Related papers: Spline-Based Stochastic Collocation Methods for Un…
Uncertainty quantification (UQ) in mathematical models is essential for accurately predicting system behavior under variability. This study provides guidance on method selection for reliable UQ across varied functional behaviors in…
Over the last few years there have been dramatic advances in our understanding of mathematical and computational models of complex systems in the presence of uncertainty. This has led to a growth in the area of uncertainty quantification as…
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based…
This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC…
We introduce stabilized spline collocation schemes for the numerical solution of nonlinear, hyperbolic conservation laws. A nonlinear, residual-based viscosity stabilization is combined with a projection stabilization-inspired linear…
We develop a general polynomial chaos (gPC) based stochastic Galerkin (SG) for hyperbolic equations with random and singular coefficients. Due to the singu- lar nature of the solution, the standard gPC-SG methods may suffer from a poor or…
We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents…
The accessibility of spatially distributed data, enabled by affordable sensors, field, and numerical experiments, has facilitated the development of data-driven solutions for scientific problems, including climate change, weather…
Over the last few years there have been dramatic advances in our understanding of mathematical and computational models of complex systems in the presence of uncertainty. This has led to a growth in the area of uncertainty quantification as…
Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
Stochastic collocation (SC) is a well-known non-intrusive method of constructing surrogate models for uncertainty quantification. In dynamical systems, SC is especially suited for full-field uncertainty propagation that characterizes the…
Partial differential equations (PDEs) are fundamental for theoretically describing numerous physical processes that are based on some input fields in spatial configurations. Understanding the physical process, in general, requires…
We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG) methods for a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain…
In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a…
Intrusive Uncertainty Quantification methods such as stochastic Galerkin are gaining popularity, whereas the classical stochastic Galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system. We apply a…
Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by…
In this work, we provide a deep investigation of a family of arbitrary high order numerical methods for hyperbolic partial differential equations (PDEs), with particular emphasis on very high order versions, i.e., with order higher than 5.…
Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional…
This article discusses the uncertainty quantification (UQ) for time-independent linear and nonlinear partial differential equation (PDE)-based systems with random model parameters carried out using sampling-free intrusive stochastic…