Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

This paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal estimation of the diffusion coefficient from non-averaged and averaged noisy magnitude data

The magnitude operation changes the signal distribution in MRI images from Gaussian to Rician. This introduces a bias that must be taken into account when estimating the apparent diffusion coefficient. Several estimators are known in the literature. In the present paper, two novel schemes are proposed. Both are based on simple least squares fitting of the measured signal, either to the median (MD) or to the maximum probability (MP) value of the Probability Density Function (PDF). Fitting to the mean (MN) or a high signal-to-noise ratio approximation to the mean (HS) is also possible. Special attention is paid to the case of averaged magnitude images. The PDF, which cannot be expressed in closed form, is analyzed numerically. A scheme for performing maximum likelihood (ML) estimation from averaged magnitude images is proposed. The performance of several estimators is evaluated by Monte Carlo (MC) simulations. We focus on typical clinical situations, where the number of acquisitions is limited. For non-averaged data the optimal choice is found to be MP or HS, whereas uncorrected schemes and the power image (PI) method should be avoided. For averaged data MD and ML perform equally well, whereas uncorrected schemes and HS are inadequate. MD provides easier implementation and higher computational efficiency than ML. Unbiased estimation of the diffusion coefficient allows high resolution diffusion tensor imaging (DTI) and may therefore help solving the problem of crossing fibers encountered in white matter tractography.     

An explanation of the unusual strength of the hydrogen bond in small water clusters

We seek to explain why the hydrogen bond possesses unusual strength in small water clusters that account for many of the complex behaviors of water. We have investigated and visualized the donation of covalent character from covalent (sigma) to hydrogen bonds by calculating the eigenvector coupling properties of quantum theory of atoms in molecules (QTAIM), stress tensor σ ( r ), and Ehrenfest Force F ( r ) on the F ( r ) molecular graph. The next-generation three-dimensional (3-D) bond-path framework sets are presented, and only the F ( r ) bond-path framework sets reproduce the earlier finding on the coupling between covalent (sigma) and hydrogen bonds that possess a degree of covalent character. Exploration of the bond-path between the covalent (sigma) and hydrogen bond's critical points provides an explanation for the previously obtained coupling results. The directional character of the covalent (sigma) and hydrogen bonds' 3-D bond-path framework sets for the F ( r ) explains differences found in the earlier results from QTAIM and the stress tensor σ ( r ).     

一般矩阵函数的变差

设Sn是n次对称群,G为Sn的子群,χ是G的次数为1的特征标.如果A是一个n阶复变矩阵,定义一般矩阵函数dχG为dχG(A)=∑σ∈Gχ(σ)∏ni=1aiσ(i).本文用lp-算子范数(1≤p≤∞)的性质证明了一般矩阵函数变差的两个不等式.     

脑胶质瘤磁共振弥散张量成像定量参数与VEGF、MMP-9表达的相关性

本研究旨在分析脑胶质瘤患者的磁共振弥散张量成像(DTI)参数与病变组织血管内皮生长因子(VEGF)和基质金属蛋白酶-9(MMP-9)表达的相关性,以及DTI参数对脑胶质瘤进行分级诊断的价值.根据病理分级将102例脑胶质瘤患者分为低级别组(47例)和高级别组(55例),均行MRI和DTI检查,定量测定表观弥散系数(ADC...     

四原子分子中Majorana算子矩阵元的新计算公式

李代数方法在研究双原子分子、三原子分子振－转光谱及相关问题等方面已被证明是一种有效方法[‘－0,并被成功推广到多原子分子[’－门．构造代数哈密顿量是此方法的关键,这就要求选择合适的     

随机R0张量互补问题的投影Levenberg-Marquardt方法

本文考虑一类离散型随机$R_0$张量互补问题,利用Fischer-Burmeister函数将问题转化为约束优化问题,并用投影Levenberg-Marquardt方法对其进行了求解。在一般的条件下得到了该方法的全局收敛性,相关的数值实验表明了该方法的有效性。     

工件的释放时间和加工时间具有一致性的单机在线排序问题研究

工件的释放时间和加工时间具有一致性, 是指释放时间大的工件其加工时间不小于释放时间小的工件的加工时间, 即若$r_{i}\geq r_{j}$, 则$p_{i}\geq p_{j}$。本文在该一致性约束下, 研究最小化最大加权完工时间单机在线排序问题, 和最小化总加权完工时间单机在线排序问题, 并分别设计出$\frac{\sqrt{5}+1}{2}$-竞争的最好可能在线算法。     

A tensor norm preserving unconditionality for -spaces

We show that, for each , there is an -tensor norm (in the sense of Grothendieck) with the surprising property that the -tensor product has local unconditional structure for each choice of arbitrary -spaces . In fact, is the tensor norm associated to the ideal of multiple -summing -linear forms on Banach spaces.

     

Evolution of hypersurfaces by the mean curvature minus an external force field

In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.