In this work, we employ the Bayesian inference framework to solve problem of estimating solution and particularly, its derivatives, which satisfy a known differential equation, from given noisy scarce observations data only. To address key issue accuracy robustness derivative estimation, use Gaussian processes jointly model solution, equation. By regarding linear equation as constraint, process...

Abstract We prove new well-posedness results for dispersion-generalized Kadomtsev–Petviashvili I equations in R 2 , which family links the classical KP-I equation with fifth or...

Abstract In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse dimensionality many different applications and have proven to be so case some methods parabolic PDEs. this paper, we revi...

In this paper, we propose a new method to specify the sequence of parameter values for a fuzzy clustering algorithm by using Q-learning. In the clustering algorithm, we employ similarities between two data points and distances from data to cluster centers as the fuzzy clustering criteria. The fuzzy clustering is achieved by optimizing an objective function which is solved by the Picard iteratio...

A new algorithm for modeling radiative transfer in inhomogeneous three-dimensional media is described. The spherical harmonics discrete ordinate method uses a spherical harmonic angular representation to reduce memory use and time computing the source function. The radiative transfer equation is integrated along discrete ordinates through a spatial grid to model the streaming of radiation. An a...

This paper studies exponential stability properties of a class of two-dimensional (2D) systems called differential repetitive processes (DRPs). Since a distinguishing feature of DRPs is that the problem domain is bounded in the “time” direction, the notion of stability to be evaluated does not require the nonlinear system defining a DRP to be stable in the typical sense. In particular, we study...

We derive a deterministic particle method for the solution of nonlinear reactiondiffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of ...

Accurate and efficient orbital propagators are critical for space situational awareness because they drive uncertainty propagation which is necessary for tracking, conjunction analysis, and maneuver detection. We have developed an adaptive, implicit Runge-Kuttabased method for orbit propagation that is superior to existing explicit methods, even before the algorithm is potentially parallelized....

These notes are a general introduction and exposition. They contain more theory and more appendices than are likely to appear in any paper but components will be published. We start with relatively well-known theory including an important variation–of– parameters formula whose (period)=(delay) form is equation (7). Next we review the Sinha–Wu [SW] method of approximation of fundamental solution...

We study an iterative low-rank approximation method for the solution of the steadystate stochastic Navier–Stokes equations with uncertain viscosity. The method is based on linearization schemes using Picard and Newton iterations and stochastic finite element discretizations of the linearized problems. For computing the low-rank approximate solution, we adapt the nonlinear iterations to an inexa...