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.

Structural variations in genome could be more critical than SNP (Single-Nucleotide Polymorphism) in causing tumor cells, immune system diseases and fatness. It is necessary and important to detect structural variations with next generation sequencing data by means of efficient and accurate algorithms. This paper introduced a new algorithm using k-mers (short fragments of genome), string algorithms and graph algorithms, which combines similarity check, variation detection with sequence assembly. Our results showed that our method has high accuracy and it is more reliable than methods using reference and more efficient than popular algorithms.

Recently sparse representation has been applied to visual tracker by modeling the target appearance using a sparse approximation over a template set, which leads to the so-called L_1 trackers as it needs to solve an ℓ_1 norm related minimization problem for many times. While these L_1 trackers showed impressive tracking accuracies, they are very computationally demanding and the speed bottleneck is the solver to ℓ_1 norm minimizations. This paper aims at developing an L_1 tracker that not only runs in real time but also enjoys better robustness than other L_1 trackers. In our proposed L_1 tracker, a new ℓ_1 norm related minimization model is proposed to improve the tracking accuracy by adding an ℓ_1 norm regularization on the coefficients associated with the trivial templates. Moreover, based on the accelerated proximal gradient approach, a very fast numerical solver is developed to solve the resulting ℓ_1 norm related minimization problem with guaranteed quadratic convergence. The great running time efficiency and tracking accuracy of the proposed tracker is validated with a comprehensive evaluation involving eight challenging sequences and five alternative state-of-the-art trackers.

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