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We present the numerical application of a meshfree algorithm for the solution of fully nonlinear PDEs by Monte Carlo simulation using branching diffusion trees coded by the nonlinearities appearing in the equation. This algorithm is applied to the numerical solution of modified and non-Newtonian Burgers equations, and to a problem with boundary conditions in fluid dynamics, by the computation of a Poiseuille flow. Our implementation uses neural networks that yield a functional space-time domain estimation, and includes numerical comparisons with the deep Galerkin (DGM) and deep backward stochastic differential equation (BSDE) methods. [ABSTRACT FROM AUTHOR]

We present an algorithm for the numerical solution of ordinary differential equations by random enumeration of the Butcher trees used in the implementation of the Runge–Kutta method. Our Monte Carlo scheme allows for the direct numerical evaluation of an ODE solution at any given time within a certain interval, without iteration through multiple time steps. In particular, this approach does not involve a discretization step size, and it does not require the truncation of Taylor series. [ABSTRACT FROM AUTHOR]

The probability hypothesis density (PHD) filter, which is used for multi-target tracking based on sensor measurements, relies on the propagation of the first-order moment, or intensity function, of a point process. This algorithm assumes that targets behave independently, an hypothesis which may not hold in practice due to potential target interactions. In this paper, we construct a second-order PHD filter based on determinantal point processes which are able to model repulsion between targets. Such processes are characterized by their first- and second-order moments, which allows the algorithm to propagate variance and covariance information in addition to first-order target count estimates. Our approach relies on posterior moment formulas for the estimation of a general hidden point process after a thinning operation and a superposition with a Poisson point process, and on suitable approximation formulas in the determinantal point process setting. The repulsive properties of determinantal point processes apply to the modeling of negative correlation between distinct measurement domains. Monte Carlo simulations with correlation estimates are provided. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Monte Carlo numerical computation of the Laplace transform of exponential Brownian functionals. In addition to the implementation of standard integral formulas, we investigate the use of various probabilistic representations. This involves in particular the simulation of the hyperbolic secant distribution and the use of several variance reduction schemes. The performance of those methods and their conditions of application are compared. [ABSTRACT FROM AUTHOR]