arXiv subjects

Akil Narayan

Publications and source records attributed to Akil Narayan.

86 records · Page 5Linked to original sources

arXiv2017

Computation of Induced Orthogonal Polynomial Distributions

We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad class of measures, which is stable for polynomial degrees up to at least degree 1000. Paired with other standard tools such as a numerical root-finding algorithm and inverse transform sampling, this provides a methodology for generating random samples from an induced orthogonal polynomial measure. Generating samples from this measure is one ingredient in optimal numerical methods for certain types of multivariate polynomial approximation. For example, sampling from induced distributions for weighted discrete least-squares approximation has recently been shown to yield convergence guarantees with a minimal number of samples. We also provide publicly-available code that implements the algorithms in this paper for sampling from induced distributions.

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arXiv2017

Offline-Enhanced Reduced Basis Method through adaptive construction of the Surrogate Parameter Domain

The Reduced Basis Method (RBM) is a popular certified model reduction approach for solving parametrized partial differential equations. One critical stage of the \textit{offline} portion of the algorithm is a greedy algorithm, requiring maximization of an error estimate over parameter space. In practice this maximization is usually performed by replacing the parameter domain continuum with a discrete "training" set. When the dimension of parameter space is large, it is necessary to significantly increase the size of this training set in order to effectively search parameter space. Large training sets diminish the attractiveness of RBM algorithms since this proportionally increases the cost of the offline {phase}. In this work we propose novel strategies for offline RBM algorithms that mitigate the computational difficulty of maximizing error estimates over a training set. The main idea is to identify a subset of the training set, a "surrogate parameter domain" (SPD), on which to perform greedy algorithms. The SPD's we construct are much smaller in size than the full training set, yet our examples suggest that they are accurate enough to represent the solution manifold of interest at the current offline RBM iteration. We propose two algorithms to construct the SPD: Our first algorithm, the Successive Maximization Method (SMM) method, is inspired by inverse transform sampling for non-standard univariate probability distributions. The second constructs an SPD by identifying pivots in the Cholesky Decomposition of an approximate error correlation matrix. We demonstrate the algorithm through numerical experiments, showing that the algorithm is capable of accelerating offline RBM procedures without degrading accuracy, assuming that the solution manifold has low Kolmogorov width.

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arXiv2016

A goal-oriented RBM-Accelerated generalized polynomial chaos algorithm

The non-intrusive generalized Polynomial Chaos (gPC) method is a popular computational approach for solving partial differential equations (PDEs) with random inputs. The main hurdle preventing its efficient direct application for high-dimensional input parameters is that the size of many parametric sampling meshes grows exponentially in the number of inputs (the "curse of dimensionality"). In this paper, we design a weighted version of the reduced basis method (RBM) for use in the non-intrusive gPC framework. We construct an RBM surrogate that can rigorously achieve a user-prescribed error tolerance, and ultimately is used to more efficiently compute a gPC approximation non-intrusively. The algorithm is capable of speeding up traditional non-intrusive gPC methods by orders of magnitude without degrading accuracy, assuming that the solution manifold has low Kolmogorov width. Numerical experiments on our test problems show that the relative efficiency improves as the parametric dimension increases, demonstrating the potential of the method in delaying the curse of dimensionality. Theoretical results as well as numerical evidence justify these findings.

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arXiv2016

Stochastic collocation methods via $L_1$ minimization using randomized quadratures

In this work, we discuss the problem of approximating a multivariate function via $\ell_1$ minimization method, using a random chosen sub-grid of the corresponding tensor grid of Gaussian points. The independent variables of the function are assumed to be random variables, and thus, the framework provides a non-intrusive way to construct the generalized polynomial chaos expansions, stemming from the motivating application of Uncertainty Quantification (UQ). We provide theoretical analysis on the validity of the approach. The framework includes both the bounded measures such as the uniform and the Chebyshev measure, and the unbounded measures which include the Gaussian measure. Several numerical examples are given to confirm the theoretical results.

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arXiv2016

A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions

In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions (PCE) to quantify uncertainty in models subject to uncertain input parameters. The standard sampling approach for recovering sparse polynomials is to use Monte Carlo (MC) sampling of the density of orthogonality. However MC methods result in poor function recovery when the polynomial degree is high. Here we propose a general algorithm that can be applied to any admissible weight function on a bounded domain and a wide class of exponential weight functions defined on unbounded domains. Our proposed algorithm samples with respect to the weighted equilibrium measure of the parametric domain, and subsequently solves a preconditioned $\ell^1$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided that demonstrate that our proposed Christoffel Sparse Approximation algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermite specific algorithms.

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arXiv2016

A Christoffel function weighted least squares algorithm for collocation approximations

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

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arXiv2015

Numerical Computation of Weil-Peterson Geodesics in the Universal Teichmüller Space

We propose an optimization algorithm for computing geodesics on the universal Teichmüller space T(1) in the Weil-Petersson ($W P$) metric. Another realization for T(1) is the space of planar shapes, modulo translation and scale, and thus our algorithm addresses a fundamental problem in computer vision: compute the distance between two given shapes. The identification of smooth shapes with elements on T(1) allows us to represent a shape as a diffeomorphism on $S^1$. Then given two diffeomorphisms on $S^1$ (i.e., two shapes we want connect with a flow), we formulate a discretized $W P$ energy and the resulting problem is a boundary-value minimization problem. We numerically solve this problem, providing several examples of geodesic flow on the space of shapes, and verifying mathematical properties of T(1). Our algorithm is more general than the application here in the sense that it can be used to compute geodesics on any other Riemannian manifold.

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arXiv2015

An Orthogonality Property of the Legendre Polynomials

We give a remarkable additional orthogonality property of the classical Legendre polynomials on the real interval $[-1,1]$: polynomials up to degree $n$ from this family are mutually orthogonal under the arcsine measure weighted by the degree-$n$ normalized Christoffel function.

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arXiv2015

Stochastic collocation on unstructured multivariate meshes

Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification (UQ) community. Techniques for least-squares regularization, compressive sampling recovery, and interpolatory reconstruction are becoming standard tools used in a variety of applications. Selection of a collocation mesh is frequently a challenge, but methods that construct geometrically "unstructured" collocation meshes have shown great potential due to attractive theoretical properties and direct, simple generation and implementation. We investigate properties of these meshes, presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.

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arXiv2014

A Reduced Radial Basis Function Method for Partial Differential Equations on irregular domains

We propose and test the first Reduced Radial Basis Function Method (R$^2$BFM) for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an optimized set of centers chosen through a reduced-basis-type greedy algorithm, and a collocation-based model reduction approach that systematically generates a reduced-order approximation whose dimension is orders of magnitude smaller than the total number of RBF centers. The resulting algorithm is efficient and accurate as demonstrated through two- and three-dimensional test problems.

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arXiv2014

Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation

We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of nodes may be added to a current sequence, producing a new sequence) and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions. When undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables. This paper establishes that a certain weighted formulation of one-dimensional Leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the Gauss quadrature nodes associated with the weight function. This property is true even for unbounded domains. We apply the Leja-sparse grid approach to several high-dimensional and problems and demonstrate that Leja sequences are often superior to more standard sparse grid constructions (e.g. Clenshaw-Curtis), at least for interpolatory metrics.

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arXiv2014

Multivariate discrete least-squares approximations with a new type of collocation grid

In this work, we discuss the problem of approximating a multivariate function by discrete least squares projection onto a polynomial space using a specially designed deterministic point set. The independent variables of the function are assumed to be random variables, stemming from the motivating application of Uncertainty Quantification (UQ). Our deterministic points are inspired by a theorem due to André Weil. We first work with the Chebyshev measure and consider the approximation in Chebyshev polynomial spaces. We prove the stability and an optimal convergence estimate, provided the number of points scales quadratically with the dimension of the polynomial space. A possible application for quantifying \textit{epistemic} uncertainties is then discussed. We show that the point set asymptotically equidistributes to the product-Chebyshev measure, allowing us to propose a weighted least squares framework, and extending our method to more general polynomial approximations. Numerical examples are given to confirm the theoretical results. It is shown that the performance of our deterministic points is similar to that of randomly-generated points. However our construction, being deterministic, does not suffer from probabilistic qualifiers on convergence results. (E.g., convergence "with high probability".)

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arXiv2012

Approximating the Weil-Petersson Metric Geodesics on the Universal Teichmüller space by Singular Solutions

We propose and investigate a numerical shooting method for computing geodesics in the Weil-Petersson ($WP$) metric on the universal Teichmüller space T(1). This space, or rather the coset subspace $\PSL_2(\R)\backslash\Diff(S^1)$, has another realization as the space of smooth, simple closed planar curves modulo translations and scalings. This alternate identification of T(1) is a convenient metrization of the space of shapes and provides an immediate application for our algorithm in computer vision. The geodesic equation on T(1) with the $WP$ metric is EPDiff($S^1$), the Euler-Poincare equation on the group of diffeomorphisms of the circle $S^1$, and admits a class of soliton-like solutions named Teichons. Our method relies on approximating the geodesic with these teichon solutions, which have momenta given by a finite linear combination of delta functions. The geodesic equation for this simpler set of solutions is more tractable from the numerical point of view. With a robust numerical integration of this equation, we formulate a shooting method utilizing a cross-ratio matching term. Several examples of geodesics in the space of shapes are demonstrated.

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arXiv2009

Deterministic Numerical Schemes for the Boltzmann Equation

This article describes methods for the deterministic simulation of the collisional Boltzmann equation. It presumes that the transport and collision parts of the equation are to be simulated separately in the time domain. Time stepping schemes to achieve the splitting as well as numerical methods for each part of the operator are reviewed, with an emphasis on clearly exposing the challenges posed by the equation as well as their resolution by various schemes.

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