Table of contents

It is shown that the unknown time-dependent right-hand side with explicitly bounded growth rate in a general parabolic equation is uniquely determined by one additional final measurement. As an important application we derive the corresponding uniqueness results for different coefficient identification problems with no smallness assumptions imposed.

A brief introduction is given to the importance and ubiquitous occurrence of inverse problems in astronomy and the potential mutual value of cross-fertilization of ideas between astronomers and inverse problem practitioners is emphasized. Astronomical inverse problems are divided into broad classes-instrumentation and data analysis, dynamics and time series, spatial distribution of sources and spectral analysis-and these are used to provide a guide to the structure of this special journal issue on inverse problems in astronomy.

The NASA/ESA Hubble Space Telescope (HST), an astronomical observatory launched in 1990 into a low-altitude space orbit, was designed to deliver near-diffraction limited images, but its optics suffered from substantial spherical aberration. The HST image restoration problem is aggravated by insufficient image sampling, by a mixture of noise sources including spatially non-stationary, non-Gaussian noise, and by the desire to quantitatively evaluate the restored data. Restoration efforts have helped to minimize the impact of the data distortions on HST's scientific return. At the end of 1993 HST was refurbished, and its optics have largely been restored to meet the design goals. Nevertheless, image restoration remains important to treat problems such as combining dithered, undersampled image frames, or to combine images with different resolutions and signal-to-noise ratios.

Imaging with mirrors is difficult at high photon energies, and this has led to the development by X- and gamma-ray astronomers of a variety of alternative approaches to forming images. Most of these involve 'coding' the sky brightness distribution, such that a single bright point source at a given position produces a complex but characteristic signature on the detector. The mathematical description of the imaging process, and the advantages and disadvantages of focusing and non-focusing approaches are discussed. The problem of image reconstruction in the focusing case is similar to that in familiar optical imaging applications, though complicated by variations in point spread function with position and energy. The authors concentrate on the inversion of coded images, which are more characteristic of high-energy astronomy. The design of the coding process, which is crucial to achieving good imaging performance, is discussed, as are techniques for image reconstruction in the case of 'perfect coding', and a variety of remedies for making the best use of imperfectly coded data.

Laser interferometric gravitational wave detectors are likely to participate in coordinated observations with each other, with resonant bar antennae, and with neutrino, gamma ray, and optical detectors. Gravitational waves carry information about the physics of their source and about cosmological parameters. In order to extract this information, one is faced with the inverse problem of reconstructing the gravitational wave signal from the noisy data acquired by a network of detectors. This problem is reviewed here and it is shown how three interferometric antennae can yield, in principle, enough data for a complete solution. The inspiral waveform of a coalescing binary system of neutron stars or black holes is particularly rich in the physics it conveys about the relativistic stars. The amplitudes of each independent polarization of the wave, together with measurements of its phase modulation, supply information that can be inverted to yield a distance to the source. The importance to fundamental physics and astronomy, of a network of detectors with the potential to solve inverse problems such as these, is discussed.

Finding potentials or force fields from given families of orbits is the type of inverse problem of dynamics discussed in this paper. We present the pertinent partial differential equations for various versions of the problem such as, for instance, for conservative or autonomous non-conservative fields in two or three dimensions, for inertial or relating frames, for one material point or, more, generally, for holonomic systems with n degrees of freedom. The notion of the family boundary curves is introduced. The role of the homogeneity of the given family or of the required potential, as well as the question of multitude of compatible pairs of orbits and potentials is discussed. Comments on the relation of the problem to problems of astronomical interest are made, at appropriate places, throughout the text.

Interesting inversion problems arise in helioseismology, the seismic study of the interior of the Sun using the frequencies of its global resonant modes of oscillation. In particular, one can use the frequencies to investigate how the rotation of the Sun varies beneath the visible surface. This review concentrates primarily on the methods and tools that have been used in the helioseismic inversion problem. In particular, an analysis of different inversion methods using the spectral properties of a generalized singular-value decomposition has proved useful in helioseismology, and this suggests ways in which the inversion of large datasets can be performed more efficiently.

The basic equation of stellar statistics connects the probability density function of a measurable quantity with the probability density of two variables, which cannot be observed directly, by the Bayes theorem of conditional probabilities. The resulting relation is a Fredholm-type integral equation of the first kind. If the two background variables are statistically independent we recover the convolution equation. The analytical solution based on the Fourier transformation is very sensitive to high-frequency noise. Eddington's solution attempts to find the unknown function in form of a series Sigma gamma _jh⁽ⁱ)(z). Malmquist's method computes the conditional probability of the unknown variable assuming that the observed variable is given. The statistical aspect of the problem is expressed if one uses Lucy's algorithm which is a particular form of the more general EM algorithm. Dolan's matrix method solves numerically the matrix equation which approximates the integral equation. Methods are superior which retain the true statistical nature of the problem.

Since April 1991, over 1000 GRBs have been observed with the Burst and Transient Source Experiment (BATSE) on board the Compton Gamma Ray Observatory (GRO), providing unprecedented information on the brightnesses, spectra and celestial coordinates for these bursts. The authors address the application of inversion techniques to this data set (in particular the brightness distribution of observed bursts) in order to constrain the physical properties of these enigmatic objects. The central method used is based on the interdependence of normalized integral moments of non-negative functions. These techniques are applied to bursts distributed both in Euclidean space and in various cosmological (non-Euclidean) spaces. The authors derive surprisingly powerful constraints on the range of intrinsic luminosities of bursts and/or on their distribution in space. They also demonstrate the formal equivalence of their technique to one involving Mellin transforms. Finally, they point out the generality of their technique, which can be applied straightforwardly to any relation amongst three quantities A, B and C of the form A=B*C, where information about only the distribution of values of A is known or observed, and information about the distribution of values of B or C is desired.

An account is given of a method of smoothing spatial inhomogeneous data sets by using wavelet reconstruction on a regular grid in an auxilliary space onto which the original data is mapped. In a previous paper by the present authors, the authors devised a method for inferring the velocity potential from the radial component of the cosmic velocity field assuming an ideal sampling. Unfortunately the sparseness of the real data (the peculiar velocities of galaxies) as well as errors of measurement requires that the velocity field is first smoothed as observed on a three-dimensional support (i.e. the galaxy positions) inhomogeneously distributed throughout the sampled volume. The wavelet formalism permits a minimal smoothing procedure to be introduced that is characterized by the variation in size of the smoothing window function. Moreover, the output-smoothed radial velocity field can be shown to correspond to a well defined theoretical quantity as long as the spatial sampling support satisfies certain criteria. The authors also argue that one should be very cautious when comparing the velocity potential derived from such a smoothed radial component of the velocity field with related quantities derived from other studies (e.g. of the density field).

A short overview is given of the principal mechanisms by which electromagnetic radiation spectra are produced in cosmic sources and how these give rise to inverse diagnostic problems of the distribution of radiating particles. Apart from black body spectra, only optically thin sources are considered and radiation by nuclear and coherent mechanisms are excluded. Problems are subdivided into: continuum spectra (synchrotron, inverse Compton, and collisional bremsstrahlung processes); spectrum line intensity problems (of plasma density and temperature structure); spectrum line profile problems, with emphasis on Doppler line broadening. The main emphasis is on indicating the physical importance of the problems and/or drawing attention to where further work on inverse problem aspects is needed.

Recent work on inverse problems in the field of spectropolarimetry is reviewed. The problems include inversions of temporal, spectral and spatial variations in the degree of polarization to determine various physical properties of the polarized systems under observation. In each case simplifying assumptions have been made regarding the physics of the system so as to make the polarization equations amenable to standard inversion techniques. The authors conclude with a short discussion of other projects which have the potential to be cast as inverse problems.

Consider an electromagnetic plane wave incident on a doubly periodic structure in R³. The inverse problem is to determine the shape of the structure from the scattered field. Uniqueness theorems are proved by applying the uniqueness theorem of Cauchy-Kowalewska, by extending Isakov's approach and using a result on local injectivity of maps between finite-dimensional spaces.

The study of the inverse scattering problem for electromagnetic waves, especially under consideration of such realistic effects as spatial inhomogeneity, dispersion, dissipativity and multiple spatial dimensions, is based on the use of such scattering waves for probing the interior of systems not readily accessible to direct examination (e.g. biological systems). The significant biological interest is concerned with the absorption and dissipation of nonionizing radiation in living organisms. As a result, one is interested in knowing the electrical properties (e.g. the dielectric constants and conductivities) of the living tissue. However, it is well known that both the dielectric constant and the conductance (i.e. the displacement and conduction current susceptibilities) are frequency dependent because living tissue is composed largely of water. In this paper, we will derive a set of nonlinear integrodifferential equations relating the displacement susceptibility and conduction current susceptibility kernels to the scattering operators (i.e. reflection and transmission operators) via the invariant imbedding techniques. From these nonlinear integrodifferential equations, we will prove theorems for the existence, uniqueness and continuous dependence on data for the direct and inverse scattering problems for both the semi-infinite medium and the finite slab.

Addresses the scattering of acoustic and electromagnetic waves from a perturbed dissipative half-space. For simplicity, the perturbation is assumed to have compact support. Section 1 discusses the application that motivated this work and explains how the scalar model used here is related to Maxwell's equations. Section 2 introduces three formulations for direct and inverse problems for the half-space geometry. Two of these formulations relate to scattering problems, and the third to a boundary value problem. Section 3 shows how the scattering problems can be related to the boundary value problem. This shows that the three inverse problems are equivalent in a certain sense. In section 4, the boundary value problem is used to outline a simple way to formulate a multi-dimensional layer stripping procedure. This procedure is unstable and does not constitute a practical algorithm for solving the inverse problem. The paper concludes with three appendices, the first two of which carry out a careful construction of solutions of the direct problems and the third of which contains a discussion of some properties of the scattering operator.

In this paper the stability of inverse problems is discussed. It is taken into account that in inverse problems the structure of the solution space is usually completely different from the structure of the data space so that the definition of stability is not trivial. We solve this problem by assuming that under experimental circumstances both the model and the data can be characterized by a finite number of parameters. In the formal definition that we present, we first compare distances in the data space and distances in the model under variations of these parameters. Second, a normalization is introduced to ensure that quantities in the solution space can be compared directly with quantities in the data space. We note that it is impossible to obtain an objective solution of stability due to the freedom one has in the choice of the norm in the solution space and in the data space. This definition is used to examine the stability of linear inverse problems as well as for the Marchenko equation and inverse problems associated with transfer-matrix methods. For the Marchenko equation it is shown that the instability arises from the nonlinearity of the inverse problem.

We consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x=1 and a solution is sought in the interval 0<x<1. This sideways heat equation is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. The heat equation is discretized by a differential-difference equation, where the time derivative has been replaced by a finite difference, and we analyse the approximation properties of time-discrete approximations using Fourier transform techniques. An error estimate is obtained for one such approximation, and it is shown that when the data error tends to zero, the error in the approximate solution tends to zero logarithmically. This error estimate also gives information about how to choose the step length in the time discretization.

In this paper the nonlinear equation m_ty=(m_yxx+m_xm_y)_x is thoroughly analysed. The Painleve test is performed yielding a positive result. The Backlund transformations are found and the Darboux-Moutard-Matveev formalism arises in the context of this analysis. The singular manifold method, based upon the Painleve analysis, is proved to be a useful tool for generating solutions. Some interesting explicit expressions for one and two solitons are obtained and analysed in such a way.

The method of Saito (1984) is modified to measure potentials in three dimensions. Uniqueness and reconstruction are established for a class of potentials that includes surface measures of certain smooth 2-mainfolds.

The Christoffel functions are used here to approximate an unknown probability density u:(0,1) to R₊ whose first m moments mu ₁,..., mu _m only are available. We obtain a sequence u^(m) of estimators which (theoretically) converges to u. We test it in several numerical examples and compare u^(m) with the probability density which matches the given moments and minimizes the functional S_w(u) identical to - integral ₀¹1n u-(dx)/ square root x(1-x).

Transient stellar mass loss has already been studied as an inverse problem by Brown and Wood (1994) and by Calvini, Bertero and Brown (1995), in the hypothesis of a uniform flow speed nu (r)= nu ₀. Here we consider a generalization of this inverse problem to an accelerated wind profile based on an empirical form for nu (r) with an adjustable acceleration parameter c₀. We deduce the two generalized convolution equations linking the time ( tau ) evolution of the equatorial mass loss rate m( tau ) and the asphericity 'shape function' u( tau ) to the observed polarization and absorption line strength variations in time. In order to perform the regularized inversion of these equations, we introduce an iterative algorithm which allows us to take into account the positivity constraint on the solutions. This method is tested on simulated data for various choices of hypothetical, m( tau ) and a( tau ) and the results are compared with those provided by Fourier deconvolution, for several values of the parameter c₀. It is found that the iterative algorithm is more efficient than the Fourier one and that, for both techniques, recoveries are less good for finite c₀ (accelerating wind) than for c₀ to infinity (impulsive acceleration at the stellar surface and then steady wind speed) due to slower sampling of the important inner-wind region.