Dimensional Minimization Problem Research Articles

Computing Bounds for Linear Functionals of Exact Weak Solutions to the Advection-Diffusion-Reaction Equation

We present a cost-effective method for computing quantitative upper and lower bounds on linear functional outputs of exact weak solutions to the advection-diffusion-reaction equation and demonstrate a simple adaptive strategy by which such outputs can be computed to a prescribed precision. The bounds are computed from independent local subproblems resulting from a standard finite element approximation of the problem. At the heart of the method lies a local dual problem by which we transform an infinite dimensional minimization problem into a finite dimensional feasibility problem. The bounds hold for all levels of refinement on polygonal domains with piecewise polynomial forcing, and the bound gap converges at twice the rate of the ${\cal H}^1$-norm of the error in the finite element solution. The bounds are valid for any choice of model problem parameters for which an equilibrating approximate solution can befound, but they become increasingly pessimistic as the parameters tend toward the advection-dominated or reaction-dominated limits.

An iterative block Arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems

A modified m-step Arnoldi algorithm proposed by Jia and Elsner for computing a few selected eigenpairs of large unsymmetric matrices is generalized to its block version. In the new method, we use new modified approximate eigenvectors obtained from a linear combination of Ritz vectors and the wasted ( m+1)th block basis vector, whose residual norms are satisfied with some ( p+1)-dimensional minimization problems. The resulting block Arnoldi algorithm is not only better than the standard m-step one both in theory and in practice but also cheaper than the standard ( m+1)-step one. The relationships among the residual norms of Ritz pairs and those of new approximate eigenpairs are analyzed. Theoretical results show that the new method can overcome the drawback of nonconvergence, which exists in the conventional one in some extent. We carry out some numerical experiments, which exhibit that the new algorithm is more efficient and works better than its counterparts.

Ideal, weakly efficient solutions for vector optimization problems

We establish existence results for finite dimensional vector minimization problems allowing the solution set to be unbounded. The solutions to be referred are ideal(strong)/weakly efficient(weakly Pareto). Also several necessary and/or sufficient conditions for the solution set to be non-empty and compact are established. Moreover, some characterizations of the non-emptiness (boundedness) of the (convex) solution set in case the solutions are searched in a subset of the real line, are also given. However, the solution set fails to be convex in general. In addition, special attention is addressed when the underlying cone is the non-negative orthant and when the semi-strict quasiconvexity of each component of the vector-valued function is assumed. Our approach is based on the asymptotic description of the functions and sets. Some examples illustrating such an approach are also exhibited.

Efficient contact solvers based on domain decomposition techniques

This paper deals with the construction of efficient algorithms for the solution of the finite dimensional constrained minimization problem arising from the finite element discretization of contact problems. Dualization techniques have been used to decrease the problem size from the large number of unknowns in the domain to the much smaller number of inequalities at the boundary. The disadvantage of this direct Schur-complement approach is the need of the inversion of the stiffness matrix. Domain decomposition techniques meet very similar requirements. A global boundary value problem is decoupled into local subproblems, and one interface problem at the (coupling) boundary. Two complementary approaches are the Dirichlet method and the Neumann method. The first one requires preconditioners for local Dirichlet problems and for the interface problem in H + 1 2 , and extension operators from the boundary into the domain. The second one needs preconditioners for local Neumann problems, and for the interface problem in H − 1 2 . Efficient multilevel algorithms for all components are available in literature. In this paper, it is shown how to use exactly these components for the construction of solvers for contact problems. New results for the analysis of convergence are presented. At least at uniformly refined meshes, we can prove optimal time complexity. Numerical results show high efficiency also on adaptively refined 3D meshes.

Flight path estimation using frequency measurements from a wide aperture acoustic array

A narrowband technique based on the acoustical Doppler effect is proposed for estimating the trajectory of a turbo-prop aircraft in level flight with constant velocity as it transits over a ground-based passive acoustic sensor array. The basic principle is to measure the temporal variation of the instantaneous frequency (IF) of the acoustic signal received by each sensor and then to minimize the sum of the squared deviations of the IF estimates from their predicted values over a sufficiently long period of time for all sensors. The technique provides estimates of the propeller blade rate and the five source motion parameters that describe the aircraft trajectory. The six dimensional minimization problem is reduced to a five dimensional maximization problem, which is solved numerically using the quasi-Newton method. A simple method is described that provides the initial parameter estimates required for the numerical maximization. The effectiveness of the motion parameter estimation technique is verified using real acoustic data recorded from a wide aperture microphone array during various transits of a turbo-prop aircraft.

Nonlinearity measures: definition, computation and applications

This paper is the first of two papers treating the quantification of open loop nonlinearity of dynamic systems. A generic definition of a nonlinearity measure is presented on the basis of the “best” linear approximation of a nonlinear system. Generalizing an earlier approach of Allgöwer, the measure can be applied both to the analysis of steady state operating points of continuously operated processes as well as to a trajectory dependent analysis of batch or other transient processes. An approximative computational strategy transferring the original infinite dimensional nested optimization problem into a convex finite dimensional minimization problem is discussed. The applications in this paper focus on operating point dependent analysis. Three continuously operated stirred tank reactor (CSTR) examples are investigated including a benchmark CSTR. The latter is also used to illustrate a computationally efficient lower bound approximation of the proposed nonlinearity measure. The additional difficulties associated with a trajectory rather than an operating point dependent analysis will be discussed in the forthcoming second part of this communication treating transient reaction processes.

Newton's Mesh Independence Principle for a Class Of Optimal Shape Design Problems

Many optimal shape design problems can be stated as infinite dimensional minimization problems. For deriving an implementable algorithm it has to be decided either to discretize the problem and to use a finite algorithm to solve the discrete problem or to state an algorithm in function space and to discretize this algorithm. This issue has yet to be addressed in the field of optimal shape design research. One big advantage for the latter procedure is a mesh independence behavior as it has been proven by Allgower et al. [ SIAM J. Numer. Anal., 23 (1986), pp. 160--169]. Since their assertions are not directly applicable to these specific kinds of problems, a modified version of their mesh independence principle is given here in order to derive more efficient algorithms for the resulting large scale problems.

A constructive method for identification of an impenetrable scatterer

This paper presents a constructive algorithm for solving the problem of reconstructing the shape of a scattering object from measurements of the scattered far field when the unknown object is illuminated by a known incident wave. The problem is recast as an optimization problem with a penalty term. The cost functional consists of a term which assesses the difference between the measured far field and the far field of the solution of the field equation to a particular surface and the penalty term which measures the error in satisfying the boundary conditions on that surface. A complete family of radiating solutions of the Helmholtz equation is employed to construct approximate solutions by solving finite dimensional minimization problems. Existence of solutions of the original problem as well as the finite dimensional approximations is established. Moreover convergence of the approximate solutions to a solution of the original problem is proven. Some preliminary numerical results are presented to indicate the viability of the method.

A limit theorem on characteristic functions via an extremal principle

A classical limit theorem on characteristic functions is proved in the paper by using duality relations between a pair of optimization problems. The primal problem in the pair is an infinite dimensional minimization problem involving the relative entropy functional

Constrained optimization of monopulse circular aperture distribution in the presence of blockage

A method of synthesizing aperture antenna excitations to obtain optimum performance in the presence of aperture blockage is presented. The problem of optimizing the performance indices such as directivity and angular sensitivity is formulated as an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -dimensional minimization problem, with constraint on the sidelobe level. In most cases, the objective function is nonlinear with multiple minima which does not yield readily to gradient methods. A simplex search method is used in the study for optimizing the performance index. The sidelobe level (SLL) constraint has been incorporated using the penalty function method. The method is applied to circular monopulse aperture distribution with circular blockage to obtain maximum directivity factor (DF) in the sum mode and maximum angular sensitivity factor (ASF) in the difference mode, with sidelobe level constraint. The effect of blockage on the maximum directivity and maximum angular sensitivity is studied for various sidelobe level constraints.

Problem solving with soap films. II

For pt.I see ibid., vol.10, p.452 (1975). It was shown in the previous article that it was possible to solve some two dimensional minimization problems by building an analogue system consisting of plates and pins and dipping the system into soap solution. The soap film produced between the plates when the system is withdrawn from the soap solution has the property that its length is minimized once the soap film has reached thermodynamic equilibrium. This article extends the method in order to solve problems requiring the determination of the minimum surface area bounded by lines in three dimensions. The specific problems examined have been chosen that the boundaries have a high degree of symmetry.