Karhunen-Loève approximation of random fields by generalized fast multipole methods

• Authors: Christoph Schwab;Radu Alexandru Todor
• Venue: Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
• Year: 2006

KL approximation of a possibly instationary random field a(ω, x) ∈ L2(Ω,dP; L∞(D)) subject to prescribed meanfield Ea(x) = ∫Ω, a (ω x) dP(ω) and covariance Va(x,x') = ∫Ω(a(ω, x) - Ea(x))(a(ω, x') - Ea(x')) dP(ω) in a polyhedral domain D ⊂ Rd is analyzed. We show how for stationary covariances Va(x,x') = ga(|x - x'|) with ga(z) analytic outside of z = 0, an M-term approximate KL-expansion aM(ω, x) of a(ω, x) can be computed in log-linear complexity. The approach applies in arbitrary domains D and for nonseparable covariances Ca. 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 aM(X, ω) of a(x, ω) has accuracy O(exp(-bM1/d)) if ga is analytic at z = 0 and accuracy O(M-k/d) if ga is Ck at zero. It is obtained in O(MN(logN)b) operations where N = O(h-d).