Regularization of nonlinear ill-posed problems by the exponential - PowerPoint PPT Presentation

Problem setting Asymptotic regularization Numerical examples Regularization of nonlinear ill-posed problems by the exponential Euler method Michael H¨ onig Lehrstuhl f¨ ur Angewandte Mathematik Heinrich Heine Universit¨ at D¨ usseldorf September 2008 Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Outline Problem setting 1 Asymptotic regularization 2 Numerical examples 3 Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Nonlinear ill-posed problems F : D ( F ) ⊂ X → Y continous and Fr´ echet differentiable X , Y real Hilbert spaces, nonlinear problem: F ( x ) = y x + the x 0 -minimum-norm solution 1 only perturbed data y δ ∈ Y , � y − y δ � Y ≤ δ available 2 problem ill-posed 3 e.g. F compact and D ( F ) weakly closed � regularization required Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Assumptions tangential cone condition 1 � � � F ( � x ) − F ( x ) − F ′ ( x )( � x − x ) � ≤ η � F ( � x ) − F ( x ) � , x , x ∈ B r ( x 0 ) � source condition 2 x 0 − x + = J ( x + ) γ w , � w � ≤ ρ, J ( x ) := F ′ ( x ) ∗ F ′ ( x ) local restriction of the derivative 3 F ′ ( x ) = R x F ′ ( x + ) � R x − I � ≤ C + � x − x + � , ∀ x ∈ B r ( x + ) w.l.o.g. � F ′ ( x ) � ≤ 1 , x ∈ B r ( x 0 ) 4 e.g. Hanke, Neubauer & Scherzer (1995), Tautenhahn (1994) Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Assumptions tangential cone condition 1 � � � F ( � x ) − F ( x ) − F ′ ( x )( � x − x ) � ≤ η � F ( � x ) − F ( x ) � , x , x ∈ B r ( x 0 ) � source condition 2 x 0 − x + = J ( x + ) γ w , � w � ≤ ρ, J ( x ) := F ′ ( x ) ∗ F ′ ( x ) local restriction of the derivative 3 F ′ ( x ) = R x F ′ ( x + ) � R x − I � ≤ C + � x − x + � , ∀ x ∈ B r ( x + ) w.l.o.g. � F ′ ( x ) � ≤ 1 , x ∈ B r ( x 0 ) 4 e.g. Hanke, Neubauer & Scherzer (1995), Tautenhahn (1994) Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Assumptions tangential cone condition 1 � � � F ( � x ) − F ( x ) − F ′ ( x )( � x − x ) � ≤ η � F ( � x ) − F ( x ) � , x , x ∈ B r ( x 0 ) � source condition 2 x 0 − x + = J ( x + ) γ w , � w � ≤ ρ, J ( x ) := F ′ ( x ) ∗ F ′ ( x ) local restriction of the derivative 3 F ′ ( x ) = R x F ′ ( x + ) � R x − I � ≤ C + � x − x + � , ∀ x ∈ B r ( x + ) w.l.o.g. � F ′ ( x ) � ≤ 1 , x ∈ B r ( x 0 ) 4 e.g. Hanke, Neubauer & Scherzer (1995), Tautenhahn (1994) Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Assumptions tangential cone condition 1 � � � F ( � x ) − F ( x ) − F ′ ( x )( � x − x ) � ≤ η � F ( � x ) − F ( x ) � , x , x ∈ B r ( x 0 ) � source condition 2 x 0 − x + = J ( x + ) γ w , � w � ≤ ρ, J ( x ) := F ′ ( x ) ∗ F ′ ( x ) local restriction of the derivative 3 F ′ ( x ) = R x F ′ ( x + ) � R x − I � ≤ C + � x − x + � , ∀ x ∈ B r ( x + ) w.l.o.g. � F ′ ( x ) � ≤ 1 , x ∈ B r ( x 0 ) 4 e.g. Hanke, Neubauer & Scherzer (1995), Tautenhahn (1994) Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Asymptotic regularization Showalter ode F ′ ( u ( t )) ∗ � � y δ − F ( u ( t )) u ( t ) t > 0 ˙ = u ( 0 ) x 0 = stopping time t ∗ chosen by discrepancy principle ( τ > 1) � F ( u ( t ∗ )) − y δ � ≤ τδ < � F ( u ( t )) − y δ � , 0 ≤ t < t ∗ analyzed by Tautenhahn (1994) u ( t ∗ ) → x + , δ → 0 as 2 γ 2 γ + 1 ) , � u ( t ∗ ) − x + � = O ( δ Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Regularization with Runge-Kutta integrators Ansatz: Application of time integration schemes for solving u ( t ) = F ′ ( u ( t )) ∗ � � y δ − F ( u ( t )) t > 0 ˙ B¨ ockmann & Pornsawad (2008): use simplified Runge-Kutta methods 1 convergence under (severe) step size restrictions 2 numerical experiments show restrictions are due to analysis 3 no proof of optimal order 4 Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Explicit Euler u ( t ) = F ′ ( u ( t )) ∗ � � y δ − F ( u ( t )) t > 0 ˙ application of explicit Euler leads to u n + 1 = u n + h n F ′ ( u n ) ∗ ( y δ − F ( u n )) nonlinear Landweber iteration for h n = 1 step size restriction 1 many (explicit) iterations 2 convergence and optimal order known 3 Hanke, Neubauer & Scherzer (1995) Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Explicit Euler u ( t ) = F ′ ( u ( t )) ∗ � � y δ − F ( u ( t )) t > 0 ˙ application of explicit Euler leads to u n + 1 = u n + h n F ′ ( u n ) ∗ ( y δ − F ( u n )) nonlinear Landweber iteration for h n = 1 step size restriction 1 many (explicit) iterations 2 convergence and optimal order known 3 Hanke, Neubauer & Scherzer (1995) Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Linearly implicit Euler u ( t ) = F ′ ( u ( t )) ∗ � � y δ − F ( u ( t )) t > 0 ˙ application of implicit Euler leads to u n + h n F ′ ( u n + 1 ) ∗ ( y δ − F ( u n + 1 )) u n + 1 = one Newton step with simplified Jacobian − J ( u n ) = − F ′ ( u n ) ∗ F ′ ( u n ) gives u n + 1 = u n + h n ( I + h n J ( u n )) − 1 F ′ ( u n ) ∗ ( y δ − F ( u n )) � Newton type method with Tikhonov Philipps as inner regularization Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Linearly implicit Euler u n + h n ( I + h n J ( u n )) − 1 F ′ ( u n ) ∗ ( y δ − F ( u n )) u n + 1 = J ( u ) F ′ ( u ) ∗ F ′ ( u ) = Newton type method with Tikhonov Philipps regularization large step sizes 1 one linear system in each timestep 2 convergence (rates) known 3 Rieder (2001) related to iteratively regularized Gauss–Newton 4 Bakushinskii (1992) Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Exponential Euler u ( t ) = F ′ ( u ( t )) ∗ � � y δ − F ( u ( t )) t > 0 ˙ exponential Euler u n + h n ϕ ( − h n J ( u n )) F ′ ( u n ) ∗ ( y δ − F ( u n )) u n + 1 = e z − 1 ϕ ( z ) = z properties solves linear problems exactly 1 “explicit” scheme 2 again: approximate Jacobian by J ( u ) = F ′ ( u ) ∗ F ′ ( u ) 3 equivalent to Newton method with regularized linear system 4 Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Exponential Euler u ( t ) = F ′ ( u ( t )) ∗ � � y δ − F ( u ( t )) t > 0 ˙ exponential Euler u n + h n ϕ ( − h n J ( u n )) F ′ ( u n ) ∗ ( y δ − F ( u n )) u n + 1 = e z − 1 ϕ ( z ) = z properties solves linear problems exactly 1 “explicit” scheme 2 again: approximate Jacobian by J ( u ) = F ′ ( u ) ∗ F ′ ( u ) 3 equivalent to Newton method with regularized linear system 4 Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Exponential Euler regularization properties: Theorem Under suitable assumptions: exponential Euler iterates u n converge to x + 1 convergence rates are of optimal order 2 Proofs: see next talk by Marlis Hochbruck Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Recapitulation linearization ✲ nonlinear problem linear problems F ′ ( u n )∆ u n = y δ − F ( u n ) F ( u ) = y δ linear Showalter regularization method ❄ nonlinear ❄ time integration nonlinear ode regularization ✲ method Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Implementation notation: h = h n , J = J ( u n ) = F ′ ( u n ) ∗ F ′ ( u n ) h . psd . computation of ϕ ( − hJ ) v by Krylov subspace methods JV m = V m T m + T m + 1 , m v m + 1 e T m T m tridiagonal, V m orthogonal ϕ ( − hJ ) v ≈ V m ϕ ( − hT m ) e 1 Druskin, Khnizhnerman (1995) Hochbruck, Lubich (1997) Michael H¨ onig Regularization by the exponential Euler method

Problem setting Asymptotic regularization Numerical examples Implementation a posteriori error estimates: (van den Eshof & Hochbruck 2006) approximation of the relative error in step m θ m = � w m − w m − 1 � w m := ϕ ( − hT m ) e 1 , � w m � approximation of the error ǫ m θ m � w m � ǫ m � 1 − θ m accuracy ǫ m = O ( δ ) sufficient Michael H¨ onig Regularization by the exponential Euler method