Reduction of multidimensional inverse problems to initial-boundary value problems in Hilbert spaces

The research was supported by the Russian Foundation for Fundamental Research (Grant 93-011-181).     

Degenerate first-order inverse problems in Banach spaces

We are concerned with an inverse problem for a degenerate linear evolution equation of first-order. Both hyperbolic and parabolic cases will be considered. We indicate sufficient conditions for the existence and uniqueness of a solution. All the results can be applied to inverse problems for equations from mathematical physics. As a possible application of the abstract theorems, some examples of partial differential equations are given.     

Constraining inverse stefan design problems

A new formulation possessing stable numerical characteristics is presented for inverse Stefan design processes. In such processes, the goal of the analysis is to design transient boundary conditions which produce the desired interfacial surface motion. This subclass of mildly ill-posed mathematical problems is amenable to the proposed solution methodology. This investigation presents a fixed-front differential formulation from which a weighted residual statement is developed. Orthogonal collocation is used to obtain numerical results illustrating the merit of imposing physical constraints in the mathematical model. These mathematical constraints can be viewed as design specifications and are available to the designer or experimentalist. The proposed methodology is flexible and can be generalized to problems involving continuous casting or crystal growth. Finally, symbolic manipulation is used for augmenting the computational methodology.     

Solution of inverse diffusion problems by operator-splitting methods

In this paper the inverse solution of the general (non-linear) diffusion problem or backward heat conduction problem. It is assumed that the direct solution can be satisfactorily modelled, for example by the finite difference method. The nature of the problem and typical approaches to its solution are briefly reviewed.An operator-splitting method is introduced as a means of solving the inverse diffusion problem. An error analysis of the method is given, particularly for the application of the method to the simple diffusion equation. The method is applied to a range of test problems to illustrate the points of the analysis and to demonstrate the properties and performance of the method.     

Solution of some inverse problems in nonequilibrium sorption dynamics

The paper considers inverse problems of sorption dynamics that determine the sorption isotherm from the output curve. Uniqueness theorems are proved for these inverse problems. Numerical solution methods are constructed. Some applications to model problems are described.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 5–15, 1986.     

Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors

The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite-dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ODE or PDE. This article gives an introduction to the framework of well-posed BIPs in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others. This framework has the advantage of ensuring uniformly well-posed inference problems independently of the finite-dimensional discretisation used for numerical solution. Recently, this framework has been extended to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of inverse problems in the class of quasi-one-dimensional functions

Solution of the inverse problem of magnetotelluric sounding in a two-dimensional medium is considered. The solution methods are based on the assumption that the distribution of electrical conductivity is representable as the sum of a slowly varying component and a small increment. This model makes it possible to introduce small parameters in the equations of the problem, which are used for the construction of a sequence of functions approximating the solution of the inverse problem.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 160–175, 1986.     

Accelerated Landweber iteration with convex penalty for linear inverse problems in Banach spaces

In recent years, Landweber iteration has been extended to solve linear inverse problems in Banach spaces by incorporating non-smooth convex penalty functionals to capture features of solutions. This method is known to be slowly convergent. However, because it is simple to implement, it still receives a lot of attention. By making use of the subspace optimization technique, we propose an accelerated version of Landweber iteration with non-smooth convex penalty which significantly speeds up the method. Numerical simulations are given to test the efficiency.     

Linearizing non-linear inverse problems and an application to inverse backscattering

We propose an abstract approach to prove local uniqueness and conditional Hölder stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization A, we need a stability estimate for A as well. That condition is satisfied in particular, if A^∗A is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and Hölder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.     

Solutions to open problems of Yang concerning inverse nodal problems

Yang [X. F. Yang, A new inverse nodal problem, Journal of Differential Equations 169 (2001), 633–653] considered a new inverse nodal problem for the Sturm-Liouville operator L(q, α, β) in L ²[0, 1]: an s-dense subset of the nodal set in (0, b) (for any fixed b ∈ ( \(\frac{1}{2}\) , 1]) determines the potential q and boundary data α, β. (1) Since the s-dense condition is stronger than the dense condition, X. F. Yang proposed an open problem “It is open if the boundary parameter α can be determined by a dense subset of the nodal set in (0, b) but not necessarily by an s-dense subset of the nodal set in (0, b).” Cheng et al. have solved this problem and shown that a dense subset of the nodal set in (0, b) completely determines the potential q and boundary data α, β. (2) Another interesting open question: “It remains open if the result holds true for b ∈ (0, \(\frac{1}{2}\) ]” is also proposed by X. F. Yang. In this paper we provide a counterexample to claim that the result does not hold true for b ∈ (0, \(\frac{1}{2}\) ), and a uniqueness theorem for b = \(\frac{1}{2}\) .     

逆极限空间上渐近伪轨跟踪性质

The connection between APOTP （asymptotic pseudo orbit tracing property） for a continuous map on a compact metric space and that for the shift map on the inverse limit space is investigated. As an application, it is showed that the shift map on Henderson pseudoarc has APOTP.     

Asymptotic arc-components of unimodal inverse limit spaces

We consider the inverse limit space (I,f) of a unimodal bonding map f as fixed bonding map. If f has a periodic turning point, then (I,f) has a finite non-empty set of asymptotic arc-components. We show how asymptotic arc-components can be determined from the kneading sequence of f. This gives an alternative to the substitution tiling space approach taken by Barge and Diamond [Ergodic Theory Dynamical Systems 21 (2001) 1333].