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Inverse problems in classical and quantum physics [Elektronische Ressource] / Andrea Amalia Almasy

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Inverse Problems in Classicaland Quantum PhysicsDissertation zur Erlangung des Grades,,Doktor der Naturwissenschaften”am Fachbereich Physikder Johannes Gutenberg-Universit¨at in MainzAndrea Amalia Almasygeb. in Sighetu Marma¸tiei, Rum¨anienMainz, den 29. Juni 2007Datum der mu¨ndlische Pru¨fung: 29. Juni 2007D77 (Diss. Universit¨at Mainz)To my family’Far better an approximate answer to the right question,which is often vague,than an exact answer to the wrong question,which can always be made precise.’John W. TukeyNotations and symbols∈ Element of⊂ Subset of≡ Identiaclly equals∼ Essentially equal to or equivalent† Hermitian conjugate+ +f , A Generalisedsolutionandgeneralisedinverserespectively0f First derivative of fhfi Mean value of fnjx =(x ) Vector in Euclidean n-dimensional space R . The com-ponents are labelled by latin letters (j =1,2,3)xxˆ = Unit vector in the direction ofx|x|μx =(x ) 4-Vector in 4-dimensional Minkowski space. The com-ponents are labelled by greek letters (μ= 0,1,2,3)n nd x,dx Volume element in Euclidean n-dimensional space R4 0 1 2 3 Volume element in 4-dimensional Minkowski spaced x=dx dx dx dxUnit operator or matrixIClosure of the domain ΩΩBoundary of the domain Ω∂ΩO(E) Terms of order EN(A) Null-space of operator AR(A) Range of operatorAD(A) Domain of operatorA∗ Adjoint of AAT Transposed of AATrace of ATrA|.| Absolute value||.

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Published 01 January 2007
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Inverse Problems in Classical
and Quantum Physics
Dissertation zur Erlangung des Grades
,,Doktor der Naturwissenschaften”
am Fachbereich Physik
der Johannes Gutenberg-Universit¨at in Mainz
Andrea Amalia Almasy
geb. in Sighetu Marma¸tiei, Rum¨anien
Mainz, den 29. Juni 2007Datum der mu¨ndlische Pru¨fung: 29. Juni 2007
D77 (Diss. Universit¨at Mainz)To my family
’Far better an approximate answer to the right question,
which is often vague,
than an exact answer to the wrong question,
which can always be made precise.’
John W. TukeyNotations and symbols
∈ Element of
⊂ Subset of
≡ Identiaclly equals
∼ Essentially equal to or equivalent
† Hermitian conjugate
+ +f , A Generalisedsolutionandgeneralisedinverserespectively
0f First derivative of f
hfi Mean value of f
njx =(x ) Vector in Euclidean n-dimensional space R . The com-
ponents are labelled by latin letters (j =1,2,3)
x
xˆ = Unit vector in the direction ofx|x|
μx =(x ) 4-Vector in 4-dimensional Minkowski space. The com-
ponents are labelled by greek letters (μ= 0,1,2,3)
n nd x,dx Volume element in Euclidean n-dimensional space R
4 0 1 2 3 Volume element in 4-dimensional Minkowski spaced x=dx dx dx dx
Unit operator or matrixI
Closure of the domain ΩΩ
Boundary of the domain Ω∂Ω
O(E) Terms of order E
N(A) Null-space of operator A
R(A) Range of operatorA
D(A) Domain of operatorA
∗ Adjoint of AA
T Transposed of AA
Trace of ATrA
|.| Absolute value
||.|| Norm defined on the spaceXXReal part of xRex
Imaginary part of xImx
Principal partPP
Summation
d(O)[O] Mass dimesion of operatorO, i.e., [O] =M
F[f](x) Fourier transform of f
(f,g) Scalar product defined on the spaceXX
Laplace operatorΔ
Nabla operator∇
δ(x) Dirac δ-function (−∞ < x <∞); δ(x) = 0 for x = 0,R∞
dxδ(x) = 1−∞
θ(x) Heaviside-function (−∞<x<∞); θ(x) = 0 for x< 0,
θ(x) = 1 for x>0
i iq ,q¯ Quark fields. Flavours are labelled by latin letters, hereα α
i, and colours by greek letters, here α
μνG Gluon field strength tensora
μγ ,γ Dirac matrices5
a Gell-Mann colour matricesλ
f Structure constant of SU(3)abc
Normal ordering of the operators A,B,...:A·B·... :
6Abbreviations
Monte CarloMC
Leading orderLO
Next-to-leading orderNLO
Singular value decompositionSVD
QCD Quantum chromodynamics
QED Quantum electrodynamics
Operator product expansionOPE
Cabibbo-Kobayashi-MaskawaCKM
Partial conservation of axial-vector currentPCAC
Electrical impedance tomographyEIT
Finite element methodFEM
Confidence levelCL
Least-squaresLS
Confidence regionCR
p.d.f. probability distribution functionContents
Introduction 11
Inverse problems 17
1 Inverse and ill-posed problems 19
1.1 Inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Some examples of inverse problems . . . . . . . . . . . . . . . . . . . 21
1.3 Ill-posed problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4 A few examples of ill-posed problems . . . . . . . . . . . . . . . . . . 29
1.5 How to cure ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Regularisation of ill-posed problems 35
2.1 The generalised solution . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Tikhonov’s regularisation method . . . . . . . . . . . . . . . . . . . . 38
2.3 Truncated SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Regularisation algorithms . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Choice of regularisation parameter . . . . . . . . . . . . . . . . . . . 44
QCD condensates from τ-decay data 47
3 The theory of τ-decays 49
3.1 Hadronic τ-decays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Leptonic τ-decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 The hadronic branching ratio R . . . . . . . . . . . . . . . . . . . . 53τ
3.4 Operator Product Expansion (OPE) . . . . . . . . . . . . . . . . . . 53
3.5 Hadronic vacuum polarisation tensor . . . . . . . . . . . . . . . . . . 56
3.6 Dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Hadronic spectral functions 61
4.1 Overview of experiments . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 The mass spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Inclusive non-strange spectral functions . . . . . . . . . . . . . . . . . 67
15 Extraction of QCD condensates 71
5.1 Condensates: general properties and previous extractions . . . . . . . 71
5.2 The method: a functional approach . . . . . . . . . . . . . . . . . . . 74
6 V−A analysis 77
V−A6.1 1-parameter fit: determination ofO . . . . . . . . . . . . . . . . . 786
V−A6.1.1 O at leading-order . . . . . . . . . . . . . . . . . . . . . . 786
V−A6.1.2 O at next-to-leading-order . . . . . . . . . . . . . . . . . . 816
V−A V−A6.2 2-parameter fit:O –O correlation . . . . . . . . . . . . . . . 836 8
6.3 Review and comparison of results . . . . . . . . . . . . . . . . . . . . 88
7 V, A and V +A analysis 91
7.1 A analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A7.1.1 1-parameter fit: determination ofO . . . . . . . . . . . . . . 934
A A7.1.2 2-parameter fit:O –O correlation . . . . . . . . . . . . . . 964 6
7.2 V and V +A analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2.1 1-parameter fits . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2.2 2-parameter fits . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 Review and comparison of results . . . . . . . . . . . . . . . . . . . . 103
The inverse conductivity problem 105
8 Electrical impedance tomography 107
8.1 The mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Modelling the electrodes . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.3 Formulation of the inverse problem . . . . . . . . . . . . . . . . . . . 112
8.4 Imaging with incomplete, noisy data . . . . . . . . . . . . . . . . . . 113
8.5 A brief history of the problem . . . . . . . . . . . . . . . . . . . . . . 113
9 The forward problem 115
9.1 Methods based on integral equations . . . . . . . . . . . . . . . . . . 116
9.2 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10 Reconstruction algorithms 121
10.1 Reconstruction from a single measurement . . . . . . . . . . . . . . . 121
10.1.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
10.1.2 An example: the unit disc . . . . . . . . . . . . . . . . . . . . 123
10.1.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.2 Reconstruction from more measurements . . . . . . . . . . . . . . . . 127
10.2.1 Reconstruction by linearisation . . . . . . . . . . . . . . . . . 127
10.2.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 130
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