Optimal approximation and error bounds in seminormed spaces

It is known that purely photoelastic procedures cannot solve the general three-dimensional stress problem. Photoelasticity furnishes data from which only the principal shears can be determined, but not the principal stresses. A new, general, and practical method of solution which has been described previously (1, 2) is reviewed briefly, and possible variations in procedure are discussed. This method combines the photoelastic data from frozen stress patterns with a numerical integration of one of the differential equations of equilibrium in Cartesian coordinates. The actual principal stresses at each point of a homogeneous and isotropic body of arbitrary shape subjected to a general system of loads are thereby determined. The method has been applied previously to a sphere subjected to diametral compressive loads of 172 lb. The present paper contains the results from a second sphere subjected to 79.6 lb which show that the degree of reproducibility of results is high. Very good agreement is also shown to exist with a theoretical solution of the same problem by Sternberg and Rosenthal (3) The paper also contains the solution of a short rectangular parallelepiped loaded through a small flat circular die. The investigation was conducted in the Photoelastic Laboratory of the Mechanics Department of Illinois Institute of Technology.

When solving nonlinear least-squares problems, it is often useful to regularize the problem using a quadratic term, a practice which is especially common in applications arising in inverse calculations. A solution method derived from a trust-region Gauss-Newton algorithm is analyzed for such applications, where, contrary to the standard algorithm, the least-squares subproblem solved at each iteration of the method is rewritten as a quadratic minimization subject to linear equality constraints. This allows the exploitation of duality properties of the associated linearized problems. This paper considers a recent conjugate-gradient-like method which performs the quadratic minimization in the dual space and produces, in exact arithmetic, the same iterates as those produced by a standard conjugate-gradients method in the primal space. This dual algorithm is computationally interesting whenever the dimension of the dual space is significantly smaller than that of the primal space, yielding gains in terms of both memory usage and computational cost. The relation between this dual space solver and PSAS (Physical-space Statistical Analysis System), another well-known dual space technique used in data assimilation problems, is explained. The use of an effective preconditioning technique is proposed and refined convergence bounds derived, which results in a practical solution method. Finally, stopping rules adequate for a trust-region solver are proposed in the dual space, providing iterates that are equivalent to those obtained with a Steihaug-Toint truncated conjugate-gradient method in the primal space.

Summary form only given: Unit commitment, one of the most critical tasks in electric power system operations, faces new challenges as the supply and demand uncertainty increases dramatically due to the integration of variable generation resources. To meet these challenges, we propose a two-stage adaptive robust unit commitment model and a practical solution method. We present a numerical study on the real-world large scale power system operated by the ISO New England. Computational results demonstrate the economic and operational advantages of our model over the traditional reserve adjustment approach. In particular, we find that by properly setting the level of conservatism in the uncertainty model, the adaptive robust model exhibits sizable savings on average dispatch cost, and significantly reduces the volatility of the dispatch cost, thus, improves the real-time reliability of the power system operation. The robust model also shows resilient performance under different probability distributions of uncertain load.

The vapor compression cooling devices operate on the same principle as heat pumps. In these types of machines thermal energy is transferred from the cold environment to the warmer side using provided power by the compressor. In this study, the effect of air flow rate on the cooling performance of vapor compression cooling devices has been investigated by simulating a designed system in AVL CRUISE™ M program. The simulated model is air-to-air cooling machine and refrigerant R134a was used as circulated gas inside the system. COP value of the system has been calculated in different working conditions and obtained results have been fully discussed. Accordingly, enthalpy variation in every element of the heat pump has been calculated in different working conditions and discussed on P-h diagram. This study presents a simulation method that is a practical solution method in the field of heat pumps, cooling machines and refrigerators which can be considered before installing device in order to have a proper prediction of the system performance.

Geophysical Inverse Theory and Regularization Problems

The main objective of this paper is optimization of second-order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi-infinite strip is considered. The boundary impedance (Neumann-to-Dirichlet map) is computed as the square root of an operator using the standard three-point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx-imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.

The most efficient practical methods of technology foresight problem solution are systematized. Formalized procedures and mathematical expert estimation methods, used for expert estimation of different-nature objects and processing of examination results, are proposed. They form the basis of an information platform for solution to foresight problems. An example of coordination of expert estimates is given.

A problem of the optimal approximation to a fuzzy tolerance relation is discussed in this paper. Specifically, our aim is to get the optimal fuzzy equivalence relation from a given fuzzy tolerance relation. Firstly, we discuss the relationship between the covering and the partition of a set. Then, we give the concept of distance between coverings (or partitions) and put forward the algorithms to get the optimal partition from a given covering. Main results include: 1) give the sufficient and necessary conditions for the optimal approximation of coverings in union sets; 2) give the necessary condition for absolutely optimal approximation of coverings; 3) based on the optimal approximation of each covering in the covering chain, the optimal fuzzy equivalence relation is obtained from the given fuzzy tolerance relation.KeywordsFuzzy Tolerance RelationEquivalence RelationQuotient SpaceOptimal Approximation

Modeling autocorrelation functions of self-similar teletraffic in communication networks based on optimal approximation in Hilbert space

The Karhunen–Loève transform (KLT) is a key ele- ment of many signal processing and communication tasks. Many recent applications involve distributed signal processing, where it is not generally possible to apply the KLT to the entire signal; rather, the KLT must be approximated in a distributed fashion. This paper investigates such distributed approaches to the KLT, where several distributed terminals observe disjoint subsets of a random vector. We introduce several versions of the distributed KLT. First, a local KLT is introduced, which is the optimal solution for a given terminal, assuming all else is fixed. This local KLT is different and in general improves upon the marginal KLT which simply ignores other terminals. Both optimal approximation and compression using this local KLT are derived. Two important special cases are studied in detail, namely, the partial observation KLT which has access to a subset of variables, but aims at reconstructing them all, and the conditional KLT which has access to side information at the decoder. We focus on the jointly Gaussian case, with known correlation structure, and on approximation and compression problems. Then, the distributed KLT is addressed by considering local KLTs in turn at the various terminals, leading to an iterative algorithm which is locally convergent, sometimes reaching a global optimum, depending on the overall correlation structure. For com- pression, it is shown that the classical distributed source coding techniques admit a natural transform coding interpretation, the transform being the distributed KLT. Examples throughout illustrate the performance of the proposed distributed KLT. This distributed transform has potential applications in sensor networks, distributed image databases, hyper-spectral imagery, and data fusion.

An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.

Generalized reflexive solutions of the matrix equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="italic"><mml:mi>AXB</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mi>D</mml:mi></mml:math> and an associated optimal approximation problem

The paper studies Lx(Rd)-norm approximations from a space spanned by a discrete set of translates of a basis function φ. Attention here is restricted to functions φ whose Fourier transform is smooth on Rd\0, and has a singularity at the origin. Examples of such basis functions are the thin-plate splines and the multiquadrics, as well as other types of radial basis functions that are employed in Approximation Theory. The above approximation problem is well-understood in the case where the setof points Ξ used for translating φ forms a lattice in Rd, and many optimal and quasi-optimal approximation schemes can already be found in the literature. In contrast, only a few, mostly specific, results are known for a set Ξ of scattered points. The main objective of this paper is to provide a general tool for extending approximation schemes that use integer translates of a basis function to the non-uniform case. We introduce a single, relatively simple, conversion method that preserves the approximation orders provided by a large number of schemes presently in the literature(more precisely, to almost all ‘stationary schemes’). In anticipation of future introduction of new schemes for uniform grids, an effort is made to impose only a few mild conditions on the function φ, which still allow fora unified error analysis to hold. In the course of the discussion here, the recent results of Buhmann, Dyn, and Levin [9] on scattered centre approximation are reproduced and improved upon.

Many optimal curve fitting and approximation problems have the same structure as certain estimation problems involving random processes. This structural correspondence has many useful consequences for curve fitting problems, including recursive algorithms and computable error bounds. The basic facts of this correspondence are reviewed and some new results on error bounds and optimal sampling are presented.

Discrete Time H∞ Model Reduction Problem