We construct quasi--Monte Carlo methods to approximate the expected values of linear functionals Petrov--Galerkin discretizations parametric operator equations which depend on a possibly infinite sequence parameters. Such problems arise in numerical solution differential and integral with random field inputs. analyze regularity solutions respect parameters terms rate decay fluctuations input field. If $p\in (0,1]$ denotes “summability exponent” corresponding affine-parametric families...

We consider a multiscale approximation scheme at scattered sites for functions in Sobolev spaces on the unit sphere $\mathbb{S}^n$. The is constructed using sequence of scaled, compactly supported radial basis restricted to A convergence theorem proved, and condition number linear system shown stay bounded by constant from level level, thereby establishing first time mathematical theory with scaled versions single function data points.

We analyze the convergence of higher order quasi--Monte Carlo (QMC) quadratures solution functionals to countably parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space $X$ admitting an unconditional Schauder basis. Such arise numerical uncertainty quantification random field inputs. Unconditional bases render inputs and solutions forward problem deterministic. show that these parametric belong class weighted Bochner spaces...

The purpose of this paper is to construct universal, auto-adaptive, localized, linear, polynomial (-valued) operators based on scattered data the (hyper) sphere $\mathbb{S}^q$ ($q\ge2$). approximation and localization properties our are studied theoretically in deterministic as well probabilistic settings. Numerical experiments presented demonstrate their superiority over traditional least squares discrete Fourier projection approximations. An essential ingredient construction quadrature...

We develop a convergence analysis of multilevel algorithm combining higher order quasi--Monte Carlo (QMC) quadratures with general Petrov--Galerkin discretizations countably affine parametric operator equations elliptic and parabolic types, extending both the first in [F. Y. Kuo, Ch. Schwab, I. H. Sloan, Found. Comput. Math., 15 (2015), pp. 411--449] single level [J. Dick et al., SIAM J. Numer. Anal., 52 (2014), 2676--2702]. cover, particular, definite as well indefinite strongly systems...

We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a ML approach based on deterministic, higher-order quasi-Monte Carlo (HoQMC) quadrature approximating the high-dimensional expectations, which arise in estimators, Petrov–Galerkin (PG) method solution underlying partial differential equation (PDE). This extends previous...

In this paper we prove the existence of random attractors for Navier–Stokes equations on 2 dimensional sphere under forcing irregular in space and time. We also deduce an invariant measure.

We analyze combined Quasi-Monte Carlo quadrature and Finite Element approximations in Bayesian estimation of solutions to countably-parametric operator equations with holomorphic dependence on the parameters as considered [Cl.~Schillings Ch.~Schwab: Sparsity Inversion Parametric Operator Equations. Inverse Problems, {\bf 30}, (2014)]. Such problems arise numerical uncertainty quantification inversion distributed uncertain inputs, such coefficients, domains or source terms boundary data. show...

Abstract The performance of high-efficiency silicon solar cells is limited by the presence bulk defects. Identification these defects has potential to improve cell and reliability. impact on minority carrier lifetime commonly measured using temperature- injection-dependent spectroscopy defect parameters, such as its energy level capture cross-section ratio, are usually extracted fitting Shockley-Read-Hall equation. We propose an alternative extraction approach machine learning trained more...

The aim of this work is to consider multiscale algorithms for solving partial differential equations (PDEs) with Galerkin methods on bounded domains. We provide results convergence and condition numbers. show how handle PDEs Dirichlet boundary conditions. also investigate in terms the mesh norms angles between subspaces better understand differences observed results. issue supports radial basis funtions overlapping our stability analysis, which has not been considered literature best knowledge.

Quasi--Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ row vector. This type integral arises, for example, from simulation normal distribution with general covariance matrix, approximation expectation value solutions PDEs random coefficients, or applications statistics. In this paper we design QMC quadrature points...

Defect parameter solution surface is a widely accepted method to determine the values of energy level and carrier capture cross sections defect from injection-dependent lifetime measurements. Results come recognition intersection points, which can be difficult when doping or temperature range measurement narrow. In this paper, we introduce new extract parameters spectroscopy data. After linearization Shockley-Read-Hall (SRH) equation, problem transformed into mathematical form, solved by...