Related papers: Solving high-dimensional eigenvalue problems using…

While Diffusion Monte Carlo (DMC) is in principle an exact stochastic method for \textit{ab initio} electronic structure calculations, in practice the fermionic sign problem necessitates the use of the fixed-node approximation and trial…

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based…

We introduce a semistochastic implementation of the power method to compute, for very large matrices, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix…

We develop diffusion-based samplers for target distributions known up to a normalising constant. To this end, we rely on the well-known diffusion path that smoothly interpolates between a simple base distribution and the target, popularised…

Fixed-node diffusion Monte Carlo (FNDMC) is a stochastic quantum many-body method that has a great potential in electronic structure theory. We examine how FNDMC satisfies exact constraints, linearity and derivative discontinuity of total…

Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…

The history of research on eigenvalue problems is rich with many outstanding contributions. Nonetheless, the rapidly increasing size of data sets requires new algorithms for old problems in the context of extremely large matrix dimensions.…

A certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients is examined. The strategy relies on computing the second order spectrum relative to subspaces…

Diffusion models (DMs) have recently shown outstanding capabilities in modeling complex image distributions, making them expressive image priors for solving Bayesian inverse problems. However, most existing DM-based methods rely on…

We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions or combinations of either for different parts…

The solution to a stochastic optimal control problem can be determined by computing the value function from a discretization of the associated Hamilton-Jacobi-Bellman equation. Alternatively, the problem can be reformulated in terms of a…

This work proposes a novel numerical scheme for solving the high-dimensional Hamilton-Jacobi-Bellman equation with a functional hierarchical tensor ansatz. We consider the setting of stochastic control, whereby one applies control to a…

A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed…

We develop exact Markov chain Monte Carlo methods for discretely-sampled, directly and indirectly observed diffusions. The qualification "exact" refers to the fact that the invariant and limiting distribution of the Markov chains is the…

In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using MC-fPINNs. We construct two neural networks $ u_{NN}(x;\theta )$ and $f_{NN}(x;\psi)$ to approximate the solution $u^{*}(x)$ and the…

Eigenvalue problems are critical to several fields of science and engineering. We expand on the method of using unsupervised neural networks for discovering eigenfunctions and eigenvalues for differential eigenvalue problems. The obtained…

We derive explicit solution representations for linear, dissipative, second-order Initial-Boundary Value Problems (IBVPs) with coefficients that are spatially varying, with linear, constant-coefficient, two-point boundary conditions. We…

Discrete time stochastic optimal control problems and Markov decision processes (MDPs), respectively, serve as fundamental models for problems that involve sequential decision making under uncertainty and as such constitute the theoretical…

Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart hyper-Bessel operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansion…

Stochastic differential equations play an important role in various applications when modeling systems that have either random perturbations or chaotic dynamics at faster time scales. The time evolution of the probability distribution of a…