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

Special inverse problems

Now we derive the operator tex2html_wrap_inline1494 for special inverse problems of partial differential equations. The derivation is in no way rigouros. We neglect subtleties such as the exact domains of definition. The essential tools are Gauss' integral theorem and Green's formula.

1. Impedance tomography.
Let tex2html_wrap_inline1510 , n = 2,3 be sufficiently regular, and let tex2html_wrap_inline1514 be a function in tex2html_wrap_inline1516 , the conductivity of the material in tex2html_wrap_inline1516 . We want to determine tex2html_wrap_inline1514 from measurements of currents tex2html_wrap_inline1460 and voltages tex2html_wrap_inline1524 , tex2html_wrap_inline1526 , on tex2html_wrap_inline1528 . The currents are usually applied at pairs of electrodes on tex2html_wrap_inline1528 .

The mathematical model is as follows. The potential tex2html_wrap_inline1376 satisfies

  equation111

We normalize tex2html_wrap_inline1376 by requiring that the mean value on tex2html_wrap_inline1528 is zero. The measurements provide us with the values of

  equation119

We have to determine tex2html_wrap_inline1514 from (3.1)-(3.2) for tex2html_wrap_inline1526 . We assume tex2html_wrap_inline1514 to be known on tex2html_wrap_inline1528 .

It is clear that (3.1) can be written in the form of (1.1) with tex2html_wrap_inline1546 , tex2html_wrap_inline1548 , tex2html_wrap_inline1550 , and tex2html_wrap_inline1514 playing the role of f. Also, (3.2) is a special case of (1.2) with tex2html_wrap_inline1556 which is simply tex2html_wrap_inline1558 with the (known) scalar product tex2html_wrap_inline1560 , and we have

displaymath1562

where tex2html_wrap_inline1376 is the solution of (3.1). Hence, (2.2) reads

displaymath1566

  equation136

    theorem145

Proof: For any z, w we have

displaymath1576

Choosing tex2html_wrap_inline1578 from (3.3) and z from (3.5) we obtain

displaymath1582

Since tex2html_wrap_inline1584 on tex2html_wrap_inline1528 an integration by parts yields

displaymath1588

or

displaymath1590

This holds for each tex2html_wrap_inline1592 vanishing on tex2html_wrap_inline1528 , verifying (3.4). tex2html_wrap_inline1596

In the light of the theorem, the iteration (1.4) proceeds as follows.

For tex2html_wrap_inline1598

eqnarray182

We see that each sweep of the iteration requires the solution of 2p boundary value problems of nearly identical shape. This is not only easy to program but also very cheap computationally, compared to other methods.

2. A parabolic inverse problem.
Consider the initial-boundary value problem

  eqnarray202

The source terms tex2html_wrap_inline1602 are assumed to be known, while the diffusion coefficient tex2html_wrap_inline1604 has to be determined from the knowledge of tex2html_wrap_inline1376 on tex2html_wrap_inline1528 :

  equation214

Again we assume tex2html_wrap_inline1514 to be known on tex2html_wrap_inline1528 .

This problem is of the form (1.1)-(1.2). We have

displaymath1614

The operator tex2html_wrap_inline1616 is given by

displaymath1618

where tex2html_wrap_inline1376 is the solution of (3.6). According to (2.2), the Fréchet derivative is

displaymath1622

  equation234

displaymath1624

   theorem247

Proof: For any z, w we have

eqnarray265

For tex2html_wrap_inline1490 , z from (3.8)-(3.9) this reads

displaymath1640

An integration by parts on the left hand side turns this into

displaymath1642

hence the theorem. tex2html_wrap_inline1596

The algorithm (1.4) is as follows: For tex2html_wrap_inline1598

Solve

displaymath1648

and

displaymath1650

Put

displaymath1652

Alternatively we may consider the boundary condition tex2html_wrap_inline1654 on tex2html_wrap_inline1656 instead of tex2html_wrap_inline1658 in (3.6), and tex2html_wrap_inline1660 on tex2html_wrap_inline1656 as data in (3.7). Then,

displaymath1664

and

displaymath1666

where now tex2html_wrap_inline1490 satisfies (3.8) with the boundary condition tex2html_wrap_inline1670 on tex2html_wrap_inline1656 . The adjoint is given by the same expression as above, but with the boundary condition z = g on tex2html_wrap_inline1656 .

3. Laser tomography: Stationary case

Here the governing equation is

  eqnarray316

where

displaymath1678

with tex2html_wrap_inline1680 the exterior normal to tex2html_wrap_inline1528 at tex2html_wrap_inline1684 . The source term q is delta-like, and the sources s are on tex2html_wrap_inline1528 . tex2html_wrap_inline1692 is assumed to be known. We want to recover a, b from knowing u on tex2html_wrap_inline1528 for sources tex2html_wrap_inline1702 . Denoting by tex2html_wrap_inline1376 the solution of (3.10) for tex2html_wrap_inline1706 we have

  eqnarray331

The information on u on tex2html_wrap_inline1528 which we want to make use of is

  equation340

Obviously, (3.11)-(3.12) is of the type of (1.1)-(1.2) with f = (a,b) and

eqnarray348

The operator tex2html_wrap_inline1714 is given by

displaymath1716

where tex2html_wrap_inline1376 is the solution to (3.11). The linearized operator is

displaymath1720

where w solves

  eqnarray365

   theorem375

Proof: For any z, w we have

eqnarray400

For z, w from (3.13)-(3.14) this implies

eqnarray410

hence the theorem. tex2html_wrap_inline1596

The algorithm is as follows:

For tex2html_wrap_inline1526

Solve

eqnarray423

and

eqnarray432

Put

eqnarray441

4. SPECT

In SPECT one has to compute the source distribution f from

  equation450

  eqnarray454

where tex2html_wrap_inline1744 .

Here, tex2html_wrap_inline1516 is a bounded domain in tex2html_wrap_inline1748 , and tex2html_wrap_inline1750 are direction vectors tex2html_wrap_inline1752 . If a is known, then (3.15)-(3.16) reduces to a linear problem which can be solved by inverting the attenuated Radon transform. We consider the case in which a is unknown, too.

(3.15)-(3.16) is of the form (1.1)-(1.2) again, with (a,f) playing the role of f and

eqnarray470

The operator tex2html_wrap_inline1762 is given by

displaymath1764

where tex2html_wrap_inline1376 is the solution of (3.15). The linearized operator is

displaymath1768

where tex2html_wrap_inline1490 solves

  equation487

   theorem493

Proof: For any z, w we have

displaymath1782

For z, w as in (3.13)-(3.17) this means

displaymath1788

hence the theorem. tex2html_wrap_inline1596

The algorithm is:
For tex2html_wrap_inline1526

Solve

  equation526

and

  equation533

Put

displaymath1794

The forward problem (3.19) can be solved by inverting the attenuated Radon transform, and (3.20) is the attenuated backprojection. Thus the algorithm is very similar to the ART algorithm in computerized tomography and in fact reduces to this algorithm if a is known, e.g. a = 0.

5. Ultrasound tomography: Time harmonic case
Here we have the Helmholtz equation

displaymath1800

where tex2html_wrap_inline1802 with the scattered wave v satisfying the radiation condition

displaymath1806

Here, f is a complex valued function which has to be recovered from knowing u for each direction tex2html_wrap_inline1812 outside the support of f. tex2html_wrap_inline1816 is a real parameter controlling the spatial resolution.

Let tex2html_wrap_inline1516 be a ball containing supp(f) in its interior. We rewrite the radiation condition as a boundary condition on tex2html_wrap_inline1528 , to wit

displaymath1824

with a linear operator B on tex2html_wrap_inline1558 . Assume that the measurements are made for finitely many directions tex2html_wrap_inline1750 , and let tex2html_wrap_inline1832 be the solution for direction tex2html_wrap_inline1834 . Then,

  eqnarray559

and f has to be determined from

  equation568

Obviously, (3.21)-(3.22) is of the form (1.1)-(1.2), with

eqnarray576

The operator tex2html_wrap_inline1838 is

displaymath1840

where tex2html_wrap_inline1842 solves (3.21). The linearized operator is given by tex2html_wrap_inline1844 , where tex2html_wrap_inline1490 is a solution of

  equation583

displaymath1848

   theorem591

Proof: For any z, w we have

eqnarray605

For the functions z, tex2html_wrap_inline1490 from (3.24)-(3.23) this yields

eqnarray617

or

displaymath1864

tex2html_wrap_inline1596

6. Ultrasound tomography: Time resolved case
Here we have the wave equation

  equation635

with initial and boundary conditions

  equation641

with tex2html_wrap_inline1868 the exterior normal on tex2html_wrap_inline1528 . The source term q is tex2html_wrap_inline1874 like and the sources s sit on tex2html_wrap_inline1528 . The problem is to recover the sound speed c in tex2html_wrap_inline1516 from the measurement of u on tex2html_wrap_inline1656 , or a part thereof, for sources tex2html_wrap_inline1702 . We assume tex2html_wrap_inline1890 with f = 0 on tex2html_wrap_inline1528 . Again, this problem is obviously a special case of (1.1), (1.2) with

displaymath1896

The linearized operator tex2html_wrap_inline1898 is given by

displaymath1900

where tex2html_wrap_inline1490 is the solution of

  eqnarray653

with tex2html_wrap_inline1376 the solution to (3.25), (3.26) for tex2html_wrap_inline1890 and tex2html_wrap_inline1706 .

   theorem669

Proof: For arbitrary w, z we have

eqnarray686

For tex2html_wrap_inline1578 from (3.27) and z as in (3.28) this reads

displaymath1926

hence

displaymath1928

tex2html_wrap_inline1596

If u is measured only on a part tex2html_wrap_inline1934 of tex2html_wrap_inline1656 , then Z has to be replaced by tex2html_wrap_inline1940 . With E the extension from tex2html_wrap_inline1934 to tex2html_wrap_inline1656 by zero, the corresponding adjoint tex2html_wrap_inline1948 is then given by tex2html_wrap_inline1950 .

7. Laser tomography in the diffusion approximation:

Time harmonic case.
Here the partial differential equation is

eqnarray716

The modulating frequency tex2html_wrap_inline1424 is fixed, and tex2html_wrap_inline1524 is the measured data for the source tex2html_wrap_inline1956 .

Putting tex2html_wrap_inline1958 we obtain

  eqnarray729

The problem is to recover tex2html_wrap_inline1514 , a from the tex2html_wrap_inline1524 . It is clear that (3.29) is of the form (1.1-2). We have

displaymath1966

tex2html_wrap_inline1434 being considered as an operator from tex2html_wrap_inline1970 into tex2html_wrap_inline1558 . We readily compute the operator tex2html_wrap_inline1974 , obtaining

displaymath1976

where tex2html_wrap_inline1490 solves

displaymath1980

displaymath1982

  theorem759

Proof: The proof is done by verification. We compute

eqnarray778

by way of Green's theorem. The integral over tex2html_wrap_inline1528 vanishes, and the integral over tex2html_wrap_inline1516 can be rewritten as

displaymath1998

which, by the differential equations for z and tex2html_wrap_inline1490 , can be written as

displaymath2004

Hence

eqnarray806

tex2html_wrap_inline1596

8. Laser tomography in the diffusion approximation:

Time resolved case

In this case the governing equation reads

  eqnarray832

The source terms tex2html_wrap_inline1602 are assumed to be tex2html_wrap_inline1874 -functions at t=0 and tex2html_wrap_inline2014 . Now the invers problem is to determine both the diffusion coeffient tex2html_wrap_inline1604 and the absorption coefficient a(x) from the given data

displaymath1496

As before this problem is of the form (1.1)-(1.2).

The operator tex2html_wrap_inline2020 is given by

displaymath1497

where tex2html_wrap_inline1376 is the solution of (3.30). According to (2.2) we calculate the linearized operator to be

displaymath1498

where tex2html_wrap_inline1490 solves the equation

  eqnarray864

  theorem874

Proof: For any z,w we have

eqnarray893

For tex2html_wrap_inline2032 from (3.31)-(3.32) this reduces to

displaymath1500

An integration by parts on the left hand side leads to

displaymath1501

This proofs the theorem. tex2html_wrap_inline1596


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

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