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Regularization by Galerkin Methods

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:

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Regularization by Galerkin Methods

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    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

    4. 4 Example: differentiation

    5. 5 Example: differentiation

    6. 6 Example: differentiation

    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

    8. 8 Projection Methods

    9. 9 Linear System of Equations

    10. 10 Regularization by Disretization

    11. 11 Theorem

    12. 12 Galerkin Method Galerkin method: for all zn ? Yn Substitute for

    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 zn ? Yn System of equations: for

    14. 14 Example: Least Squares Method Least squares method (Yn = K(Xn )) : for all zn ? Xn Substitute for

    15. 15 Example: Least Squares Method Define : Assume: for some c > 0, all x ? X Then least squares method is convergent and ?Rn? = ?n

    16. 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

    17. 17 Define : Assume: ?n Yn dense in Y range K(X) dense in Y Then dual least squares method is convergent and ?Rn? = ?n

    18. 18 Application: Symm’s Integral Equation Dirichlet problem for Laplace equation: ? ? R2 bounded domain ?? analytic boundary f ? C(??)

    19. 19 Symm’s Integral Equation Simple layer potential: solves BVP iff ? ? C(??) satisfies Symm’s equation:

    20. 20 Symm’s Integral Equation Assume ?? has parametrization for 2?-periodic analytic function ? : [0,2?] ? R2, with Then Symm’s equation transforms into: with

    21. 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}

    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:

    23. 23 Conclusions Discretisation schemes can be used as regularisation strategies Galerkin method converges iff it provides regularisation strategy Special cases of Galerkin methods: least squares method dual least squares method

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