Mathematical Background
A.5 Formulas of Vector Analysis
a twice continuously differentiable vector field,
curl curla= grad diva− 4a (A.73)
a continuously differentiable function, b continuously differentiable vector field
div (ab) =b· grada+adivb (A.74) a∈H1(B), b∈Hdiv(B)
Z
∂B
a ν·bds = Z
B
b· grada+adivbdx (A.75) a, b continuously differentiable vector fields
div (a×b) =bcurla−acurlb (A.76) a∈Hcurl(B), b∈H1(B)
Z
∂B
b·(ν×a)ds= Z
B
bcurla−acurlbdx (A.77) a continuously differentiable function,b continuously differentiable vector field
curl (ab) = acurlb−b× grada (A.78) a∈H1(B), b∈Hcurl(B)
Z
∂B
aν×bds = Z
B
acurlb−b× grada dx (A.79) a, b continuously differentiable vector fields
curl (a×b) = (a· grad )b−(b· grad )a+adivb−bdiva (A.80) a continuously differentiable vector field, then the formulas
div{Φ(x, y)a(y)} = grad Φ(x, y)·a(y) + Φ(x, y) diva(y) (A.81) curl{Φ(x, y)a(y)} = grad Φ(x, y)×a(y) + Φ(x, y) curla(y) (A.82) are true for both differentiation with respect to x and y.
divx{Φ(x, y)a(y)} = −divy{Φ(x, y)a(y)}+ Φ(x, y) divya(y) (A.83) curlx{Φ(x, y)a(y)} = −curly{Φ(x, y)a(y)}+ Φ(x, y) curlya(y) (A.84)
1.1 Basic principle of a hydrogen fuel cell . . . 3
1.2 Stack consisting of 8 fuel cells . . . 4
1.3 Fuel cell, wire grid model and measurement device for magnetic tomography 7 2.1 Discretization model with 5 points in each direction . . . 55
2.2 Magnetic field on the shell of a cylinder . . . 58
2.3 Our way of the current representation in 3d . . . 58
2.4 Simulated and reconstructed currents for a defect near the boundary . . . 59
2.5 Simulated and reconstructed currents for two defects . . . 59
2.6 Wire grid with 125 resistors . . . 60
2.7 Wire grid and simulated magnetic field . . . 60
2.8 Simulated and reconstructed currents in a small wire grid . . . 61
2.9 Comparison of a difference reconstruction with the simulated z-currents . . 61
2.10 Fuel cell and segmentation of the graphite layer . . . 62
2.11 Segmented fuel cell, measured and reconstructed currents . . . 63
2.12 Difference reconstruction with three defects . . . 63
3.1 Scene 1 . . . 85
3.2 Representation of Wj and w on the xy-plane . . . 86
3.3 Comparison of ˜wand w for a spherical test domain . . . 87
3.4 Comparison of ˜wand w for an elliptic test domain . . . 88
3.5 Comparison of ˜wand w for different test domains . . . 89
3.6 w(z) for the test domains˜ G(k)z , k = 8,4,2,6 . . . 91
3.7 Composition of the test results for dn, n= 1, . . . ,8 . . . 92
3.8 Scene 2 . . . 93
3.9 Representation of Wj and w on the xy-plane . . . 93
3.10 ˜w for test domainsG(k)z , k= 8,4,2,6 . . . 94
3.11 Composition of the results for G(k)z , k = 1, . . . ,8 . . . 95
3.12 Scene 3 . . . 100
3.13 Logarithmic values of the maximum responses for spherical test domains . 101 3.14 Logarithmic values of the maximum responses for cubic test domains . . . 101
3.15 Sampling method with spherical test domains . . . 101
3.16 Sampling method with cubic test domains . . . 101
3.17 Reconstructed domain (light blue) compared to the original (red) . . . 102 3.18 Scene 4 . . . 103 3.19 Logarithmic values of the responses for the balls . . . 104 3.20 Colored plot of ˜w=R
∂Bwa(., z(k))ds on a parallel surface of∂BRmin(0) . . 104 3.21 Reconstructed domain (light blue) compared to the original (red) . . . 105
1 Interior Dirichlet problem for Laplace’s equation . . . 27
2 Interior Neumann problem for Laplace’s equation . . . 27
3 Exterior Dirichlet problem for Laplace’s equation . . . 28
4 Exterior Neumann problem for Laplace’s equation . . . 28
5 Interior normal problem for harmonic fields . . . 29
6 Exterior normal problem for harmonic fields . . . 29
7 Interior tangential problem for harmonic fields . . . 47
8 Interior tangential problem for the reduced Stokes equation . . . 49
9 Interior Neumann problem for the impedance equation . . . 51
10 An exterior problem with Neumann condition . . . 64
11 Impedance equation with transmission and tangential condition . . . 72
12 Interior Dirichlet problem for the impedance equation . . . 73
13 Impedance equation with Neumann and tangential condition . . . 74
14 Interior Dirichlet problem for the classical Laplace equation . . . 112
15 Interior Neumann problem for the classical Laplace equation . . . 112
16 Exterior Dirichlet problem for the classical Laplace equation . . . 112
17 Exterior Neumann problem for the classical Laplace equation . . . 112
18 Interior normal problem for classical harmonic fields . . . 118
19 Exterior normal problem for classical harmonic fields . . . 118
20 Interior tangential problem for classical harmonic fields . . . 118
21 classical Interior tangential problem for the reduced Stokes equation . . . 118
22 Exterior tangential problem for classical harmonic fields . . . 118
23 classical Exterior tangential problem for the reduced Stokes equation . . . 119
[Ad] R. A. Adams: Sobolev Spaces, Academic Press, London, 1997.
[BaKo] Bank, H.T. and Kojima, F.: Boundary shape identification in two-dimensional electrostatic problems using SQUIDs, J. Inverse Ill-Posed Probl. 8 (2000), No. 5, 487-504.
[CK1] D. Colton and R. Kress: Inverse Accoustic and Electromangetic Scattering Theory, Springer-Verlag, Berlin Heidelberg, 1992.
[CK2] D. Colton and R. Kress: Integral Equation Methods in Sacttering Theory, Whiley-Interscience Publication, New York, 1983.
[Co] M. Costabel: Boundary Integral Operator on Lipschitz Domains: Elementary Re-sults, SIAM J. Math. Anal. 19 (1988), 613-626.
[EHN] H. W. Engl, M. Hanke and A. Neubacher: Regularization of Inverse Problems, Kluwer Academic Publisher, Dortrecht, 1996.
[ErPo] K. Erhard, R. Potthast: The point source method for reconstructing an inclu-sion from boundary measurements in electrical impedance tomography and acoustic scattering, Inverse Problems 19 (2003), No. 5, 1139-1157.
[GiTr] D. Gilbart and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, 2nd Edition, Springer Verlag, New York, 1983.
[GiRa] V. Girault and P.-A. Raviart: Finite Element Approximation of the Navier Stokes Equations, Springer Verlag, Berlin, 1986.
[Gri] P. Grisvard: Elliptic Problems in nonsmooth domains, Pitman Advanced Publishing Program, London, 1985.
[H¨a] P. H¨ahner: An exterior boundary-value problem for the Maxwell equations with boundary data in a Sobolev space, Proc. Roy. Soc. Edinburgh 109 A, 213-224.
[HKP] K.-H. Hauer, L. K¨uhn, R. Potthast: On uniqueness and non-uniqueness for current reconstruction from magnetic fields, Inverse Problems 21 (2005), 955-967.
[HPSW] K.-H. Hauer, R. Potthast, D. Stolten, T. W¨uster: Magnetotomography - A New Method for Analysing Fuel Cell Performance and Quality, Journal of Power Sources, to appear.
[Jost] J. Jost: Partielle Differentialgleichungen: elliptische (und parabolische) Gleichun-gen, Springer-Verlag, Berlin, 1998.
[Ker] H. Kersten: Grenz - und Sprunrelationen f¨ur die Potentiale mit quadratintegrier-barer Dichte, Res. d. Math. 3 (1980), 17-24.
[Ma] E. Martensen: Potentialtheorie, B.B Teubner, Stuttgart, 1968.
[McL] W. McLean: Strongly Elliptic System and Boundary Integral Equation, Cambridge University Press, 2000.
[MeySe] N. Meyers and J. Serrin : H=W, Proc. Nat. Acad. Sci. USA 51 (1964), 1055-1056.
[Kr] R. Kress: Linear Integral Equations, 2nd ed. Springer-Verlag, New York, 1999.
[KKP] R. Kress, L. K¨uhn and R. Potthast: Reconstruction of a current distribution from its magnetic field, Inverse Problems 18 (2002), No. 4, 1127-1146.
[K¨uPo] L. K¨uhn and R. Potthast: On the convergence of the finite integration technique for the anisotropic boundary value problem of magnetic tomography, Math. Meth.
Appl. Sci. 26 (2003), No. 9, 739-757.
[LNP] J.J.Liu, G. Nakamura, R. Potthast: A new approach and improved error analysis for reconstructing the scattered wave by the point source method, to appear
[Luke] D. R. Luke: Multifrequency inverse obstacle scattering: the point source method and generalized filtered backprojection, IMACS Math. Sim. Sci., Vol. 66 (2004), 297-314.
[LuPo] D. R. Luke, R. Potthast: The no response test - a sampling method for inverse scattering problems, SIAM J. Appl. Math. 63 (2003), No. 4, 1292-1312.
[NPS] G. Nakamura, R. Potthast, M. Sini: The convergence proof of the no-response test for localizing an inclusion, to appear.
[Ned] J.-C. Nedelec: Acoustic and Electromagnetic Equations - Integral Representations for Harmonic Problems Springer-Verlag, New York, 2001.
[Po1] R. Potthast: Point-sources and Multipoles in Inverse Scattering, Chapman and Hall, London, 2001.
[Po2] R. Potthast: A point-source method for inverse acoustic and electromagnetic ob-stacle scattering problems, IMA Journal of Appl. Math 61 (1998), 119-140.
[Po3] R. Potthast: A set-handling approach for the no-response test and related methods, IMACS Math. Sim. Sci., Vol. 66 (2004), 281-295.
[Po4] R. Potthast: Sampling and Probe Methods - An Algorithmical View, Computing, to appear.
[Po5] R. Potthast: On the convergence test of the no response test, to appear.
[Ra1] Ramon, C. Marks R. J., Nelson, A. C. and Meyer, M. G.: Resolution Enhancement of Biomagnetic Images Using the Method of Alternating Protections, IEEE Trans.
Biomed. Eng., Vol. 40 (1993), No. 4, 323-328.
[Ra2] Ramon, C., Meyer, M. G., Nelson, A. C., Spelman, F. A., Lamping, J.: Simulation Studies of Biomagnetic Computed Tomography, IEEE Trans. Biomed. Eng., Vol. 40 (1993), No. 4, 317-322.
[dRh] G. de Rham: Varietes Differentiables, Hermann (1960).
[Sa] Sarvas, J.: Basic mathematical and electromagnetic concepts of the biomagnetic in-verse problem Phys. Med. Biol. Vol. 32 (1987), No. 1, 11-22.
[Str] Stroink, G.: Cardiomagnetic Imaging. in Frontiers in Carciovascular Imaging by Zaret, B. L., Kaufman, L., Berson, A. S. and Dunn, R. A. (Eds.), Raven Press, New York, 1993.
[TiWa] Tilg, B. and Wach, P.: An iterative approach on magnetic source imaging within the human cortex - a simulation study, Int. J. Bio-Medical Computing 40 (1995), 51-57.
pers¨onliche Daten
Name Lars K¨uhn
Geburtstag 16.09.1976 Geburtsort Dresden
Eltern Kersti K¨uhn, geb. Puschbeck, DV-Administrator Reinhard K¨uhn, stellv. Studienseminarleiter Anschrift An der Koppel 160, 99100 Bienst¨adt
E-Mail Kuehn.L@gmx.de
Schulausbildung
09.1983-07.1991 45. Polytechnische Oberschule Erfurt 08.1991-06.1995 Gymnasium 7, Erfurt
Spezialschulteil mathematisch-naturwissenschaftlicher Richtung 20.06.1995 Abitur
Wehrdienst
10.1995-09.1996 verl¨angerter Grundwehrdienst in der Stabskompanie der Panzerbrigade 39 in Erfurt
Studium an der Georg August Universit¨at zu G¨ottingen 10.1996-07.2001 Diplomstudiengang Mathematik 10.1996-09.2003 Diplomstudiengang Physik Pr¨ufungen/Abschl¨usse
16.10.1998 Vordiplom Mathematik 11.02.1999 Vordiplom Physik 05.07.2001 Diplom Mathematik T¨atigkeiten
10.2000-09.2004 wissenschaftlicher Mitarbeiter in der NachwuchsforschergruppeNeue Nu-merische Verfahren zur L¨osung Inverser Probleme am Institut f¨ur Nu-merische und Angewandte Mathematik der Universit¨at G¨ottingen