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 0 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 n-th 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 n-th minimal error is of order n-r/d

• Type: Journal
• Year: 2006
• Where: JC
• Authors: Stefan Heinrich
• Keywords: General Elliptic Pde, Homogeneous Boundary Conditions, JC 2006, N-th Minimal Error