510 research outputs found
Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition
In this paper, we consider the infinite-dimensional integration problem on
weighted reproducing kernel Hilbert spaces with norms induced by an underlying
function space decomposition of ANOVA-type. The weights model the relative
importance of different groups of variables. We present new randomized
multilevel algorithms to tackle this integration problem and prove upper bounds
for their randomized error. Furthermore, we provide in this setting the first
non-trivial lower error bounds for general randomized algorithms, which, in
particular, may be adaptive or non-linear. These lower bounds show that our
multilevel algorithms are optimal. Our analysis refines and extends the
analysis provided in [F. J. Hickernell, T. M\"uller-Gronbach, B. Niu, K.
Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve
substantially on the error bounds presented there. As an illustrative example,
we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo
multilevel algorithms based on scrambled polynomial lattice rules.Comment: 31 pages, 0 figure
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
Infinite-Variate -Approximation with Nested Subspace Sampling
We consider -approximation on weighted reproducing kernel Hilbert spaces
of functions depending on infinitely many variables. We focus on unrestricted
linear information, admitting evaluations of arbitrary continuous linear
functionals. We distinguish between ANOVA and non-ANOVA spaces, where, by ANOVA
spaces, we refer to function spaces whose norms are induced by an underlying
ANOVA function decomposition. In ANOVA spaces, we provide an optimal algorithm
to solve the approximation problem using linear information. We determine the
upper and lower error bounds on the polynomial convergence rate of -th
minimal worst-case errors, which match if the weights decay regularly. For
non-ANOVA spaces, we also establish upper and lower error bounds. Our analysis
reveals that for weights with a regular and moderate decay behavior, the
convergence rate of -th minimal errors is strictly higher in ANOVA than in
non-ANOVA spaces.Comment: Submitted to the MCQMC 2022 conference proceeding
Multi-level Monte Carlo algorithms for infinite-dimensional integration on RN
AbstractWe study randomized algorithms for numerical integration with respect to a product probability measure on the sequence space RN. We consider integrands from reproducing kernel Hilbert spaces, whose kernels are superpositions of weighted tensor products. We combine tractability results for finite-dimensional integration with the multi-level technique to construct new algorithms for infinite-dimensional integration. These algorithms use variable subspace sampling, and we compare the power of variable and fixed subspace sampling by an analysis of minimal errors
Numerical Methods for PDE Constrained Optimization with Uncertain Data
Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization.
The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods
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