Direct and Inverse Optimization Problems in a Gas Deposit Model

This paper proposes an optimization model and shows that most inverse combinatorial optimization problems so far discussed can be fit into this model as special cases. We propose a Newton-type algorithm for this model under l∞ norm. This algorithm can solve the model in strongly polynomial time if the subproblem involved is solvable in strongly polynomial time for any fixed value of the parameter appearing in the subproblem, and it is shown that most particular inverse optimization problems encountered are this kind. Therefore, through this paper we show that a large group of inverse optimization problems can be handled in a uniform way and solved in strongly polynomial time.

In this paper, we develop a non‐iterative way to solve biharmonic boundary value problems by using a radial basis meshless method. This is an original application of meshless method to solving inverse problems without any iteration, since traditional numerical methods for inverse boundary value problems mainly are iterative and hence very time‐consuming. Numerical examples are presented for inverse biharmonic boundary value problems and corresponding direct problems, since solving direct problems is a preliminary step for inverse problems. All our examples of direct and inverse problems are solved within seconds in CPU time on a standard PC, which makes our proposed technique a great potential candidate for wide‐spread applications to other inverse problems. Copyright © 2004 John Wiley & Sons, Ltd.

Modeling ore generation in a magmatic context

This paper concerns the source scattering problems for acoustic wave propagation, which is governed by the two- or three-dimensional stochastic Helmholtz equation. As a source, the electric current density is assumed to be a random function driven by an additive colored noise. Given the random source, the direct problem is to determine the radiated random wave field. The inverse problem is to reconstruct statistical properties of the source from the boundary measurement of the radiated random wave field. In this work, we consider both the direct and inverse problems. We show that the direct problem has a unique mild solution via a constructive proof. Using the mild solution, we derive effective Fredholm integral equations for the inverse problem. A regularized Kaczmarz method is developed by adopting multifrequency scattering data to overcome the challenges of solving the ill-posed and large scale integral equations. Numerical experiments are presented to demonstrate the efficiency of the proposed method....

Consider an exterior problem of the three-dimensional elastic wave equation, which models the scattering of a time-harmonic plane wave by a rigid obstacle. The scattering problem is reformulated into a boundary value problem by introducing a transparent boundary condition. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

We explore direct and inverse problems for the bi-Laplace equation within a bounded, simply connected planar domain. The direct problem is recast into a system of Fredholm integral equations via Green-type identities, ensuring the existence and uniqueness through functional analysis in appropriate Sobolev spaces. For the inverse problem, we consider the reconstruction of missing Cauchy boundary data on a non accessible portion of the domain, where Riquier-Neumann-type conditions are imposed. To counteract the intrinsic ill-posedness of this data completion task, the problem is reduced to a nonlinear boundary integral system, which is stabilized through Tikhonov regularization. Numerical investigations validate the robustness and accuracy of the proposed approach, establishing its effectiveness in recovering missing boundary information for fourth-order elliptic models.

A new type of Green function approach is introduced for time-domain scattering for a stratified dissipative slab. The new Green functions are defined as mappings from the transmitted field to the split fields (i.e., the right-moving and left-moving fields) at any point within the stratified slab. Unlike the usual Green functions, the present Green functions have compact support in the time variable. The new technique is illustrated for the case of normally incident electromagnetic waves on a stratified slab where the permittivity, permeability, and conductivity vary with the depth. The linear and homogeneous partial differential equations (PDEs) for these ‘‘compact Green functions’’ are derived. The PDEs together with the initial and boundary conditions are well suited for a numerical treatment. Numerical results for the compact Green functions are presented in the direct problem, while in the inverse problem the permittivity and conductivity (or the permeability and conductivity) are reconstructed simultaneously using the reflection and transmission data for the first round trip. The present method is useful and attractive for both the direct and inverse problems due to its simplicity, high speed, and high accuracy in numerical computations.

One of the key problems of motor control is the redundancy problem, in particular how the central nervous system (CNS) chooses an action out of infinitely many possible. A promising way to address this question is to assume that the choice is made based on optimization of a certain cost function. A number of cost functions have been proposed in the literature to explain performance in different motor tasks: from force sharing in grasping to path planning in walking. However, the problem of uniqueness of the cost function(s) was not addressed until recently. In this article, we analyze two methods of finding additive cost functions in inverse optimization problems with linear constraints, so-called linear-additive inverse optimization problems. These methods are based on the Uniqueness Theorem for inverse optimization problems that we proved recently (Terekhov et al., J Math Biol 61(3):423-453, 2010). Using synthetic data, we show that both methods allow for determining the cost function. We analyze the influence of noise on the both methods. Finally, we show how a violation of the conditions of the Uniqueness Theorem may lead to incorrect solutions of the inverse optimization problem.

Determination of temperature wall functions for high Rayleigh number flows using asymptotics: A systematic approach

An inverse optimization of convection heat transfer in rectangle channels with ribbed surface based on the extremum principle of entransy dissipation

All known analytical solutions of the integral equation of the turbulent thermal boundary layer for a rotating disk have been obtained for the case of direct problem. This means finding the Nusselt number at a given distribution of the wall temperature. This distribution is described by power law and is monotone (derivative of wall temperature with respect to the radial coordinate does not change its sign). Outlined in this paper is an analytical form of non-monotone distribution of the wall temperature, which provided a new analytical solution for the turbulent Nusselt number including earlier known equations as a specific particular case. The solution is based on the integral method, which proved to be more precise than known Dorfman’s approach. Chosen for validation of the proposed method were turbulent heat transfer experiments of Northrop and Owen (1988). Predictions presented include analytical studies using inverse and direct problem solutions as well as numerical simulations using polynomial approximations of the experimental wall temperature distributions.

A model-based method has been developed for the performance simulation and conceptual design of rocket-type pulse detonation engines (PDEs). A reduced-order model (ROM) has been generated based on the high order singular value decomposition of a data tensor obtained from CFD computations. This ROM could be used to solve the direct (performance) and inverse (design) problems in the context of the early phases of pulse detonation engine design. Output performance parameters are predicted from prescribed input operation/geometry parameters in the direct problem, and vice versa in the inverse problem. The focus of this method is industrial application in situations where large parametric searches are to be performed with a reasonable level of fidelity at a low computational cost. It was found that the performance and conceptual design tool thus developed provides results that deviate, on average, by less than 10% from the CFD results. Regarding practical implementation, the method allows for shifting the heavier computational load off-line. In this way, when working on-line, the user can obtain results in less than a second for every single case. The main contribution of this study is showing that a model-based approach that combines CFD and tensor decomposition has the potential to extract a maximum of information from a given computational effort. This characteristic makes the method of interest for early design phases in the aerospace industry.

In this article, we study a class of direct and inverse coefficient problems defined for a nonlinear parabolic equation. An existence of a quasi-solution of the considered inverse problem is obtained in the appropriate class of admissible coefficients. Direct problem is solved numerically by using semi-implicit finite difference scheme and then some examples are presented related to steady-state solution of the nonlinear direct problem and ill-posedness of the inverse problem.

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane. A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases. Given the incident field, the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity. In this paper, both the direct and inverse scattering problems are discussed based on a symmetric coupling method. Variational formulations for the direct scattering problem are presented, existence and uniqueness of weak solutions are studied, and the domain derivatives of the field with respect to the cavity shape are derived. Uniqueness and local stability results are established in terms of the inverse problem.

We consider the diffraction of a time-harmonic wave incident upon a grating (or periodic) structure. We study mathematical issues that arise in the direct modeling, inverse, and optimal design problems. Particular attention is paid to the variational approach and to finite-element methods. For the direct problem various results on existence, uniqueness, and numerical approximations of solutions are presented. Convergence properties of the variational method and sensitivity to TM polarization are examined. Our recent research on inverse diffraction problems and optimal design problems is also discussed.