Related papers: On the effect of different samplings to the soluti…
Implicit sampling is a weighted sampling method that is used in data assimilation, where one sequentially updates estimates of the state of a stochastic model based on a stream of noisy or incomplete data. Here we describe how to use…
Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well…
In this work, we propose a novel sampling method for Design of Experiments. This method allows to sample such input values of the parameters of a computational model for which the constructed surrogate model will have the least possible…
Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel…
We adapt the gradient sampling algorithm to the local scoring algorithm to solve complex estimation problems based on an optimization of an objective function. This overcomes non-differentiability and non-smoothness of the objective…
Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to…
Gradient descent methods and especially their stochastic variants have become highly popular in the last decade due to their efficiency on big data optimization problems. In this thesis we present the development of data sampling strategies…
As the number of samples and dimensionality of optimization problems related to statistics an machine learning explode, block coordinate descent algorithms have gained popularity since they reduce the original problem to several smaller…
In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinley parametrized geometries. The essential ingredients of the methodology are: a Galerkin…
The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order…
This paper considers the sparse eigenvalue problem, which is to extract dominant (largest) sparse eigenvectors with at most $k$ non-zero components. We propose a simple yet effective solution called truncated power method that can…
Polynomial chaos methods have been extensively used to analyze systems in uncertainty quantification. Furthermore, several approaches exist to determine a low-dimensional approximation (or sparse approximation) for some quantity of interest…
This paper presents an algorithm for sampling random variables that allows to separation of the sampling process into subproblems by dividing the sample space into overlapping parts. The subproblems can be solved independently of each other…
We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by…
A space-time-parameters structure of parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD - a traditional dimension reduction…
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric…
This paper introduces a novel error estimator for the Proper Generalized Decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: It builds on concentration inequalities of Gaussian maps and an…
Stochastic sampling methods are arguably the most direct and least intrusive means of incorporating parametric uncertainty into numerical simulations of partial differential equations with random inputs. However, to achieve an overall error…
Sparse optimization is a central problem in machine learning and computer vision. However, this problem is inherently NP-hard and thus difficult to solve in general. Combinatorial search methods find the global optimal solution but are…
In this manuscript, we analyze the sparse signal recovery (compressive sensing) problem from the perspective of convex optimization by stochastic proximal gradient descent. This view allows us to significantly simplify the recovery analysis…