MathSciDoc: An Archive for Mathematician ∫

A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theoretical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Korteweg-de Vries-Burgers (KdVB) equation and its special cases, including the heat equation and the Burgers' equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.

In this paper we derive the stochastic differentials of the conditional central moments of the nonlinear filtering problems, especially those of the polynomial filtering problem, and develop a novel suboptimal method by solving this evolution equation. The basic idea is to augment the state of the original nonlinear system by including the original states' conditional central moments such that the augmented states form a so-called bilinear system after truncating. During our derivation, it is clear to see that the stochastic differentials of the conditional central moments of the linear filtering problem (i.e., $f$, $g$ and $h$ are all at most degree one polynomials) form a closed system automatically without truncation. This gives one reason for the existence of optimal filtering for linear problems. On the contrary, the conditional central moments form an infinite dimensional system, in general. To reduce it to a closed-form, we let all the high enough central moments to be zero, as one did in the Carleman approach \cite{GMP}. Consequently, a novel suboptimal method is developed by dealing with the bilinear system. Numerical simulation is performed for the cubic sensor problem to illustrate the accuracy and numerical stability.

The singular parabolic problem $u_t-\triangle u=\l{\frac{1+\d|\nabla u|^2}{(1-u)^2}}$ on a bounded domain $\O$ of $\Rn$ with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value $\l_\d^*>0$ such that if $0<\l<\l_\d^*$, all solutions exist globally; while for $\l>\l_\d^*$, all the solution will quench in finite time. The estimate of the quenching time in terms of large voltage $\l$ is investigated. Furthermore, the quenching set is a compact subset of $\O$, provided $\O$ is a convex bounded domain in $\mathbb{R}^n$. In particular, if the domain $\O$ is radially symmetric, then the origin is the only quenching point. We not only derive the one-side estimate of the quenching rate, but also further study the refined asymptotic behavior of the finite quenching solution.

In this paper, we construct a new suboptimal filter by deriving the Ito's stochastic differential equations of the estimation of higher order central moments satisfy and imposing 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 2-dimensional polynomial filtering problem has also been experimented.

In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $\bu = \n^\perp \Psi$, $\Psi(r,\th) = r^{\l} \psi(\th)$, for $\l >0$, we show that only trivial solutions exist in the range $0<\l<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\l>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\l}) \cap \N$. The case $\l = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity $1/3$.

The surveys in the field of nonlinear filtering (NLF) are enumerous. Most of them are application-oriented and served as the tutorials for the practioners. The local approaches, including Kalman filter and its invariants, have already been studied from various point of views, due to its off-the-shelf implementation and wide applications. However, it cannot give good estimation of the states in highly nonlinear system or with non-Gaussian initial conditional density functions. Moreover, while the local methods only approximate the mean and variance, the global ones seek the way to directly obtain the conditional density function of the states. Consequently, all the statistical information is acquired. In this survey, we shall briefly go through the local approaches and put emphases on the existing three major global approaches: finite-dimensional NLF, sequential Monte Carlo methods (particle filter) and the {Yau-Yau's} on- and off-line solver of Duncan-Mortensen-Zakai's equation {\cite{YY_08}}. The discussions are mainly from the mathematical point of view.

It is well-known that 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 hyperbolic cross 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.

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.

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 virture 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 $L^1$ sense. And the precise error has been estimated. Finally, the numerical simulation support the feasibility and efficiency of our algorithm.

It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality" in some degree. In this note, we give the error estimate of hyperbolic cross (HC) approximations with all sorts of Askey polynomials. These polynomials are useful in generalized polynomial chaos (gPC) in the field of uncertainty quantification. The exponential convergences in both regular and optimized HC approximations have been shown under the condition that the random variable depends on the random inputs smoothly in some degree. Moreover, we apply gPC to numerically solve the ordinary differential equations with slightly higher dimensional random inputs. Both regular and optimized HC have been investigated with Laguerre-chaos, Charlier-chaos and Hermite-chaos in the numerical experiment. The discussion of the connection between the standard ANOVA approximation and Galerkin approximation is in the appendix.