NDLI: Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

Content Provider	Springer Nature Link
Author	Tempone, Raúl Tamellini, Lorenzo Haji Ali, Abdul Lateef bile, Fabio
Copyright Year	2016
Abstract	We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data, and naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods, i.e., we use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDE in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Matérn model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis.
Ending Page	1605
Page Count	51
Starting Page	1555
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	6
Volume Number	16
Language	English
Publisher	Springer US
Publisher Date	2016-08-26
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Multi-level methods Economics Sparse grids Infinite dimensional integration Multivariate approximation Uncertainty quantification Linear and Multilinear Algebras, Matrix Theory Multi-level Finite element method Models, numerical methods Multi-index Stochastic Collocation Elliptic partial differential equations with random coefficients Random partial differential equations Approximation by polynomials Numerical Analysis Computer Science Stochastic Collocation methods Combination technique Applications of Mathematics Math Applications in Computer Science Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity

Multi-Index Stochastic Collocation for random PDEs

The heat modulated infinite dimensional Heston model and its numerical approximation

Surface finite element approximation of spherical Whittle--Mat\'ern Gaussian random fields

Stochastic Galerkin reduced basis methods for parametrized linear elliptic partial differential equations

Strong Approximation of Monotone Stochastic Partial Differential Equations Driven by Multiplicative Noise

On the backward Euler approximation of the stochastic Allen-Cahn equation

Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation

Elliptic equations of higher stochastic order

Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

Similar Documents

Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity

Multi-Index Stochastic Collocation for random PDEs

The heat modulated infinite dimensional Heston model and its numerical approximation

Surface finite element approximation of spherical Whittle--Mat\'ern Gaussian random fields

Stochastic Galerkin reduced basis methods for parametrized linear elliptic partial differential equations

Strong Approximation of Monotone Stochastic Partial Differential Equations Driven by Multiplicative Noise

On the backward Euler approximation of the stochastic Allen-Cahn equation

Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation

Elliptic equations of higher stochastic order

Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity