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Clustering multi-dimensional points is a fundamental task in many fields, and density-based clustering supports many applications as it can discover clusters of arbitrary shapes. This paper addresses the problem of Density-Peaks Clustering…
The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply…
In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such…
A general method for numerical computation of the thermal density matrix of a single-particle quantum system is presented. The Schrodinger equation in imaginary time tau is solved numerically by the finite difference time domain (FDTD)…
We present a framework for solving time-dependent partial differential equations (PDEs) in the spirit of the random feature method. The numerical solution is constructed using a space-time partition of unity and random feature functions.…
We establish a continuous-time framework for analyzing Deep Q-Networks (DQNs) via stochastic control and Forward-Backward Stochastic Differential Equations (FBSDEs). Considering a continuous-time Markov Decision Process (MDP) driven by a…
We propose a data-driven method to learn the time-dependent probability density of a multivariate stochastic process from sample paths, assuming that the initial probability density is known and can be evaluated. Our method uses a novel…
We approximate the solution for the time dependent Schr\"odinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of functions on rank-1 or rank-r lattice points with unitary Fourier…
We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main…
How can we tell complex point clouds with different small scale characteristics apart, while disregarding global features? Can we find a suitable transformation of such data in a way that allows to discriminate between differences in this…
We develop a numerical framework, the Deep Tangent Bundle (DTB) method, that is suitable for computing solutions of evolutionary partial differential equations (PDEs) in high dimensions. The main idea is to use the tangent bundle of an…
Temporal difference (TD) learning is a cornerstone reinforcement learning (RL) method for policy evaluation, where the goal is to estimate the value function of a Markov decision process under a fixed policy. While a substantial body of…
This paper deals with a complete bipartite matching problem with the objective of finding an optimal matching that maximizes a certain generic predefined utility function on the set of all matchings. After proving the NP-hardness of the…
The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger equation in an explicitly iterative process. However, the method requires the spatial grid…
Value functions arise as a component of algorithms as well as performance metrics in statistics and engineering applications. Computation of the associated Bellman equations is numerically challenging in all but a few special cases. A…
This paper is devoted to show the results obtained by using the magnitude-squared coherence for determining order-chaos transition in a system described by the logistic equation dynamics. For determining the power spectral density of a…
This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial…
It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm $R$-linearly converges to a Karush-Kuhn-Tucker point, provided that the…
We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line which contains dynamical correlations that change irregularly under parameter variation. Capturing…
A stochastic approach to time-dependent density functional theory (TDDFT) is developed for computing the absorption cross section and the random phase approximation (RPA) correlation energy. The core idea of the approach involves…