Karhunen-Loeve approximation of random fields by generalized fast multipole methods Dedicated to W.L. Wendland to his 70th anniversary

• Published in: Journal of computational physics Vol. 217; no. 1; pp. 100 - 122
• Main Authors: Schwab, Christoph, Todor, Radu Alexandru
• Format: Journal Article
• Language: English
• Published: 01.09.2006
• Subjects: Accuracy; Anniversaries; Approximation; Computation; Gallium; Mathematical analysis; Multipoles
• ISSN: 0021-9991
• DOI: 10.1016/j.jcp.2006.01.048

KL approximation of a possibly instationary random field a([Omega], x) L super(2)([Omega], dP; L super([infinity])(D)) subject to prescribed meanfield [image] and covariance [image] in a polyhedral domain [image] is analyzed. We show how for stationary covariances V sub()ax, x') = g sub()a|x - x'|) with g sub()az) analytic outside of z = 0, an M-term approximate KL-expansion a sub()M[Omega], x) of a([Omega], x) can be computed in log-linear complexity. The approach applies in arbitrary domains D and for nonseparable covariances C sub()a It involves Galerkin approximation of the KL eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform, possibly unstructured mesh of width h in D, plus a generalized fast multipole accelerated Krylov-Eigensolver. The approximate KL-expansion a sub()Mx, [Omega]) of a(x, [Omega]) has accuracy O(exp(-bM super(1/)d) if g sub()ais analytic at z = 0 and accuracy O(M super(-)kd if g sub()ais Ckat zero. It is obtained in O(MN(log N)b operations where N = O(h super(-)d.