Table of contents

Valuation of options and other financial derivatives critically depends on the underlying stochastic process specified for a particular market. An inverse problem of option pricing is to determine the nature of this stochastic process, namely, the distribution of expected asset returns implied by current market prices of options with different strikes. We give a rigorous mathematical formulation of this inverse problem, establish uniqueness, and suggest an efficient numerical solution. We apply the method to the S&P 500 Index and conclude that the index is negatively skewed with a higher probability of the sudden decline of the US stock market.

One of the approaches to inverse problems based upon their relations to boundary control theory (the so-called BC method) is presented. The method gives an efficient way to reconstruct a Riemannian manifold via its response operator (dynamical Dirichlet-to-Neumann map) or spectral data (a spectrum of the Beltrami - Laplace operator and traces of normal derivatives of the eigenfunctions). The approach is applied to the problem of recovering a density, including the case of inverse data given on part of a boundary. The results of the numerical testing are demonstrated.

We develop a numerical method for approximating f, given its Laplace transform g on , i.e. assuming that g is in . The method is based on the Tichonov regularization operator. This operator is approximated by a finite-dimensional displacement operator. Convergence of the method, together with some examples, will be given.

A generalization of Isakov's theorem to admissible shapes that include screens with a Robin boundary condition (in which the impedance is also an unknown) is presented. We also establish a characterization that enables us to distinguish the far field of plane screens from others. Numerical results using a quasi-Newton method are presented.

This paper discusses a novel technique of estimating gas temperatures based on impedance tomography. More specifically, assume that we have a gas funnel (e.g. doorway, window, chimney) equipped with a mesh of thin electrically conducting filaments. Furthermore, assume that the thermal and thermoelectric properties of the conducting material are known. The temperature mapping method is based on changes of the resistivity of the filaments by the changes in temperature. The inverse problem is closely related to the standard tomography problem. Due to the severe underdetermination of the problem, common inversion techniques used in computerized tomography cannot be employed here. The problem is, therefore, recast in a form of a Bayesian parameter estimation problem. Markov chain Monte Carlo methods (MCMC) are applied for exploring the posterior distribution.

The problem of tomography with a finite set of projections has been the object of many investigations. In particular the formulae for the generalized solution and the singular values have been obtained in the case of equispaced projections. These results imply that the problem of determining the generalized solution is well-posed even if it may be ill-conditioned. In this paper we derive several properties of the singular values and singular functions both in the general case and in the case of equispaced projections. We use these results to identify the singular functions related to the aliasing effects in the generalized solution and to estimate the resolution achievable when these effects have been eliminated by means of a suitable filtering. It turns out that the resolution essentially depends on the number of projections and not on the noise, if the number of projections is smaller than a certain upper limit (depending on the noise), which can be quite large. In the case of equispaced projections, the resolution coincides with that estimated by the asymptotic theory even when the number of projections is rather small.

In this paper, an approach to solving the inverse problem for the telegraph equation is examined in detail for the special case where the telegraph equation is given by

with one source generating an instant spherical wave. By splitting the wave into up-going and down-going components and observing the reflective wave (down-going component) on the surface , one can obtain q(x). The first part of this paper is to prove the existence of wave splitting (the direct problem). For the second part, a layer-stripping method and asymptotic analysis will be applied to approximate the solution of the inverse problem. The fact that the leading coefficients of the asymptotic expansion of yield q(x) is a necessary condition for the solution of the ill-posed problem which involves the inverse Dirichlet operator. The ill-posedness of recovering the reflective wave is also mentioned.

A fundamental inversion problem for the magnetic force microscopy (MFM) of a semi-infinite linear magnetic material is formulated. Magnetostatic conditions are assumed so that a magnetic scalar potential can be employed. Using a point-dipole MFM tip, a unique layer-dependent magnetic permeability can be recovered from one-dimensional force measurements. The inversion procedure uses a wavenumber-dependent kernel function K(k). These results describe the use of MFM data to extract a material property as a function of depth.

Assuming the density function is of the form and the sinogram is known for all and , we give methods for finding and . First we consider the case where the locations , are known. The value of is estimated from , where h is a function and is a non-degenerated projection direction such that for . Then we derive a method for the general case: the number m of functions, their localization and their intensity are estimated from the data . We show that this new method is more efficient than the filtered backprojection when the resolution in the variable p of the sinogram is high.

We study inverse spectral problems for second-order differential equations on a finite interval having an arbitrary number of turning points. For three classes of inverse problems we prove that they have a unique solution and we provide a procedure for constructing the solution.

In this paper the class of inverse coefficient problems for nonlinear monotone potential elliptic operators is considered. This class is characterized by the property that the coefficient of elliptic operator depends on the gradient of the solution, i.e. on . The unknown coefficient is required to belong to a set of admissible coefficients which is compact in . Using a variational approach to the nonlinear direct problem it is shown that the solution of the linearized direct problem converges to the solution of the nonlinear direct problem in -norm. For the nonlinear direct problem weak -coefficient convergence is proved. This result allows one to prove the existence of quasisolutions of inverse problems with different types of additional conditions (measured data). As an important application of the theory, an inverse elastoplastic problem for a cylindrical bar and a nonlinear Sturm - Liouville problem are considered.

Convergence and logarithmic convergence rates of the iteratively regularized Gauss - Newton method in a Hilbert space setting are proven provided a logarithmic source condition is satisfied. This method is applied to an inverse potential and an inverse scattering problem, and the source condition is interpreted as a smoothness condition in terms of Sobolev spaces for the case where the domain is a circle. Numerical experiments yield convergence and convergence rates of the form expected by our general convergence theorem.

We study the free boundary problem to identify the plasma region in a stationary tokamak machine using the magnetic flux function. Uniqueness has been proved and the free boundary has been explicitly computed in the case where the domain is an n-dimensional ball. We also derive an energy distribution identity which is applicable to identify the location of the free boundary for more general domains in two-dimensional space.

Using the matrix Riemann - Hilbert factorization approach for nonlinear evolution equations (NLEEs) integrable in the sense of the inverse scattering method, we obtain, in the solitonless sector, the leading-order asymptotics as of the solution to the Cauchy initial-value problem for the modified nonlinear Schrödinger equation, , : also obtained are analogous results for two gauge-equivalent NLEEs; in particular, the derivative nonlinear Schrödinger equation, .

An n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation , , , is considered. The problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors , where is the boundary of . A novel numerical method is derived and tested. Numerical tests are conducted for n = 2 and for ranges of parameters which are realistic for applications to early breast cancer diagnosis and the search for mines in murky shallow water using ultrafast laser pulses. The main innovation of this method lies in a new approach for a novel linearized problem (LP). Such a LP is derived and reduced to a well-posed boundary-value problem for a coupled system of elliptic partial differential equations. A principal advantage of this technique is in its speed and accuracy, since it leads to the factorization of well conditioned, sparse matrices with non-zero entries clustered in a narrow band near the diagonal. The authors call this approach the elliptic systems method (ESM). The ESM can be extended to other imaging modalities.

This paper is concerned with the nonlinear inverse acoustic problem of the construction of the density/speed of sound distribution within a domain, given spectral data on the boundary of the domain. It is assumed throughout that only a finite set of data is known and some attention is given to error-prone data. A moments approach to the solution of this problem is outlined, together with associated error estimates. Results of numerical experiments on model problems are used to demonstrate the viability of the approach and the accuracy of the error estimates. The paper is restricted to two-dimensional problems, but the technique remains applicable for higher dimensions.

This paper studies the inverse scattering problem by an open sound-hard arc. We prove a uniqueness theorem for this problem. For the foundation of Newton methods we establish the Fréchet differentiability of the far-field operator with respect to the open arc and show that the Fréchet derivative can be obtained as the solution of a hypersingular integral equation. We describe regularized Newton-type methods for approximately solving the inverse scattering problem and present numerical results for a number of differently chosen ansatz functions and arcs.

Thermal-wave slice diffraction tomography (TSDT) is a photothermal imaging technique for non-destructive evaluation (NDE), leading to the detection of subsurface cross sectional defects in opaque solids in the very-near-surface region ( - mm). An exact, Green's-function based mathematical model that represents the behaviour of a three-dimensional thermal wave is developed and correlated with a numerical reconstruction technique. The computational technique utilizes the well known Tikhonov regularization method to invert almost singular matrices resulting from the ill-posedness of the inverse thermal-wave problem for the reconstruction of thermal diffusivity cross sectional images in materials. Numerical calculations of the inverse problem are carried out using the Born approximation and simulated reconstructions in back scattering and transmission are presented.

Laser infrared photothermal radiometry is developed into a thermal-wave slice diffraction tomography (TSDT) instrumentation and measurement methodology for cross sectional imaging of subsurface defects. A mild-steel sample with artificial subsurface defects is used to test the TSDT potential. Experimental reconstructions using the Born approximation both in back-scattering and transmission modes are presented. The numerically reconstructed experimental data constitute the first experimental TSD tomograms, which were found to be limited by instrumental aperture-broadening effects.

In this paper we introduce a general regularization scheme to reconstruct solutions of nonlinear ill-posed problems F(x)=y where instead of y noisy data with are given and is a nonlinear operator between Hilbert spaces X and Y. In this regularization scheme regularized solutions are defined as a fixed point of the nonlinear equation where and stands for . Assuming certain conditions concerning the nonlinear operator F and the smoothness of the element we derive stability estimates which show that the accuracy of the regularized solutions is order optimal provided that the function and the regularization parameter have been chosen properly.

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Book review

Regularization of Inverse Problems

H W Engl, M Hanke and A Neubauer

1996 Dordrecht: Kluwer

332pp ISBN 0-7923-4157-0 £ 108.00

The book of Tikhonov and Arsenine, Solutions of Ill-Posed Problems, published in 1977, was the first general presentation of regularization theory, an approach to inverse and ill-posed problems developed by Tikhonov and his co-workers in the decade preceding the publication of the book. Since then, several books and review papers have been published on the same subject which is now universally recognized as the basic approach to inverse problems.

This book, which has appeared about twenty years after the book of Tikhonov and Arsenine, is, in my opinion, the most complete account of the mathematical theory of regularization. The authors, of course, claim that their presentation is far from being complete and this is certainly true: at the present state of the art, it would be very difficult to write a book of reasonable size covering all the most important aspects of the theory and applications. In any case, I think that this book is more complete than any other work existing on the subject.

It is a book written by mathematicians for mathematicians. The typical style of the mathematician is evident everywhere, and not only in the structure of the text, which is mainly laid out as a sequence of definitions, lemmas, theorems, etc. This is an asset for a reader with a good mathematical backgroud, especially in functional analysis, but might be a drawback for a non-mathematician mainly interested in the applications of regularization methods. The impact of the book could be greatly increased if each chapter were supplemented by a short section describing and commenting on the results in a form accessible to physicists, engineers, geophysicists, etc. Indeed the book can be considered as a primary source of information on several important topics. Amongst others I mention the order optimality of regularization methods, the a priori and a posteriori choice of the regularization parameter, the iterative and semi-iterative regularization methods, the regularization properties of conjugate gradient and maximum entropy and, finally, the regularization theory of nonlinear problems. Most of these topics are of vital importance for the applications.

The book considers inverse problems formulated in a Hilbert space setting, but this is not a substantial restriction because most inverse problems of practical interest can be formulated in such a context. After two introductory chapters, the first on examples of inverse problems and the second on basic properties of generalized inverses and spectral representations, six chapters are devoted to the regularization of linear problems, one to the discretization of linear problems, and two to the regularization of nonlinear problems.

A few words about the approach followed by the authors. After giving the general definition of regularization operators and the related concept of order optimality, a large class of regularization operators is introduced by means of the spectral filtering of the generalized inverse. This approach is used for discussing in detail the various aspects of the problem of the choice of the regularization parameter. Moreover, the classical Tikhonov (or Tikhonov - Phillips) regularization operator is discussed as a particular case. In other words, the spectral approach is preferred to the more frequently used variational approach. The latter is mainly considered in the chapter devoted to the regularization with differential operators (or with seminorms). The treatment of linear problems is completed by two chapters on iterative methods, one on the Landweber method and its various ramifications, the other on conjugate gradient. This is a short and valuable chapter where the most relevant properties of this method, especially important for the applications, are derived in a succinct way. Also the short chapter on the discretization of a linear inverse problem and the implementation of the main regularization methods should be very useful.

Finally I appreciated the two chapters devoted to the regularization of nonlinear problems. This is still a very lively research area and, even if the presentation given in the book is certainly incomplete, it can provide the reader with the appropriate background for assessment of the extensive literature existing on this topic. Moreover, results on the Tikhonov regularization of nonlinear problems are used for deriving the regularization properties of the maximum entropy method, and this is another important point which recommends this book over other existing works.

M Bertero Università di Genova