A novel iterative integration regularization method for ill-posed inverse problems

Abstract

This paper proposes a new iterative integration regularization method for robust solution of ill-posed inverse problems. The proposed method is motivated from the fact that inversion of a positive definite matrix can be expressed in an integral form. Then, the development of the proposed method is mainly twofold. Firstly, two ways—including the linear iteration and the exponential ($2^j$) iteration—are invoked to compute the integral, of which the exponential iteration is often preferred due to its fast convergence. Secondly, after stability analysis, the proposed method is shown able to filter out the undesired effect of relatively small singular values, while preserving the desired terms of relatively large singular values, i.e., the proposed method has the guaranteed regularization effect. Numerical examples on three typical ill-posed problems are conducted with detailed comparison to some usual direct and iterative regularization methods. Final results have highlighted the proposed method: (a) due to the iterative nature, the proposed method often turns out to be more efficient than the conventional direct regularization methods including the Tikhonov regularization and the truncated singular value decomposition (TSVD), (b) the proposed method converges much faster than the Landweber method and (c) the regularization effect is guaranteed in the proposed method, while may not be in the conjugate gradient method for least squares problem (CGLS).

Appendix

To get the matrix ${\mathbf {A}}$ for force reconstruction in structural systems, the Newmark method for solution of the dynamic equation is called for $k=1,2,\ldots ,N_\mathrm{{T}}$,

$$\begin{aligned} \left\{ \begin{array}{l} {\mathbf {M}} \ddot{{\varvec{u}}}(t_{k+1}) + {\mathbf {C}} \dot{{\varvec{u}}} (t_{k+1})+{\mathbf {K}} {{\varvec{u}}}(t_{k+1})={\mathbf {G}} {\varvec{f}}(t_{k+1}), \\ \dot{{\varvec{u}}} (t_{k+1})=\dot{{\varvec{u}}} (t_{k})+[\gamma \ddot{{\varvec{u}}} (t_{k+1})+(1-\gamma )\dot{{\varvec{u}}} (t_{k})] \Delta t, \\ {{\varvec{u}}} (t_{k+1})={{\varvec{u}}} (t_{k})+\dot{{\varvec{u}}} (t_{k}) \Delta t +[\beta \ddot{{\varvec{u}}} (t_{k+1})+(1/2-\beta ) \ddot{{\varvec{u}}} (t_{k})] \Delta t^2, \end{array} \right. \end{aligned}$$

(71)

where $\gamma =1/2,\beta =1/4$ are fixed in this paper, $\mathbf {M,C,K}$ are the mass, damping and stiffness matrices, ${\varvec{u}}(t)$ denotes the displacements, ${\varvec{f}}(t)$ collects all independent forces and ${\mathbf {G}}$ describes the spatial distributions of the forces. Simple manipulations on Eq. (71) can yield the following state equation

$$\begin{aligned} \left( \begin{array}{l} {\varvec{u}}(t_{k+1}) \\ \dot{{\varvec{u}}} (t_{k+1}) \end{array}\right) ={\mathbf {B}} \left( \begin{array}{l} {\varvec{u}}(t_{k}) \\ \dot{{\varvec{u}}} (t_{k}) \end{array}\right) +{\mathbf {Z}}_1 {\mathbf {G}} {\varvec{f}}(t_{k})+{\mathbf {Z}}_2 {\mathbf {G}} {\varvec{f}}(t_{k+1}), \end{aligned}$$

(72)

with

$$\begin{aligned} \begin{aligned} {\mathbf {B}}&=\left( \begin{array}{cc} \gamma {\mathbf {K}} &{} \gamma {\mathbf {C}}+{\mathbf {M}} /\Delta t\\ 2\beta {\mathbf {K}} +2{\mathbf {M}}/\Delta t^2 &{} 2 \beta {\mathbf {C}} \\ \end{array} \right) ^{-1}\\&\quad \left( \begin{array}{cc} -(1-\gamma ) {\mathbf {K}} &{} -(1-\gamma ) {\mathbf {C}}+{\mathbf {M}}/\Delta t \\ -(1-2\beta ) {\mathbf {K}} +2{\mathbf {M}}/\Delta t^2 &{} -(1-2 \beta ) {\mathbf {C}}+ 2{\mathbf {M}}/\Delta t\\ \end{array} \right) \\ {\mathbf {Z}}_1&=\left( \begin{array}{cc} \gamma {\mathbf {K}} &{} \gamma {\mathbf {C}}+{\mathbf {M}}/\Delta t \\ 2\beta {\mathbf {K}} +2{\mathbf {M}}/\Delta t^2 &{} 2 \beta {\mathbf {C}} \\ \end{array} \right) ^{-1}\left( \begin{array}{c} (1-\gamma ){\mathbf {I}} \\ (1-2\beta ){\mathbf {I}} \\ \end{array} \right) \\ {\mathbf {Z}}_2&=\left( \begin{array}{cc} \gamma {\mathbf {K}} &{} \gamma {\mathbf {C}}+{\mathbf {M}}/\Delta t \\ 2\beta {\mathbf {K}} +2{\mathbf {M}}/\Delta t^2 &{} 2 \beta {\mathbf {C}} \\ \end{array} \right) ^{-1}\left( \begin{array}{c} \gamma {\mathbf {I}} \\ 2\beta {\mathbf {I}} \\ \end{array} \right) \end{aligned} \end{aligned}$$

(73)

where ${\mathbf {I}}$ denotes the identity matrix, $\Delta t=t_{k+1}-t_k$. The measured data ${\varvec{d}}(t_k),\forall k$ are often of the form

$$\begin{aligned} {\varvec{d}}(t_k)={\mathbf {O}} \left( \begin{array}{c} {\varvec{u}}(t_k) \\ \dot{{\varvec{u}}}(t_k) \\ \end{array} \right) +{\mathbf {D}} {\mathbf {G}}{\varvec{f}}(t_k), \end{aligned}$$

(74)

where $\mathbf {O,D}$ are observation matrices. Setting the initial states ${\varvec{u}}(t_1)=\dot{{\varvec{u}}}(t_1)={\varvec{0}}$ and combining (72) with (74), it is acquired that

$$\begin{aligned} \begin{aligned} {\varvec{b}}&={\mathbf {A}}{\varvec{x}} \\ {\varvec{b}}&=[{\varvec{d}}(t_1);{\varvec{d}}(t_2);...;{\varvec{d}}(t_{N_T})],{\varvec{x}} =[{\varvec{f}}(t_1);{\varvec{f}}(t_2);...;{\varvec{f}}(t_{N_T})] \\ {\mathbf {A}}&=\left( \begin{array}{ccccc} \mathbf {DG} &{} &{} &{} &{} \\ \mathbf {OZ_1G} &{} \mathbf {DG+OZ_2G} &{} &{} &{} \\ \\ \vdots &{} \vdots &{} &{} \ddots &{} \\ \mathbf {OB}^{N_T-3}\mathbf {Z_1G}&{} \mathbf {OB}^{N_T-4}\mathbf {Z_1G+OB}^{N_T-3}\mathbf {Z_2G} &{} \cdots &{} \mathbf {DG+OZ_2G}&{} \\ \mathbf {OB}^{N_T-2}\mathbf {Z_1G}&{} \mathbf {OB}^{N_T-3}\mathbf {Z_1G+OB}^{N_T-2}\mathbf {Z_2G} &{} \cdots &{} \mathbf {OZ_1G+OBZ_2G} &{} \mathbf {DG+OZ_2G} \\ \end{array} \right) . \end{aligned} \end{aligned}$$

(75)