85 research outputs found
Application of Hierarchical Matrix Techniques To The Homogenization of Composite Materials
In this paper, we study numerical homogenization methods based on integral
equations. Our work is motivated by materials such as concrete, modeled as
composites structured as randomly distributed inclusions imbedded in a matrix.
We investigate two integral reformulations of the corrector problem to be
solved, namely the equivalent inclusion method based on the Lippmann-Schwinger
equation, and a method based on boundary integral equations. The fully
populated matrices obtained by the discretization of the integral operators are
successfully dealt with using the H-matrix format
A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems
In this paper, we propose a method for the approximation of the solution of
high-dimensional weakly coercive problems formulated in tensor spaces using
low-rank approximation formats. The method can be seen as a perturbation of a
minimal residual method with residual norm corresponding to the error in a
specified solution norm. We introduce and analyze an iterative algorithm that
is able to provide a controlled approximation of the optimal approximation of
the solution in a given low-rank subset, without any a priori information on
this solution. We also introduce a weak greedy algorithm which uses this
perturbed minimal residual method for the computation of successive greedy
corrections in small tensor subsets. We prove its convergence under some
conditions on the parameters of the algorithm. The residual norm can be
designed such that the resulting low-rank approximations are quasi-optimal with
respect to particular norms of interest, thus yielding to goal-oriented order
reduction strategies for the approximation of high-dimensional problems. The
proposed numerical method is applied to the solution of a stochastic partial
differential equation which is discretized using standard Galerkin methods in
tensor product spaces
Certified dimension reduction in nonlinear Bayesian inverse problems
We propose a dimension reduction technique for Bayesian inverse problems with
nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation
noise. The likelihood function is approximated by a ridge function, i.e., a map
which depends non-trivially only on a few linear combinations of the
parameters. We build this ridge approximation by minimizing an upper bound on
the Kullback--Leibler divergence between the posterior distribution and its
approximation. This bound, obtained via logarithmic Sobolev inequalities,
allows one to certify the error of the posterior approximation. Computing the
bound requires computing the second moment matrix of the gradient of the
log-likelihood function. In practice, a sample-based approximation of the upper
bound is then required. We provide an analysis that enables control of the
posterior approximation error due to this sampling. Numerical and theoretical
comparisons with existing methods illustrate the benefits of the proposed
methodology
Méthodes de réduction de modèle pour les équations paramétrées -- Applications à la quantification d’incertitude.
Model order reduction has become an inescapable tool for the solution of high dimensional parameter-dependent equations arising in uncertainty quantification, optimization or inverse problems. In this thesis we focus on low rank approximation methods, in particular on reduced basis methods and on tensor approximation methods.The approximation obtained by Galerkin projections may be inaccurate when the operator is ill-conditioned. For projection based methods, we propose preconditioners built by interpolation of the operator inverse. We rely on randomized linear algebra for the efficient computation of these preconditioners. Adaptive interpolation strategies are proposed in order to improve either the error estimates or the projection onto reduced spaces. For tensor approximation methods, we propose a minimal residual formulation with ideal residual norms. The proposed algorithm, which can be interpreted as a gradient algorithm with an implicit preconditioner, allows obtaining a quasi-optimal approximation of the solution.Finally, we address the problem of the approximation of vector-valued or functional-valued quantities of interest. For this purpose we generalize the 'primal-dual' approaches to the non-scalar case, and we propose new methods for the projection onto reduced spaces. In the context of tensor approximation we consider a norm which depends on the error on the quantity of interest. This allows obtaining approximations of the solution that take into account the objective of the numerical simulation.Les méthodes de réduction de modèle sont incontournables pour la résolution d'équations paramétrées de grande dimension qui apparaissent dans les problèmes de quantification d'incertitude, d'optimisation ou encore les problèmes inverses. Dans cette thèse nous nous intéressons aux méthodes d'approximation de faible rang, notamment aux méthodes de bases réduites et d'approximation de tenseur.L'approximation obtenue par projection de Galerkin peut être de mauvaise qualité lorsque l'opérateur est mal conditionné. Pour les méthodes de projection sur des espaces réduits, nous proposons des préconditionneurs construits par interpolation d'inverse d'opérateur, calculés efficacement par des outils d'algèbre linéaire "randomisée". Des stratégies d'interpolation adaptatives sont proposées pour améliorer soit les estimateurs d'erreur, soit les projections sur les espaces réduits. Pour les méthodes d'approximation de tenseur, nous proposons une formulation en minimum de résidu avec utilisation de norme idéale. L'algorithme de résolution, qui s'interprète comme un algorithme de gradient avec préconditionneur implicite, permet d'obtenir une approximation quasi-optimale de la solution.Enfin nous nous intéressons à l'approximation de quantités d'intérêt à valeur fonctionnelle ou vectorielle. Nous généralisons pour cela les approches de type "primale-duale" au cas non scalaire, et nous proposons de nouvelles méthodes de projection sur espaces réduits. Dans le cadre de l'approximation de tenseur, nous considérons une norme dépendant de l'erreur en quantité d'intérêt afin d'obtenir une approximation de la solution qui tient compte de l'objectif du calcul
An adaptive transport framework for joint and conditional density estimation
We propose a general framework to robustly characterize joint and conditional
probability distributions via transport maps. Transport maps or "flows"
deterministically couple two distributions via an expressive monotone
transformation. Yet, learning the parameters of such transformations in high
dimensions is challenging given few samples from the unknown target
distribution, and structural choices for these transformations can have a
significant impact on performance. Here we formulate a systematic framework for
representing and learning monotone maps, via invertible transformations of
smooth functions, and demonstrate that the associated minimization problem has
a unique global optimum. Given a hierarchical basis for the appropriate
function space, we propose a sample-efficient adaptive algorithm that estimates
a sparse approximation for the map. We demonstrate how this framework can learn
densities with stable generalization performance across a wide range of sample
sizes on real-world datasets
Randomized residual-based error estimators for parametrized equations
International audienceWe propose a randomized a posteriori error estimator for reduced order approximations of parametrized (partial) differential equations. The error estimator has several important properties: the effectivity is close to unity with prescribed lower and upper bounds at specified high probability; the estimator does not require the calculation of stability (coercivity, or inf-sup) constants; the online cost to evaluate the a posteriori error estimator is commensurate with the cost to find the reduced order approximation; the probabilistic bounds extend to many queries with only modest increase in cost. To build this estimator, we first estimate the norm of the error with a Monte-Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g. user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. In order to have a fast-to-evaluate estimator, model order reduction methods can be used to approximate the random dual solutions. Here, we propose a greedy algorithm that is guided by a scalar quantity of interest depending on the error estimator. Numerical experiments on a multi-parametric Helmholtz problem demonstrate that this strategy yields rather low-dimensional reduced dual spaces
Gradient-based dimension reduction of multivariate vector-valued functions
Multivariate functions encountered in high-dimensional uncertainty
quantification problems often vary most strongly along a few dominant
directions in the input parameter space. We propose a gradient-based method for
detecting these directions and using them to construct ridge approximations of
such functions, in the case where the functions are vector-valued (e.g., taking
values in ). The methodology consists of minimizing an upper
bound on the approximation error, obtained by subspace Poincar\'e inequalities.
We provide a thorough mathematical analysis in the case where the parameter
space is equipped with a Gaussian probability measure. The resulting method
generalizes the notion of active subspaces associated with scalar-valued
functions. A numerical illustration shows that using gradients of the function
yields effective dimension reduction. We also show how the choice of norm on
the codomain of the function has an impact on the function's low-dimensional
approximation
Nonlinear dimension reduction for surrogate modeling using gradient information
We introduce a method for the nonlinear dimension reduction of a
high-dimensional function , . Our
objective is to identify a nonlinear feature map
, with a prescribed intermediate
dimension , so that can be well approximated by for some
profile function . We propose to build the
feature map by aligning the Jacobian with the gradient ,
and we theoretically analyze the properties of the resulting . Once is
built, we construct by solving a gradient-enhanced least squares problem.
Our practical algorithm makes use of a sample and builds both and on adaptive downward-closed
polynomial spaces, using cross validation to avoid overfitting. We numerically
evaluate the performance of our algorithm across different benchmarks, and
explore the impact of the intermediate dimension . We show that building a
nonlinear feature map can permit more accurate approximation of than a
linear , for the same input data set
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