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

Shukai Du; Nailin Du

doi:10.1515/cmam-2019-0173

Article

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

Shukai Du and Nailin Du

Published/Copyright: April 15, 2020

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From the journal Computational Methods in Applied Mathematics Volume 20 Issue 4

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 .

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Received: 2019-11-12

Revised: 2020-03-18

Accepted: 2020-03-18

Published Online: 2020-04-15

Published in Print: 2020-10-01

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Articles in the same Issue

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

Keywords for this article

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