相关论文: An Adaptive-rank Approach with Greedy Sampling for…

High-order semi-Lagrangian methods for kinetic equations have been under rapid development in the past few decades. In this work, we propose a semi-Lagrangian adaptive rank (SLAR) integrator in the finite difference framework for linear…

Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision…

We introduce novel dynamical low-rank methods for solving large-scale matrix differential equations, motivated by algorithms from randomized numerical linear algebra. In terms of performance (cost and accuracy), our methods overperform…

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal…

It has recently been demonstrated that dynamical low-rank algorithms can provide robust and efficient approximation to a range of kinetic equations. This is true especially if the solution is close to some asymptotic limit where it is known…

We extend our previous work on a semi-Lagrangian adaptive rank (SLAR) integrator, in the finite difference framework for nonlinear Vlasov-Poisson systems, to the general high-order tensor setting. The proposed scheme retains the high-order…

Greedy-GQ is an off-policy two timescale algorithm for optimal control in reinforcement learning. This paper develops the first finite-sample analysis for the Greedy-GQ algorithm with linear function approximation under Markovian noise. Our…

Adaptive or dynamic signal sampling in sensing systems can adapt subsequent sampling strategies based on acquired signals, thereby potentially improving image quality and speed. This paper proposes a Bayesian method for adaptive sampling…

We consider the neural sparse representation to solve Boltzmann equation with BGK and quadratic collision model, where a network-based ansatz that can approximate the distribution function with extremely high efficiency is proposed.…

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…

Learning of low-rank matrices is fundamental to many machine learning applications. A state-of-the-art algorithm is the rank-one matrix pursuit (R1MP). However, it can only be used in matrix completion problems with the square loss. In this…

Adaptive regularization methods pre-multiply a descent direction by a preconditioning matrix. Due to the large number of parameters of machine learning problems, full-matrix preconditioning methods are prohibitively expensive. We show how…

Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and…

We introduce a novel adaptive Gaussian Process Regression (GPR) methodology for efficient construction of surrogate models for Bayesian inverse problems with expensive forward model evaluations. An adaptive design strategy focuses on…

This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only…

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…

We develop a mass-conserving, adaptive-rank solver for the 1D1V Wigner-Poisson system. Our work is motivated by applications to the study of the stopping power of $\alpha$ particles at the National Ignition Facility (NIF). In this regime,…

In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [S. Y. Cho, et al., Conservative semi-Lagrangian schemes for…

In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose…

We propose a high order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of…