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Inverse problems and optimal design have come of age as a consequence of the availability of better, more accurate, and more efficient simulation packages. Many of these simulators, which can run on small workstations, can capture the complicated behavior of the physical systems they are modeling, and have become commonplace tools in engineering and science. There is a great desire to use them as part of a process by which measured field data are analyzed or by which design of a product is automated. A major obstacle in doing precisely this is that one is ultimately confronted with a large-scale optimization problem. This volume contains expository articles on both inverse problems and design problems formulated as optimization. Each paper describes the physical problem in some detail and is meant to be accessible to researchers in optimization as well as those who work in applied areas where optimization is a key tool. What emerges in the presentations is that there are features about the problem that must be taken into account in posing the objective function, and in choosing an optimization strategy. In particular there are certain structures peculiar to the problems that deserve special treatment, and there is ample opportunity for parallel computation. THIS IS BACK COVER TEXT!!! Inverse problems and optimal design have come of age as a consequence of the availability of better, more accurate, and more efficient, simulation packages. The problem of determining the parameters of a physical system from



Space Mapping Optimization for Engineering Design

This contribution describes the Space Mapping (SM) technique which is relevant to engineering optimization. The SM technique utilizes different models of the same physical object. Similarly to approximation, interpolation, variable-complexity, response surface modeling, surrogate models, and related techniques, the idea is to replace computationally intensive simulations of an accurate model by faster though less accurate evaluations of another model. In contrast, SM attempts to establish a mapping between the input parameter spaces of those models. This mapping allows us to redirect the optimization-related calculations to the fast model while preserving the accuracy and confidence offered by a few well-targeted evaluations of the accurate model. The SM technique has been successfully applied in the area of microwave circuit design. In principle, however, it is applicable to a wide range of problems where models of different complexity and computational intensity are available, although insight into engineering modeling in the specific application area might be essential.
John W. Bandler, Radek M. Biernacki, Shaohua Chen, Ronald H. Hemmers, Kaj Madsen

An Inverse Problem in Plasma Physics: the Identification of the Current Density Profile in a Tokamak

This paper deals with the numerical identification of the plasma current density in a Tokamak, from experimental measurements. This problem consists in the identification of a non-linearity in a semi-linear 2D elliptic equation from Cauchy boundary measurements and from integrals of the magnetic field over several chords. A simplified problem, corresponding to the cylindrical approximation, is first solved by a SQP algorithm, where the non-linearity is decomposed in a basis of B-splines and where Tikhonov regularization is used, the regularizing parameter being determined by a cross-validation procedure. An alternative approach based on the decomposition of the non-linearity in a basis of compactly supported wavelets and the use of a new scalar product in the HH2 space has enabled to solve the identification problem in a robust way, without using Tikhonov regularization Finally several test-cases are presented for the determination of the current density in the real toroidal configuration from magnetic, interferometric and polarimetric measurements.
J. Blum, H. Buvat

Duality for Inverse Problems in Wave Propagation

A general dual formulation for inequality constrained optimization problems applies directly to inverse problems for multi-experiment data fitting. In the case of inverse problems in wave propagation, proper choice of the multi-experiment consistency constraint yields a dual problem with better convexity properties than the “primal” or straightforward data fitting formulation. The plane wave detection problem, a very simple inverse problem in wave propagation, provides a transparent framework in which to illustrate these ideas.
Mark S. Gockenbach, William W. Symes

Piecewise Differentiable Minimization for Ill-Posed Inverse Problems

Based on minimizing a piecewise differentiable l p function subject to a single inequality constraint, this paper discusses algorithms for a discretized regularization problem for ill-posed inverse problems. We examine computational challenges of solving this regularization problem. Possible minimization algorithms such as the steepest descent method, iteratively weighted least squares (IRLS) method and a recent globally convergent affine scaling Newton approach are considered. Limitations and efficiency of these algorithms are demonstrated using the geophysical traveltime tomographic inversion and image restoration applications.
Yuying Li

The Use of Optimization in the Reconstruction of Obstacles from Acoustic or Electromagnetic Scattering Data

We consider some three dimensional time harmonic acoustic and electromagnetic scattering problems for bounded simply connected obstacles. We consider the following inverse problem: from the knowledge of several far field patterns generated by the obstacle when hit by known incoming waves and from the knowledge of some a-priori information about the obstacle, i.e. boundary impedance, shape symmetry, etc., reconstruct the shape or the shape and the impedance of the obstacle. There are a large number of effective numerical methods to solve the direct problem associated with this inverse problem, but techniques to solve the inverse problem are still in their infancy. We reformulate the inverse problem as two different unconstrained optimization problems. We present a review of results obtained by the authors on the inverse problem and we give some ideas concerning the solution of the direct problem by efficient parallel algorithms.
Pierluigi Maponi, Maria Cristina Recchioni, Francesco Zirilli

Design of 3D-Reflectors for Near Field and Far Field Problems

We report about a project concerning the computer-aided design of reflectors in 3 where the light coming from a point source is reflected in such a way that the illumination intensity distribution of the outgoing light on either a given plane or a given sphere can be prescribed. We derive a mathematical model and a numerical algorithm based on the iterative solution of a certain minimization problem Finally, we present numerical results showing that the algorithm works in practice.
Andreas Neubauer

Optimal Die Shape and Ram Velocity Design for Metal Forging

A primary objective in metal forming is designing the geometry of the workpiece and dies in order to achieve a part with a given shape and microstructure. This problem is usually handled by extensive trial and error—using simulations, or test forgings, or both—until an acceptable final result is obtained. The goal of this work is to apply optimization techniques to this problem. Expensive function evaluations make computational efficiency of prime importance. As a practical matter, off-the-shelf software is used for both process modeling and optimization. While this approach makes it possible to put together a working package relatively quickly, it brings several problems of its own. The algorithm is applied to a example of real current interest—the forging of an automobile engine valve from a high performance material—with some success.
Linda D. Smith, Jordan M. Berg, James C. Malas

Eigenvalues in Optimum Structural Design

Eigenvalues frequently appear in structural analysis. The most common cases are vibration frequencies and eigenvalues in the form of load magnitudes in structural stability analysis. In structural design optimization, the eigenvalues may appear either as objective function or as constraint functions. For example maximizing the eigenvalue representing the load magnitude subject to a constraint on structural weight.
Free vibration frequencies and load magnitudes in stability analysis are computed by solving large and sparse generalized symmetric eigenvalue problems. Eigenvalue constraints can therefore be represented using matrix inequalities as opposed to directly referring to the eigenvalues themselves. Since eigenvalues are nonsmooth functions of the design parameters, it is desirable to pose the constraints using matrix inequalities since this makes it possible to use a barrier transformation giving a smooth optimization formulation.
An overview of different structural design problems where eigenvalues appear as either constraints or objective function is given. In particular, it is described how barrier methods are useful for eigenvalue constraints. The more difficult case of unsymmetric matrices is also considered. An important application is structural optimization subject to aeroelasticity constraints which is briefly discussed.
Ulf Torbjörn Ringertz

Optimization Issues in Ocean Acoustics

Ocean acoustics, e.g., SONAR, has traditionally been used for the detection and localization of targets such as submarines or schools of fish. However, more recently the community has focused on the use of acoustics to probe the ocean itself and its boundaries. Ocean acoustic tomography for the purpose of estimating temperature structure throughout a large ocean volume is a fine example of such an application. This a a case where theory, propagation modeling, technology, and computer capabilities have all reached a sufficient level of maturity that such a large scale inverse problem becomes tractable. Other applications include the estimation of shallow water bottom properties such as sediment thicknesses and densities, of under-ice reflectivity, and more generally the estimation of any parameters which influence the acoustic propagation. However, the community continues to struggle with such basic issues as how to pose each problem properly so as to guarantee uniqueness for the solution and how to find that optimizing solution. The essential difficulty in finding solutions arises not only because the search space for the unknowns can be extremely large, but also because that space is usually highly non-convex thereby preventing the use of simple gradient based searches. Methods in use to find solutions include simulated annealing, genetic algorithms, and tailored search algorithms based on examinations of the solution space itself.
A. Tolstoy

Gradient Methods in Inverse Acoustic and Electromagnetic Scattering

The problem of determining the complex permittivity or sound speed in a bounded inhomogeneity imbedded in a homogeneous medium from scattered field measurements exterior to the inhomogeneity is considered. A number of methods of attacking this problem, all based on minimizing the difference between an integral representation of the scattered field and the measured data, are described. These include Born, Newton-Kantorovich and distorted Born methods. The main part of the paper will be devoted to a description of a gradient type algorithm which is used to minimize a cost functional in which two objective functions are sought simultaneously. The error which is minimized is a bilinear form involving the product of two functions. This special form of the nonlinearity is retained in the algorithm. A number of numerical results will be presented which illustrate the effectiveness and limitations of the approach.
P. M. Van Den Berg, R. E. Kleinman

Atmospheric Data Assimilation Based on the Reduced Hessian Successive Quadratic Programming Algorithm

Mesoscale weather forecasting cannot be improved until a better data assimilation is obtained. Four dimensional variational analysis (4DVAR) provides the most elegant framework for data assimilation. One of the most critical issues of applying 4DVAR to weather prediction is how efficiently these variational problems can be solved. We introduce the reduced Hessian SQP algorithm for these problems and obtain an adjoint reduced Hessian SQP method, which is quadratically convergent since the exact reduced Hessian is used.
Y. F. Xie


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