We interpret steady linear statistical inverse problems as artificial dynamic systems with white noise and introduce a stochastic differential equation (SDE) system where the inverse of the ending time $T$ naturally plays the role of the squared noise level. The time-continuous framework then allows us to apply classical methods from data assimilation, namely the Kalman-Bucy filter and 3DVAR, and to analyze their behaviour as a regularization method for the original problem. Such treatment offers some connections to the famous asymptotical regularization method, which has not yet been analyzed in the context of random noise. We derive error bounds for both methods in terms of the mean-squared error under standard assumptions and discuss commonalities and differences between both approaches. If an additional tuning parameter $α$ for the initial covariance is chosen appropriately in terms of the ending time $T$, one of the proposed methods gains order optimality. Our results extend theoretical findings in the discrete setting given in the recent paper Iglesias et al. (2017). Numerical examples confirm our theoretical results.
In this paper, we study a fractional-order variant of the asymptotical regularization method, called {\it Fractional Asymptotical Regularization (FAR)}, for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range $(1,2)$ yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical rea…
In this paper, we consider the asymptotical regularization with convex constraints for nonlinear ill-posed problems. The method allows to use non-smooth penalty terms, including the L1-like and the total variation-like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy. Under certain conditions we give convergence properties of the methods. Moreover, we propose Runge-Kutta type methods to discrete the initial value problems to construct new type iterative regularization methods.
We introduce Stochastic Asymptotical Regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in science and engineering. We prove the regularizing properties of SAR with regard to mean-square convergence. We also show that SAR is an optimal-order regularization method for linear ill-posed problems provided that the terminating time of SAR is chosen according to the smoothness of the solution. This result is proven for both a priori and a posteriori sto…
We propose a new space-variant regularization term for variational image restoration based on the assumption that the gradient magnitudes of the target image distribute locally according to a half-Generalized Gaussian distribution. This leads to a highly flexible regularizer characterized by two per-pixel free parameters, which are automatically estimated from the observed image. The proposed regularizer is coupled with either the $L_2$ or the $L_1$ fidelity terms, in order to effectively deal with additive white Gaussian noise or impulsive noises such as, e.g, additive white Laplace and salt …
Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Department of Mathematics University of Crete（克里特大学数学系）；Institute for Applied and Computational Mathematics Foundation for Research and Technology-Hellas（希腊研究与技术基金会应用计算数学研究所）；Department of Mathematics University of Sussex（萨塞克斯大学数学系）
In this work, we propose a new criterion for choosing the regularization parameter in Tikhonov regularization when the noise is white Gaussian. The criterion minimizes a lower bound of the predictive risk, when both data norm and noise variance are known, and the parameter choice involves minimizing a function whose solution depends only on the signal-to-noise ratio. Moreover, when neither noise variance nor data norm is given, we propose an iterative algorithm which alternates between a minimization step of finding the regularization parameter and an estimation step of estimating signal-to-no…
We propose a novel automatic parameter selection strategy for variational imaging problems under Poisson noise corruption. The selection of a suitable regularization parameter, whose value is crucial in order to achieve high quality reconstructions, is known to be a particularly hard task in low photon-count regimes. In this work, we extend the so-called residual whiteness principle originally designed for additive white noise to Poisson data. The proposed strategy relies on the study of the whiteness property of a standardized Poisson noise process. After deriving the theoretical properties t…
In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a linear vanishing damping term, which can be viewed not only as an extension of the asymptotical regularization, but also as a continuous analog of the Nesterov's acceleration scheme. New iterative regularization methods are derived from this continuous model in combination with damped symplectic numerical schemes. The regularization property as well as convergenc…
Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität (WWU) Münster（应用与数值数学研究所，明斯特西法利西亚·威廉大学）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）；Cantab Capital institute, Faculty of Mathematics, University of Cambridge（剑桥大学数学学院坎塔资本研究所）
We consider standard tracking-type, distributed elliptic optimal control problems with $L^2$ regularization, and their finite element discretization. We are investigating the $L^2$ error between the finite element approximation $u_{\varrho h}$ of the state $u_\varrho$ and the desired state (target) $\bar{u}$ in terms of the regularization parameter $\varrho$ and the mesh size $h$ that leads to the optimal choice $\varrho = h^4$. It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble-Pasciak CG methods that allow us …
Upon the recent development of the quasi-reversibility method for terminal value parabolic problems in \cite{Nguyen2019}, it is imperative to investigate the convergence analysis of this regularization method in the stochastic setting. In this paper, we positively unravel this open question by focusing on a coupled system of Dirichlet reaction-diffusion equations with additive white Gaussian noise on the terminal data. In this regard, the approximate problem is designed by adding the so-called perturbing operator to the original problem and by exploiting the Fourier reconstructed terminal data…
