Infinite-dimensional Formulation Research Articles

ON HAMILTONIAN FORMULATIONS AND CONSERVATION LAWS FOR PLATE THEORIES OF VEKUA-AMOSOV TYPE

Some variants of the generalized Hamiltonian formulation of the plate theory of I. N. Vekua – A. A. Amosov type are presented. The infinite dimensional formulation with one evolution variable, or an “instantaneous” formalism, as well as the de Donder – Weyl one are considered, and their application to the numerical simulation of shell and plate dynamics is briefly discussed. The main conservation laws are formulated for the general plate theory of Nth order, and the possible motion integrals are introduced

Conceptual inconsistencies in finite-dimensional quantum and classical mechanics

Utilizing operational dynamic modeling [D. I. Bondar et al., Phys. Rev. Lett. 109, 190403 (2012)], we demonstrate that any finite-dimensional representation of quantum and classical dynamics violates the Ehrenfest theorems. Other peculiarities are also revealed, including the nonexistence of the free particle and ambiguity in defining potential forces. Non-Hermitian mechanics is shown to have the same problems. This work compromises a popular belief that finite-dimensional mechanics is a straightforward discretization of the corresponding infinite-dimensional formulation.

Computation of Correlated Equilibrium with Global-Optimal Expected Social Welfare

In this paper, we propose an algorithm which computes the correlated equilibrium with global-optimal (i.e., maximum) expected social welfare for single stage polynomial games. We first derive tractable primal/dual semidefinite programming (SDP) relaxations for an infinite-dimensional formulation of correlated equilibria. We give an asymptotic convergence proof, which ensures solving the sequence of relaxations leads to solutions that converge to the correlated equilibrium with the highest expected social welfare. Finally, we give a dedicated sequential SDP algorithm and demonstrate it in a wireless application with numerical results.

Modeling the Forward Surface of Mortality

Longevity risk constitutes an important risk factor for insurance companies and pension plans. For its analysis, but also for evaluating mortality-contingent structured financial products, modeling approaches allowing for uncertainties in mortality projections are needed. One model class that has attracted interest in applied research as well as among practitioners are forward mortality models, which are defined based on forecasts of survival probabilities as can be found in generation life tables and infer dynamics on the entire age/term structure, or forward surface, of mortality. However, thus far, there has been little guidance on identifying suitable specifications and their properties. The current paper provides a detailed analysis of forward mortality models driven by a finite-dimensional Brownian motion. In particular, after discussing basic properties, we present an infinite-dimensional formulation, and we examine the existence of finite-dimensional realizations for time-homogeneous deterministic...

A Green's function approach to Fisher information analysis and preconditioning in microwave tomography

The Fisher Information Integral Operator (FIO) and related sensitivity analysis is formulated in a variational framework that is suitable for analytical Green's function and gradient-based approaches in microwave tomography. The main application considered here is for parameter sensitivity analysis and related preconditioning for gradient-based quasi-Newton inverse scattering algorithms. In particular, the Fisher information analysis can be used as a basic principle yielding a systematic approach to robust preconditioning, where the diagonal elements of the FIO kernel are used as targets for sensitivity equalization. The infinite-dimensional formulation has several practical advantages over the finite-dimensional Fisher Information Matrix (FIM) analysis approach. In particular, the FIO approach avoids the need of making a priori assumptions about the underlying discretization of the material such as the shape, orientation and positions of the assumed image pixels. Furthermore, the integral operator and its spectrum can be efficiently approximated by using suitable quadrature methods for numerical integration. The eigenfunctions of the integral operator, corresponding to the identifiable parameters via the significant eigenvalues and the corresponding Cramér–Rao bounds, constitute a suitable global basis for sensitivity and resolution analysis. As a generic numerical example, a two-dimensional inverse electromagnetic scattering problem is analysed and illustrates the spectral decomposition and the related resolution analysis. As an application example in microwave tomography, a simulation study has been performed to illustrate the parameter sensitivity analysis and to demonstrate the effect of the related preconditioning for gradient-based quasi-Newton inverse scattering algorithms.

Finite element synthesis of structural or acoustic receptances in view of practical design applications

The present study addresses the modal synthesis of dynamic responses or receptances, which can be considered as a major issue in vibratory or acoustic design. A new method termed “method of orthocomplement” offers to increase the accuracy of modal superposition. Two apparently independent domains are considered: first, the numerical control of round-off errors in finite-dimensional models; second, the regularization of modal boundary discontinuities in infinite-dimensional formulations. Concerning the finite-dimensional systems, using the proposed method in conjunction with pseudo-inversion theorems, proves that “inertia relief corrections” remedy modal truncation problems in singular floating systems, just as static corrections do for simply supported systems. The proof is based on hitherto unpublished expressions of inertial contributions in terms of orthogonal projectors. It is worth noting that such expressions present by themselves some technical interest. Modal formulae being delivered in closed form, a thorough analysis of errors becomes possible. Special “spectrograms” are introduced to appraise the convergence of ordinary or modified modal sums. When applied to infinite-dimensional modal boundary discontinuities, the method of orthocomplement eliminates the Gibbs oscillations that affect the ordinary spectral sums, and produces the same corrected formulae as it did for finite-dimensional examples. Effective solutions for the computation of vibratory or acoustic boundary receptances, are developed in that way. The paper once again illustrates how orthogonal projectors can contribute to the determination of static corrections. Future applications to coupling problems, model reduction and modal synthesis are briefly presented at the end of the text.

Dynamic snap-through of an elastic shell subjected to a pulsed load

Dynamic instability “in the large” is investigated for an elastic shallow shell subjected to a pulsed load (PL) and a more general non-linear elastic continuous conservative system with potential energy of the “square of the norm plus a weakly continuous functional” kind with Rayleigh friction and a given initial velocity. An energy approach /1, 2/ developed in /3–6/ is used to analyse the dynamic snap-through (DS) of the non-linear elastic system subjected to a stationary step load. The problem is considered in an exact infinite-dimensional formulation. By using the concept of an equilibrium stability trough and reserve /7/, definitions are given of the dynamic stability of a system, the critical PL of its DS and the astatic critical PL. The latter is determined from the stationary problem and yields the lower bound for those PL values for which DS occurs. It is established that a necessary condition for the DS of a system subjected to a PL is the existence of a saddle point of the potential energy for the same load-free system at the boundary of a stable zero equilibrium trough. It is proved that this necessary condition is satisfied for sufficiently thin strictly convex shells of revolution with movable and fixed hinge-support as well as for a certain class of arbitrary strictly convex shells with a movable hinge support. Therefore, the stability reserve has a graphic mechanical meaning as an exact upper bound of the kinetic energy which can be added to the system at rest so that it does not reach the least saddle points among the energy heights leading from a zero equilibrium trough to troughs of other equilibria. Results of computer calculations of the critical PL of dynamic snap-through and astatic snap-through for spherical and conical shells are presented. Depending on the boundary conditions, the lower limits are determined for the ratio between the shell rise and its thickness, for which DS is possible under the action of a PL. Note that the reasoning associated with estimating the kinetic energy needed to overcome the energy barrier in the problem of DS under the action of a PL on a system with two degrees of freedom, obtained by the Bubnov-Galerkin method from the vibrations equations for an elastic arch, were first applied in /1/. The extensions to finite-dimensional models with a large number of degrees of freedom made by different authors are reflected in /2, 8, 9/.