A High-Quality Preconditioning Technique for Multi-Length-Scale Symmetric Positive Definite Linear Systems

We study preconditioning techniques used in conjunction with the conjugate gradient method for solving multi-length-scale symmetric positive definite linear systems originating from the quantum Monte Carlo simulation of electron interaction of correlated materials. Existing preconditioning techniques are not designed to be adaptive to varying numerical properties of the multi-length-scale systems. In this paper, we propose a hybrid incomplete Cholesky （HIC） preconditioner and demonstrate its adaptivity to the multi-length-scale systems. In addition, we propose an extension of the compressed sparse column with row access （CSCR） sparse matrix storage format to efficiently accommodate the data access pattem to compute the HIC preconditioner. We show that for moderately correlated materials, the HIC preconditioner achieves the optimal linear scaling of the simulation. The development of a linear-scaling preconditioner for strongly correlated materials remains an open topic.

BANDED M-MATRIX SPLITTING PRECONDITIONER FOR RIESZ SPACE FRACTIONAL REACTION-DISPERSION EQUATION

Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme.However,after estimating the condition number of the coefficient matrix of the discretized scheme,we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small.To overcome this deficiency,we further develop an effective banded M-matrix splitting preconditioner for the coefficient matrix.Some properties of this preconditioner together with its preconditioning effect are discussed.Finally,Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.

Efficient image representation for object recognition via pivots selection

Patch-level features are essential for achieving good performance in computer vision tasks. Besides well- known pre-defined patch-level descriptors such as scalein- variant feature transform （SIFT） and histogram of oriented gradient （HOG）, the kernel descriptor （KD） method [1] of- fers a new way to ＂grow-up＂ features from a match-kernel defined over image patch pairs using kernel principal compo- nent analysis （KPCA） and yields impressive results. In this paper, we present efficient kernel descriptor （EKD） and efficient hierarchical kernel descriptor （EHKD）, which are built upon incomplete Cholesky decomposition. EKD au- tomatically selects a small number of pivot features for gener- ating patch-level features to achieve better computational effi- ciency. EHKD recursively applies EKD to form image-level features layer-by-layer. Perhaps due to parsimony, we find surprisingly that the EKD and EHKD approaches achieved competitive results on several public datasets compared with other state-of-the-art methods, at an improved efficiency over KD.