In this paper, a brief introduction of the nonlinear filtering problems and a review of the quasi-implicit Euler method are presented. The major contribution of this paper is that we propose a nonnegativity-preserving algorithm of Yau-Yau method for solving high-dimensional nonlinear filtering problems by applying quasi-implicit Euler method with discrete sine transform. Furthermore, our algorithms are directly applicable on the compact difference schemes, so that the number of spatial points can be substantially reduced and retain the same accuracy. Numerical results indicate that the proposed algorithm is capable of solving up to six-dimensional nonlinear filtering problems efficiently and accurately.

It is well known that the nonlinear filter has important applications in military, engineering and commercial industries. In this paper, we propose efficient and accurate numerical algorithms for the realization of the Yau-Yau method for solving nonlinear filtering problems by using finite difference schemes. The Yau-Yau method reduces the nonlinear filtering problem to the initial-value problem of Kolmogorov equations. We first solve this problem by the implicit Euler method, which is stable in most cases, but costly. Then, we propose a quasi-implicit Euler method which is feasible for acceleration by fast Fourier transformations. Furthermore, we propose a superposition technique which enables us to deal with the nonlinear filtering problem in an off-time process and thus, save a large amount of computational cost. Next, we prove that the numerical solutions of Kolmogorov equations by our schemes are always nonnegative in each iteration. Consequently, our iterative process preserves the probability density functions. In addition, we prove convergence of our schemes under some mild conditions. Numerical results show that the proposed algorithms are efficient and promising.

An n dimensional minimal submanifold $\sum$  of $R^{n+m}$ is called nonparametric if $\sum$  can be represented as the graph of a vector-valued function $f : D \subset \mathbb{ R}^n \to \mathbb{R}^n$. This note provides a sufficient condition for the stability of such $\sum$ in terms of the norm of the differential $df$.

Let $\Omega $ be a bounded $C^2$ domain in $R^n$ and $\phi : \partial \Omega \to \mathbb{R}^m$ be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map $f : \Omega \to \mathbb{R}^m$ with $ f |_{\partial \Omega}$ and with the graph of $f$ a minimal submanifold in $R^{n+m}$. For $m = 1$, the Dirichlet problem was solved more than thirty years ago by Jenkins-Serrin [13] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension $m$. We prove if $\psi : \bar{\Omega} \to \mathbb{R}^m$ satisfies $8n\delta sup_{\Omega}||D^2 \psi| + \sqrt{2} sup_{\partial \Omega} |D\psi| <1$, then the Dirichlet problem for $ \psi |_{\partial \Omega}$ is solvable in smooth maps. Here $\delta$ is the diameter of $\Omega$ . Such a condition is necessary in view of an example of Lawson-Osserman [15]. In order to prove this result, we study the associated parabolic system and solve the Cauchy-Dirichlet problem with $\psi$ as initial data.

Watermarking algorithms provide a way of hiding or embedding some bits of information in a watermark. In the case of watermarking a 3D model, many algorithms employ a so-called indexed localization scheme. In this paper, we propose an optimization framework with two new steps for such watermarking algorithms to improve their capacity and invisibility. The first step is to find an optimal layout of invariant units to improve capacity. The second step is to rearrange the correspondence between the watermark units and the invariant units to improve invisibility. Experimental tests show that by using this framework, the capacity and invisibility of watermarking algorithms can be greatly improved.