Table of contents

The initial/boundary value problem on the semiline for the Burgers equation u_t=u_xx+2u_xu, with the initial and boundary data u(x,0)=u₀(x), x>0, and u_x(0,t)=F(t), t>0, is reduced to an integrodifferential equation in one independent variable. This equation is, however, non-linear.

A highly regarded method for solving discrete ill-posed problems is the regularisation method due to Tikhonov (1963). The author uses the generalised SVD to derive perturbation bounds for the regularised solution when both the matrix and the right-hand side are perturbed.

For pt.I see ibid., vol.3, p.195 (1987). The authors have shown that the resolution of a confocal scanning microscope can be improved by recording the full image at each scanning point and then inverting the data. These analyses were restricted to the case of coherent illumination. They investigate, along similar lines, the incoherent case, which applies to fluorescence microscopy. They investigate the one-dimensional and two-dimensional square-pupil problems and they prove, by means of numerical computations of the singular value spectrum and of the impulse response function, that for a signal-to-noise ratio of, say 10%, it is possible to obtain an improvement of approximately 60% in resolution with respect to the conventional incoherent light confocal microscope. This represents a working bandwidth of 3.5 times the Rayleigh limit.

The author considers the reflection of a time-harmonic electromagnetic field at a perfect conductor and treats the inverse problem to determine the shape of the object from the knowledge of the electric far-field pattern. To solve this ill-posed problem approximately the author applies two methods which were introduced respectively by Kirsch and Kress (1987) and by Colton and Monk (1986) for solving the parallel problem for the reflection of acoustic waves at a sound soft obstacle.

The method of maximum statistical entropy (to be distinguished from the more usual maximum configurational entropy) is explained for a general inverse problem. Application to the biomagnetic situation then yields a solution which, in the limit as the available information becomes zero, gives the minimum-power current distribution. The method is developed in detail for the case of a uniform sphere.

The problem of reconstructing the complex index of refraction of a weakly scattering object from a limited data set is addressed within the framework of diffraction tomography (DT). This problem is cast into a form that includes the well studied limited-view problem of conventional computed tomography (CT) as a special case, obtained in the limit of vanishing wavelength. The theory is developed in detail for the case of plane-wave probes (parallel-beam case) in a manner completely parallel to that usually employed in studies of the limited-view problem in CT. An integral equation formulation is employed that leads directly to a number of results that include a theorem that states that any reconstruction that is convolutionally related to the exact object function can always be implemented in the form of a filtered backpropagation algorithm.

The authors consider non-linear ill-posed problems in a Hilbert space setting, they show that Tikhonov regularisation is a stable method for solving non-linear ill-posed problems and give conditions that guarantee the convergence rate O( square root delta ) for the regularised solutions, where delta is a norm bound for the noise in the data. They illustrate these conditions for several examples including parameter estimation problems. In an appendix, they study the connection between the ill-posedness of a non-linear problem and its linearisation and show that this connection is rather weak. A sufficient condition for ill-posedness is given in the case that the non-linear operator is compact.

The author studies Tikhonov regularisation as a stable method for approximating the solutions of non-linear ill-posed problems. The authors gives conditions that guarantee the best possible rate O( delta ²3/) for the regularised solutions, where delta is a norm bound for the noise in the data, in the infinite-dimensional setting, and illustrates these conditions for several examples including parameter estimation. The author also presents results on convergence and convergence rates for Tikhonov regularisation combined with the finite-dimensional approximation.

It is shown how the theory of integrable systems for loop groups as derived from zero-curvature equations and classical r-matrices extends to Banach-Lie groups of invertible bounded operators in Hilbert space. An abstract version of the Miura transformation and modified equations is equally considered.

Solution methods for integrable systems are considered from the point of view of asymptotic modules. An asymptotic module is a set of functions depending on a complex variable k and on some auxiliary real variable x_l. This set admits both an asymptotic structure for which some points of the Riemann k-sphere appear as essential singularities and a module structure over some ring of x_l differential operators. These conditions imply compatibility relations between asymptotic coefficients which may lead to a solution of some integrable model.

In a statistical inverse theory both the unknown quantity and the measurement are random variables. The solution of the inverse problem is then the conditional distribution of the unknown variable with the measurement supposed to be known. Both variables often have their values in spaces of functions or generalised functions while the statistical theory of conditional distributions has only been fully developed for Polish spaces. Also, the mappings representing the solution of linear inverse problems with Gaussian priors and noises are usually only defined on a subset of the spaces used. This problem has previously been correctly handled only for Hilbert-space-valued variables. The existence of a regular version of the conditional distribution of random variables with values in spaces of generalised functions is shown and the inverse problem is solved in the linear Gaussian case.

The tau function, introduced by the 'Kyoto School' as a central element in the description of soliton equation hierarchies, is identified with the determinant of a family of linear operators solving linear, constant-coefficient PDE in the hierarchy variables, for the Kadomtsev-Petviashvili (KP) hierarchy. For Gel'fand-Levitan-Marchenko integral operators, the tau function is the Fredholm determinant; the author gives a new proof of this fact. He shows further that the determinant of a suitably chosen family of finite-dimensional matrices is a tau function which gives rise to rational solutions of the KP equation. Finally, he proves the 'vertex operator identity' for tau functions in the Fredholm determinant case.

A finite-difference solution is demonstrated for an inverse problem of determining a control function p(t) in the parabolic partial differential equation u_t=u_xx+pu+f(x,t), 0<x<1;0<t<or=T.

Many one-dimensional inverse scattering problems can be formulated as a two-component wave system inverse problem, including inverse problems for lossless and absorbing acoustic and dielectric media. The advantage of doing so is that well known signal processing algorithms with good numerical stability properties can be used to reconstruct such media from either reflection or transmission responses to impulsive or harmonic sources. If the system is asymmetric, i.e. has different reflectivity functions in different directions, transmission data as well as reflection data are required. The author summarises algorithms for a wide variety of one-dimensional inverse problems, derives some new ones, and presents a simple framework that reveals much about these problems.