Inverse Problems In Hilbert Spaces Research Articles

A new second-order dynamical method for solving linear inverse problems in Hilbert spaces

A new second-order dynamic method (SODM) is proposed for solving ill-posed linear inverse problems in Hilbert spaces. The SODM can be viewed as a combination of Tikhonov regularization and second-order asymptotical regularization methods. As a result, a double-regularization-parameter strategy is adopted. The regularization properties of SODM are demonstrated under both a priori and a posteriori stopping rules. In the context of time discretization, we propose several iterative schemes with different choices of damping parameters. A truncated discrepancy principle is employed as the stop criterion. Finally, numerical experiments are performed to show the efficiency of the SODM: on the whole, compared with the classical Tikhonov method and the first-order dynamical-system method, the SODM leads to more-accurate approximate solutions while requiring fewer-iterative numbers.

On Convergence Rates of Proximal Alternating Direction Method of Multipliers

In this paper we consider from two different aspects the proximal alternating direction method of multipliers (ADMM) in Hilbert spaces. We first consider the application of the proximal ADMM to solve well-posed linearly constrained two-block separable convex minimization problems in Hilbert spaces and obtain new and improved non-ergodic convergence rate results, including linear and sublinear rates under certain regularity conditions. We next consider the proximal ADMM as a regularization method for solving linear ill-posed inverse problems in Hilbert spaces. When the data is corrupted by additive noise, we establish, under a benchmark source condition, a convergence rate result in terms of the noise level when the number of iterations is properly chosen.

Mann-type approximation scheme for solving a new class of split inverse problems in Hilbert spaces

ABSTRACT In this paper, we introduce and study a new class of split inverse problems, named split hierarchical monotone variational inclusion problem with multiple output sets in real Hilbert spaces. By using the inertial technique and self-adaptive step size strategy, we propose and analyze a new Mann-type iterative method for solving the problem. The convergence analysis of the proposed iterative method under some suitable conditions is studied. Also, we show that the sequence of iterates generated by this method converges strongly to a minimum-norm solution of the problem. As theoretical applications, we apply our results to approximate the solutions of other classes of split inverse problems. Finally, we present some numerical experiments to illustrate the practical potential and advantages of our proposed method.

Iterative Runge-Kutta-Type Methods with Convex Penalty for Inverse Problems in Hilbert Spaces

Iterative Runge-Kutta-Type Methods with Convex Penalty for Inverse Problems in Hilbert Spaces

Bayesian inverse problems with heterogeneous variance

AbstractWe consider inverse problems in Hilbert spaces under correlated Gaussian noise, and use a Bayesian approach to find their regularized solution. We focus on mildly ill‐posed inverse problems with fractional noise, using a novel wavelet‐based vaguelette–vaguelette approach. It allows us to apply sequence space methods without assuming that all operators are simultaneously diagonalizable. The results are proved for more general bases and covariance operators. Our primary aim is to study posterior contraction rate in such inverse problems over Sobolev classes and compare it to the derived minimax rate. Secondly, we study effect of plugging in a consistent estimator of variances in sequence space on the posterior contraction rate. This result is applied to the problem with error in forward operator. Thirdly, we show that empirical Bayes posterior distribution with a plugged‐in maximum marginal likelihood estimator of the prior scale contracts at the optimal rate, adaptively, in the minimax sense.

A Fast Data-Driven Iteratively Regularized Method with Convex Penalty for Solving Ill-Posed Problems

We propose a new iterative regularization method for solving inverse problems in Hilbert spaces. The iterative process of the proposed method combines classical iterative regularization format and Data-Driven approach. Data-Driven technique is based on the idea of deep learning to estimate the interior of a black box through a training set, so as to solve problems better and faster in some cases. In order to capture the special feature of solutions, convex functions are utilized to be penalty terms. Algorithmically, the two-point gradient acceleration strategy based on homotopy perturbation method is applied to the iterative scheme, which makes the method have satisfactory acceleration effect. We provide convergence analysis of the method under standard assumptions for iterative regularization methods. Finally, several numerical experiments are presented to show the effectiveness and acceleration effect of our method.

A Study of an Iteratively-Regularized Simplified Landweber Iteration for Nonlinear Inverse Problems in Hilbert Spaces

Lanweber-type methods are a well-known regularization methods to solve linear and nonlinear ill-posed problems. In this article, we consider a simplified form of Landweber method, say, an iteratively-regularized simplified Landweber iteration for solving nonlinear ill-posed problems. We study a detailed convergence analysis of the method under standard conditions on the nonlinearity and the rate of convergence under a Hölder-type source condition. Finally, numerical simulations are performed to validate the performance of the method.

Convergence analysis of inexact Newton–Landweber iteration under Hölder stability

In this paper, we focus on a class of inverse problems with Lipschitz continuous Fréchet derivatives both in Hilbert spaces and Banach spaces. The convergence and convergence rate of the inexact Newton–Landweber method (INLM) for such problems are presented under some assumptions. For the inverse problems in Hilbert spaces, we revisit the convergence result and the convergence rate of the INLM under Lipschitz condition and Hölder stability. Furthermore, the INLM for nonlinear inverse problems in Banach spaces is also considered. By using a Hölder stability corresponding to the Bregman distance, we derive the convergence property and convergence rate of the method.

A Fast Averaged Kaczmarz Iteration with Convex Penalty for Inverse Problems in Hilbert Spaces

A Fast Averaged Kaczmarz Iteration with Convex Penalty for Inverse Problems in Hilbert Spaces

Stochastic gradient descent for linear inverse problems in Hilbert spaces

We investigate stochastic gradient decent (SGD) for solving full infinite dimensional ill-posed problems in Hilbert spaces. We allow for batch-size versions of SGD where the randomly chosen batches incur noise fluctuations. Based on the corresponding bias-variance decomposition we provide bounds for the root mean squared error. These bounds take into account the discretization levels, the decay of the step-size, which is more flexible than in existing results, and the underlying smoothness in terms of general source conditions. This allows to apply SGD to severely ill-posed problems. The obtained error bounds exhibit three stages of the performance of SGD. In particular, the pre-asymptotic behavior can be well seen. Some numerical studies verify the theoretical predictions.

Levenberg–Marquardt method with general convex penalty for nonlinear inverse problems

We consider a Levenberg–Marquardt method for solving nonlinear inverse problems in Hilbert spaces. The proposed method uses general convex penalty terms to reconstruct nonsmooth solutions of inverse problems. Instead of an a priori choice, the regularization parameter in each iteration is chosen by solving an equation which depends on the residual. We utilize the discrepancy principle to terminate the iteration and give the convergence results. In addition, numerical simulations are presented to test the performance of the method.

Convex and Differentiable Formulation for Inverse Problems in Hilbert Spaces with Nonlinear Clipping Effects

Convex and Differentiable Formulation for Inverse Problems in Hilbert Spaces with Nonlinear Clipping Effects

Эффективные алгоритмы вычисления глобальной и локальной апостериорной оценки точности решений линейных некорректных задач

We consider extremal problems introduced and investigated earlier by the author for calculating global and local a posteriori error estimates of approximate solutions to ill-posed inverse problems. For linear inverse problems in Hilbert spaces, they consist in maximization of quadratic functionals with two quadratic constraints. The article shows how under certain conditions these problems can be reduced to a problem of maximization of a special (written analytically) differentiable functional with one constraint. New algorithms for calculating global and local a posteriori error estimates based on the solution of these problems are proposed. Their effectiveness is illustrated by numerical experiments on a posteriori error estimation of solutions to the model two-dimensional inverse problem of potential continuation. Experiments show that the proposed algorithms give a posteriori error estimates close to the true error values. Proposed algorithms for global a posteriori error estimation turn out to be more rapid (3 to 5 times) than the previously known algorithms.

On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems

In this work, we analyze the regularizing property of the stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method randomly chooses one equation from the nonlinear system to obtain an unbiased stochastic estimate of the gradient, and then performs a descent step with the estimated gradient. It is a randomized version of the classical Landweber method for nonlinear inverse problems, and it is highly scalable to the problem size and holds significant potentials for solving large-scale inverse problems. Under the canonical tangential cone condition, we prove the regularizing property for a priori stopping rules, and then establish the convergence rates under suitable sourcewise condition and range invariance condition.

Irregularization accelerates iterative regularization

When iterative methods are employed as regularizers of inverse problems, a main issue is the trade-off between smoothing effects and computation time, related to the convergence rate of iterations. Very often, faster methods obtain less accuracy. A new acceleration strategy is presented here, inspired by a choice of penalty terms formerly proposed in 2012 by Huckle and Sedlacek in the context of Tikhonov regularization by direct solvers. More precisely, we consider a special penalty term endowed with high regularization capabilities, and we apply it by using the opposite sign, that is negative, to its regularization parameter. This unprecedented choice leads to an “irregularization” phenomenon, which speeds up the underlying basic iterative method. The speeding up effects of the negative valued penalty term can be controlled through a sequence of decreasing coefficients as the iterations proceed in order to prevent noise amplification, tuning the weight of the correction term which generates the anti-regularization behavior. Filter factor expansion and convergence are analyzed in the simplified context of linear inverse problems in Hilbert spaces, by considering modified Landweber iterations as a first case study.

Preconditioned alternating direction method of multipliers for inverse problems with constraints

We propose a preconditioned alternating direction method of multipliers (ADMM) to solve linear inverse problems in Hilbert spaces with constraints, where the feature of the sought solution under a linear transformation is captured by a possibly non-smooth convex function. During each iteration step, our method avoids solving large linear systems by choosing a suitable preconditioning operator. In case the data is given exactly, we prove the convergence of our preconditioned ADMM without assuming the existence of a Lagrange multiplier. In case the data is corrupted by noise, we propose a stopping rule using information on noise level and show that our preconditioned ADMM is a regularization method; we also propose a heuristic rule when the information on noise level is unavailable or unreliable and give its detailed analysis. Numerical examples are presented to test the performance of the proposed method.

Alternating Direction Method of Multipliers for Linear Inverse Problems

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.

Conjugate Gradient Method in Hilbert and Banach Spaces to Enhance the Spatial Resolution of Radiometer Data

In this paper, a new computer-time effective iterative method is proposed to enhance the spatial resolution of microwave radiometer data in both Hilbert and Banach spaces. The reconstruction method is based on the conjugate gradient (CG) method. CG is a well-known method in electromagnetics, where it is commonly used to address inverse problems in Hilbert spaces. However, the method has never been applied to inverse problems related to spatial resolution enhancement of microwave remotely sensed measurements. In this paper, CG is adapted to the microwave radiometer case and exploited to reconstruct the brightness field at enhanced spatial resolution. Then, CG is, for the first time, extended to Banach spaces by means of duality maps and then applied to enhance the radiometer spatial resolution. Experiments undertaken on both simulated and actual radiometer data confirm the soundness of the proposed approach in Banach spaces and demonstrate that CG is able to provide a reconstruction accuracy similar to the conventional Landweber method, but with a significantly reduced processing time.

Tikhonov and Landweber convergence rates: characterization by interpolation spaces

Algebraic convergences rates of (iterated) Tikhonov regularization for linear inverse problems in Hilbert spaces are characterized by the membership of the exact solution to intermediate spaces produced by the K-method of real interpolation. Similar results are obtained for the Landweber iteration.

Oracle-type posterior contraction rates in Bayesian inverse problems

We discuss Bayesian inverse problems in Hilbert spaces. The focus is on a fast concentration of the posterior probability around the unknown true solution as expressed in the concept of posterior contraction rates. This concentration is dominated by a parameter which controls the variance of the prior distribution. Previous results determine posterior contraction rates based on known solution smoothness. Here we show that an oracle-type parameter choice is possible. This is done by relating the posterior contraction rate to the root mean squared estimation error. In addition we show that the tail probability, which usually is bounded by using the Chebyshev inequality, has exponential decay, at least for a priori parameter choices. These results implement the exponential concentration of Gaussian measures in Hilbert spaces.