Inverse scattering and spectral problems for PDEs
Valery Serov, Markus Harju, Georgios Fotopoulos, Urpo Kyllönen, Teemu Tyni
We study the partial recovery of a potential from limited scattering data. More precisely, let q be a potential appearing in the Schrödinger equation in ℝⁿ (we are also planning to consider more general partial differential operators, in particular, the Schrödinger operator with magnetic potential). Our main interest is to locate the points of discontinuity of q from limited data. It is well known that the full (or general) scattering data A(k,θ,θ′)for k ϵ R, θ′,θ ϵ Sn-1, where A is the scattering amplitude, is overdetermined. Therefore, we turn our attention to backscattering, fixed angle and fixed energy data. The first two of this datum are formally well-determined in any dimension whereas the latter only in dimension two. We find them also suitable for practical applications.
The method used for such a recovery is called the Born approximation. In this approach the unknown potential is essentially the inverse Fourier transform of the scattering amplitude A. The main idea of the technique is to prove that the difference between the Born approximation and the true potential is smoother than the potential itself - maybe even continuous. This means that the main singularities (such as jumps across a domain) can be recovered from qB, the Born approximation. One would also like to obtain this theoretical result locally for widest possible class of potentials, i.e. q ϵ Lploc with p as small as possible.
Our current work is devoted to the generalization of the Born approximation to the nonlinear Schrödinger operator in dimensions one and two. In the linear setting we consider the two-dimensional case with backscattering and fixed angle data. The goal is to improve the best known results of Ola, Päivärinta and Serov (Comm. PDE. 26, 2001, no. 3-4, 697-715) and Ruiz and Vargas (Comm. PDE. 30, 2005, no. 1-3, 67-96) from Sobolev space Hscomp to Lebesgue space Lploc. We are also devising numerical implementations of the theoretical results.
Concerning the inverse spectral problems we study first the spectral properties of elliptic partial differential operators with singular coefficients in bounded domains in Rn. We will also study the generalization of the n-dimensional Borg-Levinson theorem (see Päivärinta and Serov, Adv. in Appl. Math. 29, 2002, no. 4, 509-520) for such operators, that is, we study the uniqueness theorem by the spectral data - the Dirichlet eigenvalues and normal derivatives of the eigenfunctions at the boundary.
Fortran Linear Inverse Problem Solver (FLIPS)
FLIPS is a Fortran 90/95 module for solving large scale (statistical) linear systems of form
m = Ax + ε,
where m is called the measurement, A is the direct theory matrix, ε is the error and x is called the unknown.
Depending on the form and the properties of the direct theory matrix A, FLIPS is able to calculate the exact solution vector x, the shortest solution vector x0 in the solution space or the solution x0 in the least squares sense (See FLIPS manual for comprehensive explanation). The error is usually given as the standard deviation of the measurements.
What makes FLIPS unique among the multitude of linear solvers is that the data (measurements, errors and theory matrix) can be fed into the FLIPS system one row at the time. FLIPS uses so called Givens rotations to rotate the fed data into so called target matrix and target vector which have fixed size and form. After the rotations are made, the data can be discarded. When all the data is fed into the system, we are left with equation
Y = Rx0,
where Y and R are the before mentioned target vector and matrix, respectively, and x0 is the exact or approximate solution of the original problem. Moreover, the target matrix R is upper triangular which makes the above equation very fast to solve using (for example) back substitution.
Since the target matrix and vector have fixed sizes (if the system has N unknowns, the size of target vector Y is N and the size of target matrix R is NxN) FLIPS is especially suitable for large overdetermined systems.
Another useful feature of FLIPS is the ability to marginalize away any number of unknowns at any time. It is also possible to add new unknowns to the system.
For more information, see the FLIPS page.
Statistical inverse theory
Statistical inverse theory is a method for reasoning from ignorance towards knowledge given indirect and noisy observation of the unknown. The starting point is presenting our state of ignorance as a probability distribution. It describes the uncertainties that we have about the behavior of the unknown. The knowledge -- also a probability distribution -- is obtained via the Bayes formula, where data governs the updating.
Statistical inverse theory has three basic questions:
- THE PROBLEM OF PRIOR: How the prior distribution is constructed from our incomplete knowledge?
- INVERSION: What the Bayes formula gives as the posterior distribution?
- THE PROBLEM OF POSTERIOR: What reasonable claims about the unknown can be extracted from the posterior probability distribution?
These questions are studied in infinite-dimensional limit cases.
I have studied the transmission problem for the electromagnetic scattering by a dissipative chiral obstacle in achiral surrounding. The transmission problem can be reduced to a single integral equation over the boundary of the obstacle with one unknown tangential vector field. The equation is uniquely solvable except for some values of the material parameters and the frequency. Another research topic has been the propagation of TM-waves in a planar waveguide with a nonlinear film surrounded by semi-infinite linear media. This work also contains numerical examples.
Last updated: 23.6.2016