Related papers: Spline-Based Stochastic Collocation Methods for Un…
In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations…
This paper is concerned with generalized polynomial chaos (gPC) approximation for a general system of quasilinear hyperbolic conservation laws with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of…
Deterministic uncertainty quantification (UQ) in deep learning aims to estimate uncertainty with a single pass through a network by leveraging outputs from the network's feature extractor. Existing methods require that the feature extractor…
In this work we adapt classical residual-based stabilization techniques to the spline collocation setting. Inspired by the Streamline-Upwind-Petrov-Galerkin and Pressure-Stabilizing-Petrov-Galerkin methods, our stabilized collocation…
We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin…
We first investigate the structure of the systems derived from the gPC based stochastic Galerkin method for the nonlinear hyperbolic systems with random inputs. This method adopts a generalized Polynomial Chaos (gPC) approximations in the…
This work considers the problem of numerically approximating statistical moments of a Quantity of Interest (QoI) that depends on the solution of a linear parabolic partial differential equation. The geometry is assumed to be random and is…
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and…
In the paper, we propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for solving Stokes and Navier-Stokes equations. We start with a detailed explanation of the method for the…
We present a finite volume method that is applicable to hyperbolic PDEs including spatially varying and semilinear nonconservative systems. The spatial discretization, like that of the well-known Clawpack software, is based on solving…
The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic…
This paper extends deterministic notions of Strong Stability Preservation (SSP) to the stochastic setting, enabling nonlinearly stable numerical solutions to stochastic differential equations (SDEs) and stochastic partial differential…
In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media.…
This paper presents a nonparametric statistical modeling method for quantifying uncertainty in stochastic gradient systems with isotropic diffusion. The central idea is to apply the diffusion maps algorithm to a training data set to produce…
Generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accuracy, PDE solutions need to have high regularity in the random…
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…
We present a novel uncertainty quantification approach for high-dimensional stochastic partial differential equations that reduces the computational cost of polynomial chaos methods by decomposing the computational domain into…
We describe a computational framework linking Uncertainty Quantification (UQ) methods for continuum problems depending on random parameters with Equation-Free (EF) methods for performing continuum deterministic numerics by acting directly…
Spatially distributed problems are often approximately modelled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g. concentrations). The derivation of accurate such PDEs starting from finer…
We present a novel physics-constrained polynomial chaos expansion as a surrogate modeling method capable of performing both scientific machine learning (SciML) and uncertainty quantification (UQ) tasks. The proposed method possesses a…