Inverse Principles Research Articles

Shape Optimal Design Using InverseVariational Principles

At present, the optimal shape design of structures is of great importance. Problems of this type mostly lead to very complicated nonlinear systems, and therefore new optimization solutions are being sought. One very efficient tool offers inverse variational principles. It starts with formulation of a variational principle under the assumption that the volume (in the 2D area) of the domain variables is constant. It can easily be shown that then the boundary density of potential energy in the optimal state should be constant too. As the differences of the density of the energy at the boundary nodal points may be very large, the new positions of the boundary nodal points are determined in a special scale. Since inverse variational principles, in connection with the boundary element method, seem to be very prospective for solutions of geometric behavior of homogeneous, and also of partly homogeneous (e.g. phase-wise homogeneous), media, this complex study is presented. To prove the ability of the procedure proposed, two examples are solved at the end of this paper. They are selected in such a way that a comparison with statistically determined and statistically undetermined slender beams, obeying Bernoulli-Euler assumptions, is possible. The examples themselves solve an optimization of stretched plates.

Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions

We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings. The results obtained provide complete characterizations of the properties under consideration in a general setting of closed-graph multifunctions in finite dimensions. They ensure new information even in the classical cases of smooth single-valued mappings as well as multifunctions with convex graphs.

The generalized maximum entropy principle

Generalizations of the maximum entropy principle (MEP) of E.T. Jaynes (1957) and the minimum discrimination information principle (MDIP) of S. Kullback (1959) are described. The generalizations have been achieved by enunciating the entropy maximization postulate and examining its consequences. The inverse principles which are inherent in the MEP and MDIP are made quite explicit. Several examples are given to illustrate the power and scope of the generalized maximum entropy principle that follows from the entropy maximization postulate.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Michell's Problem

FOR the last two hundred years the attention of logicians and mathematicians has been directed to the inverse principles of the theory of probability, in which we reason from known events to possible causes. Two different methods of calculation are in use, which give approximately the same results. According to the celebrated theorem of James Bernoulli, “If a sufficiently large number of trials is made, the ratio of the favourable to the unfavourable events will not differ from the ratio of their respective probabilities beyond a certain limit in excess or defect, and the probability of keeping within these limits, however small, can be made as near certainty as we please by taking a sufficiently large number of trials.” The inverse use of this theorem is much more important and much more liable to objection and difficulties than the direct use. In the words of De Morgan, “When an event has happened, and may have happened in two or three different ways, that way which is most likely to bring about the event, is most likely to have been the cause.”