1. Regularization by Galerkin Methods Hans Groot

2. 2 Overview In previous talks about inverse problems: well-posedness worst-case errors regularization strategies

3. 3 Overview In this talk: Introduction Projection methods Galerkin methods Symm’s integral equation Conclusions

4. 4 Differentiation: Inverse problem → integration: Example: differentiation Introduction Symm’s Integral EquationProjection MethodsGalerkin MethodsConclusions

5. 5 Given perturbation y  of y : Integration ill-posed: Example: differentiation Introduction Symm’s Integral EquationProjection MethodsGalerkin MethodsConclusions

6. 6 Interpolation : Numerical integral does not blow up: Example: differentiation Introduction Symm’s Integral EquationProjection MethodsGalerkin MethodsConclusions

7. 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 EquationIntroductionGalerkin MethodsConclusions Projection Methods

8. 8 Let:  X n ⊂ X, Y n ⊂ Y n -dimensional subspaces  Q n : Y → Y n projection operator Projection Method:  Given y ∈ Y, solve Q n Kx n = Q n y for x n ∈ X n Projection Methods Symm’s Integral EquationIntroductionGalerkin MethodsConclusions Projection Methods

9. 9 Let:    Then Linear System of Equations Symm’s Integral EquationIntroductionGalerkin MethodsConclusions Projection Methods

10. 10 Regularization by Disretization General assumptions:  ∪ n X n dense in X  Q n K | X n : X n → Y n one-to-one Definition:  R n ≔ (Q n K | X n ) -1 Q n : Y → X n Convergence:  x n = R n Kx → x (n → ∞) Symm’s Integral EquationIntroductionGalerkin MethodsConclusions Projection Methods

11. 11 Under given assumptions: convergence iff R n is regularization strategy: for some c > 0, all n ∈ ℕ error estimate: Theorem Symm’s Integral EquationIntroductionGalerkin MethodsConclusions Projection Methods

12. 12 Galerkin Method Galerkin method: for all z n ∈ Y n Substitute for Symm’s Integral EquationIntroductionProjection MethodsConclusions Galerkin Methods

13. 13 Error Estimates Approximate right-hand side y  ∈ Y, ∥ y - y  ∥ ≤  : Equation: Error estimate: Approximate right-hand side   ∈ Y, |  -   | ≤  : Equation: Error estimate: for all z n ∈ Y n System of equations: for Symm’s Integral EquationIntroductionProjection MethodsConclusions Galerkin Methods

14. 14 Example: Least Squares Method Least squares method ( Y n = K ( X n ) ) : for all z n ∈ X n Substitute for Symm’s Integral EquationIntroductionProjection MethodsConclusions Galerkin Methods

15. 15 Example: Least Squares Method Define : Assume: for some c > 0, all x ∈ X  Then least squares method is convergent and ∥ R n ∥ ≤  n Symm’s Integral EquationIntroductionProjection MethodsConclusions Galerkin Methods

16. 16 Example: Dual Least Squares Method Dual least squares method ( X n = K * ( Y n )) : - with K * : Y → X adjoint of K - for all z n ∈ Y n Substitute for Symm’s Integral EquationIntroductionProjection MethodsConclusions Galerkin Methods

17. 17 Define : Assume:  ∪ n Y n dense in Y  range K(X) dense in Y  Then dual least squares method is convergent and ∥ R n ∥ ≤  n Example: Dual Least Squares Method Symm’s Integral EquationIntroductionProjection MethodsConclusions Galerkin Methods

18. 18 Application: Symm’s Integral Equation Dirichlet problem for Laplace equation:   ⊂ ℝ 2 bounded domain  ∂  analytic boundary  f ∈ C( ∂  ) Galerkin MethodsIntroductionProjection MethodsConclusions Symm’s Integral Equation

19. 19 Symm’s Integral Equation  Simple layer potential:  solves BVP iff  ∈ C( ∂  ) satisfies Symm’s equation: Galerkin MethodsIntroductionProjection MethodsConclusions Symm’s Integral Equation

20. 20 Symm’s Integral Equation  Assume ∂  has parametrization for 2  -periodic analytic function  : [0,2  ] → ℝ 2, with  Then Symm’s equation transforms into: with Galerkin MethodsIntroductionProjection MethodsConclusions Symm’s Integral Equation

21. 21 Application: Symm’s Integral Equation  Define K : H r ( 0, 2  ) → H r +1 ( 0, 2  ) and g ∈ H r ( 0, 2  ), r ≥ 0 by  Define X n = Y n = { :  j ∈ ℂ } Galerkin MethodsIntroductionProjection MethodsConclusions Symm’s Integral Equation

22. 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: Galerkin MethodsIntroductionProjection MethodsConclusions Symm’s Integral Equation

23. 23 Conclusions Discretisation schemes can be used as regularisation strategies Galerkin method converges iff it provides regularisation strategy Special cases of Galerkin methods: oleast squares method odual least squares method Symm’s Integral EquationIntroductionProjection MethodsGalerkin Methods Conclusions