Xue Luo

Hermite Spectral Method to 1-D Forward Kolmogorov Equation and Its Application to Nonlinear Filtering Problems

In this paper, we investigate the Hermite spectral method (HSM) to numerically solve the forward Kolmogorov equation (FKE). A useful guideline of choosing the scaling factor of the generalized Hermite functions is given in this paper. It greatly improves the resolution of HSM. The convergence rate of HSM to FKE is analyzed in the suitable function space and has been verified by the numerical simulation. As an important application and our primary motivation to study the HSM to FKE, we work on the implementation of the nonlinear filtering (NLF) problems with a real-time algorithm developed by S.-T. Yau and the second author in 2008. The HSM to FKE is served as the off-line computation in this algorithm. The translating factor of the generalized Hermite functions and the moving-window technique are introduced to deal with the drifting of the posterior conditional density function of the states in the on-line experiments. Two numerical experiments of NLF problems are carried out to illustrate the feasibility of our algorithm. Moreover, our algorithm surpasses the particle filters as a real-time solver to NLF.

Complete Real Time Solution of the General Nonlinear Filtering Problem Without Memory

It is well known that the nonlinear filtering problem has important applications in both military and civil industries. The central problem of nonlinear filtering is to solve the Duncan-Mortensen-Zakai (DMZ) equation in real time and in a memoryless manner. In this paper, we shall extend the algorithm developed previously by S.-T. Yau and the second author to the most general setting of nonlinear filterings, where the explicit time-dependence is in the drift term, observation term, and the variance of the noises could be a matrix of functions of both time and the states. To preserve the off-line virtue of the algorithm, necessary modifications are illustrated clearly. Moreover, it is shown rigorously that the approximated solution obtained by the algorithm converges to the real solution in the L1 sense. And the precise error has been estimated. Finally, the numerical simulation support the feasibility and efficiency of our algorithm.

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function ψ(x, y) which is the number of positive integers ≤ x and free of prime factors > y. Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional (n ≥ 3) real right-angled simplices. In this paper, we prove this Number Theoretic Conjecture for n = 5. As an application, we give a sharp estimate of Dickman-De Bruijn function ψ(x, y) for 5 ≤ y < 13.

Hermite Spectral Method with Hyperbolic Cross Approximations to High-Dimensional Parabolic PDEs

It is well known that the sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality” in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized HC approximations has been shown. Moreover, the error estimate of Hermite spectral method to high-dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.

Direct Method for Time-Varying Nonlinear Filtering Problems

This paper discusses how to solve a filtering problem for a class of continuous nonlinear time-varying systems via the Duncan–Mortensen–Zakai (DMZ) equation. In this paper, the original DMZ equation is changed into the Kolmogorov forward equation (KFE) by exponential transformations in each time interval, and then, under some assumptions, the KFE can be transformed into a time-varying Schrödinger equation, which can be solved explicitly. The novelty of this paper lies in how to transform the KFE into the Schrödinger equation. As a direct application, the results of the paper “Nonlinear filtering and time varying Schrodinger equation” are extended for time-varying Yau systems.

Novel suboptimal filter via higher order central moments

In this paper, we construct a new suboptimal filter by deriving the Itos stochastic differential equations of the estimation of higher order central moments, satisfy, and impose some conditions to form a closed system. The essentially infinite-dimensional cubic sensor problem has been investigated in detail numerically to illustrate the reasonableness of the imposed conditions, and the numerical experiments support our discussion. A two-dimensional polynomial filtering problem has also been experimented.

On classification of toric surface codes of low dimension

This work is a natural continuation of our previous work \cite{yz}. In this paper, we give a complete classification of toric surface codes of dimension less than or equal to 6, except a special pair,

A novel algorithm to solve the robust DMZ equation in real time

C_{P_6^{(4)}}

Suboptimal linear estimation for continuous–discrete bilinear systems

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Time-dependent Hermite-Galerkin spectral method and its applications

C_{P_6^{(5)}}