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Feature extraction from persistence diagrams, as a tool to enrich machine learning techniques, has received increasing attention in recent years. In this paper we explore an adaptive methodology to localize features in persistent diagrams,…
Gradient descent is a popular algorithm in optimization, and its performance in convex settings is mostly well understood. In non-convex settings, it has been shown that gradient descent is able to escape saddle points asymptotically and…
An efficient multigrid framework is developed for the time marching of steady-state compressible flows with a spatially high-order ($p$-order polynomial) modal discontinuous Galerkin method. The core algorithm that based on a global…
This is the second part of our error analysis of the stabilized Lagrange-Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi-Pitk\"aranta's…
We analyze the performance of the alternating direction method of multipliers (ADMM) to track, in a decentralized manner, a solution of a stochastic sequence of optimization problems parametrized by a discrete time Markov process. The main…
Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps…
The aim of this paper is to apply a high-order discontinuous-in-time scheme to second-order hyperbolic partial differential equations (PDEs). We first discretize the PDEs in time while keeping the spatial differential operators…
A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials…
We present a stochastic descent algorithm for unconstrained optimization that is particularly efficient when the objective function is slow to evaluate and gradients are not easily obtained, as in some PDE-constrained optimization and…
We present the second-order multidimensional Staggered Grid Hydrodynamics Residual Distribution (SGH RD) scheme for Lagrangian hydrodynamics. The SGH RD scheme is based on the staggered finite element discretizations as in [Dobrev et al.,…
A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of…
The accuracy of solving partial differential equations (PDEs) on coarse grids is greatly affected by the choice of discretization schemes. In this work, we propose to learn time integration schemes based on neural networks which satisfy…
In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured…
For uncertainty propagation of highly complex and/or nonlinear problems, one must resort to sample-based non-intrusive approaches [1]. In such cases, minimizing the number of function evaluations required to evaluate the response surface is…
In [Heimann, Lehrenfeld, Preu{\ss}, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165] new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and…
Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and…
We present a nonlinear stabilized Lagrange-Galerkin scheme for the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi-Pitk\"aranta's stabilization method for the conforming linear…
Most numerical schemes proposed for solving BGK models for rarefied gas dynamics are based on the discrete velocity approximation. Since such approach uses fixed velocity grids, one must secure a sufficiently large domain with fine velocity…
This paper focuses on solving the capacitated arc routing problem with time-dependent service costs (CARPTDSC), which is motivated by winter gritting applications. In the current literature, exact algorithms designed for CARPTDSC can only…
We present a new numerical method for transporting arbitrary sets in a velocity field. The method computes a deformation mapping of the domain and advects particular sets by function composition with the map. This also allows for the…