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.
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.
This paper is concerned with a mean-reversion trading rule. In
contrast to most market models treated in the literature, the underlying market
is solely determined by a two-state Markov chain. The major advantage of
such Markov chain model is its striking simplicity and yet its capability of
capturing various market movements. The purpose of this paper is to study
an optimal trading rule under such a model. The objective of the problem
under consideration is to nd a sequence stopping (buying and selling) times
so as to maximize an expected return. Under some suitable conditions, explicit
solutions to the associated HJ equations (variational inequalities) are obtained.
The optimal stopping times are given in terms of a set of threshold levels. A
verication theorem is provided to justify their optimality. Finally, a numerical
example is provided to illustrate the results.
This paper considers regularized block multiconvex optimization, where the feasible set and objective
function are generally nonconvex but convex in each block of variables. It also accepts nonconvex
blocks and requires these blocks to be updated by proximal minimization. We review some interesting
applications and propose a generalized block coordinate descent method. Under certain
conditions, we show that any limit point satisfies the Nash equilibrium conditions. Furthermore, we
establish global convergence and estimate the asymptotic convergence rate of the method by assuming
a property based on the Kurdyka-Lojasiewicz inequality. The proposed algorithms are tested on
nonnegative matrix and tensor factorization, as well as matrix and tensor recovery from incomplete
observations. The tests include synthetic data and hyperspectral data, as well as image sets from
the CBCL and ORL databases. Compared to the existing state-of-the-art algorithms, the proposed
algorithms demonstrate superior performance in both speed and solution quality. The MATLAB
code of nonnegative matrix/tensor decomposition and completion, along with a few demos, are
accessible from the authors' homepages.