Monte Carlo methods. Vol. 1: basics
Monte Carlo methods. Vol. 1: basics
Information-based complexity
Monte Carlo complexity of global solution of integral equations
Journal of Complexity
Monte Carlo complexity of parametric integration
Journal of Complexity
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
The randomized information complexity of elliptic PDE
Journal of Complexity
The randomized information complexity of elliptic PDE
Journal of Complexity
The randomized information complexity of elliptic PDE
Journal of Complexity
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We study the randomized approximation of weakly singular integral operators. For a suitable class of kernels having a standard type of singularity and being otherwise of finite smoothness, we develop a Monte Carlo multilevel method, give convergence estimates and prove lower bounds which show the optimality of this method and establish the complexity. As an application we obtain optimal methods for and the complexity of randomized solution of the Poisson equation in simple domains, when the solution is sought on subdomains of arbitrary dimension.