This paper introduces a novel framework for the study of the total-variation model for image denoising. In the model, the denoised image is the proximity operator of the total-variation evaluated at a given noisy image. The totalvariation can be viewed as the composition of a convex function (the 1 norm for the anisotropic total-variation or the 2 norm for the isotropic totalvariation) with a linear transformation (the first-order difference operator). These two facts lead us to investigate the proximity operator of the composition of a convex function with a linear transformation. Under the assumption that the proximity operator of a given convex function (e.g., the 1 norm or the 2 norm) can be readily obtained, we propose a fixed-point algorithm for computing the proximity operator of the composition of the convex function with a linear transformation. We then specialize this fixed-point methodology to the total-variation denoising models. The resulting algorithms are compared with the Goldstein-Osher split-Bregman denoising algorithm. An important advantage of the fixed-point framework leads us to a convenient analysis for convergence of the proposed algorithms as well as a platform for us to develop efficient numerical algorithms via various fixed-point iterations. Our numerical experience indicates that the methods proposed here perform favorably.

We introduce the notion of reproducing kernel Banach spaces (RKBS) and study special semiinner-product RKBS by making use of semi-inner-products and the duality mapping. Properties of an RKBS and its reproducing kernel are investigated. As applications, we develop in the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, support vector machines, and kernel principal component analysis. In particular, existence, uniqueness and representer theorems are established.

Sequence alignment is a long standing problem in bioinformatics. The Basic Local Alignment Search Tool (BLAST) is one of the most popular and fundamental alignment tools. The explosive growth of biological sequences calls for speedup of sequence alignment tools such as BLAST. To this end, we develop high speed BLASTN (HS-BLASTN), a parallel and fast nucleotide database search tool that accelerates MegaBLAST—the default module of NCBI-BLASTN. HS-BLASTN builds a new lookup table using the FMD-index of the database and employs an accurate and effective seeding method to find short stretches of identities (called seeds) between the query and the database. HS-BLASTN produces the same alignment results as MegaBLAST and its computational speed is much faster than MegaBLAST. Specifically, our experiments conducted on a 12-core server show that HS-BLASTN can be 22 times faster than MegaBLAST and exhibits better parallel performance than MegaBLAST. HS-BLASTN is written in C++ and the related source code is available at https://github.com/chenying2016/queries under the GPLv3 license.

We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition.

We propose a preconditioned alternating projection algorithm (PAPA) for solving the maximum a posteriori (MAP) emission computed tomography (ECT) reconstruction problem. Specifically, we formulate the reconstruction problem as a constrained convex optimization problem with the total variation (TV) regularization. We then characterize the solution of the constrained convex optimization problem and show that it satisfies a system of fixed-point equations defined in terms of two proximity operators raised from the convex functions that define the TV-norm and the constrain involved in the problem. The characterization (of the solution) via the proximity operators that define two projection operators naturally leads to an alternating projection algorithm for finding the solution. For efficient numerical computation, we introduce to the alternating projection algorithm a preconditioning matrix (the EM-preconditioner) for the dense system matrix involved in the optimization problem. We prove theoretically convergence of the preconditioned alternating projection algorithm. In numerical experiments, performance of our algorithms, with an appropriately selected preconditioning matrix, is compared with performance of the conventional MAP expectation-maximization (MAP-EM) algorithm with TV regularizer (EM-TV) and that of the recently developed nested EM-TV algorithm for ECT reconstruction. Based on the numerical experiments performed in this work, we observe that the alternating projection algorithm with the EM-preconditioner outperforms significantly the EM-TV in all aspects including the convergence speed, the noise in the reconstructed images and the image quality. It also outperforms the nested EM-TV in the convergence speed while providing comparable image quality.

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This paper studies higher-order finite volume methods for solving elliptic boundary value problems. We develop a general framework for construction and analysis of higher-order finite volume methods. Specifically, we establish the boundedness and uniform ellipticity of the bilinear forms for the methods, and show that they lead to an optimal error estimate of the methods. We prove that the uniform local-ellipticity of the family of the bilinear forms ensures its uniform ellipticity. We then establish necessary and sufficient Communicated by Aihui Zhou. 192 Z. Chen et al.conditions for the uniform local-ellipticity in terms of geometric requirements on the meshes of the domain of the differential equation, and provide a general way to investigate the mesh geometric requirements for arbitrary higher-order schemes. Several useful examples of higher-order finite volume methods are presented to illustrate the mesh geometric requirements.

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