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

Abstract

KL approximation of a possibly instationary random field a( ω, x) ∈ L 2( Ω, d P; L ∞( D)) subject to prescribed meanfield E a ( x ) = ∫ Ω a ( ω , x ) d P ( ω ) and covariance V a ( x , x ′ ) = ∫ Ω ( a ( ω , x ) - E a ( x ) ) ( a ( ω , x ′ ) - E a ( x ′ ) ) d P ( ω ) in a polyhedral domain D ⊂ R d is analyzed. We show how for stationary covariances V a ( x, x′) = g a (| x − x′|) with g a ( z) analytic outside of z = 0, an M-term approximate KL-expansion a M ( ω, x) of a( ω, x) can be computed in log-linear complexity. The approach applies in arbitrary domains D and for nonseparable covariances C 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 M ( x, ω) of a( x, ω) has accuracy O(exp(− bM 1/ d )) if g a is analytic at z = 0 and accuracy O( M − k/ d ) if g a is C k at zero. It is obtained in O( MN(log N) b ) operations where N = O( h − d ).