Volume 24, Issue 6

The necessary and sufficient conditions for solvability of the inverse problem about two-spectra for finite order real Jacobi matrices with zero-diagonal elements are established. An explicit procedure of reconstruction of the matrix from the two-spectra is given.

In this paper we consider the inverse problem of simultaneously reconstructing the interface where the jump of the conductivity occurs and the Robin parameter for a transmission problem with piecewise constant conductivity and Robin-type transmission conditions on the interface. We propose a reconstruction method based on a shape optimization approach and compare the results obtained using two different types of shape functionals. The reformulation of the shape optimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup combined with a function space parameterization technique. The reconstruction is then performed by means of an iterative algorithm based on a conjugate shape gradient method combined with a level set approach. To conclude we give and discuss several numerical examples.

The paper deals with the nonlinear inverse source problem of identifying an unknown time-dependent point source occurring in a two-dimensional evolution advection-dispersion-reaction equation with spatially varying velocity field and dispersion tensor. The originality of this study consists in establishing a constructive identifiability theorem that leads to develop an identification method using only significant boundary observations and operating other than the classic optimization approach. To this end, we derive two dispersion-current functions that have the main property to be of orthogonal gradients which yield identifiability of the elements defining the involved unknown source from some boundary observations related to the associated state. Provided the velocity field fulfills the so-called no-slipping condition, the required boundary observations are reduced to only recording the state on the outflow boundary and its flux on the inflow boundary of the monitored domain. We establish an identification method that uses those boundary records (1) to localize the sought source position as the unique solution of a nonlinear system defined by the two dispersion-current functions, (2) to give lower and upper bounds of the total amount loaded by the unknown time-dependent source intensity function, (3) to transform the identification of this latest into solving a deconvolution problem. Some numerical experiments on a variant of the surface water BOD pollution model are presented.

We investigate the null space of Fredholm integral operators of the first kind with TD:=∫ℬD(x)k(x,⋅)dx,${{TD}\mathrel{\mathop{:}}=\int_{\mathcal{B}}D(x)k(x,\cdot\,)\,\mathrm{d}x},$ where ℬ${{\mathcal{B}}}$ is a ball, the integral kernel satisfies k⁢(x,y)=∑n=0∞cn⁢|x|ln|y|n-2+q⁢Pn(q)⁢(x|x|⋅y|y|),x∈ℬ,y∈ℝq∖ℬ,${k(x,y)=\sum_{n=0}^{\infty}c_{n}\frac{|x|^{l_{n}}}{|y|^{n-2+q}}P_{n}^{(q)}% \biggl{(}\frac{x}{|x|}\cdot\frac{y}{|y|}\biggr{)},\quad x\in\mathcal{B},\,y\in% \mathbb{R}^{q}\setminus\mathcal{B}},$ where (cn)${(c_{n})}$ and (ln)${(l_{n})}$ are sequences with particular constraints, and the Pn(q)${P_{n}^{(q)}}$ are Gegenbauer polynomials. We first discuss the case of a 3-dimensional ball in detail, where the Pn(3)=Pn${P_{n}^{(3)}=P_{n}}$ are Legendre polynomials, and then derive generalizations for the q -dimensional ball. The discussed class includes some important tomographic inverse problems in the geosciences and in medical imaging. Amongst others, uniqueness constraints are proposed and compared. One result is that information on the radial dependence of D is lost in T⁢D${{TD}}$. We are also able to generalize a famous result on the null space of Newton’s gravitational potential operator to the ℝq${\mathbb{R}^{q}}$. Moreover, we characterize the orthonormal basis of the derived singular value decomposition of T as eigenfunctions of a differential operator and as basis functions of a particular Sobolev space. Our results give further insight to the interconnections of magnetic field inversion on the one side and gravitational/electric field inversion on the other side.

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert’s method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.

The aim of this paper is to reconstruct local volatility from market prices of binary options. In the case of the space-dependent volatility, we obtain the stable linearization and the available integral equation to identify local volatility from observable data of binary options. We achieve the reconstruction of local volatility by numerical simulation.

Finite difference (FD) solutions of wave equations have been proven useful in exploration seismology. To yield reliable and interpretable results, the numerically induced error should be minimized over a range of frequencies and angles of propagation. Grid dispersion is one of the key numerical problems and there exist some methods to solve this problem in the literature. Traditionally, the spatial FD operator coefficients are only determined in the spatial domain; however, the wave equation is solved in the temporal and spatial domain simultaneously. Recently, some methods based on the joint temporal-spatial domains have been proposed to address this problem. Variable length coefficients methods are proposed in the literature to improve efficiency while preserving accuracy by using longer operators in the low velocity regions and shorter operators in the high velocity regions. To cope with the ill-conditioning of the linear system induced by long stencil FD operators, we study in this paper a regularizing simplified least-squares model to minimize the phase velocity error in the joint temporal-spatial domain with a variable length of coefficients. Different from our previous study, we determine FD coefficients on the regular grid instead of on the staggered grid. Though the regular grid FD methods are less precise, however, with a little increase of the operator length, the precision can be improved. Stability of the numerical solutions is enhanced by the regularization. Numerical simulations made on one-dimensional to three-dimensional examples show that our scheme needs shorter operators and preserves accuracy compared with the previous methods.

We solve numerically the side Cauchy problem for a 1-D parabolic equation. The initial condition is unknown. This is an ill-posed problem. The main difference with previous results is that our equation is quasilinear, whereas known publications on this topic work only with linear PDEs. The key idea is to minimize a weighted Tikhonov functional with the Carleman Weight Function (CWF) in it. Roughly, given a reasonable bounded set of any size in a reasonable Hilbert space, one can choose the parameter of the CWF in such a way that this functional becomes strictly convex on that set.

A local boundary value problem for an inhomogeneous biharmonic equation in a rectangular domain is considered. Boundary conditions are given on the whole boundary of the domain. It is shown that the problem turns out to be self-adjoint. And herewith the problem is ill-posed. An example is constructed demonstrating that the solution stability to the problem is violated. Necessary and sufficient conditions of the existence of a strong solution to the investigated problem are found. The idea of the method is that the solution to the problem is constructed in the form of an expansion on eigenfunctions of this self-adjoint problem. This problem has an isolated point of continuous spectrum in zero. It is shown that there exists a series (a sequence) of eigenvalues converging to zero. Asymptotics of these eigenvalues is found. Namely this asymptotics defines a reason for the ill-posedness of the investigated problem. A space of well-posedness for the investigated problem is constructed.