++ b/setup.py @@ -1,4 +1,15 @@ # Third Party -import setuptools +from setuptools import setup -setuptools.setup() \ No newline at end of file +setup( + name='sparselib', + version='1.0.0', + description='Sparse spectral discretization for linear advection', + author='Parker Ewen', + author_email='parker.ewen5441@gmail.com', + packages=['sparselib'], + install_requires=['matplotlib', + 'numpy', + 'scipy', + 'tqdm'], +) diff --git a/src/sparselib/__init__.py b/sparselib/__init__.py similarity index 65% rename from src/sparselib/__init__.py rename to sparselib/__init__.py index 0342cfb..0d697d3 100644 --- a/src/sparselib/__init__.py +++ b/sparselib/__init__.py @@ -1,2 +1,3 @@ from .spgrid import SparseGrid +from .specgal import SpectralGalerkin from .gridContainer import GridContainer diff --git a/sparselib/gridContainer.py b/sparselib/gridContainer.py new file mode 100644 index 0000000..1e2f897 --- /dev/null +++ b/sparselib/gridContainer.py @@ -0,0 +1,37 @@ +import copy +from .spgrid import SparseGrid + +class GridContainer(): + def __init__(self, params, logging=False): + """ + Constructs a container for the sparse grid classes. + Each element of the container contains a single SparseGrid + class. By cvonvention, the first grid in the container is + corresponds to the state uncertainty. + + :param params: The solver parameters + :type params: SolverParams struct + """ + # Initialize first empty grid object + grid = SparseGrid(params.domain, params.max_level, + params.dim, logging=logging) + grid.build() + + # Deepcopy grid object to container + self.grids = [copy.deepcopy(grid) + for _ in range(params.dim + 1)] + + def fit(self, params): + """ + Calls the fit functions in each container element. + + :param params: The solver parameters + :type params: SolverParams struct + """ + + if len(params.funcs) != len(self.grids): + raise RuntimeError('There are {} functions defined but {} sparse grids'.format( + len(params.funcs), len(self.grids))) + + # Fit each sparse grid to the corresponding function + [grid.fit(f) for grid, f in zip(self.grids, params.funcs)] diff --git a/sparselib/specgal.py b/sparselib/specgal.py new file mode 100644 index 0000000..8afbbea --- /dev/null +++ b/sparselib/specgal.py @@ -0,0 +1,112 @@ +import os +import time +import numpy as np +from tqdm import tqdm +import matplotlib.pyplot as plt +from scipy.linalg import pinv, expm + +import sparselib + + +class SpectralGalerkin: + def __init__(self, params, logging=False): + """ + Class for spectral Galerkin method. + + ... + + Attributes + ---------- + + Methods + ------- + + """ + self.logging = logging + + self.container = sparselib.GridContainer(params, logging) + self.container.fit(params) + + self.spgridUncertainty = self.container.grids[0] + self.spgridVectorField = self.container.grids[1:] + + self.__compute_galerkin_matrix() + + def solve(self, t): + """ + Propagates the probability density approximation coefficients using the + spectral Galerkin approach. + + :param t: Propagation time + :type t: float + """ + if self.logging: + print("Propagating...") + t0 = time.time() + + # Get the weights of the Fourier basis + coeffs = self.spgridUncertainty.weights + + # Compute Galerkin matrix + G = -(self.B) / np.power(2 * np.pi, self.spgridUncertainty.dim) + + # Solve ODE + new_coeffs = expm(G * t) @ np.array(coeffs) + + # Set new coeffs as spgrid weights + self.spgridUncertainty.weights = new_coeffs + + if self.logging: + t1 = time.time() + print('Propagation time: ', t1 - t0) + + def eval(self, x, container_id=0): + return self.container.grids[container_id].eval(x) + + def __compute_galerkin_matrix(self): + """ + Calculates the galerkin matrix. + """ + # Matrices containing sparse grid weights for vector + # fields. + D1 = np.zeros((self.spgridUncertainty.N, + self.spgridUncertainty.dim), + dtype=np.complex128) + if self.spgridUncertainty.dim == 1: + D1 = self.spgridVectorField[0].weights + else: + for l in range(self.spgridUncertainty.dim): + dl1 = self.spgridVectorField[l].weights + D1[:, l] = dl1 + + N = self.spgridUncertainty.N + hc = np.rint(self.spgridUncertainty.hyperCross).astype('int64') + + tuplook = {} + for i in range(N): + tuplook[hc[i, :].tobytes()] = i + + # Compute Galerkin matrix for uncertainty propagation + B = np.zeros((N, N), dtype=np.complex128) + cf = 1j * 0.5 * (2 * np.pi)**(self.spgridUncertainty.dim) + + if self.logging: + pbar = tqdm(total=N*N) + + for i in range(N): + I = hc[i,:] + for j in range(N): + J = hc[j,:] + K_candidate = J - I + vals_maybe_nz = cf * (I + J) + k_candidate = tuplook.get(K_candidate.tobytes()) + if k_candidate is not None: + B[j, i] = np.dot(vals_maybe_nz, D1[k_candidate,:]) + + if self.logging: + pbar.update(1) + + if self.logging: + pbar.close() + + self.B = B diff --git a/sparselib/spgrid.py b/sparselib/spgrid.py new file mode 100644 index 0000000..96810c1 --- /dev/null +++ b/sparselib/spgrid.py @@ -0,0 +1,224 @@ +import itertools +import numpy as np +import matplotlib.pyplot as plt + +from tqdm import tqdm +from matplotlib import cm +from scipy.interpolate import griddata + + +class SparseGrid: + def __init__(self, domain, max_level, dim, logging=False): + """ + Sparse grid class using a Fourier basis defined with a hyperbolic + cross. + + Attributes + ---------- + + + Methods + ------- + + """ + self.dim = dim + self.domain = domain + self.max_level = max_level + + self.N = [] # Number of basis functions + self.invA = [] # Matrix for fitting coeffs + self.hyperCross = [] # Hyperbolic Cross + self.sparseGrid = [] # Corresponding Sparse Grid + + self.logging = logging + + def build(self): + """ + Builds a hyperbolic cross defined in [1]. First builds the hyperbolic + cross (2.1) and then builds the corresponding sparse grid (2.3) used + for computing the weights of the Fourier basis functions. + + [1] MICHAEL DOHLER, STEFAN KUNIS, AND DANIEL POTTS, "NONEQUISPACED HYPERBOLIC + CROSS FAST FOURIER TRANSFORM", 2010. + """ + + self.__buildHyperCross() + self.__buildSparseGrid(fullGrid=False) + self.__buildAMatrix() + + def fit(self, f): + """ + Fits sparse grid to given function using the lienar sparse grid interpolation + scheme. + + :param f: function to fit + :type f: function handle + """ + self.weights = np.matmul(self.invA, f(self.sparseGrid)) + + def eval(self, x): + """ + Evaluates the sparse grid using the chosen basis at particular points x. + Uses the chosen basis to build the A matrix (Eq. 16 of [1]) and the stores + basis weights to compute the interpolation at each point. + + [1] Richard Dennis, "Using a Hyperbolic Cross to Solve Non-linear + Macroeconomic Models", 2021. + + :param x: Interpolation points + :type x: np.array (NxD) + """ + N = x.shape[0] # Number of eval points + M = self.hyperCross.shape[0] # Number of basis functions + + A = np.zeros(shape=(N, M), dtype=np.complex128) + for idx, k in enumerate(self.hyperCross): + A[:, idx] = self.__basis(x, k) + + return np.matmul(A, self.weights) + + def __buildHyperCross(self): + """ + Builds a hyperbolic cross defined in [1]. + + [1] MICHAEL DOHLER, STEFAN KUNIS, AND DANIEL POTTS, "NONEQUISPACED HYPERBOLIC + CROSS FAST FOURIER TRANSFORM", 2010. + + Parameters: + d (int): Dimension of the space. + N (int): Refinement level. + + Returns: + List of tuples representing the indices in the hyperbolic cross. + """ + def add_index_recursive(dim, max_product, current_product=1, current_index=[]): + if dim == 0: + # Base case: If we have filled all dimensions, yield the index + yield tuple(current_index) + else: + for i in range(0, n+1): + new_product = current_product * max(i+1, 1) + if new_product <= max_product: + # Recurse with the next dimension + yield from add_index_recursive(dim - 1, max_product, new_product, current_index + [i]) + + # Generate all indices for the given dimension and refinement level + n = 2**(self.max_level-1) + indices = list(add_index_recursive(self.dim, n)) + + self.hyperCross = self.__generate_combinations( + np.unique(np.asarray(indices), axis=0)) + self.hyperCross = np.unique(np.asarray(self.hyperCross), axis=0) + self.N = self.hyperCross.shape[0] + + def __buildSparseGrid(self, fullGrid=False): + """ + Builds the sparse grid (2.3) used for computing the weights of the Fourier + basis functions defined in [1]. + + [1] MICHAEL DOHLER, STEFAN KUNIS, AND DANIEL POTTS, "NONEQUISPACED HYPERBOLIC + CROSS FAST FOURIER TRANSFORM", 2010. + """ + + # Compute translation and scaling + translation = self.domain[0] + scaling = self.domain[1] - self.domain[0] + + sparseGridLevel = self.max_level + 1 + + # Builds the corresponding sparse grid for interpolation over hypercross + nLevel = [np.linspace(0, sparseGridLevel, sparseGridLevel+1) for _ in range(self.dim)] + fullGridLevels = np.array(np.meshgrid(*nLevel)).T.reshape(-1, self.dim) + level_sums = np.sum(fullGridLevels, axis=1) + + if fullGrid: + sparseGridLevels = fullGridLevels + else: + sparseGridLevels = (fullGridLevels[level_sums <= np.max(sparseGridLevel)]) + + sparseGridLevels = sparseGridLevels[np.argsort(sparseGridLevels.sum(axis=1)),:] + + # Initialize sparse interpolation grid + if self.logging: + print("Building sparse grid...") + pbar = tqdm(total=len(sparseGridLevels)) + + weights = [] + pts = np.zeros((1, self.dim)) + for level in sparseGridLevels: + h = scaling * 2**(-np.asarray(level)) + idxs = [np.arange(2**(np.asarray(level)[d])) for d in range(self.dim)] + idxs = list(itertools.product(*idxs)) + + # Sparse grid points + pt = np.asarray(np.asarray(idxs) * np.asarray(h) + translation) + pts = np.vstack((pts, pt)) + + # Corresponding weight for each point for quadrature + weight = np.prod(2**(-level)) + weights.extend([weight] * len(idxs)) + + if self.logging: + pbar.update(1) + + if self.logging: + pbar.close() + + self.sparseGrid = np.unique(pts[1:, :], axis=0) + self.sparseGridWeights = np.array(weights) + + def __buildAMatrix(self): + """ + Pre-builds the matrix used to fit Fourier coefficients + """ + + N = self.sparseGrid.shape[0] # Number of eval points + M = self.hyperCross.shape[0] # Number of basis functions + + A = np.zeros(shape=(N, M), dtype=np.complex128) + if self.logging: + print("Prebuilding matrices...") + pbar = tqdm(total=M) + + for idx, k in enumerate(self.hyperCross): + A[:, idx] = self.__basis(self.sparseGrid, k) + + if self.logging: + pbar.update(1) + + if self.logging: + pbar.close() + + self.invA = np.linalg.pinv(A) + + def __generate_combinations(self, pairs): + results = [] + for pair in pairs: + # Create all combinations of negative and positive values for the current pair + signs = list(itertools.product((1, -1), repeat=len(pair))) + combinations = [tuple(s * x for s, x in zip(sign, pair)) for sign in signs] + results.extend(combinations) + return results + + def __basis(self, x, k): + """ + Basis function for sparse grid function space. + + :param x: Evaluation points + :type x: (NxD) np.array + :param k: Evaluation frequency + :type k: float + + :returns: Basis function evaluation at x + :rtype: float + """ + if not isinstance(k, list) and not isinstance(k, np.ndarray): + k = [k] + if not isinstance(x, list) and not isinstance(x, np.ndarray): + x = np.array([[x]]) + + val = 1 + for d in range(self.dim): + val *= np.exp(1j * k[d] * x[:,d]) + + return val \ No newline at end of file diff --git a/src/sparselib/gridContainer.py b/src/sparselib/gridContainer.py deleted file mode 100644 index d54af3b..0000000 --- a/src/sparselib/gridContainer.py +++ /dev/null @@ -1,9 +0,0 @@ -from spgrid import SparseGrid - -class GridContainer(): - def __init__(self, params): - # Sparse grid for uncertainty - self.grids = [SparseGrid(params.domain, params.max_level, params.dim) for _ in range(params.dim + 1)] - - def fit(self): - [grid.fit(f) for grid, f in zip(self.grids, params.funcs)] diff --git a/src/sparselib/spgrid.py b/src/sparselib/spgrid.py deleted file mode 100644 index 42d0139..0000000 --- a/src/sparselib/spgrid.py +++ /dev/null @@ -1,227 +0,0 @@ -import warnings -import itertools -import numpy as np -import matplotlib.pyplot as plt - -import matplotlib -import os -from matplotlib import cm -from scipy.interpolate import griddata -import config -from config import Domain - - -# matplotlib.use('Qt5Agg') - -class SparseGrid: - def __init__(self, domain, max_level, dim): - """ - Sparse grid class using a Fourier basis defined with a hyperbolic - cross. - - Attributes - ---------- - - - Methods - ------- - - """ - self.dim = dim - self.domain = domain - self.max_level = max_level - - self.N = [] - - self.hyperCross = [] # Hyperbolic Cross - self.sparseGrid = [] # Corresponding Sparse Grid - - # self.build_sparse_grid() - - def build_sparse_grid(self): - """ - Builds a hyperbolic cross defined in [1]. First builds the hyperbolic - cross (2.1) and then builds the corresponding sparse grid (2.3) used - for computing the weights of the Fourier basis functions. - - [1] MICHAEL DOHLER, STEFAN KUNIS, AND DANIEL POTTS, "NONEQUISPACED HYPERBOLIC - CROSS FAST FOURIER TRANSFORM", 2010. - """ - - # Compute translation and scaling relative to [0,1]^d - translation = self.domain[0, 0] - scaling = self.domain[0, 1] - self.domain[0, 0] - - # Define constants - n = 2**(self.max_level-1) - - # Build full tensor grid levels - nLevel = [np.linspace(0, n, n+1) for _ in range(self.dim)] - fullCross = np.array(np.meshgrid(*nLevel)).T.reshape(-1, self.dim) - - # Choose subset of grid levels satisfying sparsity condition - k_mix = np.prod(np.maximum(fullCross + 1, np.ones_like(fullCross)), axis=1) - - # Compute hypercross of Fourier frequencies - self.hyperCross = self.generate_combinations((fullCross[k_mix <= np.max(n)])) - self.hyperCross = np.unique(self.hyperCross, axis=0) - - self.N = self.hyperCross.shape[0] - - # Builds the corresponding sparse grid for interpolation over hypercross - nLevel = [np.linspace(0, k, k+1) for _ in range(self.dim)] - fullGridLevels = np.array(np.meshgrid(*nLevel)).T.reshape(-1, self.dim) - level_sums = np.sum(fullGridLevels, axis=1) - sparseGridLevels = (fullGridLevels[level_sums <= np.max(k)]) - sparseGridLevels = sparseGridLevels[np.argsort(sparseGridLevels.sum(axis=1)),:] - - # Initialize sparse interpolation grid - pts = np.zeros((1, self.dim)) - grid = [{'level': np.asarray(level)} for level in sparseGridLevels] - for level in grid: - h = scaling * 2**(-level['level']) - idxs = [np.arange(2**(level['level'][d])) for d in range(self.dim)] - idxs = list(itertools.product(*idxs)) - - pt = np.asarray(np.asarray(idxs) * np.asarray(h) + translation) - pts = np.vstack((pts, pt)) - - self.sparseGrid = np.unique(pts[1:, :], axis=0) - - def fit_sparse_grid(self, f): - """ - Fits sparse grid to given function using the sparse grid interpolation - scheme. - - :param f: function to fit - :type f: function handle - """ - - # Function values at all node points - N = self.sparseGrid.shape[0] # Number of eval points - M = self.hyperCross.shape[0] # Number of basis functions - - A = np.zeros(shape=(N, M), dtype=np.complex128) - - for idx, k in enumerate(self.hyperCross): - A[:, idx] = self.basis(self.sparseGrid, k) - - self.weights = np.matmul(np.linalg.pinv(A), f(self.sparseGrid)) - - def generate_combinations(self, pairs): - results = [] - for pair in pairs: - # Create all combinations of negative and positive values for the current pair - signs = list(itertools.product((1, -1), repeat=len(pair))) - combinations = [tuple(s * x for s, x in zip(sign, pair)) for sign in signs] - results.extend(combinations) - return results - - def eval(self, x): - """ - Evaluates the sparse grid using the chosen basis at particular points x. - Uses the chosen basis to build the A matrix (Eq. 16 of [1]) and the stores - basis weights to compute the interpolation at each point. - - [1] Richard Dennis, "Using a Hyperbolic Cross to Solve Non-linear - Macroeconomic Models", 2021. - - :param x: Interpolation points - :type x: np.array (NxD) - """ - N = x.shape[0] # Number of eval points - M = self.hyperCross.shape[0] # Number of basis functions - - A = np.zeros(shape=(N, M), dtype=np.complex128) - for idx, k in enumerate(self.hyperCross): - A[:, idx] = self.basis(x, k) - - return np.matmul(A, self.weights) - - def basis(self, x, k): - """ - Basis function for sparse grid function space. - - :param x: Evaluation points - :type x: (NxD) np.array - :param k: Evaluation frequency - :type k: float - - :returns: Basis function evaluation at x - :rtype: float - """ - if not isinstance(k, list) and not isinstance(k, np.ndarray): - k = [k] - if not isinstance(x, list) and not isinstance(x, np.ndarray): - x = np.array([[x]]) - - val = 1 - for d in range(self.dim): - val *= np.exp(1j * k[d] * x[:,d]) - - return val - - def basis_der(self, x, k): - """ - Derivative of basis function for sparse grid function space. - - :param x: Evaluation points - :type x: (NxD) np.array - - - :returns: Basis function derivative evaluation at x - :rtype: float - """ - if not isinstance(k, list) and not isinstance(k, np.ndarray): - k = [k] - if not isinstance(x, list) and not isinstance(x, np.ndarray): - x = np.array([[x]]) - - val = 1 - for d in range(self.dim): - val *= 1j * k[d] * np.exp(1j * k[d] * x[:,d]) - return val - - def marginalize(self, dims, prob=False): - """ - Compute marginal distribution of dims, integrating out - additional dimensions from total probability density function. - - :param dims: The dimensions to compute marginals for - :type dims: list(int) - """ - print("Marginalizing...") - if not len(dims) == 2: - print("Can only marginalize for 2 dimensions at the moment...") - return - - # Initialize new 1D sparse grid - spgrid1D = SparseGrid(config.get_inner_product_domain()) - - M = 100 - nLevel = [np.linspace(self.domain[dims[0], 0], self.domain[dims[0], 1], M), - np.linspace(self.domain[dims[1], 0], self.domain[dims[1], 1], M)] - coordinates = np.array(np.meshgrid(*nLevel)).T.reshape(-1, 2) - - interp = np.zeros((coordinates.shape[0],)) - for val in spgrid1D.sparseGrid: - extended_coordinates = np.hstack((coordinates, val * np.ones((coordinates.shape[0],1)))) - - if prob: - interp += np.power(np.real(self.eval(extended_coordinates)), 2) - else: - interp += np.real(self.eval(extended_coordinates)) - - interp /= spgrid1D.sparseGrid.shape[0] - - xq, yq = np.meshgrid(np.arange(self.domain[dims[0], 0], self.domain[dims[0], 1], 0.01), - np.arange(self.domain[dims[1], 0], self.domain[dims[1], 1], 0.01)) - - zq = griddata(coordinates[:,dims], interp, (xq, yq)) - fig = plt.figure(figsize =(14, 14)) - ax = plt.axes(projection ='3d') - surf = ax.plot_surface(xq, yq, zq, cmap=cm.inferno, - linewidth=1, antialiased=True) - plt.show() - - return