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State estimation aims at approximately reconstructing the solution $u$ to a parametrized partial differential equation from $m$ linear measurements, when the parameter vector $y$ is unknown. Fast numerical recovery methods have been…
We propose a new model-order reduction framework to poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, the solution manifold exhibits a slowly decaying Kolmogorov…
We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are…
In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the…
In this paper we investigate the optimal partition approach for multiparametric conic linear optimization (mpCLO) problems in which the objective function depends linearly on vectors. We first establish more useful properties of the…
In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of…
Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the…
We examine nonlinear dynamical systems of ordinary differential equations or differential algebraic equations. In an uncertainty quantification, physical parameters are replaced by random variables. The inner variables as well as a quantity…
This paper investigates model reduction methods for efficiently approximating the solution of parameter-dependent PDEs with a multi-parameter vector $\vec{\mu} \in \mathbb{R}^p$. In cases where the Kolmogorov $N$-width decays fast enough,…
Piecewise-linear nonlinear systems appear in many engineering disciplines. Prediction of the dynamic behavior of such systems is of great importance from practical and theoretical viewpoint. In this paper, a data-driven model order…
Traditional projection-based reduced-order modeling approximates the full-order model by projecting it onto a linear subspace. With a fast-decaying Kolmogorov $n$-width of the solution manifold, the resulting reduced-order model (ROM) can…
We consider the problem of solving a large-scale system of linear equations in a distributed or federated manner by a taskmaster and a set of machines, each possessing a subset of the equations. We provide a comprehensive comparison of two…
We develop a novel deep learning technique, termed Deep Orthogonal Decomposition (DOD), for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations. The approach consists in the construction…
We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension…
Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables.…
Leveraging nonlinear parametrizations for model reduction can overcome the Kolmogorov barrier that affects transport-dominated problems. In this work, we build on the reduced dynamics given by Neural Galerkin schemes and propose to…
Many signal processing problems can be solved by maximizing the fitness of a segmented model over all possible partitions of the data interval. This letter describes a simple but powerful algorithm that searches the exponentially large…
We examine nonlinear Kolmogorov partial differential equations (PDEs). Here the nonlinear part of the PDE comes from its Hamiltonian where one maximizes over all possible drift and diffusion coefficients which fall within a…
We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…