Band 20, Heft 3

Abstract. For a given two-dimensional image, we define the horizontal and vertical increments at a pixel location to be the difference between the intensity values at that pixel and at the neighboring pixels to the right and above, respectively. For a typical image, it makes intuitive sense that the increments will usually be near zero, corresponding to areas of smooth variation in image intensity, but will often have large magnitude, corresponding to edges where sharp intensity changes occur. In this paper, we explore the use of the Laplace increment model, in which the increments are assumed to be independent and identically distributed Laplace random variables – a distribution with heavy tails allowing for large increment values – with zero mean. The prior constructed from the Laplace increment model is very similar to the total variation (TV) prior. We perform a theoretical analysis of its properties, which shows that the Laplace prior yields a regularization scheme with regularized solutions contained in the space of bounded variation, just as for the TV prior. Moreover, numerical experiments indicate that the Laplace prior yields reconstructions that are qualitatively very similar to those obtained using TV.

Abstract. In this paper we study the inverse problem of determining three unknown source terms in the linearized Lotka–Volterra competition-diffusion system of three species with variable coefficients from measured output data in the form of Dirichlet and the final time overdetermination boundary conditions. We use Hasanov's approach based on quasi-solution method and adjoint problem solution for inverse source problems. Further we provide the Fréchet gradient of the cost functional through the solution of the adjoint problem and several necessary and sufficient results to prove the existence of a quasi-solution and also unicity of the considered inverse problem.

Abstract. We prove new global Hölder-logarithmic stability estimates for the Gel'fand inverse problem at fixed energy in dimension . Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. Comparisons with preceding results in this direction are given.

Abstract. We consider an inverse problem of parameter identification for a parabolic equation. The underlying practical example is the reconstruction of the unknown drift in the extended Black–Scholes option pricing model. Using a priori information about the unknown solution (i.e. its Lipschitz constant), we provide a solution to this non-linear ill-posed problem, as well as an error estimate. Other types of a priori information may be used (for example, monotonicity and/or convexity of the unknown solution).

Abstract. This paper deals with numerical inversion for space-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation with additional boundary measurement. An implicit difference scheme for the forward problem is presented based on discretization of Caputo fractional derivative, and numerical stability and convergence of the linear system are proved with the help of matrix analysis. An optimal perturbation regularization algorithm is introduced to determine the space-dependent diffusion coefficient numerically in different approximate space, and numerical inversions are carried out by several numerical examples. Furthermore, impacts of the fractional order, approximate space, regularization parameter, numerical differential step, and initial iterations on the inversion algorithm are analyzed showing that the inversion is of numerical uniqueness, and the inversion algorithm is efficient at least to the inverse problem here. The order of fractional derivative not only represents a global property of the forward problem, but also indicates ill-posedness of the corresponding inverse problem.

Abstract. In this paper we develop a fast multiscale Galerkin method solving ill-posed integral equations with not exactly given input data via Tikhonov regularization. The method leads to fast solutions of discrete Tikhonov regularization. The convergence rates of the Tikhonov regularization are achieved by using a modified discrepancy principle. Finally, numerical experiments are given to illustrate the efficiency of the method.