Analytic regularity and collocation approximation for elliptic PDEs with Random domain deformations
TL;DR: It is shown that the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables can be analytically extended to a well defined region in C^N with respect to the random variables.
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Abstract: 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 onto a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in C^N with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and variance of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments.
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References
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
TL;DR: A rigorous convergence analysis is provided and exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space is demonstrated, under some regularity assumptions on the random input data.
•Book
Analytic functions of several complex variables
Robert C. Gunning,Hugo Rossi +1 more
- 01 Jan 1965
TL;DR: The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century after initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable as discussed by the authors.
1.4K
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
TL;DR: This work demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates, indicating for which problems the sparse grid stochastic collocation method is more efficient than Monte Carlo.
•Book
Function theory of several complex variables
Steven G. Krantz
- 01 Jan 1982
TL;DR: In this article, the authors introduce the subject of harmonic analysis and its applications, including the solution of the Levi problem, the zero set of a holomorphic function, and the invariant metrics.
•Book
Analytic functions of several complex variables
Carl Ludwig Siegel,P. T. Bateman +1 more
- 01 Jan 1962
TL;DR: In this article, the authors discuss the properties of holomorphic functions of complex vectors in more detail, including their properties with respect to their properties in the context of complex vector models.
1.3K