Qinian Jin

Qinian Jin, Ulrich Tautenhahn

We consider the computation of stable approximations to the exact solution $x^†$ of nonlinear ill-posed inverse problems $F(x)=y$ with nonlinear operators $F:X\to Y$ between two Hilbert spaces $X$ and $Y$ by the Newton type methods $$ x_{k+1}^δ=x_0-g_{α_k} (F'(x_k^δ)^*F'(x_k^δ)) F'(x_k^δ)^* (F(x_k^δ)-y^δ-F'(x_k^δ)(x_k^δ-x_0)) $$ in the case that only available data is a noise $y^δ$ of $y$ satisfying $\|y^δ-y\|\le δ$ with a given small noise level $δ>0$. We terminate the iteration by the discrepancy principle in which the stopping index $k_δ$ is determined as the first integer such that $$ \|F(x_{k_δ}^δ)-y^δ\|\le τδ<\|F(x_k^δ)-y^δ\|, \qquad 0\le k<k_δ$$ with a given number $τ>1$. Under certain conditions on $\{α_k\}$, $\{g_α\}$ and $F$, we prove that $x_{k_δ}^δ$ converges to $x^†$ as $δ\to 0$ and establish various order optimal convergence rate results. It is remarkable that we even can show the order optimality under merely the Lipschitz condition on the Fréchet derivative $F'$ of $F$ if $x_0-x^†$ is smooth enough.

Qinian Jin

In recent years Landweber(-Kaczmarz) method has been proposed for solving nonlinear ill-posed inverse problems in Banach spaces using general convex penalty functions. The implementation of this method involves solving a (nonsmooth) convex minimization problem at each iteration step and the existing theory requires its exact resolution which in general is impossible in practical applications. In this paper we propose a version of Landweber-Kaczmarz method in Banach spaces in which the minimization problem involved in each iteration step is solved inexactly. Based on the $\varepsilon$-subdifferential calculus we give a convergence analysis of our method. Furthermore, using Nesterov's strategy, we propose a possible accelerated version of Landweber-Kaczmarz method. Numerical results on computed tomography and parameter identification in partial differential equations are provided to support our theoretical results and to demonstrate our accelerated method.

Yuling Jiao, Qinian Jin, Xiliang Lu et al.

In this paper we propose an iterative method using alternating direction method of multipliers (ADMM) strategy to solve linear inverse problems in Hilbert spaces with general convex penalty term. When the data is given exactly, we give a convergence analysis of our ADMM algorithm without assuming the existence of Lagrange multiplier. In case the data contains noise, we show that our method is a regularization method as long as it is terminated by a suitable stopping rule. Various numerical simulations are performed to test the efficiency of the method.

Qinian Jin

We generalize the heuristic parameter choice rule of Hanke-Raus for quadratic regularization to general variational regularization for solving linear as well as nonlinear ill-posed inverse problems in Banach spaces. Under source conditions formulated as variational inequalities, we obtain a posteriori error estimates in term of Bregman distance. By imposing certain conditions on the random noise, we establish four convergence results; one relies on the source conditions and the other three do not depend on any source conditions. Numerical results are presented to illustrate the performance.

Qinian Jin

In this paper we consider the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed inverse problems. Under merely Lipschitz condition, we prove that this method together with an a posteriori stopping rule defines an order optimal regularization method if the solution is regular in some suitable sense.

Qinian Jin, Ulrich Tautenhahn

We consider a class of inexact Newton regularization methods for solving nonlinear inverse problems in Hilbert scales. Under certain conditions we obtain the order optimal convergence rate result.

Qinian Jin, Ulrich Tautenhahn

For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting operator monotonicity of certain functions and interpolation techniques in variable Hilbert scales, we study these methods under general smoothness conditions. Order optimal error bounds are given in case the regularization parameter is chosen either {\it a priori} or {\it a posteriori} by the discrepancy principle. For realizing the discrepancy principle, some fast algorithm is proposed which is based on Newton's method applied to some properly transformed equations.

Qinian Jin, Peter Mathe

The authors study statistical linear inverse problems in Hilbert spaces. Approximate solutions are sought within a class of linear one-parameter regularization schemes, and the parameter choice is crucial to control the root mean squared error. Here a variant of the Raus{Gfrerer rule is analyzed, and it is shown that this parameter choice gives rise to error bounds in terms of oracle inequalities, which in turn provide order optimal error bounds (up to logarithmic factors). These bounds can only be established for solutions which obey a certain self-similarity structure. The proof of the main result relies on some auxiliary error analysis for linear inverse problems under general noise assumptions, and this may be interesting in its own.

Qinian Jin

Inexact Newton regularization methods have been proposed by Hanke and Rieder for solving nonlinear ill-posed inverse problems. Every such a method consists of two components: an outer Newton iteration and an inner scheme providing increments by regularizing local linearized equations. The method is terminated by a discrepancy principle. In this paper we consider the inexact Newton regularization methods with the inner scheme defined by Landweber iteration, the implicit iteration, the asymptotic regularization and Tikhonov regularization. Under certain conditions we obtain the order optimal convergence rate result which improves the suboptimal one of Rieder. We in fact obtain a more general order optimality result by considering these inexact Newton methods in Hilbert scales.

Min Zhong, Wei Wang, Qinian Jin

In this paper, we propose and analyze a two-point gradient method for solving inverse problems in Banach spaces which is based on the Landweber iteration and an extrapolation strategy. The method allows to use non-smooth penalty terms, including the L^1 and the total variation-like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy in practical applications. The design of the method involves the choices of the step sizes and the combination parameters which are carefully discussed. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

Qinian Jin

We consider the inexact Newton methods $$ x_{n+1}^\d=x_n^\d-g_{\a_n}(F'(x_n^\d)^* F'(x_n^\d)) F'(x_n^\d)^* (F(x_n^\d)-y^\d) $$ for solving nonlinear ill-posed inverse problems $F(x)=y$ using the only available noise data $y^\d$ satisfying $\|y^\d-y\|\le \d$ with a given small noise level $\d>0$. We terminate the iteration by the discrepancy principle $$ \|F(x_{n_\d}^\d)-y^\d\|\le τ\d<\|F(x_n^\d)-y^\d\|, \qquad 0\le n<n_\d $$ with a given number $τ>1$. Under certain conditions on $\{\a_n\}$ and $F$, we prove for a large class of spectral filter functions $\{g_\a\}$ the convergence of $x_{n_\d}^\d$ to a true solution as $\d\rightarrow 0$. Moreover, we derive the order optimal rates of convergence when certain Hölder source conditions hold. Numerical examples are given to test the theoretical results.

Ruixue Gu, Qinian Jin

In this paper we develop a stochastic heavy ball method for solving ill-posed inverse problems. The method updates the iterate using only a randomly selected equation at each iteration step while incorporating a momentum term into the process. To facilitate fast convergence, we propose an adaptive strategy for selecting the step size and the momentum coefficient. Inspired by the spirit of the discrepancy principle, we introduce an {\it a posteriori} stopping rule for our adaptive stochastic heavy ball method. This rule avoids the need to compute residuals of all equations in the system at every iteration or at fixed frequency intervals, thereby enhancing computational efficiency and practicality. Additionally, convex penalty functions are employed to capture the specific features of the desired solutions. Under suitable conditions, we establish almost sure convergence as well as convergence in expectation. Extensive numerical experiments are conducted to evaluate the performance of the proposed method, demonstrating its efficiency and promising potential for solving large-scale ill-posed problems.