Parallel mesh methods for tension splines

This paper addresses the problem of shape preserving spline interpolation formulated as a differential multipoint boundary value problem (DMBVP for short). Its discretization by mesh method yields a five-diagonal linear system which can be ill-conditioned for unequally spaced data. Using the superposition principle we split this system in a set of tridiagonal linear systems with a diagonal dominance. The latter ones can be stably solved either by direct (Gaussian elimination) or iterative methods (SOR method and finite-difference schemes in fractional steps) and admit effective parallelization. Numerical examples illustrate the main features of this approach.     

线性系统稳定性与最优性的关系(Ⅱ)

我们在[1]中提出并讨论了与R.E.Kalman的最优控制反问题不同的另一类线性最优控制的反问题:任给一渐近稳定的定常线性系统和一个非负二次型性能指标,问是否可以从该稳定系统中分解出一个状态反馈,使得这个状态反馈就是给定指标下的最优控制。本文将上述问题加以推广,对线性离散系统和线性时变系统得出了相应的结论,进而得出[1]中提出的渐近稳定系统和最优系统之间的对应关系是一切线性系统的内在特征。     

A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems

We study a class of time-domain decomposition-based methods for the numerical solution of large-scale linear quadratic optimal control problems. Our methods are based on a multiple shooting reformulation of the linear quadratic optimal control problem as a discrete-time optimal control (DTOC) problem. The optimality conditions for this DTOC problem lead to a linear block tridiagonal system. The diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time-subintervals. This motivates the application of block Gauss–Seidel (GS)-type methods for the solution of the block tridiagonal systems. Numerical experiments show that the spectral radii of the block GS iteration matrices are larger than one for typical applications, but that the eigenvalues of the iteration matrices decay to zero fast. Hence, while the GS method is not expected to convergence for typical applications, it can be effective as a preconditioner for Krylov-subspace methods. This is confirmed by our numerical tests.A byproduct of this research is the insight that certain instantaneous control techniques can be viewed as the application of one step of the forward block GS method applied to the DTOC optimality system.     

Tikhonov solutions of approximately given systems of linear algebraic equations under finite perturbations of their matrices

The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.     

超定线性方程极大极小解的一种近似算法

本文通过构造一个无约束凸规划问题,建立了求超定线性方程组的极大极小解的一种近似算法,证明了算法的收剑性,并给出了初步的数值结果.     

Models in solar water heating—a critical comparison

A problem is discussed concerning the stochastic modelling of a system for collecting and storing solar energy. Three models are compared. These are: (i) a trivial, zero-order model; (ii) a linear (first-order) approximation to the system controls; and (iii) a full computer simulation. The Law of Diminishing Returns is demonstrated by this comparison, for although the computer simulation, involving great computational and programming effort is not matched by the linear models, considerable information is available, for little effort, from the linear approximation. Further it is shown that the use of this linear approximation can greatly assist a computer simulation since its use as a control variate can drastically reduce the variance.     

An approximation theory of matrix rank minimization and its application to quadratic equations

Matrix rank minimization problems are gaining plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization problem can be approximated to any level of accuracy via continuous optimization (especially, linear and nonlinear semidefinite programming) problems. One of the main results in this paper shows that if the feasible set of the problem has a minimum rank element with the least Frobenius norm, then any accumulation point of solutions to the approximation problem, as the approximation parameter tends to zero, is a minimum rank solution of the original problem. The tractability under certain conditions and convex relaxation of the approximation problem are also discussed. An immediate application of this theory to the system of quadratic equations is presented in this paper. It turns out that the condition for such a system without a nonzero solution can be characterized by a rank minimization problem, and thus the proposed approximation theory can be used to establish some sufficient conditions for the system to possess only zero solution.     

A Comparison of Preconditioners for the Steklov–Poincaré Formulation of the Fluid-Structure Coupling in Hemodynamics

A Fluid–Structure Interaction (FSI) problem can be reinterpreted as a heterogeneous problem with two subdomains. It is possible to describe the coupled problem at the interface between the fluid and the structure, yielding a nonlinear Steklov–Poincaré problem. The linear system can be linearized by Newton iterations on the interface and the resulting linear problem can be solved by the preconditioned GMRES method. In this work we investigate the behavior of preconditioners of Neumann–Neumann and Dirichlet–Neumann type. We find that, in the context of hemodynamics, the Dirichlet– Neumann, i.e., using Dirichlet boundary conditions on the fluid side and Neumann on the structure side, outperforms the Neumann–Neumann method, except when a weighting is used such that it basically reduces to the Dirichlet–Neumann method. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nudelman interpolation and the band method

In previous work the authors developed a new addition of the band method based on a Grassmannian approach for solving a completion/extension problem in a general, abstract framework. This addition allows one to obtain a linear fractional representation of all solutions of the abstract completion problem from special extensions which are not necessarily band extensions (for the positive case) or triangular extensions (for the contractive case). In this work we extend this framework to a somewhat more general setting and show how one can obtain formulas for the required special extensions from solutions of a system of linear equations. As an application we show how the formalism can be applied to the bitangential Nevanlinna-Pick interpolation problem, a case which, up to now, was not amenable to the band method.The first author was partially supported by National Science Foundation grant DMS-9500912.     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

Iterative solution of a coupled mixed and standard Galerkin discretization method for elliptic problems

In this paper, we consider approximation of a second‐order elliptic problem defined on a domain in two‐dimensional Euclidean space. Partitioning the domain into two subdomains, we consider a technique proposed by Wieners and Wohlmuth [9] for coupling mixed finite element approximation on one subdomain with a standard finite element approximation on the other. In this paper, we study the iterative solution of the resulting linear system of equations. This system is symmetric and indefinite (of saddle‐point type). The stability estimates for the discretization imply that the algebraic system can be preconditioned by a block diagonal operator involving a preconditioner for H (div) (on the mixed side) and one for the discrete Laplacian (on the finite element side). Alternatively, we provide iterative techniques based on domain decomposition. Utilizing subdomain solvers, the composite problem is reduced to a problem defined only on the interface between the two subdomains. We prove that the interface problem is symmetric, positive definite and well conditioned and hence can be effectively solved by a conjugate gradient iteration. Copyright © 2001 John Wiley & Sons, Ltd.     

The algebraic regulator problem from the state-space point of view

We study the algebraic aspects of the regulator problem, using some new ideas in the state-space (“geometric”) approach to feedback design problems for linear multi- variable systems. Necessary and sufficient conditions are given for the solvability of a general version of this problem, requiring output stability, internal stability, and disturbance decoupling as well. An algorithm is given by which these conditions can be verified from the system parameters.     

Frequency characteristics of flexibly vibration discrete-continuous mechatronic system

The purpose of this paper is formulating of problem of flexibly vibrating mechatronic system. The main approach of the subject was to formulate the problem in the form of set of differential equation of motion and state equation of considered mechatronic model of object. The considered flexibly vibrating mechanical system is a continuous beam, clamped at one of its end. Integral part of mechatronic system is a transducer, extorted by harmonic voltage. In the paper the linear mechanical subsystem and linear electric subsystem of mechatronic system has been considered. The methods of analysis and obtained results can be base on design and investigation for this type of mechatronic systems. The mechatronic system formed from mechanical and electric subsystems with electromechanical bondage has been considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of Complementarity Problems via Piecewise Linearization

My master thesis concerns the solution linear complementarity problems (LCP). The Lemke algorithm, the most commonly used algorithm for solving a LCP until this day, was compared with the piecewise Newton method (PLN algorithm). The piecewise Newton method is an algorithm to solve a piecewise linear system on the basis of damped Newton methods. The linear complementarity problem is formulated as a piecewise linear system for the applicability of the PLN algorithm. Then, different application examples will be presented, solved with the PLN algorithm. As a result of the findings (of my master thesis) it can be assumed that – under the condition of coherent orientation – the PLN-algorithm requires fewer iterations to solve a linear complementarity problem than the Lemke algorithm. The coherent orientation for piecewise linear problems corresponds for linear complementarity problems to the P-matrix-property. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An efficient linear solver for the hybridized discontinuous Galerkin method

Discretizing partial differential equations by an implicit solving technique ultimately leads to a linear system of equations that has to be solved. The number of globally coupled unknowns is especially large for discontinuous Galerkin (DG) methods. It can be reduced by using hybridized discontinuous Galerkin (HDG) methods, but still efficient linear solvers are needed. It has been shown that, if hierarchical basis functions are used, a hierarchical scale separation (HSS) ansatz can be an efficient solver. In this work, we couple the HDG method with an HSS solver to solve a scalar nonlinear problem. It is validated by comparing the results with results obtained by GMRES with ILU(3) preconditioning as linear solver. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fast sensitivity analysis of defective system

Sensitivity analysis of linear vibration system is of wide interest. In this paper, sensitivity analysis based on non-defective system and defective system is summarized in all cases. Specially, for the defective systems, a fast method for the perturbation problem of state vectors is constructed in terms of the theories of generalized eigenvectors and adjoint matrices. By this method, the state vector derivatives can be expressed by a linear combination of generalized eigenvectors. The expansion coefficients can be obtained without solving large-scale equations based on eigensolutions of original system. Numerical results demonstrate the effectiveness and the stability of the method.     

Parametric linear programming and cluster analysis

In the cluster analysis problem one seeks to partition a finite set of objects into disjoint groups (or clusters) such that each group contains relatively similar objects and, relatively dissimilar objects are placed in different groups. For certain classes of the problem or, under certain assumptions, the partitioning exercise can be formulated as a sequence of linear programs (LPs), each with a parametric objective function. Such LPs can be solved using the parametric linear programming procedure developed by Gass and Saaty [(Gass, S., Saaty, T. (1955), Naval Research Logistics Quarterly 2, 39–45)]. In this paper, a parametric linear programming model for solving cluster analysis problems is presented. We show how this model can be used to find optimal solutions for certain variations of the clustering problem or, in other cases, for an approximation of the general clustering problem.     

Coefficient quantization for frames in Banach spaces

Let (ei) be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces.     

Control of asynchronous spectrum of linear systems with a two-sided dependence of blocks of complete column rank

We consider a linear periodic control system with a two-sided dependence of blocks of complete column rank in the nonstationary component of the coefficient matrix in the critical case. In this case, the nontrivial intersection of vector subspaces formed by linear spans of the columns in the blocks can be arbitrary. We assume that the control is given in the form of feedback linear in the state variables and is periodic with the period of the system. We derive necessary and sufficient conditions for the solvability of the control problem for the asynchronous spectrum, that is, the problem of finding a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies.     

Deriving crisp and consistent priorities for fuzzy AHP-based multicriteria systems using non-linear constrained optimization

Fuzzy optimization models are used to derive crisp weights (priority vectors) for the fuzzy analytic hierarchy process (AHP) based multicriteria decision making systems. These optimization models deal with the imprecise judgements of decision makers by formulating the optimization problem as the system of constrained non linear equations. Firstly, a Genetic Algorithm based heuristic solution for this optimization problem is implemented in this paper. It has been found that the crisp weights derived from this solution for fuzzy-AHP system, sometimes lead to less consistent or inconsistent solutions. To deal with this problem, we have proposed a consistency based constraint for the optimization models. A decision maker can set the consistency threshold value and if the solution exists for that threshold value then crisp weights can be derived, otherwise it can be concluded that the fuzzy comparison matrix for AHP is not consistent for the given threshold. Three examples are considered to demonstrate the effectiveness of the proposed method. Results with the proposed constraint based fuzzy optimization model are more consistent than the existing optimization models.