regularization by galerkin methods hans groot. 2 overview in previous talks about inverse problems:...
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2
Overview
• In previous talks about inverse problems:well-posednessworst-case errorsregularization strategies
3
Overview
• In this talk:IntroductionProjection methodsGalerkin methodsSymm’s integral equationConclusions
4
• Differentiation:
• Inverse problem → integration:
Example: differentiation
, 00
0
ttdssytxt
t
)()(
00 , tttxdt
dty )()(
Introduction Symm’s Integral EquationProjection Methods Galerkin Methods Conclusions
5
• Given perturbation y of y:
• Integration ill-posed:
Example: differentiation
10
11210
0
0
2
2
)(
)(
)(
exp
exp
tty
tty
dsytx
tt
t
tss
00
2
2 ,
ttyty
tt
exp)(
Introduction Symm’s Integral EquationProjection Methods Galerkin Methods Conclusions
6
• Interpolation:
• Numerical integral does not blow up:
Example: differentiation
n
i
t
titt
t
ti
n
itt
dsswy
dsswytxii
1
1
00
2
20
0
2
2
)(
)()(
exp
exp
)(expexp twyy i
n
itttt ii
1
2
2
2
2
Introduction Symm’s Integral EquationProjection Methods Galerkin Methods Conclusions
7
Inverse Problems
• Let: X, Y Hilbert spaces K : X → Y linear, bounded, one-to-one
mapping
• Inverse Problem: Given y ∈ Y, solve Kx = y for x ∈ X
Symm’s Integral EquationIntroduction Galerkin Methods ConclusionsProjection Methods
8
• Let: Xn ⊂ X, Yn ⊂ Y n-dimensional subspaces
Qn : Y → Yn projection operator
• Projection Method: Given y ∈ Y, solve QnKxn = Qny for xn ∈ Xn
Projection Methods
Symm’s Integral EquationIntroduction Galerkin Methods ConclusionsProjection Methods
9
• Let:
• Then
Linear System of Equations
y,...,yYx,...,xX nnnn ˆˆˆˆ 11 ,
yAxKQyyQ i
n
iijjni
n
iin ˆˆˆ
11
,
i
n
jjij
n
jjjn Axx
11
ˆ
y Q KxQnnn
Symm’s Integral EquationIntroduction Galerkin Methods ConclusionsProjection Methods
10
Regularization by Disretization
• General assumptions: ∪n Xn dense in X QnK|Xn
: Xn → Yn one-to-one
• Definition: Rn ≔ (QnK|Xn
)-1Qn : Y → Xn
• Convergence: xn = RnKx → x (n → ∞)
Symm’s Integral EquationIntroduction Galerkin Methods ConclusionsProjection Methods
11
• Under given assumptions:•convergence iff Rn is regularization
strategy:
for some c > 0, all n ∈ ℕ•error estimate:
Theorem
cKRn x Kx R nn )(
xzcxx nXznnn
min
)(1
Symm’s Integral EquationIntroduction Galerkin Methods ConclusionsProjection Methods
12
Galerkin Method• Galerkin method:
for all zn ∈ Yn
• Substitute
for
)()( nnn zyzKx , ,
i
n
jjijA
1
)ˆ()ˆˆ( iiijij yyyxKA , , ,
:
n
jjjn xx
1ˆ
Symm’s Integral EquationIntroduction Projection Methods ConclusionsGalerkin Methods
13
Error Estimates• Approximate right-hand side y ∈ Y, ∥y - y ∥ ≤ :
• Equation:
• Error estimate:
• Approximate right-hand side ∈ Y, | - | ≤ :
• Equation:
• Error estimate:
for all zn ∈ Yn
• System of equations:
for
)()( nnn zyzKx , ,
xKxRRxx nnn
n
jjjni
n
jjij xxA
11ˆ with
xKxRRrxx nnnn
Symm’s Integral EquationIntroduction Projection Methods ConclusionsGalerkin Methods
14
Example: Least Squares Method
• Least squares method (Yn = K(Xn )) :
for all zn ∈ Xn
• Substitute
for
)()(nnn
KzyKzKx , ,
)ˆ()ˆˆ(iiijij
xKyxKxKA , , ,
:
n
jjjn xx
1ˆ i
n
jjijA
1
Symm’s Integral EquationIntroduction Projection Methods ConclusionsGalerkin Methods
15
Example: Least Squares Method
• Define :
• Assume:
for some c > 0, all x ∈ X Then least squares method is convergent
and ∥Rn∥ ≤ n
Kz X z z nnnnn1max ,: :
x czxKzx nnnXz nn
)(min
Symm’s Integral EquationIntroduction Projection Methods ConclusionsGalerkin Methods
16
Example: Dual Least Squares Method
• Dual least squares method (Xn = K* (Yn )) :
- with K* : Y → X adjoint of K
- for all zn ∈ Yn
• Substitute
for
nnnnnuKxzyzuKK ** )()( , , ,
)ˆ()ˆˆ( ***iiijij
yKyyKyKA , , ,
:
n
jjjn xx
1ˆ i
n
jjijA
1
Symm’s Integral EquationIntroduction Projection Methods ConclusionsGalerkin Methods
17
• Define :
• Assume: ∪n Yn dense in Y range K(X) dense in Y
Then dual least squares method is convergent and ∥Rn∥ ≤ n
zK Y zz nnnnn 1*max ,: :
Example: Dual Least Squares Method
Symm’s Integral EquationIntroduction Projection Methods ConclusionsGalerkin Methods
18
Application: Symm’s Integral Equation
• Dirichlet problem for Laplace equation:
⊂ ℝ2 bounded domain ∂ analytic boundary f ∈ C(∂)
on in (BVP)
fuu 0
Galerkin MethodsIntroduction Projection Methods ConclusionsSymm’s Integral Equation
19
Symm’s Integral Equation
Simple layer potential:
solves BVP iff ∈ C(∂) satisfies Symm’s equation:
xydsyxyxu , )(ln)(1)(
xxfydsyxy , )()(ln)(1
Galerkin MethodsIntroduction Projection Methods ConclusionsSymm’s Integral Equation
20
Symm’s Integral Equation Assume ∂ has parametrization
for 2-periodic analytic function : [0,2] → ℝ2, with
Then Symm’s equation transforms into:
with
])( 20 ,[ , ssx
][)()()(ln)( )(
20
2
0
1 , , ttfdssts
]0)( 20 ,[ , ss
])()(:)( 20)( ,[ , ssss
Galerkin MethodsIntroduction Projection Methods ConclusionsSymm’s Integral Equation
21
Application: Symm’s Integral Equation
Define K : Hr(0,2) → Hr+1(0,2) and g ∈ Hr(0,2), r ≥ 0 by
Define Xn = Yn = { : j ∈ ℂ}
)( )(:)( tftg
dsststK
2
0
)()(ln)(1:)(
n
nj
ijtj e
Galerkin MethodsIntroduction Projection Methods ConclusionsSymm’s Integral Equation
22
Application: Symm’s Integral Equation• Approximate right-hand side g ∈ Y, ∥g - g ∥
≤ :• (Bubnov-)Galerkin method:
• Least squares method:
• Dual least squares method:
• Error Estimate:
)()( nnnnXn KgKK , , :
)()( nnnnXn gK , , :
, , , : nnnnnn
Xn
KgKK ~)()~(
rHr
Ln nnc
2
Galerkin MethodsIntroduction Projection Methods ConclusionsSymm’s Integral Equation
23
Conclusions
• Discretisation schemes can be used as regularisation strategies
• Galerkin method converges iff it provides regularisation strategy
• Special cases of Galerkin methods:o least squares methodo dual least squares method
Symm’s Integral EquationIntroduction Projection Methods Galerkin Methods Conclusions