A Factorization of Least-Squares Projection Schemes for Ill-Posed Problems

Shukai Du; Nailin Du

doi:10.1515/cmam-2019-0173

Published by De Gruyter April 15, 2020

A Factorization of Least-Squares Projection Schemes for Ill-Posed Problems

Shukai Du and Nailin Du

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2019-0173

Showing a limited preview of this publication:

Abstract

We give a factorization formula to least-squares projection schemes, from which new convergence conditions together with formulas estimating the rate of convergence can be derived. We prove that the convergence of the method (including the rate of convergence) can be completely determined by the principal angles between T † ⁢ T ⁢ ( X n ) and T * ⁢ T ⁢ ( X n ) , and the principal angles between X n ∩ ( 𝒩 ⁢ ( T ) ∩ X n ) ⊥ and ( 𝒩 ⁢ ( T ) + X n ) ∩ 𝒩 ⁢ ( T ) ⊥ . At the end, we consider several specific cases and examples to further illustrate our theorems.

Keywords: Least-Squares Projection Method; Ill-Posed Problems; Generalized Inverses

MSC 2010: 47A58; 47A52; 15A09

A Facts on Projection

In this appendix, we collect some facts on projection from [17, 9] and [24, Lemma 2.13].

Proposition A.1.

Let S be a projection on Hilbert space H; then

( I - S - S * ) 2 = I + ( S - S * ) * ⁢ ( S - S * ) , so I - S - S * is invertible,
P ℛ ⁢ ( S ) = - S ⁢ ( I - S - S * ) - 1 , P 𝒩 ⁢ ( S ) = ( I - S ) ⁢ ( I - S - S * ) - 1 , so P 𝒩 ⁢ ( S ) - P ℛ ⁢ ( S ) and P 𝒩 ⁢ ( S ) + P ℛ ⁢ ( S ) are invertible,
S = P ℛ ⁢ ( S ) ⁢ ( P ℛ ⁢ ( S ) - P 𝒩 ⁢ ( S ) ) - 1 = ( P ℛ ⁢ ( S ) - P 𝒩 ⁢ ( S ) ) - 1 ⁢ P ℛ ⁢ ( S * ) ,
S † = ( P ℛ ⁢ ( S ) - P 𝒩 ⁢ ( S ) ) ⁢ P ℛ ⁢ ( S ) = P ℛ ⁢ ( S * ) ⁢ ( P ℛ ⁢ ( S ) - P 𝒩 ⁢ ( S ) ) = P ℛ ⁢ ( S * ) ⁢ P ℛ ⁢ ( S ) .

Proposition A.2.

Let Q , S be projections on H with ∥ ( Q - S ) 2 ∥ < 1 ; then there exists U , V ∈ B ⁢ ( H ) such that

U ⁢ V = V ⁢ U = I , S = U ⁢ Q ⁢ U - 1 , Q = V ⁢ S ⁢ V - 1 , ℛ ⁢ ( Q ) = ℛ ⁢ ( Q ⁢ S ) , ℛ ⁢ ( S ) = ℛ ⁢ ( S ⁢ Q ) .

Specifically, when Q , S are orthogonal projections, V = U * .

Proposition A.3.

Let P , Q be orthogonal projections on a Hilbert space H; then

∥ P - Q ∥ = max ⁡ { ∥ ( I - Q ) ⁢ P ∥ , ∥ ( I - P ) ⁢ Q ∥ } ≤ 1 .

Furthermore, if ∥ P - Q ∥ < 1 , then ∥ P - Q ∥ = ∥ ( I - Q ) ⁢ P ∥ = ∥ ( I - P ) ⁢ Q ∥ .

Proposition A.4.

Let S be a projection on a Hilbert space H. Then

∥ P ℛ ⁢ ( S ) - P ℛ ⁢ ( S * ) ∥ = ∥ P 𝒩 ⁢ ( S ) ⁢ P ℛ ⁢ ( S ) ∥ = ∥ P ℛ ⁢ ( S ) ⁢ P 𝒩 ⁢ ( S ) ∥ < 1 .

If in addition, S ≠ 0 , then ∥ S ∥ = ( 1 - ∥ P R ⁢ ( S ) - P R ⁢ ( S * ) ∥ 2 ) - 1 2 . Consequently, if S ≠ 0 and S ≠ I , then ∥ S ∥ = ∥ I - S ∥ .

Proposition A.5.

Let H be a Hilbert space, and let { H n } be a sequence of closed subspace of H; then

{ P H n } ⁢ is strongly convergent ⇔ s - lim n → ∞ ⁡ H n = w - lim ~ n → ∞ ⁡ H n ,

and when { P H n } is strongly convergent, we have

s - lim n → ∞ P H n = P M , 𝑤ℎ𝑒𝑟𝑒 M : = s - lim n → ∞ H n .

References

[1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Robert E. Krieger, Huntington, 1980. Search in Google Scholar

[2] G. Bruckner and S. Pereverzev, Self-regularization of projection methods with a posteriori discretization level choice for severely ill-posed problems, Inverse Problems 19 (2003), no. 1, 147–156. 10.1088/0266-5611/19/1/308Search in Google Scholar

[3] D. Chu, L. Lin, R. C. E. Tan and Y. Wei, Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill-posed problems, Numer. Linear Algebra Appl. 18 (2011), no. 1, 87–103. 10.1002/nla.702Search in Google Scholar

[4] N. Du, Finite-dimensional approximation settings for infinite-dimensional Moore–Penrose inverses, SIAM J. Numer. Anal. 46 (2008), no. 3, 1454–1482. 10.1137/060661120Search in Google Scholar

[5] H. W. Engl, On the convergence of regularization methods for ill-posed linear operator equations, Improperly Posed Problems and Their Numerical Treatment (Oberwolfach 1982), Internat. Schriftenreihe Numer. Math. 63, Birkhäuser, Basel (1983), 81–95. 10.1007/978-3-0348-5460-3_6Search in Google Scholar

[6] H. W. Engl and C. W. Groetsch, Projection-regularization methods for linear operator equations of the first kind, Special Program on Inverse Problems, Proc. Centre Math. Anal. Austral. Nat. Univ. 17, Australian National University, Canberra (1988), 17–31. Search in Google Scholar

[7] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Search in Google Scholar

[8] H. W. Engl and A. Neubauer, An improved version of Marti’s method for solving ill-posed linear integral equations, Math. Comp. 45 (1985), no. 172, 405–416. 10.2307/2008133Search in Google Scholar

[9] A. Galántai, Projectors and Projection Methods, Adv. Math. (Dordrecht) 6, Kluwer Academic, Boston, 2004. 10.1007/978-1-4419-9180-5Search in Google Scholar

[10] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University, Baltimore, 2012. Search in Google Scholar

[11] C. W. Groetsch, Generalized Inverses of Linear Operators: Representation and Approximation, Marcel Dekker, New York, 1977. Search in Google Scholar

[12] C. W. Groetsch, On a regularization-Ritz method for Fredholm equations of the first kind, J. Integral Equations 4 (1982), no. 2, 173–182. Search in Google Scholar

[13] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Res. Notes in Math.105, Pitman, Boston, 1984. Search in Google Scholar

[14] C. W. Groetsch and M. Hanke, Regularization by projection for unbounded operators arising in inverse problems, Inverse Problems and Applications to Geophysics, Industry, Medicine and Technology (Ho Chi Minh City 1995), Publ. HoChiMinh City Math. Soc. 2, HoChiMinh City Mathematical Society, Ho Chi Minh City (1995), 61–70. Search in Google Scholar

[15] U. Hämarik, E. Avi and A. Ganina, On the solution of ill-posed problems by projection methods with a posteriori choice of the discretization level, Math. Model. Anal. 7 (2002), no. 2, 241–252. 10.3846/13926292.2002.9637196Search in Google Scholar

[16] B. Kaltenbacher, Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems, Inverse Problems 16 (2000), no. 5, 1523–1539. 10.1088/0266-5611/16/5/322Search in Google Scholar

[17] T. Kato, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995. 10.1007/978-3-642-66282-9Search in Google Scholar

[18] U. Khyamarik, Self-regularization in solving ill-posed problems by projection methods, Tartu Riikl. Ül. Toimetised (1988), no. 833, 91–96. Search in Google Scholar

[19] S. Kindermann, Projection methods for ill-posed problems revisited, Comput. Methods Appl. Math. 16 (2016), no. 2, 257–276. 10.1515/cmam-2015-0036Search in Google Scholar

[20] J. T. King and A. Neubauer, A variant of finite-dimensional Tikhonov regularization with a posteriori parameter choice, Computing 40 (1988), no. 2, 91–109. 10.1007/BF02247939Search in Google Scholar

[21] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Appl. Math. Sci. 120, Springer, New York, 1996. 10.1007/978-1-4612-5338-9Search in Google Scholar

[22] R. Kress, Linear Integral Equations, 2nd ed., Appl. Math. Sci. 82, Springer, New York, 1999. 10.1007/978-1-4612-0559-3Search in Google Scholar

[23] Z.-C. Li, H.-T. Huang and Y. Wei, Ill-conditioning of the truncated singular value decomposition, Tikhonov regularization and their applications to numerical partial differential equations, Numer. Linear Algebra Appl. 18 (2011), no. 2, 205–221. 10.1002/nla.766Search in Google Scholar

[24] D. Nailin, The basic principles for stable approximations to orthogonal generalized inverses of linear operators in Hilbert spaces, Numer. Funct. Anal. Optim. 26 (2005), no. 6, 675–708. 10.1080/01630560500377329Search in Google Scholar

[25] M. T. Nair, Linear Operator Equations. Approximation and Regularization, World Scientific, Hackensack, 2009. 10.1142/7055Search in Google Scholar

[26] R. Plato and G. Vainikko, On the regularization of projection methods for solving ill-posed problems, Numer. Math. 57 (1990), no. 1, 63–79. 10.1007/BF01386397Search in Google Scholar

[27] T. I. Seidman, Nonconvergence results for the application of least-squares estimation to ill-posed problems, J. Optim. Theory Appl. 30 (1980), no. 4, 535–547. 10.1007/BF01686719Search in Google Scholar

[28] R. D. Spies and K. G. Temperini, Arbitrary divergence speed of the least-squares method in infinite-dimensional inverse ill-posed problems, Inverse Problems 22 (2006), no. 2, 611–626. 10.1088/0266-5611/22/2/014Search in Google Scholar

[29] G. Wang, Y. Wei and S. Qiao, Generalized Inverses: Theory and Computations, 2nd ed., Dev. Math. 53, Springer, Singapore, 2018. 10.1007/978-981-13-0146-9Search in Google Scholar

[30] Y. Wei, P. Xie and L. Zhang, Tikhonov regularization and randomized GSVD, SIAM J. Matrix Anal. Appl. 37 (2016), no. 2, 649–675. 10.1137/15M1030200Search in Google Scholar

[31] Q. Xu, Y. Wei and Y. Gu, Sharp norm-estimations for Moore–Penrose inverses of stable perturbations of Hilbert C * -module operators, SIAM J. Numer. Anal. 47 (2010), no. 6, 4735–4758. 10.1137/090755576Search in Google Scholar

[32] K. Yosida, Functional Analysis, 6th ed., Grundlehren Math. Wiss. 123, Springer, Berlin, 1980. Search in Google Scholar

Received: 2019-11-12

Revised: 2020-03-18

Accepted: 2020-03-18

Published Online: 2020-04-15

Published in Print: 2020-10-01

A Factorization of Least-Squares Projection Schemes for Ill-Posed Problems

Abstract

A Facts on Projection

Proposition A.1.

Proposition A.2.

Proposition A.3.

Proposition A.4.

Proposition A.5.

References

Journal and Issue

Articles in the same Issue