Determining surface temperature and heat flux by a wavelet dual least squares method

We apply a wavelet dual least squares method to a general sideways parabolic equation for determining surface temperature and surface heat flux. Connecting Meyer wavelet bases with a special project method dual least squares method, we can obtain a regularized solution. Meanwhile, order optimal error estimates between the approximate solution and exact solution are proved.     

Multiple Penalty Regression: Fitting and Extrapolating a Discrete Incomplete Multi-way Layout

The d iscrete multi-way layout is an abstract data type associated with regression, experimental designs, digital images or videos, spatial statistics, gene or protein chips, and more. The factors influencing response can be nominal or ordinal. The observed factor level combinations are finitely discrete and often incomplete or irregularly spaced. This paper develops low risk biased estimators of the means at the observed factor level combinations; and extrapolates the estimated means to larger discrete complete layouts. Candidate penalized least squares (PLS) estimators with multiple quadratic penalties express competing conjectures about each of the main effects and interactions in the analysis of variance decomposition of the means. The candidate PLS estimator with smallest estimated quadratic risk attains, asymptotically, the smallest risk over all candidate PLS estimators. In the theoretical analysis, the dimension of the regression space tends to infinity. No assumptions are made about the unknown means or about replication.     

Low Rank Approximation of a Hankel Matrix by Structured Total Least Norm

The structure preserving rank reduction problem arises in many important applications. The singular value decomposition (SVD), while giving the closest low rank approximation to a given matrix in matrix L ₂ norm and Frobenius norm, may not be appropriate for these applications since it does not preserve the given structure. We present a new method for structure preserving low rank approximation of a matrix, which is based on Structured Total Least Norm (STLN). The STLN is an efficient method for obtaining an approximate solution to an overdetermined linear system AX B, preserving the given linear structure in the perturbation [E F] such that (A + E)X = B + F. The approximate solution can be obtained to minimize the perturbation [E F] in the L _p norm, where p = 1, 2, or . An algorithm is described for Hankel structure preserving low rank approximation using STLN with L _p norm. Computational results are presented, which show performances of the STLN based method for L ₁ and L ₂ norms for reduced rank approximation for Hankel matrices.