Solving stochastic partial differential equations based on the experimental data

Solving stochastic partial differential equations based on the experimental data

Babuška, Ivo; Liu, Kang-Man; Tempone, Raúl "Solving stochastic partial differential equations based on the experimental data."

Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday. Math. Models Methods Appl. Sci. 13 (2003), no. 3, 415–444.

Babuška, Ivo; Liu, Kang-Man; Tempone, Raúl
Covariance; Karhunen Loeve expansion; stationary random function; principle component analysis
2003
We consider a stochastic linear elliptic boundary value problem whose stochastic coefficient a(x, ω) is expressed by a finite number NKL of mutually independent random variables, and transform this problem into a deterministic one. We show how to choose a suitable NKL which should be as low as possible for practical reasons, and we give the a priori estimates for modeling error when a(x, ω) is completely known. When a random function a(x, ω) is selected to fit the experimental data, we address the estimation of the error in this selection due to insufficient experimental data. We present a simple model problem, simulate the experiments, and give the numerical results and error estimates.

2003