Related papers: Optimal piecewise linear data compression for solu…

Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of…

Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging. One solution is to use Riemannian…

This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces…

We formulate the problem of performing optimal data compression under the constraints that compressed data can be used for accurate classification in machine learning. We show that this translates to a problem of minimizing the mutual…

Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space -…

We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an…

Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks…

We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme)…

In this paper, we introduce the proper latent decomposition (PLD) as a generalization of the proper orthogonal decomposition (POD) on manifolds. PLD is a nonlinear reduced-order modeling technique for compressing high-dimensional data into…

The problem of minimizing a separable convex function under linearly coupled constraints arises from various application domains such as economic systems, distributed control, and network flow. The main challenge for solving this problem is…

This work introduces a new approach to reduce the computational cost of solving partial differential equations (PDEs) with convection-dominated solutions: model reduction with implicit feature tracking. Traditional model reduction…

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized {nonlinear} elliptic partial differential equations (PDEs). CB-pMOR is designed to deal with large-scale problems for which full-order…

Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…

Partial differential equations can be used to model many problems in several fields of application including, e.g., fluid mechanics, heat and mass transfer, and electromagnetism. Accurate discretization methods (e.g., finite element or…

The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall…

The task of finding the optimal compression of a polyline with straight-line segments and arcs is performed in many applications, such as polyline compression, noise filtering, and feature recognition. Optimal compression algorithms find…

Subspace clustering aims to group data points into multiple clusters of which each corresponds to one subspace. Most existing subspace clustering approaches assume that input data lie on linear subspaces. In practice, however, this…

This paper proposes a method for solving multivariate regression and classification problems using piecewise linear predictors over a polyhedral partition of the feature space. The resulting algorithm that we call PARC (Piecewise Affine…

A slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the…

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of…