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.

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, 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, 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.