225 research outputs found
Shape optimization for quadratic functionals and states with random right-hand sides
In this work, we investigate a particular class of shape optimization
problems under uncertainties on the input parameters. More precisely, we are
interested in the minimization of the expectation of a quadratic objective in a
situation where the state function depends linearly on a random input
parameter. This framework covers important objectives such as tracking-type
functionals for elliptic second order partial differential equations and the
compliance in linear elasticity. We show that the robust objective and its
gradient are completely and explicitly determined by low-order moments of the
random input. We then derive a cheap, deterministic algorithm to minimize this
objective and present model cases in structural optimization
Bernoulli free boundary problems under uncertainty: the convex case
The present article is concerned with solving Bernoulli's exterior free boundary problem in case of an interior boundary which is random. We provide a new regularity result on the map that sends a parametrization of the inner boundary to a parametrization of the outer boundary. Moreover, by assuming that the interior boundary is convex, also the exterior boundary is convex, which enables to identify the boundaries with support functions and to determine their expectations. We in particular construct a confidence region for the outer boundary based on Aumann's expectation and provide a numerical method to compute it
Multiresolution kernel matrix algebra
We propose a sparse arithmetic for kernel matrices, enabling efficient
scattered data analysis. The compression of kernel matrices by means of
samplets yields sparse matrices such that assembly, addition, and
multiplication of these matrices can be performed with essentially linear cost.
Since the inverse of a kernel matrix is compressible, too, we have also fast
access to the inverse kernel matrix by employing exact sparse selected
inversion techniques. As a consequence, we can rapidly evaluate series
expansions and contour integrals to access, numerically and approximately in a
data-sparse format, more complicated matrix functions such as and
. By exploiting the matrix arithmetic, also efficient Gaussian process
learning algorithms for spatial statistics can be realized. Numerical results
are presented to quantify and quality our findings
Uncertainty quantification for random domains using periodic random variables
We consider uncertainty quantification for the Poisson problem subject to
domain uncertainty. For the stochastic parameterization of the random domain,
we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J.
Numer. Anal., 2020) in which a countably infinite number of independent random
variables enter the random field as periodic functions. We develop lattice
quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the
solution to the Poisson problem subject to domain uncertainty. These QMC rules
can be shown to exhibit higher order cubature convergence rates permitted by
the periodic setting independently of the stochastic dimension of the problem.
In addition, we present a complete error analysis for the problem by taking
into account the approximation errors incurred by truncating the input random
field to a finite number of terms and discretizing the spatial domain using
finite elements. The paper concludes with numerical experiments demonstrating
the theoretical error estimates.Comment: 38 pages, 3 figure
Uncertainty quantification for random domains using periodic random variables
We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates
Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
We provide a general discussion of Smolyak's algorithm for the acceleration
of scientific computations. The algorithm first appeared in Smolyak's work on
multidimensional integration and interpolation. Since then, it has been
generalized in multiple directions and has been associated with the keywords:
sparse grids, hyperbolic cross approximation, combination technique, and
multilevel methods. Variants of Smolyak's algorithm have been employed in the
computation of high-dimensional integrals in finance, chemistry, and physics,
in the numerical solution of partial and stochastic differential equations, and
in uncertainty quantification. Motivated by this broad and ever-increasing
range of applications, we describe a general framework that summarizes
fundamental results and assumptions in a concise application-independent
manner
A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data
Stochastic collocation methods for approximating the solution of partial
differential equations with random input data (e.g., coefficients and forcing
terms) suffer from the curse of dimensionality whereby increases in the
stochastic dimension cause an explosion of the computational effort. We propose
and analyze a multilevel version of the stochastic collocation method that, as
is the case for multilevel Monte Carlo (MLMC) methods, uses hierarchies of
spatial approximations to reduce the overall computational complexity. In
addition, our proposed approach utilizes, for approximation in stochastic
space, a sequence of multi-dimensional interpolants of increasing fidelity
which can then be used for approximating statistics of the solution as well as
for building high-order surrogates featuring faster convergence rates. A
rigorous convergence and computational cost analysis of the new multilevel
stochastic collocation method is provided, demonstrating its advantages
compared to standard single-level stochastic collocation approximations as well
as MLMC methods. Numerical results are provided that illustrate the theory and
the effectiveness of the new multilevel method
Wavelet boundary element methods – Adaptivity and goal-oriented error estimation
This article is dedicated to the adaptive wavelet boundary element method. It computes an approximation to the unknown solution of the boundary integral equation under consideration with a rate , whenever the solution can be approximated with this rate in the setting determined by the underlying wavelet basis. The computational cost scale linearly in the number of degrees of freedom. Goal-oriented error estimation for evaluating linear output functionals of the solution is also considered. An algorithm is proposed that approximately evaluates a linear output functional with a rate , whenever the primal solution can be approximated with a rate and the dual solution can be approximated with a rate , while the cost still scale linearly in . Numerical results for an acoustic scattering problem and for the point evaluation of the potential in case of the Laplace equation are reported to validate and quantify the approach
Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial
This article provides a high-level overview of some recent works on the
application of quasi-Monte Carlo (QMC) methods to PDEs with random
coefficients. It is based on an in-depth survey of a similar title by the same
authors, with an accompanying software package which is also briefly discussed
here. Embedded in this article is a step-by-step tutorial of the required
analysis for the setting known as the uniform case with first order QMC rules.
The aim of this article is to provide an easy entry point for QMC experts
wanting to start research in this direction and for PDE analysts and
practitioners wanting to tap into contemporary QMC theory and methods.Comment: arXiv admin note: text overlap with arXiv:1606.0661
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