The randomized information complexity of elliptic PDE

We study the information complexity in the randomized setting of solving a general elliptic PDE of order 2m in a smooth, bounded domain Q ⊂ Rd with smooth coefficients and homogeneous boundary conditions. The solution is sought on a smooth submanifold M ⊆ Q of dimension O≤d1≤d, the right-hand side is supposed to be in Cr (Q), the error is measured in the L∞(M) norm. We show that the nth minimal error is (up to logarithmic factors) of order n-min((r+2m)/d1,r/d+1/2).For comparison, in the deterministic setting the nth minimal error is of order n-r/d, for all d1.

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