Regularization method and Bayesian inverse method are two dominating ways for solving inverse problems generated from various fields, e.g., seismic exploration and medical imaging. The two methods are related with each other by the MAP estimates of posterior probability distributions. Considering this connection, we construct a prior probability distribution with several hyper-parameters and provide the relevant Bayes' formula, then we propose a corresponding adaptive augmented regularization model (AARM). According to the measured data, the proposed AARM can adjust its form to various regularization models at each discrete point of the estimated function, which makes the characterization of local smooth properties of the estimated function possible. By proposing a modified Bregman iterative algorithm, we construct an alternate iterative algorithm to solve the AARM efficiently. In the end, we provide some numerical examples which clearly indicate that the proposed AARM can generates a favorable result for some examples compared with several Tikhonov and Total-Variation regularization models.
Many problems in Science and Engineering give rise to linear integral equations of the first kind with a smooth kernel. Discretization of the integral operator yields a matrix, whose singular values cluster at the origin. We describe the approximation of such matrices by adaptive cross approximation, which avoids forming the entire matrix. The choice of the number of steps of adaptive cross approximation is discussed. The discretized right-hand side represents data that commonly are contaminated by measurement error. Solution of the linear system of equations so obtained is not meaningful beca…
We modify the total-variation-regularized image segmentation model proposed by Chan, Esedoglu and Nikolova [SIAM Journal on Applied Mathematics 66, 2006] by introducing local regularization that takes into account spatial image information. We propose some techniques for defining local regularization parameters, based on the cartoon-texture decomposition of the given image, on the mean and median filters, and on a thresholding technique, with the aim of preventing excessive regularization in piecewise-constant or smooth regions and preserving spatial features in nonsmooth regions. We solve the…
We consider machine-learning of time-dependent quantities of interest derived from solution trajectories of parabolic partial differential equations. For large-scale or long-time integration scenarios, where using a full order model (FOM) to generate sufficient training data is computationally prohibitive, we propose an adaptive hierarchy of intermediate Reduced Basis reduced order models (ROM) to augment the FOM training data by certified ROM training data required to fit a kernel model.
University of Innsbruck（因斯布鲁克大学）；University of Idaho（爱达荷大学）
We propose aNETT (augmented NETwork Tikhonov) regularization as a novel data-driven reconstruction framework for solving inverse problems. An encoder-decoder type network defines a regularizer consisting of a penalty term that enforces regularity in the encoder domain, augmented by a penalty that penalizes the distance to the data manifold. We present a rigorous convergence analysis including stability estimates and convergence rates. For that purpose, we prove the coercivity of the regularizer used without requiring explicit coercivity assumptions for the networks involved. We propose a possi…
A data scalable augmented Lagrangian KKT preconditioner for large scale inverse problems
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）
Ensemble Kalman inversion (EKI) is a derivative-free optimizer aimed at solving inverse problems, taking motivation from the celebrated ensemble Kalman filter. The purpose of this article is to consider the introduction of adaptive Tikhonov strategies for EKI. This work builds upon Tikhonov EKI (TEKI) which was proposed for a fixed regularization constant. By adaptively learning the regularization parameter, this procedure is known to improve the recovery of the underlying unknown. For the analysis, we consider a continuous-time setting where we extend known results such as well-posdeness and …
B-spline models are a powerful way to represent scientific data sets with a functional approximation. However, these models can suffer from spurious oscillations when the data to be approximated are not uniformly distributed. Model regularization (i.e., smoothing) has traditionally been used to minimize these oscillations; unfortunately, it is sometimes impossible to sufficiently remove unwanted artifacts without smoothing away key features of the data set. In this article, we present a method of model regularization that preserves significant features of a data set while minimizing artificial…
DARE: A Deformable Adaptive Regularization Estimator for Learning-Based Medical Image Registration
Department of Mathematics, University of Innsbruck（数学系，因斯布鲁克大学）；Department of Radiology, Medical University of Innsbruck（放射学系，因斯布鲁克医学院）；Neuroimaging Research Core Facility, Medical University of Innsbruck（神经影像研究核心设施，因斯布鲁克医学院）
The cubic regularization method (CR) and its adaptive version (ARC) are popular Newton-type methods in solving unconstrained non-convex optimization problems, due to its global convergence to local minima under mild conditions. The main aim of this paper is to develop a momentum-accelerated adaptive cubic regularization method (ARCm) to improve the convergent performance. With the proper choice of momentum step size, we show the global convergence of ARCm and the local convergence can also be guaranteed under the \KL property. Such global and local convergence can also be established when inex…
We consider solving nonlinear optimization problems with a stochastic objective and deterministic equality constraints. We assume for the objective that its evaluation, gradient, and Hessian are inaccessible, while one can compute their stochastic estimates by, for example, subsampling. We propose a stochastic algorithm based on sequential quadratic programming (SQP) that uses a differentiable exact augmented Lagrangian as the merit function. To motivate our algorithm design, we first revisit and simplify an old SQP method \citep{Lucidi1990Recursive} developed for solving deterministic problem…
Adaptive multi-penalty regularization based on a generalized Lasso path
Department of Mathematical Sciences, Norwegian University of Science and Technology（挪威科学与技术大学数学科学系）；Section for Computing and Software, Simula Research Laboratory AS（Simula研究实验室计算与软件部门）
