373 research outputs found
Sparse approximation of multilinear problems with applications to kernel-based methods in UQ
We provide a framework for the sparse approximation of multilinear problems
and show that several problems in uncertainty quantification fit within this
framework. In these problems, the value of a multilinear map has to be
approximated using approximations of different accuracy and computational work
of the arguments of this map. We propose and analyze a generalized version of
Smolyak's algorithm, which provides sparse approximation formulas with
convergence rates that mitigate the curse of dimension that appears in
multilinear approximation problems with a large number of arguments. We apply
the general framework to response surface approximation and optimization under
uncertainty for parametric partial differential equations using kernel-based
approximation. The theoretical results are supplemented by numerical
experiments
Wavelet-Fourier CORSING techniques for multi-dimensional advection-diffusion-reaction equations
We present and analyze a novel wavelet-Fourier technique for the numerical
treatment of multidimensional advection-diffusion-reaction equations based on
the CORSING (COmpRessed SolvING) paradigm. Combining the Petrov-Galerkin
technique with the compressed sensing approach, the proposed method is able to
approximate the largest coefficients of the solution with respect to a
biorthogonal wavelet basis. Namely, we assemble a compressed discretization
based on randomized subsampling of the Fourier test space and we employ sparse
recovery techniques to approximate the solution to the PDE. In this paper, we
provide the first rigorous recovery error bounds and effective recipes for the
implementation of the CORSING technique in the multi-dimensional setting. Our
theoretical analysis relies on new estimates for the local a-coherence, which
measures interferences between wavelet and Fourier basis functions with respect
to the metric induced by the PDE operator. The stability and robustness of the
proposed scheme is shown by numerical illustrations in the one-, two-, and
three-dimensional case
Multi-Index Monte Carlo: When Sparsity Meets Sampling
We propose and analyze a novel Multi-Index Monte Carlo (MIMC) method for weak
approximation of stochastic models that are described in terms of differential
equations either driven by random measures or with random coefficients. The
MIMC method is both a stochastic version of the combination technique
introduced by Zenger, Griebel and collaborators and an extension of the
Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles.
Inspired by Giles's seminal work, we use in MIMC high-order mixed differences
instead of using first-order differences as in MLMC to reduce the variance of
the hierarchical differences dramatically. This in turn yields new and improved
complexity results, which are natural generalizations of Giles's MLMC analysis
and which increase the domain of the problem parameters for which we achieve
the optimal convergence, Moreover, in MIMC, the
rate of increase of required memory with respect to is independent
of the number of directions up to a logarithmic term which allows far more
accurate solutions to be calculated for higher dimensions than what is possible
when using MLMC.
We motivate the setting of MIMC by first focusing on a simple full tensor
index set. We then propose a systematic construction of optimal sets of indices
for MIMC based on properly defined profits that in turn depend on the average
cost per sample and the corresponding weak error and variance. Under standard
assumptions on the convergence rates of the weak error, variance and work per
sample, the optimal index set turns out to be the total degree (TD) type. In
some cases, using optimal index sets, MIMC achieves a better rate for the
computational complexity than the corresponding rate when using full tensor
index sets..
Dynamically Orthogonal Approximation for Stochastic Differential Equations
In this paper, we set the mathematical foundations of the Dynamical Low Rank
Approximation (DLRA) method for high-dimensional stochastic differential
equations. DLRA aims at approximating the solution as a linear combination of a
small number of basis vectors with random coefficients (low rank format) with
the peculiarity that both the basis vectors and the random coefficients vary in
time. While the formulation and properties of DLRA are now well understood for
random/parametric equations, the same cannot be said for SDEs and this work
aims to fill this gap. We start by rigorously formulating a Dynamically
Orthogonal (DO) approximation (an instance of DLRA successfully used in
applications) for SDEs, which we then generalize to define a parametrization
independent DLRA for SDEs. We show local well-posedness of the DO equations and
their equivalence with the DLRA formulation. We also characterize the explosion
time of the DO solution by a loss of linear independence of the random
coefficients defining the solution expansion and give sufficient conditions for
global existence.Comment: 32 page
Partitioned Algorithms for Fluid-Structure Interaction Problems in Haemodynamics
We consider the fluid-structure interaction problem arising in haemodynamic applications. The finite elasticity equations for the vessel are written in Lagrangian form, while the Navier-Stokes equations for the blood in Arbitrary Lagrangian Eulerian form. The resulting three fields problem (fluid/ structure/ fluid domain) is formalized via the introduction of three Lagrange multipliers and consistently discretized by p-th order backward differentiation formulae (BDFp). We focus on partitioned algorithms for its numerical solution, which consist in the successive solution of the three subproblems. We review several strategies that all rely on the exchange of Robin interface conditions and review their performances reported recently in the literature. We also analyze the stability of explicit partitioned procedures and convergence of iterative implicit partitioned procedures on a simple linear FSI problem for a general BDFp temporal discretization
Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method
In this work, we tackle the problem of minimising the
Conditional-Value-at-Risk (CVaR) of output quantities of complex differential
models with random input data, using gradient-based approaches in combination
with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the
framework of multi-level Monte Carlo for parametric expectations and propose
modifications of the MLMC estimator, error estimation procedure, and adaptive
MLMC parameter selection to ensure the estimation of the CVaR and sensitivities
for a given design with a prescribed accuracy. We then propose combining the
MLMC framework with an alternating inexact minimisation-gradient descent
algorithm, for which we prove exponential convergence in the optimisation
iterations under the assumptions of strong convexity and Lipschitz continuity
of the gradient of the objective function. We demonstrate the performance of
our approach on two numerical examples of practical relevance, which evidence
the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods
for fixed design computations of output expectations.Comment: 26 pages, 18 figures, 1 table, Related to arXiv:2208.07252, Data
available at https://zenodo.org/record/719344
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