Regularization by Galerkin Methods

Hans Groot

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