Journal: Mathematical Models and Methods in Applied Sciences

Abbreviation

Math. Models Methods Appl. Sci.

Publisher

World Scientific

Journal Volumes

ISSN

0218-2025
1793-6314

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Publications1 - 10 of 33
  • N-term Wiener Chaos Approximation Rates for elliptic PDEs with lognormal Gaussian random inputs
    Item type: Journal Article
    Ha-Hoang, Viet A.; Schwab, Christoph (2014)
    Mathematical Models and Methods in Applied Sciences
  • Multilevel approximation of parametric and stochastic PDES
    Item type: Journal Article
    Zech, Jakob; Düng, Dinh; Schwab, Christoph (2019)
    Mathematical Models and Methods in Applied Sciences
    We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Assuming a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric families are obtained as approximate solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. 103 (2015) 400–428], for example by means of stable, finite-dimensional approximations. We discuss in detail nonlinear Petrov–Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The present results unify and generalize earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov–Galerkin and discrete Petrov–Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel computational strategy to localize sequences of nested index sets for the anisotropic Smolyak interpolation in parameter space is developed which realizes best -term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in the integrands’ gpc expansions by symmetries of quadratures and the probability measure [J.Z̃ech and Ch.S̃chwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich (2017)]. Several examples illustrating the abstract theory include domain uncertainty quantification (UQ) for general, linear, second-order, elliptic advection–reaction–diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.
  • Multilevel higher-order quasi-Monte Carlo Bayesian estimation
    Item type: Journal Article
    Dick, Josef; Gantner, Robert N.; Le Gia, Quoc T.; et al. (2017)
    Mathematical Models and Methods in Applied Sciences
  • Statistical solutions of hyperbolic systems of conservation laws: Numerical approximation
    Item type: Journal Article
    Fjordholm, Ulrik S.; Lye, Kjetil; Mishra, Siddhartha; et al. (2020)
    Mathematical Models and Methods in Applied Sciences
  • PARTICLE SIMULATIONS OF MORPHOGENESIS
    Item type: Journal Article
    Koumoutsakos, Petros; Bayati, Basil; Milde, Florian; et al. (2011)
    Mathematical Models and Methods in Applied Sciences
  • Stochastic Galerkin Discretization of the Log-Normal isotropic Diffusion Problem
    Item type: Journal Article
    Gittelson, Claude Jeffrey (2010)
    Mathematical Models and Methods in Applied Sciences
  • High-Order Galerkin Approximations for Parametric Second-Order Elliptic Partial Differential Equations
    Item type: Journal Article
    Nistor, Victor; Schwab, Christoph (2013)
    Mathematical Models and Methods in Applied Sciences
  • Combined dynamics of magnetization and particle rotation of a suspended superparamagnetic particle in the presence of an orienting field: Semi-analytical and numerical solution
    Item type: Journal Article
    Kröger, Martin; Ilg, Patrick (2022)
    Mathematical Models and Methods in Applied Sciences
    The magnetization dynamics of suspended superparamagnetic particles is governed by internal Neel relaxation as well as Brownian diffusion of the whole particle. We here present semi-analytical and numerical solutions of the kinetic equation, describing the combined rotation of particle orientation and magnetization. The solutions are based on an expansion of the joint probability density into a complete set of bipolar harmonics, leading to a coupled set of ordinary differential equations for the expansion coefficients. Extending previous works, we discuss the spectrum of relaxation times as well as convergence and limits of applicability of the method. Furthermore, we also provide the numerical scheme in electronic form, so that readers can readily implement and use the model.
  • Multilevel approximation of Gaussian random fields: Fast simulation
    Item type: Journal Article
    Herrmann, Lukas; Kirchner, Kristin; Schwab, Christoph (2020)
    Mathematical Models and Methods in Applied Sciences
  • A fully divergence-free finite element method for magnetohydrodynamic equations
    Item type: Journal Article
    Hiptmair, Ralf; Li, Lingxiao; Mao, Shipeng; et al. (2018)
    Mathematical Models and Methods in Applied Sciences
Publications1 - 10 of 33