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Next: Linerarization Up: Numerical Solution of Bilinear Previous: Numerical Solution of Bilinear

Introduction

This note is concerned with a quite down-to-earth approach to the numerical solution of inverse problems. These problems often have a bilinear structure. We assume that our inverse problem is governed by a family of bilinear equations

  equation20

where tex2html_wrap_inline1360 is linear, tex2html_wrap_inline1362 bilinear, tex2html_wrap_inline1364 , tex2html_wrap_inline1366 is linear. Here, tex2html_wrap_inline1368 is the Hilbert space of states, F the Hilbert space of parameters, and tex2html_wrap_inline1372 another Hilbert space. The subscript j refers to a projection direction or source position in transmission tomography or to a electrode pair in impedance tomography. Considering only one of the equations (1.1) at a time is an essential feature of our method. We assume that (1.1) is uniquely solvable for tex2html_wrap_inline1376 if f is given. We consider the problem of determining f from the additional equations

  equation25

tex2html_wrap_inline1382 , tex2html_wrap_inline1384 , tex2html_wrap_inline1386 and tex2html_wrap_inline1388 are exactly as tex2html_wrap_inline1390 , tex2html_wrap_inline1392 , tex2html_wrap_inline1394 , tex2html_wrap_inline1396 in (1.1) with tex2html_wrap_inline1372 replaced by a third Hilbert space tex2html_wrap_inline1400 , the space of observations. Let tex2html_wrap_inline1402 be the solution of (1.1) for a certain tex2html_wrap_inline1404 . Then our problem amounts to solving the equations

  equation30

displaymath1406

for f.

A standard method for solving the nonlinear problem (1.3) is the Newton or Gauss-Newton method. These methods necessitate the computation of the operators tex2html_wrap_inline1410 , where tex2html_wrap_inline1412 is the Fréchet derivative of tex2html_wrap_inline1414 . For problems of practical relevance this is totally out of question. An obvious way to circumvent this difficulty would be to compute a minimal norm solution to just one of the linearized equations

displaymath1416

i.e.

displaymath1418

and to update f according to tex2html_wrap_inline1422 with a relaxation factor tex2html_wrap_inline1424 . However, the computational burden for tex2html_wrap_inline1426 is still by far too high. Thus we go one step further, replacing tex2html_wrap_inline1426 by an easy to evaluate operator, e.g. by the identity operator. Then, one step of our algorithm, transforming the approximation tex2html_wrap_inline1430 into a new approximation tex2html_wrap_inline1432 , reads:

  eqnarray43

If tex2html_wrap_inline1434 is a linear operator and tex2html_wrap_inline1426 is simply the diagonal of tex2html_wrap_inline1438 , this iteration is the well-known algebraic reconstruction technique of computerized tomography, or the (block) Kaczmarz method, for solving underdetermined linear systems. We shall prove that (1.4) converges subject to natural assumptions for tex2html_wrap_inline1440 provided that tex2html_wrap_inline1426 is positive definite and tex2html_wrap_inline1444 . In the general nonlinear case, we do not have any convergence results. But it is easy to see what the method actually does. Suppose the problem (1.3) has a solution f - not necessarily unique. Then, in F,

displaymath1450

The factor of tex2html_wrap_inline1424 is

eqnarray64

where the dots stand for higher order terms in tex2html_wrap_inline1454 . Thus, if tex2html_wrap_inline1426 is positiv definit and tex2html_wrap_inline1424 sufficiently small, we can expect tex2html_wrap_inline1460 to be closer to f than tex2html_wrap_inline1464 , provided tex2html_wrap_inline1466 . The decrease of the error is best possible if

displaymath1468

However, the emphasis of this paper is not so much on theoretical questions such as convergence. Rather we describe the method to some detail for a variety of inverse problems for partial differential equations. We also present numerical examples which demonstrate that the method is quite easy to implement and, compared to other methods, very efficient. Its success depends of course on the degree of ill-posedness of the inverse problem which varies very much within the class of problems treated here. Also, the method is quite plausible, making use of vivid phenomena such as backpropagation. The method is closely related to the method of adjoint fields which is used in the engineering literature [1].

We do not claim any originality of our approach. Similar methods have been used in thousends of papers. A typical example is [2]. An implementation for ultrasound tomography is given in [4].


next up previous
Next: Linerarization Up: Numerical Solution of Bilinear Previous: Numerical Solution of Bilinear

Frank Wuebbeling
Mon Sep 7 11:40:50 MET DST 1998