A tensor product matrix approximation problem in quantum physics

We consider a matrix approximation problem arising in the study of entanglement in quantum physics. This notion represents a certain type of correlations between subsystems in a composite quantum system. The states of a system are described by a density matrix, which is a positive semidefinite matrix with trace one. The goal is to approximate such a given density matrix by a so-called separable density matrix, and the distance between these matrices gives information about the degree of entanglement in the system. Separability here is expressed in terms of tensor products. We discuss this approximation problem for a composite system with two subsystems and show that it can be written as a convex optimization problem with special structure. We investigate related convex sets, and suggest an algorithm for this approximation problem which exploits the tensor product structure in certain subproblems. Finally some computational results and experiences are presented.     

On backward errors of structured polynomial eigenproblems solved by structure preserving linearizations

We derive explicit computable expressions of structured backward errors of approximate eigenelements of structured matrix polynomials including symmetric, skew-symmetric, Hermitian, skew-Hermitian, even and odd polynomials. We determine minimal structured perturbations for which approximate eigenelements are exact eigenelements of the perturbed polynomials. We also analyze structured pseudospectra of a structured matrix polynomial and establish a partial equality between unstructured and structured pseudospectra. Finally, we analyze the effect of structure preserving linearizations of structured matrix polynomials on the structured backward errors of approximate eigenelements and show that structure preserving linearizations which minimize structured condition numbers of eigenvalues also minimize the structured backward errors of approximate eigenelements.     

Composite beams with shape memory alloy fibres

We are concerned with the bending problem of fibrous composite beams in which fibres are made of shape memory alloys. These are alloys that may undergo a stress‐induced martensitic phase transformation. The matrix is treated as an elastic medium, and perfect bonding between matrix and fibres is supposed. In our model, the beam is decomposed into layers and the hysteretic behaviour of the shape memory fibres is taken into account. The boundary value problem is formulated in the form of an evolution variational inequality which, after finite element discretization, can be solved incrementally as a sequence of linear complementarity problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of effective properties for CFRP curing coupled to viscoleasticity based on a three-scale framework

Our work presents a three-scale model for temperature-dependent visco-elastic effects accompanied by curing, which are important phenomena in a resin transfer molding (RTM) process. The effective bulk quantities in dependence on the degree of cure are obtained by homogenization for a representative unit cell (micro-RVE) on the heterogeneous microscale. To this end, an analytic solution is derived by extension of the composite spheres model [1]. Voigt and Reuss bounds resulting from the assumption of a homogeneous matrix proposed in [2] are compared to the effective quantities. During curing, the periodic mesostructure defined by a visco-elastic polymeric matrix and linear-thermo-elastic fibres is taken into account as a representative unit cell (meso-RVE) subjected to thermo-mechanical loading on the mesoscale. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evaluating policies based on their long term average cost

Many decision problems can be characterized by a set of possible states and a cost associated with each possible state transition, hi this paper we discuss how to select a policy from a set of possible policies in the long term. If the cost matrix is not available the transition matrix can be used to compare expected return times to states. In our setting the transition matrix is defined by use of linguistic terms and as a consequence, the expected return times are fuzzy. In the case where the cost matrix is available, fuzzy average costs are computed. The resulting fuzzy quantities are compared by introducing the concept of minimizing sets. Finally, we look at the case where the transition takes place from a state to a state that is known to be an element of some subset of states, but we do not know which one. We use the Dempster–Shafer theory [Shafer 1976] together with techniques of Norton [Norton 1988] and Smetz [Smetz 1976] to approximate the transition probabilities     

Prewavelet analysis of the heat equation

Summary. We consider the heat equation in a smooth domain of R with Dirichlet and Neumann boundary conditions. It is solved by using its integral formulation with double-layer potentials, where the unknown , the jump of the solution through the boundary, belongs to an anisotropic Sobolev space. We approximate by the Galerkin method and use a prewavelet basis on , which characterizes the anisotropic space. The use of prewavelets allows to compress the stiffness matrix from to when N is the size of the matrix, and the condition number of the compressed matrix is uniformly bounded as the initial one in the prewavelet basis. Finally we show that the compressed scheme converges as fast as the Galerkin one, even for the Dirichlet problem which does not admit a coercive variational formulation. Received April 16, 1999 / Published online August 2, 2000     

Thermo-chemical modelling of the formation of a composite material

We consider a composite material composed of carbon or glass fibres included in a resin which becomes solid when it is heated up (the reaction of reticulation).

A mathematical model of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period ? >0, thus we get a coupled system of nonlinear partial differential equations.

First we prove the existence and uniqueness of a solution by using a fixed point theorem and we obtain a priori estimates. Then we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero as well as the estimates for the difference of the exact and the approximate solutions.     

Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow

Dirac-harmonic maps couple a second order harmonic map type system with a first nonlinear Dirac equation. We consider approximate Dirac-harmonic maps \(\{(\phi _n,\psi _n)\}\), that is, maps that satisfy the Dirac-harmonic system up to controlled error terms. We show that such approximate Dirac-harmonic maps defined on a Riemann surface, that is, in dimension 2, continue to satisfy the basic properties of blow-up analysis like the energy identity and the no neck property. The assumptions are such that they hold for solutions of the heat flow of Dirac-harmonic maps. That flow turns the harmonic map type system into a parabolic system, but simply keeps the Dirac equation as a nonlinear first order constraint along the flow. As a corollary of the main result of this paper, when such a flow blows up at infinite time at interior points, we obtain an energy identity and the no neck property.     

Geometric error of finite volume schemes for conservation laws on evolving surfaces

This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of \(\mathbb {R}^3\) . We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal.     

The accuracy of alternative stochastic models of participation in group discussion†

Four related models of participation in group discussion are presented and compared in the accuracy with which they predict proportional participation, mean run length, mean recurrence time and the variances of runs and recurrences. For some quantities, such as proportional participation and mean run length, the most restricted model does virtually as well as the least restricted model; for other quantities it does not. None of the models does a good job of predicting variances of runs or recurrences.     

Some results on the controllability of forward stochastic heat equations with control on the drift

In this paper, we establish the null/approximate controllability for forward stochastic heat equations with control on the drift. The null controllability is obtained by a time iteration method and an observability estimate on partial sums of eigenfunctions for elliptic operators. As a consequence of the null controllability, we obtain the observability estimate for backward stochastic heat equations, which leads to a unique continuation property for backward stochastic heat equations, and hence the desired approximate controllability for forward stochastic heat equations. It deserves to point out that one needs to introduce a little stronger assumption on the controller for the approximate controllability of forward stochastic heat equations than that for the null controllability. This is a new phenomenon arising in the study of the controllability problem for stochastic heat equations.     

无约束优化问题的对角稀疏拟牛顿法

对无约束优化问题提出了对角稀疏拟牛顿法,该算法采用了Armijo非精确线性搜索,并在每次迭代中利用对角矩阵近似拟牛顿法中的校正矩阵,使计算搜索方向的存贮量和工作量明显减少,为大型无约束优化问题的求解提供了新的思路．在通常的假设条件下,证明了算法的全局收敛性,线性收敛速度并分析了超线性收敛特征。数值实验表明算法比共轭梯度法有效,适于求解大型无约束优化问题．     

Computational Aspects of Bayesian Spectral Density Estimation

Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the nonsparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We propose to sample from the approximate posterior (i.e., the prior times the approximate likelihood), and then to recover the exact posterior through importance sampling. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We explain why the approximate posterior may typically be multimodal, and we derive a Sequential Monte Carlo sampler based on an annealing sequence to sample from that target distribution. Performance of the overall approach is evaluated on simulated and real datasets. In addition, for one real-world dataset, we provide some numerical evidence that a Bayesian approach to semiparametric estimation of spectral density may provide more reasonable results than its frequentist counterparts. The article comes with supplementary materials, available online, that contain an Appendix with a proof of our main Theorem, a Python package that implements the proposed procedure, and the Ethernet dataset.     

Numerical analysis of a model of cure process for composites

We consider a composite material composed of fibres included in a resin which becomes solid when it is heated up (reaction of reticulation). The mathematical modelling of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. In this paper, we are interested in the computation of approximate solutions. We propose a family of discretized problems depending on two parameters (β₁, β₂) ε [0, 1]² which split the linear and non‐ linear terms in implicit and explicit parts. We prove the stability and convergence of the discretization for any (β₁, β₂) ε [½, 1 ] × [0, 1]. We present also some numerical results. Copyright © 2006 John Wiley & Sons, Ltd.     

Cotangent bundles for “matrix algebras converge to the sphere”

In the high-energy quantum-physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact quantum metric spaces, and convergence is with respect to a quantum Gromov–Hausdorff-type distance. More recently I have dealt with corresponding statements in the literature about vector bundles on spheres and matrix algebras. But physicists want, even more, to treat structures on spheres (and other spaces) such as Dirac operators, Yang–Mills functionals, etc., and they want to approximate these by corresponding structures on matrix algebras. In preparation for understanding what the Dirac operators should be, we determine here what the corresponding “cotangent bundles” should be for the matrix algebras, since it is on them that a “Riemannian metric” must be defined, which is then the information needed to determine a Dirac operator. (In the physics literature there are at least 3 inequivalent suggestions for the Dirac operators.)     

Approximate conservation laws of nonlinear perturbed heat and wave equations

We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.     

Approximating Lyapunov exponents and Sacker–Sell spectrum for retarded functional differential equations

We consider Lyapunov exponents and Sacker–Sell spectrum for linear, nonautonomous retarded functional differential equations posed on an appropriate Hilbert space. A numerical method is proposed to approximate such quantities, based on the reduction to finite dimension of the evolution family associated to the system, to which a classic discrete QR method is then applied. The discretization of the evolution family is accomplished by a combination of collocation and generalized Fourier projection. A rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra. The efficacy of the results is illustrated with some numerical examples.     

On the balancing principle for some problems of Numerical Analysis

We discuss a choice of weight in penalization methods. The motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution. One example of such improvement is a regularization, where a penalization substitutes an ill-posed problem for a well-posed one. In modern numerical methods for PDEs a penalization is used, for example, to enforce a continuity of an approximate solution on non-matching grids. A choice of penalty weight should provide a balance between error components related with convergence and stability, which are usually unknown. In this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components. It is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates. Finally, we successfully apply our strategy for self-regularization of Volterra-type severely ill-posed problems, such as the sideways heat equation, and for the choice of a weight in interior penalty discontinuous approximation on non-matching grids. Numerical experiments on a series of model problems support theoretical results.     

Structured Low Rank Approximation of a Bezout Matrix

The task of determining the approximate greatest common divisor (GCD) of more than two univariate polynomials with inexact coefficients can be formulated as computing for a given Bezout matrix a new Bezout matrix of lower rank whose entries are near the corresponding entries of that input matrix. We present an algorithm based on a version of structured nonlinear total least squares (SNTLS) method for computing approximate GCD and demonstrate the practical performance of our algorithm on a diverse set of univariate polynomials. The work is partially supported by a National Key Basic Research Project of China 2004CB318000 and Chinese National Science Foundation under Grant 10401035.     

Fuzzy modelling of a moving grate biomass furnace for simulation and control purposes

Takagi–Sugeno (TS) fuzzy models are developed for a moving grate biomass furnace for the purpose of simulating and predicting the main process output variables, which are heat output, oxygen concentration of flue gas, and temperature of flue gas. Numerous approaches to modelling biomass furnaces have been proposed in the literature. Usually their objective is to simulate the furnace as accurately as possible. Hence, very complex model architectures are utilized which are not suited for applications like model predictive control. TS fuzzy models are able to approximate the global non-linear behaviour of a moving grate biomass furnace by interpolating between local linear, time-invariant models. The fuzzy partitions of the individual TS fuzzy models are constructed by an axis-orthogonal, incremental partitioning scheme. Validation results with measured process data demonstrate the excellent performance of the developed fuzzy models.