Semi-discrete equations in Banach spaces: The approximate inverse approach

Transcription

1 Semi-discrete equations in Banach spaces: The approximate inverse approach Andreas Rieder Thomas Schuster Saarbrücken Frank Schöpfer Oldenburg FAKULTÄT FÜR MATHEMATIK INSTITUT FÜR ANGEWANDTE UND NUMERISCHE MATHEMATIK SFB 1173 KIT University of the State of Baden-Württemberg and National Research Center of the Helmholtz Association

2 AI in Banach spaces: pros and cons Pros: No geometric assumptions on Banach spaces. Discretization effects included into the analysis. Cons: Some components may not be easily available in concrete applications. Abstract concept; not easily presented. 2 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

3 Setting the stage X, Y real Banach spaces A L(X,Y), R(A) R(A) inverse problem: Af = g where g R(A) Remarks: The inverse problem is ill-posed in the sense of Nashed. There is no bounded inner inverse for A: B L(Y,X): ABA = A. Inner inverses are equation solvers: f = Bg solves the inverse problem. In the Hilbert space setting A + (Moore-Penrose inverse) is an inner inverse which is bounded iff R(A) = R(A). 3 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

4 Modelling including Measurement Process Ψ n : Y R n observation operator (linear, bounded) Example: ψ i Y sensitivity profile of i-th detector Ψ n g := ( ψ 1,g Y Y,..., ψ n,g Y Y ) Semi-discrete problem (SDP): A n f n = g n (No discretization!) where A n = Ψ n A L(X,R n ), g n = Ψ n g, f n X SDP is highly under-determined! If X is uniformly convex then the minimum norm solution of SDP is well defined and unique. 4 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

5 The key concept: computing moments Instead of trying to find a solution f n X of SDP we aim at computing moments: e d,i,f n X X for mollifiers e d,i X, i = 1,...,d. Then, we recover an approximate solution E d f n of SDP by d E d f n := e d,i,f n X X b d,i where {b d,i } d i=1 X. i=1 {e d,i } d i=1 X and {b d,i } d i=1 X have to be intertwined such that E d satisfies the mollifier property lim E dw w X = 0 for any w X. d 5 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

6 Mollifier property: an example X = (C([0,1]), ), X BV (0,1): f µ,f X X := Define E d f = d e d,i,f X X b d,i where e d,i,f X X = 2 d i=0 1 0 (i+1)/d (i 1)/d f(x)dµ(x). f(x)dx. 1 b d,i e d,i Then, i d lim f E df = 0 for any f C(0,1). d Moreover, f E d f C E d α f C α (0,1), 0 < α 1. 6 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

7 Reconstruction kernels: the basic idea To evaluate E d f n we need to calculate the moments e d,i,f n X X. Let the equations have solutions. Then, A nv n d,i = e d,i, i = 1,...,d, (1) and defining we obtain e d,i,f n X X = vd,i n,a nf n 2 = vd,i n,g n 2 d à n,d : R n X, à n,d α := vd,i n,α 2b d,i à n,d A n = E d. i=1 Ãn,d approximate inverse of A n, v n d,i reconstruction kernel However, (1) cannot be expected to have solutions. (LOUIS, MAASS 1990, LOUIS 1996) 7 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

8 Surrogate kernels from the underlying continuous setting Let A be injective. Given ε > 0 and e X. If X is reflexive, R(A ) is dense in X and we find v Y : A v e X ε. If X is arbitrary, R(A ) is weak- dense in X. Let f X denote the (unique) solution of Af = g. Then, we find v = v(f) Y : A v e,f X X ε f X. We call v a continuous kernel with inaccuracy ε (related to A, f, and e). IMPORTANT 8 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

9 Surrogate kernels from the underlying continuous setting Let A be injective. Given ε > 0 and e X. If X is reflexive, R(A ) is dense in X and we find v Y : A v e X ε. If X is arbitrary, R(A ) is weak- dense in X. Let f X denote the (unique) solution of Af = g. Then, we find v = v(f) Y : A v e,f X X ε f X. We call v a continuous kernel with inaccuracy ε (related to A, f, and e). 8 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

10 Continuous kernels: an example X = (C([0,1]), ), X BV (0,1) A: X X, Af(x) = x 0 f(t)dt A : X X, A µ(ξ) = µ(1)ξ Aµ(ξ). Let η X with 1 0 Then, v(x) = η(x) satisfies η(t)dt = 1 and η(1) = 0. Define e(x) := Aη(x). A v(ξ) = v(1)ξ Av(ξ) = η(1)ξ +Aη(ξ) = e(ξ). Let η γ = χ [z γ,z+γ[ /(2γ) for a fixed z ]0,1[ with γ > 0 suff. small. Then, e γ,f X X = v γ,af X X = = 1 Af(z +γ) Af(z γ) 2γ 0 Af(t)dη γ (t) γ 0 (Af) (z) = f(z). 9 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

11 Surrogate kernels from the underlying continuous setting (cont d) Having continuous kernels v d,i satisfying A v d,i e d,i,f X X ε d,i f X, i = 1,...,d, at our disposal we define the approximate inverse with surrogate kernels à n,d α := d Φ n v d,i,α 2 b d,i. i=1 Here, Φ n : Y R n is linear and continuous and needs to be defined such that Ãn,dA n f f X = 0 for any f X. lim n d 10 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

12 Interlude: overview of the involved operators X A Ψ n Φ n Y R n Y d b d,i e d,i = E d span{b d,1,...,b d,d } X i=0 A n à n,d = d i=0 b d,i Φ n v d,i E d Ãn,dA n 11 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

13 Convergence Theorem: If all building blocks are chosen adequately then Ãn,dA n f f X f E d f X +σ(d) ( ρ n max v d,i Y + max ε ) d,i 1 i d 1 i d where ρ n 0 as n. Four error components: 1. Discretization error in X by mollification operator, 2. Discretization error in Y by observation operator and 3. Error due to the surrogate kernels: ρ n = ρ n (Af,Ψ n,φ n ), 4. Accuracy of the continuous kernels (A v d,i e d,i ). d i=1 α i b d,i X σ(d) max 1 i d α i, α R d 12 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

14 Convergence (cont d) Corollary: Choosing d = d(n) = d(n,f) such that d(n) as n as well as σ ( d(n) ) ρ n max v d(n),i Y 0 1 i d(n) as n and we have the convergence σ ( d(n) ) max 1 i d(n) ε d(n),i 0 as n lim Ãn,d(n)A n f f X = 0. n 13 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

15 Regularization property The noise model: (Ψ δ ny) k = (Ψ n y) k +δ k, δ k δ, y Y, (2) where δ > 0 is the noise level. Theorem: Adopt all previous assumptions. Let d = d(n) = d(n,f) as n such that convergence holds in the noise-free setting. If n = n δ such that n δ when δ 0 as well as δσ ( d(n δ ) ) nδ max Φ n δ v d(nδ ),i 2 0 as δ 0 1 i d(n δ ) then { lim sup Ãnδ,d(n δ 0 δ )Ψ δ n δ Af f } X : Ψ δ n δ fulfills (2) = / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

16 Example: the integration operator X = (C([0,1]), ), A: X X, Af(x) = x 0 f(t)dt Let ψ n,0 = χ ]0,1] and ψ n,k = χ [k/n,1], k = 1,...,n. Then, ψ n,k X and (Ψ n g) k = ψ n,k,g X X = g( k n ). A n : X R n, (A n f) k = k n 0 f(t)dt, k = 0,...,n. d Ãn,dw = Φ n v d,i,w 2 b d,i, v d,i = e d,i, i=0 Φ n : Y R n+1 is given by b d,i i d e d,i (Φ n v) k = v,b n,k X X 15 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

17 Example: the integration operator (cont d) Ã n,d w = d Φ n v d,i,w 2 b d,i, v d,i = e d,i, b d,i e d,i i=0 If n = d then Φ n v d,i,w 2 is easily evaluated to be w 1 w 0 : k = 0, Ã n,n w( k n ) = Φ nv n,k,w 2 = n (w k+1 w k 1 )/2 : k = 1,...,n 1, w n w n 1 : k = n. Compare with the continuous setting v n,k,af X X = Af(k+1 n ) Af(k 1 2 n n ). i d 16 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

18 Example: the integration operator (cont d) We have that ( Ãn,dA n f f f E d f +σ(d) }{{} = 1 n 1 {}}{ ρ n Setting d(n) = n 1 γ for one 0 < γ < 1 yields max 1 i d v d,i X }{{} d lim f Ãn,d(n)A n f = 0 for any f C(0,1). n + max ε ) d,i 1 i d }{{} = 0 Consequence We can recover about n moments of f from n observations of Af. Remark In a comparable Hilbert space setting we can only recover about n 2/3 moments from n observations. 17 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

19 Example: the integration operator (cont d) (Ψ δ ng) k = g( k n )+δ k, δ k δ (3) Data error: δ n max 1 i d(n) Φ nv d(n),i 2 δ nd Setting d(n) = n 1 γ, 0 < γ < 1, and n δ δ γ 1 3/2 γ yields { lim sup Ãnδ,d(n δ 0 δ )Ψ δ n δ Af f } : Ψ δ n δ fulfills (3) = / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015

20 What you should recall from this talk Approximate Inverse requires no geometric assumptions on Banach spaces and discretization effects are included into the analysis. Th. Schuster, A. Rieder, F. Schöpfer The approximate inverse in action IV: semi-discrete equations in a Banach space setting Inverse Problems 28 (2012) 19 / 19 c Andreas Rieder Semi-discrete equations in Banach spaces AIP2015, Helsinki, May 2015