| At present, the inverse source problem has became a focus of many investigators engaged in science field, it has practical applications in medicine, geological detecting and other areas. In this paper we discuss the inverse problem of recovering the source function from the boundary data of the solution to poisson equation. We discovery that the locations of a finite number of point sources are the poles or branch points of a complex function which has a special form. According to the Cauchy formula, we can convert the poles and branch points to a polynomial's zeros, then use an algebraic algorithm to solve the problem.At first, we give the mathematical model of the inverse source problem. Assume thatΩ(?) Rn(n = 2,3) is a ball,Ω=Ω0∪Ω1∪Ω2, whereΩ0,Ω1,Ω2 are three disjoint homogeneous connected layers, with spherical boundaries and constant conductivitiesσi i.e,σ=σi, z∈Ωi(i = 0,1,2). Use Bn represent the unit ball in Rn, letΩ0 = Bn,Γn = (?)Bn,Γ= (?)Ω. Assume that the source term F be of the formhere, aj∈Ω0,βj∈Rn, m2∈N represent the location, strength and number of dipole sources respectively. Correspondingly, bk∈Ω0,λk∈R, m0∈N are the location, strength and number of monopole sources. Then the potential u created by these point sources is a solution of The inverse source problem we consider here is: suppose the boundary conditionφand the measurement data u|Γ* on a subsetΓ* of boundaryΓis known, find F verifying form (1.4) such that (1.5) holds.However, problem (1.5) can be reduced. u is harmonic inΩ2 andΩ1, and inΩ\ {aj, bk}, u is Holder continuous, then we can transform the boundary problem inΩinto the boundary problem inΩ0 = Bn(n = 2,3)here we letσ0 = 1. Consequently, the inverse source problem reduced to: suppose thatφ|Γn and measurement data g = u|Γn is known, find aj, bk such that (2.2) holds.Let u = u2 is the solution to (2.2) for n = 2. Theorem 1 gives us the expression of u by complex function.Theorem 1. There exists a function (?) analytic in B2 such that, if we definethenMoreover, ifthen f is given onΓ2 (up to an additive constant) byfor every z∈Γ2, whereξ0∈Γ2 is fixed once and for all. u is the real part of the complex function f, the poles and branch points of f are the locations of dipole and monopole sources. In addition, the values of f onΓ2 can be calculated by the boundary condition g andφ. So the inverse problem of determining the source locations is equal to the problem of finding the singularities of f from it's boundary values onΓ2.In chapter three we just right discuss the algebraic algorithm of inverting the singularities of function f. But the idea in this paper only can be used to recovery the situations that the dipole or monopole sources present individuality. For this reason, we respectively describe the two cases.Give some priority conditions at first. Assume that we know an upper bound N of the number m of point sources, the strengthβj,λk are large enough and point sources are well-separated, i.e, suppose there existδ> 0, d > 0 such that(a) .We suppose that there are only dipole sources in B2. Assume m = m1+m2 is the total number of point sources in area B2, then m2 = 0,m = m1. For non-negative integer n, define . On one hand, according to theorem 2.1, cn can be obtained by integrating f onΓ2; on the other hand, (?) is analytic in B2, from the Cauchy formula, .Theorem 2. Suppose that the sequence If l1,..., lm satisfies the generating equationthen a1, a2 ..., am are solutions of The converse is also true. Furthermore, if (3.1) holds, then it holds for all n.Theorem 3. Let cn be as in theorem 2 and letthenIn particular,Theorem 2 and 3 suggest that if we know m, form solving equation (3.2) we can obtain a1, a2, ..., am. Hence, finding the locations of dipole sources can be transformed to looking for the zeros of a polynomial. The computing process is as follows.SP1. Measure boundary data g(z)|Γ2,φ(z)|Γ2, computeSP2. Compute integrationSP3. Calculate Dn, n = N, N - 1, ..., find an integer m such that SP4. With m found in the previous step, solve linear equationsto get l1,. ..,lm.SP5. Find zeros a1, a2,..., am of the polynomial equationZeros are singularities ,and also sources' locations.SP6. Solve the equationto findβ1,...,βm.(b).If there are only monopole sources. Then m = m1,Theorem 4. Assume that B2 is the unit disc in complex plane,Γ2 is the boundary. Let f(z) =λlog(z - b), withλ∈C, b∈B2\Γ2.ThenDefine , from theorem 4 and compatibility condition (2.4), we know that . At this time , cn also can be computed by the boundary conditions. Following the case of dipole sources, finding the branch points of complex function f still be equal to solving a polynomial. We notice that: if there are not a monopole source at the origin, thenand if a monopole point source appears at the origin, we just as well let bm = 0, thenThis moment the method in sp3 which be used to judge the number of point sources is no longer complete suitable, we have to make a further revise:SP3'. Compute Dn,n = N, N - 1, ..., find integer m such thatChapter four has exhibited a few simple numerical examples, which verify the accuracy of the method in this paper.We still first test the case of dipole sources. Numerical results display that: for determining the dipole source locations and strength, the effect of this method is excellent. Even though some "false sources" emerge, but their strength are far smaller than the lower bound of source strength which is allowed in priority conditions. Therefore, we ought to add the last work to the process from step 1 to step 6.SP7. Among computed poles discard those at which the residues are less than |. Then we suppose that there are just monopole sources in B2. If d/2≤|bk|(k = 1,...,m), i.e, all of the point sources have certain distance from the origin, the method to recovery the monopoles that not near the origin has high accuracy. If there exist a point sources at or nearing the origin, we force to compute by this algorithm, analyze the results ,and find that: although approximate number of branch points is not accurate enough, even less than the practical number ,but just miss one; except the missing one, the inversion of the others is feasible, moreover that is not difficult to notice that the branch point which has lost is just the one at or around the origin. So long as analyze the existence of the branch point at or near the origin, we are able to recovery all of the monopole sources. Therefore, for monopole sources, the job afterward isSP7' . Among computed point sources discard those at which the strengthes less thanδ/2. Verify condition∑k=1mλk = 0, if it's not true, then add an monopole sources at origin, and the sum of the strength of it and the other computed point sources is equal to zero.The idea in this paper is also suitable for computing other inverse problems, such as EIT. If want to consider the 3-D point sources, some people project them to 2-D points for convenience. We approach here 2-D cases, describe an algorithm of recovering dipole and monopole sources. But if they appear together, it's waiting for further investigation. Moreover, the method involve solving algebraic equations and is instable when the noise exists, so improving the algorithm is also valuable.
|
PDF Full Text Request