Introduction Proposals for rules Numerical results On solution of linear problems by the extrapolated Tikhonov method Uno Hämarik, Reimo Palm, Toomas Raus September 7, 2009 Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results The problem ◮ We consider an operator equation Au = f 0 , where A : X → F is a linear bounded operator between real Hilbert spaces, f 0 ∈ R ( A ) ⇒ ∃ solution u ∗ ∈ H . ◮ Instead of exact data f 0 , noisy data f are available. ◮ Knowledge of � f 0 − f � : ◮ Case 1: exact noise level δ : � f 0 − f � ≤ δ ◮ Case 2: approximate noise level δ : lim � f 0 − f � /δ ≤ C as δ → 0, with unknown constant C ◮ Case 3: no information about � f 0 − f � Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Motivation of extrapolation ◮ Tikhonov approximation: u α = ( α I + A ∗ A ) − 1 A ∗ f . Extrapolated approximations are linear combinations of u α with different α . Examples: v 2 ,α = 2 u α − u 2 α , v 3 ,α = 8 3 u α/ 2 − 2 u α + 1 3 u 2 α . ◮ Error estimate: if � f − f 0 � ≤ δ and u ∗ ∈ R (( A ∗ A ) p / 2 ) , then for proper α � u α − u ∗ � ≤ const δ p / ( p + 1 ) ( p ≤ 2 ) � v 2 ,α − u ∗ � ≤ const δ p / ( p + 1 ) ( p ≤ 4 ) � v 3 ,α − u ∗ � ≤ const δ p / ( p + 1 ) ( p ≤ 6 ) Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Approximate solutions ◮ Tikhonov method: v 1 ,α = u α = ( α I + A ∗ A ) − 1 A ∗ f ◮ Extrapolated Tikhonov approximations, computed on grid α i = q i , i = 0, 1, . . . ( q < 1, we used q = 0 . 9): v 2 ,α i = ( 1 − q ) − 1 ( u α i − qu α i − 1 ) . ◮ For choice of parameter α i we also compute v 3 ,α i = ( 1 − q ) − 2 � ( 1 + q ) − 1 u α i + 1 − qu α i + q 3 ( 1 + q ) − 1 u α i − 1 � , v 4 ,α i = ( 1 − q ) − 3 ( 1 + q ) − 1 � ( 1 + q + q 2 ) − 1 u α i + 1 − qu α i + q 3 u α i − 1 − q 6 ( 1 + q + q 2 ) − 1 u α i − 2 � Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Rules for choice of parameters in v k ,α i for exact noise level δ ◮ We compute v k ,α i and r k , i = Av k ,α i − f for α 0 = 1, α 1 = q , α 2 = q 2 , . . . until some condition is satisfied. ◮ Approach 1. ME-rule: choose k = k ME as the first k with d ME ( k ) := ( r k , 0 + r k + 1 , 0 , r k + 1 , 0 ) ≤ δ. 2 � r k + 1 , 0 � ◮ Approach 2. We choose α i in v k ,α i ( k ∈ { 1 , 2 } ) by rules: ◮ α D is α i with first i for which � r k , i � ≤ δ . ◮ α ME is α i with first i for which ( r k , i , r k + 1 , i ) / � r k + 1 , i � ≤ δ. ◮ α R2, τ is α i with first i for which √ α i � v k ,α i − v k + 1 ,α i � 2 ( 1 + α � A � − 2 ) τ d R2, τ ( α i ) := ( v k ,α i − v k + 1 ,α i , v k + 1 ,α i − v k + 2 ,α i ) 1 / 2 ≤ b k δ with b 1 = 0 . 3, b 2 = 0 . 2; 0 ≤ τ ≤ 1. Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Minimum strategy for choice of α in case of exact noise level δ ◮ α ME ≥ α opt := argmin {� u α − u ∗ � , α ≥ 0 } , computations suggested to use α MEe = min ( 0 . 5 α ME , 0 . 6 α 1 . 07 ME ) , which is good in case � f − f 0 � = δ ◮ α R2e = 0 . 5 α R2,1/2 is good in case � f − f 0 � < δ ◮ In both cases α MR2e = min ( α MEe , α R2e ) chooses the best of α MEe and α R2e . Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Rule for choice of α in case of approximate noise level δ Rule DM 1) find α as the first α i for which √ α i � v k ,α i − v k + 1 ,α i � ≤ c 1 δ , c 1 = const (we used c 1 = 0 . 001 . . . 0 . 02); 2) find α i = argmin d R2,1 ( α i ) α c 2 − 0 . 5 on [ α, 1 ] . We used i c 2 = 0 . 03 . . . 0 . 14. If the first condition is not fulfilled up to α i = 10 − 30 , then α = 10 − 30 . Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Convergence and convergence rate ◮ Approach 1. ME-rule gives k = k ME with property � v k ,α 0 − u ∗ � < � v k − 1 ,α 0 − u ∗ � for k = 1, 2, . . . , k ME , � v k ME ,α 0 − u ∗ � → 0 ( δ → 0). If u ∗ ∈ R (( A ∗ A ) p / 2 ) , then � v k ME ,α 0 − u ∗ � ≤ const δ p / ( p + 1 ) for all p > 0. ◮ Approach 2. Convergence v k ,α → u ∗ ( δ → 0) is guaranteed for choice of α by rules ME and R2 (and also by rule DM, if lim � f 0 − f � /δ ≤ C as δ → 0). If u ∗ ∈ R (( A ∗ A ) p / 2 ) , the rules ME and R2 (and DM if c 1 ≥ 0 . 24) guarantee � v k ,α − u ∗ � ≤ const δ p / ( p + 1 ) ( p ≤ 2 k ). ◮ If the parameter choice rule does not use δ , no convergence v k ,α → u ∗ ( δ → 0) is guaranteed. Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Heuristic rules not using noise level δ The following rules are modifications of the quasioptimality criterion and the rule from [Brezinski, Rodriguez, Seatzu, 2008, Numer. Algor. 49:85–104]. ◮ Rules QC, R2C and BRSC : find α i as the minimizer of ψ ( α i ) (QC: ψ ( α i ) = � v k ,α i − v k + 1 ,α i � , R2C: ψ ( α i ) = d R2,1 ( α i ) , BRSC: ψ ( α i ) = � r k , i � 2 / ( α i � v k ,α i � ) ) on the interval [ α, 1 ] , where α is the largest α i , for which the value of ψ ( α i ) is C times (QC, R2C: C = 5; BRSC: C = 3) larger than its value at the minimum point. ◮ Rules DR21 and BRS1 : choose α i as the largest local minimizer of functions � r k , i � d R2,1 ( α i ) α 0 . 4 and i � r k , i � 2 / ( α � v k ,α i � ) α 0 . 7 , respectively. i Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Test problems ◮ Test problems: 10 problems from P. C. Hansen’s Regularization tools + additional problems hilbert , gauss , lotkin , moler , pascal , prolate from paper [Brezinski, Rodriguez, Seatzu, 2008, Numer. Algor. 49:85–104]. ◮ Besides solution u ∗ also smoother solution u ∗ , p = ( A ∗ A ) p / 2 u ∗ with f 0 = Au ∗ , p , p = 2 was used. ◮ The problems were normalized, so that norms of the operator and the right hand side were 1. ◮ For perturbed data we took f = f 0 + ∆ , � ∆ � = 0 . 5, 10 − 1 , . . . , 10 − 6 with 10 different perturbations ∆ generated by computer. Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Data in Tables 1–7 In the following tables we present averages (in Table 1 also maximums) of error ratios � u α − u ∗ � / e opt , where e opt = min {� u α − u ∗ � : α ≥ 0 } . ◮ In Tables 1–5 we use δ = d � f 0 − f � with d = 1, 1 . 3, 2 (in Table 1) or d = 0 . 01, 0 . 1, 0 . 5, 1, 2, 10, 100 (in Tables 2, 3; in Table 4 additionally d = 0 . 3, 30 and in Table 5 d = 0 . 03, 0 . 3, 4, 30); d ≥ 1 corresponds to overestimation of noise level. Rules of Tables 6, 7 do not use δ . ◮ Columns vR2e, vMEe, vMR2e, vQC, vR2C, vBRSC, vQ1, vBRS1 show averages and maximums of error ratios � v 2 ,α − u ∗ � / e opt for rules R2e, MEe, MR2e, QC, R2C, BRSC, Q1, BRS1. Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
Introduction Proposals for rules Numerical results Table 1, averages and maximums d p D MEe R2e MR2e vMEe vR2e vMR2e 1 0 1.24 1.18 1.38 1.18 1.17 1.45 1.29 1 2 2.68 1.20 1.17 1.13 0.69 0.79 0.77 Averages 1.3 0 1.78 1.66 1.44 1.43 1.60 1.41 1.39 1.3 2 3.66 3.26 1.25 1.25 1.41 0.77 0.77 2 0 2.15 1.96 1.56 1.56 1.87 1.50 1.49 2 2 4.72 4.43 1.53 1.53 2.06 0.78 0.78 1 0 5.82 5.18 16 4.96 5.39 25 25 Maximums 1 2 25 119 46 29 399 595 595 1.3 0 20 19 16 16 21 19 19 1.3 2 4e3 3e3 241 241 723 500 500 2 0 25 22 17 17 21 19 19 2 2 5e3 3e3 844 844 850 277 277 Uno Hämarik, Reimo Palm, Toomas Raus On solution of linear problems by the extrapolated Tikhonov method
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