Quasi-one-dimensional method for the two-dimensional magnetotelluric sounding inverse problem

The two-dimensional magnetotelluric sounding inverse problem is solved by a quasi-one-dimensional method. A linearization method is proposed for one-dimensional inverse problems. The appearance of false conducting structures in the solution of the inverse problem is considered and some counter-measures are suggested. The quasi-one-dimensional method is applicable to many-dimensional magnetotelluric sounding inverse problems if the electrical conductivity is a priori known to vary slowly along the Earth’s surface. __________ Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 5–11, 2005.     

An inverse problem of determining the implied volatility in option pricing

In the Black-Scholes world there is the important quantity of volatility which cannot be observed directly but has a major impact on the option value. In practice, traders usually work with what is known as implied volatility which is implied by option prices observed in the market. In this paper, we use an optimal control framework to discuss an inverse problem of determining the implied volatility when the average option premium, namely the average value of option premium corresponding with a fixed strike price and all possible maturities from the current time to a chosen future time, is known. The issue is converted into a terminal control problem by Green function method. The existence and uniqueness of the minimum of the control functional are addressed by the optimal control method, and the necessary condition which must be satisfied by the minimum is also given. The results obtained in the paper may be useful for those who engage in risk management or volatility trading.     

Criteria for the functional controllability and invertibility of non-linear systems

In the development of investigations on inverse problems [1, 2], criteria for the functional controllability and invertibility of non-linear systems of equations with an output are obtained. The solution is based on the construction of an inverse system for which the input action of the initial system is the output. An identification problem is considered which corresponds to the problem of invertibility with an unknown initial state. The properties of λ-invertibility and λ-identifiability, which arise in cases when the output signal is known in a set of trajectories, are investigated.     

Estimates for the inverse of a matrix

The problem of finding bounds for the elements of the inverse of a matrix satisfying various known regularity conditions is considered. The results are obtained in such a way as to apply automatically to partitioned matrices.     

On the direct and inverse problems in fluid-structural interaction

This paper is concerned with the direct and inverse scattering problems in fluid-structure interaction. The scattering problem in the fluid-structure interaction can be simply described as follows: an acoustic wave propagates in the fluid domain of infinite extent where a bounded elastic body is immersed. The direct problem is to determine the scattered pressure and velocity fields in the fluid domain as well as the displacement fields in the elastic body, while the inverse problem is to reconstruct the shape of the elastic scatterer from a knowledge of the far field pattern of the fluid pressure or from the measured scattered fluid pressure field. As is well known, the inverse problems are generally nonlinear and highly ill-posed. For treating inverse problem of this kind, we reformulate the problem as a nonlinear optimization problem including special regularization terms. The precise formulation of the nonlinear objective functional will depend on the approaches of the direct problem. In this paper, the direct problem is reformulated by introducing an artificial boundary and the corresponding inverse problem will be analyzed. Some of the basic results are summarized without proofs. The latter are available in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inverse unitary eigenproblems and related orthogonal functions

Summary. This paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions. In particular, the inverse eigenvalue problems for unitary Hessenberg matrices and for Schur parameter pencils are considered. The Szeg? recursion is known to be identical to the Arnoldi process and can be seen as an algorithm for solving an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils. It is shown that solving this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only O() arithmetic operations as compared with O() operations needed for algorithms that ignore the structure of the problem. Received April 3, 1995 / Revised version received August 29, 1996     

On the Main Aspects of the Inverse Conductivity Problem

We consider a nonlinear inverse problem for an elliptic partial differential equation known as the Calder{\''o}n problem or the inverse conductivity problem. Based on several results, we briefly summarize them to motivate this research field. We give a general view of the problem by reviewing the available results for $C^2$ conductivities. After reducing the original problem to the inverse problem for a Schr\"odinger equation, we apply complex geometrical optics solutions to show its uniqueness. After extending the ideas of the uniqueness proof result, we establish a stable dependence between the conductivity and the boundary measurements. By using the Carleman estimate, we discuss the partial data problem, which deals with measurements that are taken only in a part of the boundary.     

Robust inverse optimal control for flexible structures

An inverse optimal control problem is formulated to develop robust control laws for purely oscillatory systems. The optimal control solution requires output feedback with specified constraints, leading to robustness with respect to unmodeled modes and a large class of parameter variations. The robustness properties are proved directly from known properties of control laws resulting from quadratic performance indices. The control laws are useful for poorly damped flexible structures.This research was supported by the Office of Naval Research, Contract No. N00014-77-C-0247.     

Solution of the inverse magnetotelluric sounding problem by means of a method using the synthesized magnetic field

A method of solution of the inverse magnetotelluric sounding problem is considered. The method uses the known frequency-dependent magnetic field on the Earth’s surface. The magnetic field on the Earth’s surface is found with the help of a special technique from the known impedance on the Earth’s surface. The method is applied to solve typical inverse problems and provides for efficient determination of final solutions.     

Inverse problems of reconstructing parameters of the Navier-Stokes system

An inverse problem of reconstructing parameters not known a priori of the dynamical system described by the boundary-value problem for the Navier-Stokes system is considered. The reconstruction is based on one piece of admissible information or another about the motion of the dynamical system (solution of the corresponding boundary-value problem). In particular, one of the problems considered is the inverse problem consisting of reconstruction of the a priori unknown right-hand side of the Navier-Stokes system. The right-hand side characterizes the density of exterior mass forces acting on the system. This problem, as well as many other similar problems, is ill-posed. Two methods are proposed for its solution: the statistical method and the dynamical method. These methods use different initial information. In solving the problem by using the statistical method, initial information for the solution is the results of approximate measurement (in one sense or another) of the motion of the dynamical system on a given interval of time. Here, the reconstruction is performed after the corresponding interval of time. For solution of the problem by this method, the concepts and constructions of open-loop control theory are used. In solving the problem by using the dynamical method, initial information for its solution is the results of approximate (in one sense or another) measurements of the current states of the system, which are dynamically obtained by the observer. Here, the reconstruction is dynamically performed during the process. For solution of the problem by the dynamical method, the concepts and constructions of the dynamical regularization method based on positional control theory are used. Also, the author considers various modifications and regularizations of the methods for solution of problems proposed that are based on one piece of a priori information or another about the desired solution and solvability conditions of the problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Dependence of the solution of a semi‐discrete von Mises equation with respect to the pressure for an inverse problem

We study precisely the solution of a semi‐discrete von Mises problem with known pressure to generalize a method for an inverse problem: the problem with known displacement. We prove the boundedness of the diffusive term, the uniqueness and the continuity of the solution with respect to the pressure and a coercivity property. However, the estimate of the diffusive term is not sharp enough for the inverse problem. We highlight a special behaviour of this term due to the combination of the nonlinearity with the degeneracy of the diffusion coefficient. Copyright © 2013 John Wiley & Sons, Ltd.     

On possibility of obtaining linear accuracy evaluation of approximate solutions to inverse problems

We consider a concept of linear a priori estimate of the accuracy for approximate solutions to inverse problems with perturbed data. We establish that if the linear estimate is valid for a method of solving the inverse problem, then the inverse problem is well-posed according to Tikhonov. We also find conditions, which ensure the converse for the method of solving the inverse problem independent on the error levels of data. This method is well-known method of quasi-solutions by V. K. Ivanov. It provides for well-posed (according to Tikhonov) inverse problems the existence of linear estimates. If the error levels of data are known, a method of solving well-posed according to Tikhonov inverse problems is proposed. This method called the residual method on the correctness set (RMCS) ensures linear estimates for approximate solutions. We give an algorithm for finding linear estimates in the RMCS.     

AN ADAPTIVE INVERSE ITERATION FEM FOR THE INHOMOGENEOUS DIELECTRIC WAVEGUIDES

We introduce an adaptive finite element method for computing electromagnetic guided waves in a closed, inhomogeneous, pillared three-dimensional waveguide at a given frequency based on the inverse iteration method. The problem is formulated as a generalized eigenvalue problems. By modifying the exact inverse iteration algorithm for the eigenvalue problem, we design a new adaptive inverse iteration finite element algorithm. Adaptive finite element methods based on a posteriori error estimate are known to be successful in resolving singularities of eigenfunctions which deteriorate the finite element convergence. We construct a posteriori error estimator for the electromagnetic guided waves problem. Numerical results are reported to illustrate the quasi-optimal performance of our adaptive inverse iteration finite element method.     

On the inverse problem at fixed energy in the "Iterative" range

The scattering amplitude by a spherically symmetric potential at fixed energy is given in the Born approximation by a filtered Fourier transform, whose inverse is not unique. It is well known that matrix methods enable one to study exactly the problem at fixed energy in classes of potentials parametrised by sequences of numbers. In the range of potentials (or of phase shifts) where these methods can be managed by iteration, Born case is a limit. This article is a brief survey of the inverse problem (scattering amplitude?→?potential?) recalling how the nonuniqueness predicted in the Born approximation appears in these exact methods, showing henceforth that the inverse problem ill-posedness corresponds to physical features of the potential on which experiments at finite energy are unable to give information.     

Stability of inverse problems in an infinite slab with partial data

In this paper, we study the stability of two inverse boundary value problems in an infinite slab with partial data. These problems have been studied by Li and Uhlmann for the case of the Schrödinger equation and by Krupchyk, Lassas, and Uhlmann for the case of the magnetic Schrödinger equation. Here, we quantify the method of uniqueness proposed by Li and Uhlmann and prove a log–log stability estimate for the inverse problems associated to the Schrödinger equation. The boundary measurements considered in these problems are modeled by partial knowledge of the Dirichlet-to-Neumann map: in the first inverse problem, the corresponding Dirichlet and Neumann data are known on different boundary hyperplanes of the slab; in the second inverse problem, they are known on the same boundary hyperplane of the slab.     

Pure pursuit navigation on Riemannian manifolds

Here, the Pure Pursuit navigation on Riemannian manifolds is studied and the time optimal trajectories of this process are found. It has been shown that these trajectories are geodesics of a non-Randers type Finsler metric, known in the literature as Matsumoto metric. A geometrical approach to kinematics in control theory is illustrated and the inverse problem is formulated.     

Uniqueness and stability in determining the heat radiative coefficient,the initial temperature and a boundary coefficient in a parabolic equation

We consider an inverse parabolic problem. We prove that the heat radiative coefficient, the initial temperature and a boundary coefficient can be simultaneously determined from the final overdetermination, provided that the heat radiative coefficient is a priori known in a small subdomain. Moreover we establish a stability estimate for this inverse problem.     

Error estimates for the approximate identification of a constant coefficient from boundary flux data

A priori error estimates are derived for the simplest finite difference and finite element approximations to an inverse problem in which it is desired to identify an unknown constant coefficient in a differential equation whose general form is known.     

Determination of the coefficient of thermal conductivity and heat capacity per unit volume in a multilayer medium

We consider the inverse problem of simultaneously determining two time-dependent thermophysical characteristics—the coefficient of thermal conductivity and the heat capacity per unit volume—for a body having the shape of a layer situated between two other layers with known thermophysical characteristics. The necessary measurements are carried out on their outside boundaries. The problem is reduced to a system of nonlinear equations for which the existence of a solution is established by using Schauder's fixed-point theorem. We find conditions that guarantee that the solution of the inverse problem is unique. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 153–159.     

On the Inverse Scattering Transform for the Kadomtsev-Petviashvili Equation

The initial value problem of the Kadomtsev-Petviashvili equation for one choice of sign in the equation has been recently investigated in the literature. Here we consider the other choice of sign. We introduce suitable eigenfunctions which though bounded are not analytic in the spectral parameter. This, in contrast to the known case, prevents us from formulating the inverse problem as a nonlocal Riemann-Hilbert boundary value problem. Nevertheless a suitable formulation is given and a formal solution is constructed via a linear integral equation.