Journal: SIAM Journal on Numerical Analysis

Abbreviation

SIAM j. numer. anal.

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SIAM

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ISSN

0036-1429
1095-7170

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Publications1 - 10 of 60
  • Adaptive Petrov-Galerkin Methods for First Order Transport Equations
    Item type: Journal Article
    Dahmen, Wolfgang; Huang, Chunyan; Schwab, Christoph; et al. (2012)
    SIAM Journal on Numerical Analysis
    We propose stable variational formulations for certain linear, unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper. The central objective is to develop for such classes adaptive solution concepts with provable error reduction. To adaptively resolve $anisotropic $ solution features such as propagating singularities, the presently proposed variational formulations allow, in particular, the employment of trial spaces spanned by directional representation systems. Since such systems, typically given as frames, are known to be stable only in $L_2$, special emphasis is placed on $L_2$-stable formulations. The proposed stability concept is based on perturbations of certain “ideal” test spaces in Petrov--Galerkin formulations; see also [L. F. Demkowicz and J. Gopalakrishnan, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1558--1572], [L. Demkowicz and J. Gopalakrishnan, Numer. Methods Partial Differential Equations, 27 (2011), pp. 70--105], [J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and V. Calo, J. Comput. Phys., 230 (2011), pp. 2406--2432]. We propose a general strategy for realizing the resulting Petrov--Galerkin schemes based on an Uzawa iteration circumventing an excessively expensive computation of corresponding test basis functions. Moreover, based on this iteration, we develop adaptive solution concepts with provable error reduction. The results are illustrated by numerical experiments.
  • Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients
    Item type: Journal Article
    Kuo, Frances Y.; Schwab, Christoph; Sloan, Ian H. (2012)
    SIAM Journal on Numerical Analysis
    In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably infinite number of terms in a Karhunen--Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC method to estimate expected values of linear functionals of the exact or approximate solution of the PDE, with the expected value considered as an infinite dimensional integral in the parameter space corresponding to the randomness induced by the random coefficient. The error analysis, arguably the first rigorous application of the modern theory of QMC in weighted spaces, exploits the regularity with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). In the weighted-space theory of QMC methods, “weights,” describing the varying difficulty of different subsets of the variables, are introduced in order to make sure that the high-dimensional integration problem is tractable. It turns out that the weights arising from the present analysis are of a nonstandard kind, being of neither product nor order dependent form, but instead a hybrid of the two---we refer to these as “product and order dependent weights,” or “POD weights” for short. These POD weights are of a simple enough form to permit a component-by-component construction of a randomly shifted lattice rule that has optimal convergence properties for the given weighted space setting. If the terms in the expansion for the random coefficient have an appropriate decay property, and if we choose POD weights that minimize a certain upper bound on the error, then the solution of the PDE belongs to the joint function space needed for the analysis, and the QMC error (in the sense of a root-mean-square error averaged over shifts) is of order $O(N^{-1 + \delta})$ for arbitrary $\delta$ > 0, where $N$ denotes the number of sampling points in the parameter space. Moreover, this convergence rate and slower convergence rates under more relaxed conditions are achieved under conditions similar to those found recently by Cohen, De Vore, and Schwab [Found. Comput. Math., 10 (2010), pp. 615--646] for the same model by best $N$-term approximation. We analyze the impact of a finite element (FE) discretization on the overall efficiency of the scheme, in terms of accuracy versus overall cost, with results that are comparable to those of the best $N$-term approximation.
  • Exponential convergence of Gauss-Jacobi quadratures for singular integrals over simplices in arbitrary dimension
    Item type: Journal Article
    Chernov, Alexey; Schwab, Christoph (2012)
    SIAM Journal on Numerical Analysis
  • Constraint-Preserving Upwind Methods for Multidimensional Advection Equations
    Item type: Journal Article
    Torrilhon, M.; Fey, M. (2004)
    SIAM Journal on Numerical Analysis
  • A Coercive Combined Field Integral Equation for Electromagnetic Scattering
    Item type: Journal Article
    Buffa, A.; Hiptmair, R. (2004)
    SIAM Journal on Numerical Analysis
  • Dual-primal FETI algorithms for edge element approximations
    Item type: Journal Article
    Toselli, Andrea; Vasseur, Xavier (2005)
    SIAM Journal on Numerical Analysis
  • Optimal Mesh Size for Inverse Medium Scattering Problems
    Item type: Journal Article
    Ammari, Habib; Chow, Yat Tin; Liu, Keji (2020)
    SIAM Journal on Numerical Analysis
    An optimal mesh size of the sampling region can help to reduce computational burden in practical applications. In this work, we investigate optimal choices of mesh sizes for the identifications of medium obstacles from either the far-field or near-field data in two and three dimensions. The results would have applications in the reconstruction process of inverse scattering problems.
  • Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities
    Item type: Journal Article
    Kornhuber, Ralf; Schwab, Christoph; Wolf, Maren-Wanda (2014)
    SIAM Journal on Numerical Analysis
    Multilevel Monte Carlo finite element methods (MLMC-FEMs) for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of “pathwise” weak formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for the classical MC method and novel MLMC-FEMs. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC-FEMs then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experiments.
  • Mesh-independent operator preconditioning for boundary elements on open curves
    Item type: Journal Article
    Hiptmair, Ralf; Jerez-Hanckes, Carlos; ​Urzúa-Torres, Carolina (2014)
    SIAM Journal on Numerical Analysis
  • Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification
    Item type: Journal Article
    Dick, Josef; Feischl, Michael; Schwab, Christoph (2019)
    SIAM Journal on Numerical Analysis
Publications1 - 10 of 60