Related papers: Robust discretization in quantized tensor train fo…
In this article, we design an original solver based on Quantized Tensor Trains (QTT) for linear elliptic equations with heterogeneous coefficient field, that allows for extremely fine meshes. It can achieve full-field simulations in…
We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical…
We present an efficient and robust numerical algorithm for solving the two-dimensional linear elasticity problem that combines the Quantized Tensor Train format and a domain partitioning strategy. This approach makes it possible to solve…
Quantized tensor trains (QTTs) are a multiscale computational framework that can potentially reduce the computational cost of solving partial differential equations and initial value problems by making low-rank approximations. However, its…
In the last two decades, increased need for high-fidelity simulations of the time evolution and propagation of forces in granular media has spurred renewed interest in discrete element method (DEM) modeling of frictional contact. Force…
Accurately solving high-dimensional partial differential equations (PDEs) remains a central challenge in computational mathematics. Traditional numerical methods, while effective in low-dimensional settings or on coarse grids, often…
Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial…
The goal of this paper is to develop a numerical algorithm that solves a two-dimensional elliptic partial differential equation in a polygonal domain using tensor methods and ideas from isogeometric analysis. The proposed algorithm is based…
We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include…
Nonlinear filtering with correlated noise leads to a Duncan-Mortensen-Zakai (DMZ) equation in the form of a stochastic partial differential equation (SPDE). Unlike the independent noise case, the presence of correlation prevents the…
The numerical solution of kinetic equations is challenging due to the high dimensionality of the underlying phase space. In this paper, we develop a dynamical low-rank method based on the projector-splitting integrator in tensor-train (TT)…
We develop an unconditionally energy-stable tensor-product space-time discretization framework for the solution of a linear kinetic transport equation in one space dimension. The kinetic equation is a simplified model of radiative transfer…
Spectral methods provide highly accurate numerical solutions for partial differential equations, exhibiting exponential convergence with the number of spectral nodes. Traditionally, in addressing time-dependent nonlinear problems, attention…
We introduce a fully discrete scheme to solve a class of high-dimensional Mean Field Games systems. Our approach couples semi-Lagrangian (SL) time discretizations with Tensor-Train (TT) decompositions to tame the curse of dimensionality. By…
We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion partial differential equations (PDEs) in one dimension. Specifically, we show that, independently of the scale of…
When solving the time-dependent radiative transport equation (RTE), implicit time discretization is often employed for its robustness and stability. This results in a sequence of steady-state RTEs with identical cross-sections but varying…
In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely…
Solving partial differential equations of highly featured problems represents a formidable challenge, where reaching high precision across multiple length scales can require a prohibitive amount of computer memory or computing time.…
We present a quantum-inspired solver for the one-dimensional Gross-Pitaevskii equation in the Quantics Tensor-Train (QTT) representation. By evolving the system entirely within a low-rank tensor manifold, the method sidesteps the memory and…
We propose a framework for discrete scientific data compression based on the tensor-train (TT) decomposition. Our approach is tailored to handle unstructured output data from discrete element method (DEM) simulations, demonstrating its…