By John W. Bandler, Radek M. Biernacki (auth.), Lorenz T. Biegler, Thomas F. Coleman, Andrew R. Conn, Fadil N. Santosa (eds.)
Inverse difficulties and optimum layout have come of age as a result of the provision of higher, extra actual, and extra effective simulation programs. lots of those simulators, which may run on small workstations, can seize the complex habit of the actual structures they're modeling, and became typical instruments in engineering and technological know-how. there's a nice wish to use them as a part of a procedure wherein measured box info are analyzed or during which layout of a product is computerized. an important situation in doing accurately this can be that one is eventually faced with a large-scale optimization challenge. This quantity includes expository articles on either inverse difficulties and layout difficulties formulated as optimization. each one paper describes the actual challenge in a few element and is intended to be available to researchers in optimization in addition to those that paintings in utilized parts the place optimization is a key device. What emerges within the shows is that there are gains in regards to the challenge that needs to be taken into consideration in posing the target functionality, and in determining an optimization method. specifically there are specific buildings odd to the issues that deserve targeted remedy, and there's abundant chance for parallel computation. this is often again disguise TEXT!!! Inverse difficulties and optimum layout have come of age because of the supply of higher, extra exact, and extra effective, simulation applications. the matter of making a choice on the parameters of a actual method from
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Additional info for Large-Scale Optimization with Applications: Part I: Optimization in Inverse Problems and Design
Thus we obtain, in the typical case in which Xof 0, * H(f*) = 0, iI'(f*) < 0. This allows the efficient location of f* . A simple illustration of this concept is provided by linear regression in the plane. Given vectors a, ~ E R 2 , we are to find the closest vector to a on the line determined by ~T X = 0. The "primal" problem is and the dual formulation seeks the smallest zero of A short calculation shows that otherwise whence f* = I~T al. Note that H has the properties discussed above. t. 4).
The characteristics which prevent effective application of local optimization to the primal approach will be evident. The third section defines the notion of duality used here for an arbitrary constrained optimization problem, and the fourth section specializes this notion to multi-experiment inverse problems and the coherence constraint. A further specialization to separable prob- DUALITY FOR INVERSE PROBLEMS IN WAVE PROPAGATION 39 lems is useful: the plane wave detection problem, in common with many inverse problems in wave propagation, separates the model parameter into linear and nonlinear categories.
These plots should be compared to Figure 7; obviously it is simple to minimize J f for a small value of f. The function fI is plotted in Figure 9. This function is almost linear; it is easy to find the desired root using Newton's method; recall that the derivative of fI is available, since it is related to the Lagrange multiplier for the optimization problem defining if. As indicated in Figure 9, only a few iterations are required to find f* to good accuracy. It should be noted that although, for this example, J f becomes more nonconvex as f increases, we can expect to have a very good starting point for minimizing J f after we have done it once; moreover, the first minimization, for a small value of f, is simple.
Large-Scale Optimization with Applications: Part I: Optimization in Inverse Problems and Design by John W. Bandler, Radek M. Biernacki (auth.), Lorenz T. Biegler, Thomas F. Coleman, Andrew R. Conn, Fadil N. Santosa (eds.)