Journal of Applied Mathematics and Statistical Applications

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Research Article - Journal of Applied Mathematics and Statistical Applications (2018) Volume 1, Issue 2

A numerical solution of inverse scattering problem with non-over-determined data

*Corresponding Author:
Alexander Ramm
Department of Mathematics, Kansas State University, Manhattan, USA
Phone : (785)532-0580
Fax : (785)532-0546
E-mail: [email protected]

Accepted date: July 05, 2018

Citation: Ramm A, Van C. A numerical algorithm for solving 3D inverse scattering problem with non-over-determined data. J Appl Math Statist Appl. 2018;2(1):5-7.

Abstract

The 3D inverse scattering problem with non-over-determined scattering data is considered. The data are the scattering amplitude A(β, α0, k) for all β ∈ S2β, where S2β is an open subset of the unit sphere S2 in R3, α∈ S2 is fixed, and K ∈ (a, b),0 ≤ a < b. In this paper, a numerical method is given for solving the inverse scattering problem with non-over-determined scattering data and the numerical results are presented.

Keywords

Scattering theory, Obstacle scattering, Uniqueness theorem, Non-over-determined scattering data.

Introduction

Let us first consider the direct scattering problem:

equation , (1)

equation , (2)

equation (3)

where α, β ϵ S2 are the directions of the incident wave and scattered wave correspondingly, S2 is the unit sphere k2 > 0 is energy, k > 0 is a constant, A(β, α, k) is the scattering amplitude (scattering data), which can be measured, and q(x) ϵ Q, where Q is a set of C1-smooth real-valued compactly supported functions, q = 0 for maxj |x| ≥ R, R > 0 is a constant. By D the support of q is denoted.

The direct scattering problem (1) - (3) has a unique solution [1].

Consider now the inverse scattering problem

Find the potential q(x) ϵ Q from the scattering data A(β, α, k). The uniqueness of the inverse scattering problem with fixedenergy data (k = k0 > 0 is fixed) is proved by Ramm [1]: q(x) ϵ Q is uniquely determined by the scattering data A(β, α, k0) for a fixed k = k0 > 0 and all α, β ϵ S2. Ramm also gave a method for solving the inverse scattering problem with fixed-energy data and obtained an error estimate for the solution for exact data and also for noisy data [1], Chapter 5.

In this paper, we give a numerical implementation of the method proposed by A. G. Ramm, for solving the inverse scattering problem with non-over-determined data, that is, finding q(x) ϵ Q from the scattering data A(β, k) := A(β, α0, k) for a fixed α0 ϵ S2, all β ϵ S2 and all k ϵ (a, b), 0 ≤ a < b. The basic uniqueness theorem for this problem is proved, see also [2-7]. In Section 2, the idea of this numerical method is described. This idea and the method described in Section 2 belong to A. G. Ramm. In Section 3, the DSM algorithm used to solve ill-posed linear system is described. In Section 4, the numerical procedure is presented and in Section 5, some examples of the numerical inversion are given.

The essential novel features of this inversion method are:

• The data are non-over-determined. So these are minimal scattering data that allow one to uniquely recover the potential.

• The inverse scattering problem is highly non-linear. Nevertheless, our method reduces the inversion to stable solution of linear algebraic system (8).

• The numerical difficulty comes from the fact that system (8) is highly ill-conditioned.

Inversion method

Let q ϵ Q. The unique solution to (1) - (3) solves the integral equation:

equation (4)

k > 0 is a constant, α0 ϵ S2 is fixed. This equation is uniquely solvable for u under our assumptions on q. Let h(x, k) = q(x)u(x, k). Then (4) implies

equation .(5)

Equations (3) and (4) yield the exact formula for the scattering amplitude:

equation . (6)

From equation (5) one derives the formula for q(x) if h(k, x) is found:

equation. (7)

The idea of our inversion method

By the uniqueness theorem from [6], equation (6) has a unique solution h(x, k). Solve numerically equation (6) for h(x, k). To do this, discretize (6) and get a linear algebraic system for hpm:= h(yp, km). If hpm are found, then q(xp) are found by formula (7), see also formula (9) below.

Let us partition D into P small cubes with volume Δp, 1 ≤ p ≤ P. Let yp be any point inside the small cube Δp. Choose P different numbers km ϵ (a, b), 1 ≤ m ≤ P, and P different vectors βj ϵ S2, 1 ≤ j ≤ P. Then discretize (6) and get a linear algebraic system for finding hpm:

equation. (8)

Where hpm:= h(yp, km). Solve the linear algebraic system (8) numerically, then use equation (7) to find the values of the unknown potential q(xp):

equation (9)

Remark 1: Note that the right hand side of (9) should not depend on m or j. This independence is an important requirement in the numerical solution of our inverse scattering problem, a compatibility condition for the data. This requirement is automatically satisfied for the limiting integral equation (6), see formula (7).

The values of q(yp) essentially determine the unknown potential q(x) if P is large and q is continuous. The potential q is unique by the uniqueness theorem in [6].

Note that one can choose βj and km so that the determinant of the system (8) is not equal to zero [7], so that the system (8) is uniquely solvable, but the numerical difficulty is unavoidable: the system (8) is very ill-conditioned because it comes from an integral equation (6) of the first kind with an analytic kernel. We use the dynamical system method (DSM) from [2] to solve stably the ill-conditioned system (8).

Dynamical System Method (DSM)

Equation (8) with noisy data is a linear algebraic system of the form

Muδ = fδ (10)

where M is the matrix of the size P × P and fδ is the noisy data, that is, the noisy values of -4πA(βj, km) in equation (8), || fδ - f || = δ for a fixed m and 1 ≤ j ≤ P. To solve this ill-posed system we use the Dynamical Systems Method (DSM) from [2]:

equation (11)

where T := M *M, M* is the adjoint matrix and a(t) > 0 is a nonincreasing function such that a(t)0 as t. Equation (11) has a unique solution

equation (12)

To use the DSM method, we need to choose a(t) and find a stopping time tδ so that uδ(tδ) approximates the solution of Au = f, so that lim δ → 0 ||uδ - u|| = 0, where the norm is in equation .

Choice of a(t) and tδ:

In [2], necessary conditions for a(t) are: a(t) is a nonincreasing function and limt →∞ a(t)=0 . In our experiments, we choose equation. Consider a step function ã(t) approximating a(t) :

equation, (13)

where an = a(tn) and we choose tn such that tn+1 - tn = hn, hn = qn, q = 2. From equation (12), one computes un = u(tn) by:

equation . (14)

and one gets an iterative method to solve (12):

equation. (15)

To choose tδ, we use a relaxed discrepancy principle [2]:

equation (16)

where the norm is the vector norm: equation . One stops the iterative method (15) when condition (16) is satisfied.

Numerical Procedure and Results

In practice, one can measure the noisy scattering data experimentally. For our numerical experiments, we need to construct the noisy scattering data A(βj, km).

Constructing noisy scattering data

Given a potential q(x), let us first construct the exact scattering data Ã(βj, km). We partition D into P small cubes and discretize equation (4) to get

equation. (17)

One solves this linear system for u(xp, km) assuming that q(x) is known. Then the exact scattering data are calculated by the following formula:

equation. (18)

Then one can randomly perturb each Ã(βj, km), 1≤ j ≤ P byequation= const > 0 to get the noisy scattering data A(βj, km) = Ã(βj, km) ± equation where the plus or minus sign is randomly choosen. Thenequation , for a fixed k is the noise level of the data in equation (10).

Numerical procedure

The following steps are implemented in each experiment:

• Choose D, α0, P, q(x), km, and equation.

• Use (18) to find the exact scattering data Ã(βj, km), 1≤ j ≤ P. Try different values of km so that the determinant of the system in (8) is not zero. Let k be the found value of km.

• Use the procedure in Section 4.1 to get the noisy scattering data A(βj, k), 1≤ j ≤ P.

• Solve the linear algebraic system (8) to get hpm = h(yp, km). Here we use the DSM method in Section 3 with the noise:

equation.

• Calculate the potential equation by formula (9):

, 1 ≤ p ≤ P

See Remark 1 below formula (9).

• Find the relative error of the estimate of the potential:

equation. (19)

Numerical Results

In these experiments, we choose D to be the unit cube centered at the origin, with sides parallel to coordinate planes, P = 1000, α0 = (1, 0, 0), and 50 ≤ km ≤ 100 is chosen so that the determinant of the system in (8) is not zero. The condition number of A is ̴1013.

In the following graphs, the x-axis is the index of the collocation points (this index varies from 1 to 1000) and the y-axis is the value of the potential.

Constant potential with compact support

The inverse scattering problem with constant in D potential is used to test our inversion method. In this experiment, we take q(x) = 10. The results are obtained as in Table 1 and Figure 1.

applied-mathematics-statistical-applications-potential

Figure 1: Constructed potential vs original constant potential q(x) =10 when equation = 0.01.

Constant Potential q(x) = 10
 equation equation  Relative Error
.04  0.4348  0.0566
.02  0.2174  0.0037
.01  0.1087  0.00065

Table 1: Numerical results for constant potential.

Potentialequation

In this experiment, we take equation. The results are obtained as in Table 2 and Figure 2.

applied-mathematics-statistical-applications-potential

Figure 2: Constructed potential vs original potential equation when equation = 0.01.

Potential equation
 equation equation     Relative Error
.04  0.0806  0.1284
.02  0.0403  0.0547
.01  0.0201  0.0367

Table 2: Numerical results for the potential equation

Conclusion

In this paper, a numerical method is given for solving the inverse scattering problem with non-over-determined scattering data and the numerical results are presented.

References