Nonlocal Optimization Research Articles

Lemniscate representation for inversion of gravity and magnetic data through nonlocal optimization

A method of representing surfaces and volumes by a set of geometric points and a small set of auxiliary parameters, based on generalization of Bernoulli’s notion of lemniscates is introduced. It provides for easy generation and modification of surfaces and volumes, which could be connected, disjoint or even with very irregular boundaries. This allows solving geophysical inversion problems, without constraining the anomalous volumes to some ideal or simple forms. This is illustrated by the example of joint inversion of gravity and magnetic data sets attributable to two-dimensional anomalous bodies. A nonlocal optimization algorithm calledsegmented Hamming scan is used for inversion. Comparison with nonlinear least-squares algorithm shows the advantages of the chosen approach. The concepts ofdesideratal andprocedural detours are illustrated.

Bernstein preprocessing for nonlinear least squares inversion of gravity and magnetic data

Nonlinear, nonlocal and adaptive optimization algorithms, now readily available, as applied to parameter estimation problems, require that the data to be inverted should not be very noisy. If they are so, the algorithm tends to fit them, rather than smoothening the noise component out. Here, use of Bernstein polynomials is proposed to prefilter noise out, before inversion with the help of a sophisticated optimization algorithm. Their properties are described. Inversion of gravity and magnetic data for basement depth estimation, singly and jointly, and without and after Bernstein-preprocessing is conducted to illustrate that the inversion of Bernstein-preprocessed gravity data alone may be slightly superior to the joint inversion of gravity and magnetic data.

Nonlocal Optimization Models

Nonlocal Optimization Models

Optimization, relaxation and Young measures

We review the use of Young measures in analyzing relaxed and generalized formulations for typical problems of optimization including variational principles, optimal control problems, models in materials science, optimal design problems and nonlocal optimization problems.

Bi-parental product algorithm for coded waveform design in radar

The problem of obtaining long sequences with finite alphabet and peaky aperiodic auto-correlation is important in the context of radar, sonar and system identification and is called the coded waveform design problem, or simply the signal design problem in this limited context. It is good to remember that there are other signal design problems in coding theory and digital communication. It is viewed as a problem of optimization. An algorithm based on two operational ideas is developed. From the earlier experience of using the eugenic algorithm for the problem of waveform design, it was realised that rather than random but multiple mutations, all the first-order mutations should be examined to pick up the best one. This is called Hamming scan, which has the advantage of being locally complete, rather than random. The conventional genetic algorithm for non-local optimization leaves out the anabolic role of chemistry of allowing quick growth of complexity. Here, the Hamming scan is made to operate on the Kronecker or Chinese product of two sequences with best-known discrimination values, so that one can go to large lengths and yet get good results in affordable time. The details of the ternary pulse compression sequences obtained are given. They suggest the superiority of the ternary sequences.

Genetic algorithms in seismic waveform inversion

SUMMARY Recently a new class of methods, to solve non-linear optimization problems, has generated considerable interest in the field of Artificial Intelligence. These methods, known as genetic algorithms, are able to solve highly non-linear and non-local optimization problems and belong to the class of global optimization techniques, which includes Monte Carlo and Simulated Annealing methods. Unlike local techniques, such as damped least squares or conjugate gradients, genetic algorithms avoid all use of curvature information on the objective function. This means that they do not require any derivative information and therefore one can use any type of misfit function equally well. Most iterative methods work with a single model and find improvements by perturbing it in some fashion. Genetic algorithms, however, work with a group of models simultaneously and use stochastic processes to guide the search for an optimal solution. Both Simulated Annealing and genetic algorithms are modelled on natural optimization systems. Simulated Annealing uses an analogy with thermodynamics; genetic algorithms have an analogy with biological evolution. This evolution leads to an efficient exchange of information between all models encountered, and allows the algorithm to rapidly assimilate and exploit the information gained to find better data fitting models. To illustrate the power of genetic algorithms compared to Monte Carlo, we consider a simple multidimensional quadratic optimization problem and show that its relative efficiency increases dramatically as the number of unknowns is increased. As an example of their use in a geophysical problem with real data we consider the non-linear inversion of marine seismic refraction waveforms. The results show that genetic algorithms are inherently superior to random search techniques and can also perform better than iterative matrix inversion which requires a good starting model. This is primarily because genetic algorithms are able to combine both local and global search mechanisms into a single efficient method. Since many forward and inverse problems involve solving an optimization problem, we expect that the genetic approach will find applications in many other geophysical problems; these include seismic ray tracing, earthquake location, non-linear data fitting and, possibly seismic tomography.

22] Modeling DNA structures: Molecular mechanics and molecular dynamics

Model building studies may be used to supplement structurally low resolution experimental data with detailed three-dimensional hypothetical atomic models. Because of the strong relation between structure and function in biological molecules such models may give a consistent, integral view of a wealth of experimental data. In most cases such models will predict the outcome of certain experiments. The outcome of these experiments will often either confirm the model may be used for further refinement or even demand a major revision of the model. Coordinates obtained from X-ray fiber diffraction data or in special cases single-crystal data may provide the elements for DNA or RNA model building. Local and nonlocal optimization may be used to refine these structures and to evaluate their statistical significance as estimated by a chosen force field. Appreciable progress using nonlocal optimization procedures can only be expected if the dimensionality of the problem can be reduced sufficiently to the relevant degrees of freedom. Taking advantage of structural symmetries may critically improve the convergence while refining the target molecule or its building blocks. Monte Carlo and molecular dynamics methods allow one to calculate averaged quantities. In addition, molecular dynamics provides time evolutions of certain averages. During the simulation of certain physical properties of molecules a huge amount of data will be generated. They will provide many answers, but these answers may not always apply to the original question. So what type of questions will be reliably answered by a force field? Relatively safe answers concern the local geometry of the molecules. If a conformation leads to strong distortions of bond distances or angles or to close van der Waals contacts, this conformation can safely be rejected. Optimizing such unfavorable structures energetically may lead to structures showing how to avoid such distortions. More difficult are energetical questions: which of two conformers is more stable, or what is the free energy of the substrate in the active site? One cannot always be sure that the force field provides the correct answer. Therefore, one should pose only those questions which can be checked experimentally. Because of the many possible answers, the experiment may benefit by starting with a choice proposed by the simulation. The application of this procedure to curved DNA and the DNA four-way junction was successful.

Modeling DNA Structures

For a molecule of biological importance, one expects a strong correlation between the three-dimensional structure and its biological function(s). Molecular simulations allow the prediction of physical properties of macromolecules. Many of these properties are closely related to the molecular structure. Model-building studies may thus supplement structurally low-resolution experimental data with detailed three-dimensional hypothetical atomic models. Such studies may give a consistent integral view of a wealth of experimental data. In most cases, such models will predict the outcome of certain experiments. Their actual results will often either confirm the model, be used for further refinement, or demand a major revision. Empirical force-fields provide a large amount of physicochemical knowledge concerning structural and other physical properties about various classes of molecules. They give reasonable bond distances and angles and prevent short van der Waals contacts. Difficulties arise for the prediction of large-scale structural elements. These are not only determined by short-range interactions, but also result from long-range electrostatic, hydration, and hydrophobic forces. In the case of the ionically driven DNA B-Z transition, the point of transition as a function of ionic strength and size can be correctly predicted (33). More must be done for a better understanding of the hydration forces (6). What type of questions will be reliably answered by a force-field? Relatively safe answers concern the local geometry of the molecules. If a conformation leads to strong distortions of bond distances or angles or to close van der Waals contacts, it can safely be rejected. Optimizing such unfavorable structures energetically may lead to structures showing how to avoid such distortions. More difficult are energetic questions: Which of two conformers is more stable, or what is the free energy of the substrate in the active site (63)? One cannot always be sure that the force-field provides the correct answers. Therefore, one should concentrate on questions that can be checked experimentally. The application of such concepts to model curved DNA (19, 60) and the DNA four-way junction (61) provides promising results. To explore the knowledge contained in the force-fields, several methods have been proposed. Taking advantage of structural symmetries may improve critically the convergence, while refining the target molecule or its building blocks. A numerically stable derivative for the torsion potential has been proposed. The optimization method of conjugated gradients (see Section II,B) is a powerful tool to find the way downhill toward a local minimum. To surmount barriers and escape local minima requires nonlocal optimization procedures.(ABSTRACT TRUNCATED AT 400 WORDS)

Discussion of the Paper by Professor Efron and Dr Morris

Journal of the Royal Statistical Society: Series B (Methodological)Volume 35, Issue 3 p. 403-421 DiscussionFree Access Discussion of the Paper by Professor Efron and Dr Morris First published: July 1973 https://doi.org/10.1111/j.2517-6161.1973.tb00969.xAboutPDF ToolsExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat References in the Discussion Birnbaum, A. (1969). Concepts of statistical evidence. In Philosophy, Science and Method ( S. Morgenbesser, P. Suppes and N. White, eds), pp. 112– 143. New York: st Martin's Press. Brown, L. D. (1966). On the admissibility of invariant estimators of one or more location parameters. Ann. Math. Statist., 37, 1087– 1136. Brown, L. D. (1971). Non-local asymptotic optimality of appropriate likelihood ratio tests. Ann. Math. Statist., 42, 1206– 1240. Copas, J. B. (1972). Empirical Bayes methods and the repeated use of a standard. Biometrika, 59, 349– 360. Leonard, T. (1972). Bayesian methods for binomial data. Biometrika, 59, 581– 589. Lindley, D. V. (1971). The estimation of many parameters. In Foundations of Statistical Inference ( V. P. Godambe and D. A. Sprott, eds), pp. 435– 455. Toronto: Holt, Rinehart & Winston. Mosteixer, F. and Tukey, J. W. (1968). Data analysis, including statistics. In The Handbook of Social Psychology ( G. Lindsey and E. Aronson, eds), Vol. 2. Reading, Mass.: Addison-Wesley. Smith, A. F. M. (1973a). A general Bayesian linear model. J. R. Statist. Soc. B, 35, 67– 75. Smith, A. F. M. (1973b). Bayes estimates in one-way and two-way models. Biometrika, 60, 319– 329. Zidek, J. V. (1969). A presentation of Bayes invariant procedures in terms of Haar measure. Ann. Inst. Statist. Maths., 21, 291– 308. Volume35, Issue3July 1973Pages 403-421 This article also appears in:Discussion Papers ReferencesRelatedInformation

Non-Local Asymptotic Optimality of Appropriate Likelihood Ratio Tests

Non-Local Asymptotic Optimality of Appropriate Likelihood Ratio Tests

On a class of non-local optimization problems which embeds one-dimensional lubrication problems

A class of non-local optimization problems is presented which embeds a large number of 1-dimensional optimum design lubrication problems. It is shown that optimality may actually be demonstrated directly when the non-local formulation is used.