Related papers: Dynamical Low-Rank Approximation for Stochastic Di…
It is shown that the computational efficiency of the discrete least-squares (DLS) approximation of solutions of stochastic elliptic PDEs is improved by incorporating a reduced-basis method into the DLS framework. The goal is to recover the…
Low-Rank Adaptation (LoRA) has proven effective in reducing computational costs while maintaining performance comparable to fully fine-tuned foundation models across various tasks. However, its fixed low-rank structure restricts its…
Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation…
We present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each…
Low-rank plus diagonal (LRPD) decompositions provide a powerful structural model for large covariance matrices, simultaneously capturing global shared factors and localized corrections that arise in covariance estimation, factor analysis,…
Time-dependent kinetic models are ubiquitous in computational science and engineering. The underlying integro-differential equations in these models are high-dimensional, comprised of a six--dimensional phase space, making simulations of…
The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational…
A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
Computing numerical solutions of the thermal radiative transfer equations on a finely resolved grid can be costly due to high computational and memory requirements. A numerical reduced order method that has recently been applied to a wide…
Low-rank regularization (LRR) has been widely applied in various machine learning tasks, but the associated optimization is challenging. Directly optimizing the rank function under constraints is NP-hard in general. To overcome this…
We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and…
It is known that standard stochastic Galerkin methods face challenges when solving partial differential equations (PDEs) with random inputs. These challenges are typically attributed to the large number of required physical basis functions…
Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is…
This work considers the low-rank approximation of a matrix $A(t)$ depending on a parameter $t$ in a compact set $D \subset \mathbb{R}^d$. Application areas that give rise to such problems include computational statistics and dynamical…
Distributional reinforcement learning (DRL) enhances the understanding of the effects of the randomness in the environment by letting agents learn the distribution of a random return, rather than its expected value as in standard RL. At the…
Dynamic Rank Reinforcement Learning (DR-RL) approximations rely on static rank assumptions, limiting their flexibility across diverse linguistic contexts. Our method dynamically modulates ranks based on real-time sequence dynamics,…
The recent introduction of the Least-Squares Support Vector Regression (LS-SVR) algorithm for solving differential and integral equations has sparked interest. In this study, we expand the application of this algorithm to address systems of…
We present a novel solution method for It\^o stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation between two successive intervals. The main…
Predicting how the microstructure governs the mechanical response of heterogeneous materials is essential for optimizing design and performance. Yet this task remains difficult due to the complex, high dimensional nature of microstructural…