We propose a computational framework named iterative local adaptive majorize-minimization (I-LAMM) to simultaneously control algorithmic complexity and statistical error when fitting high dimensional models. I-LAMM is a two-stage algorithmic implementation of the local linear approximation to a family of folded concave penalized quasi-likelihood. The first stage solves a convex program with a crude precision tolerance to obtain a coarse initial estimator, which is further refined in the second stage by iteratively solving a sequence of convex programs with smaller precision tolerances. Theoretically, we establish a phase transition: the first stage has a sublinear iteration complexity, while the second stage achieves an improved linear rate of convergence. Though this framework is completely algorithmic, it provides solutions with optimal statistical performances and controlled algorithmic complexity for a large family

Surface based 3D shape analysis plays a fundamental role in computer vision and medical imaging. This work proposes to use optimal mass transport map for shape matching and comparison, focusing on two important applications including surface registration and shape space. The computation of the optimal mass transport map is based on Monge-Brenier theory, in comparison to the conventional method based on Monge-Kantorovich theory, this method significantly improves the efficiency by reducing computational complexity from O(n ² ) to O(n). For surface registration problem, one commonly used approach is to use conformal map to convert the shapes into some canonical space. Although conformal mappings have small angle distortions, they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis. This work proposes to compose the

This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three dimensional photonic crystals with face centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a preconditioner. We also demonstrate how two forms of matrix-vector multiplications, which are the most costly part of the inverse Lanczos method, can be computed by fast Fourier transformation due to the eigendecomposition to significantly reduce the computation cost. Integrating all of these findings and techniques, we obtain a fast eigenvalue solver. The solver has been implemented by MATLAB and successfully solves each of a set of $5.184$ million dimension eigenvalue problems within $50$ to $104$ minutes on a workstation with two Intel Quad-C ore Xeon X5687 3.6 GHz CPUs.

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Solutions for many problems of interest exhibit singular behaviors at domain corners or points where boundary condition changes type. For this type of problems, direct spectral methods with usual polynomial basis functions do not lead to a satisfactory convergence rate. We develop in this paper a M\"untz-Galerkin method which is based on specially tuned M\"untz polynomials to deal with the singular behaviors of the underlying problems. By exploring the relations between Jacobi polynomials and M\"untz polynomials, we develop efficient implementation procedures for the M\"untz-Galerkin method, and provide optimal error estimates. As examples of applications, we consider the Poisson equation with mixed Dirichlet-Neumann boundary conditions, whose solution behaves like $O(r^{1/2})$ near the singular point, and demonstrate that the M\"untz-Galerkin method greatly improves the rates of convergence of the usual spectral method.