Analytic regularity and collocation approximation for PDEs with random domain deformations

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables. The elliptic problem is remapped on to a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in C with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and standard deviation of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments.

[1]  Thomas Gerstner,et al.  Dimension–Adaptive Tensor–Product Quadrature , 2003, Computing.

[2]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[3]  Reinhold Schneider,et al.  Sparse second moment analysis for elliptic problems in stochastic domains , 2008, Numerische Mathematik.

[4]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[5]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[6]  Ilse C. F. Ipsen,et al.  Perturbation Bounds for Determinants and Characteristic Polynomials , 2008, SIAM J. Matrix Anal. Appl..

[7]  David London,et al.  A note on matrices with positive definite real part , 1981 .

[8]  Fabio Nobile,et al.  An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[9]  Julia Charrier,et al.  Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients , 2012, SIAM J. Numer. Anal..

[10]  Erich Novak,et al.  High dimensional polynomial interpolation on sparse grids , 2000, Adv. Comput. Math..

[11]  Eberhard Freitag,et al.  Analytic Functions of Several Complex Variables , 2011 .

[12]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[13]  R. Tempone,et al.  Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison , 2011 .

[14]  Ilse C. F. Ipsen,et al.  Determinant Approximations , 2011, 1105.0437.

[15]  N. Levenberg,et al.  Function theory in several complex variables , 2001 .

[16]  Daniel M. Tartakovsky,et al.  Stochastic analysis of transport in tubes with rough walls , 2006, J. Comput. Phys..