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We consider a minimization model with total variational regularization, which can be reformulated as a saddle-point problem and then be efficiently solved by the primal–dual method. We utilize the consistent finite element method to discretize the saddle-point reformulation; thus possible jumps of the solution can be captured over some adaptive mes...
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Citations
... For this purpose, it suffices to apply the primal-dual method in [8], because it does not require specific initial iterates and its resulting subproblems are easier than the original model. The primal-dual method in [8] and its variants have been widely applied in various areas such as PDEs [35,36], imagining processing [8], statistical learning [20], and inverse problems [6,10,54,57]. ...
We consider a general class of nonsmooth optimal control problems with partial differential equation (PDE) constraints, which are very challenging due to its nonsmooth objective functionals and the resulting high-dimensional and ill-conditioned systems after discretization. We focus on the application of a primal-dual method, with which different types of variables can be treated individually and thus its main computation at each iteration only requires solving two PDEs. Our target is to accelerate the primal-dual method with either larger step sizes or operator learning techniques. For the accelerated primal-dual method with larger step sizes, its convergence can be still proved rigorously while it numerically accelerates the original primal-dual method in a simple and universal way. For the operator learning acceleration, we construct deep neural network surrogate models for the involved PDEs. Once a neural operator is learned, solving a PDE requires only a forward pass of the neural network, and the computational cost is thus substantially reduced. The accelerated primal-dual method with operator learning is mesh-free, numerically efficient, and scalable to different types of PDEs. The acceleration effectiveness of these two techniques is promisingly validated by some preliminary numerical results.
... Example 6.3. This example we consider is a benchmark problem as seen in [2,4,29]. Let Ω = (0, 1) 2 and denote its center by x Ω . ...
Solution methods for the nonlinear partial differential equation of the Rudin-Osher-Fatemi (ROF) and minimum-surface models are fundamental for many modern applications. Many efficient algorithms have been proposed. First order methods are common. They are popular due to their simplicity and easy implementation. Some second order Newton-type iterative methods have been proposed like Chan-Golub-Mulet method. In this paper, we propose a new Newton-Krylov solver for primal-dual finite element discretization of the ROF model. The method is so simple that we just need to use some diagonal preconditioners during the iterations. Theoretically, the proposed preconditioners are further proved to be robust and optimal with respect to the mesh size, the penalization parameter, the regularization parameter, and the iterative step, essentially it is a parameter independent preconditioner. We first discretize the primal-dual system by using mixed finite element methods, and then linearize the discrete system by Newton\textquoteright s method. Exploiting the well-posedness of the linearized problem on appropriate Sobolev spaces equipped with proper norms, we propose block diagonal preconditioners for the corresponding system solved with the minimum residual method. Numerical results are presented to support the theoretical results.
... Along this line of study, many welldeveloped methods in the theory of finite element method are applied to this problem (c.f. Lai and Messi,28 Matamba Messi, 29 Bartels,30,31 Bartels and Milicevic,32 Chen and Tai, 33 Feng and Prohl, 34 Feng et al., 35 Litvinov et al., 36 Stamm and Wihler, 36 Tian and Yuan, 37 and Xu et al. 38 ). It is worth noting that all these works employ either piecewise-constant or piecewise-linear finite elements. ...
Total variation smoothing methods have been proven to be very efficient at discriminating between structures (edges and textures) and noise in images. Recently, it was shown that such methods do not create new discontinuities and preserve the modulus of continuity of functions. In this paper, we propose a Galerkin–Ritz method to solve the Rudin–Osher–Fatemi image denoising model where smooth bivariate spline functions on triangulations are used as approximating spaces. Using the extension property of functions of bounded variation on Lipschitz domains, we construct a minimizing sequence of continuous bivariate spline functions of arbitrary degree, d, for the TV-L² energy functional and prove the convergence of the finite element solutions to the solution of the Rudin, Osher, and Fatemi model. Moreover, an iterative algorithm for computing spline minimizers is developed and the convergence of the algorithm is proved.
Based on the Fenchel duality we build a primal-dual framework for minimizing a general functional consisting of a combined L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} data-fidelity term and a scalar or vectorial total variation regularisation term. The minimization is performed over the space of functions of bounded variations and appropriate discrete subspaces. We analyze the existence and uniqueness of solutions of the respective minimization problems. For computing a numerical solution we derive a semi-smooth Newton method on finite element spaces and highlight applications in denoising, inpainting and optical flow estimation.
A model with the L ² and total variational (TV) regularization terms for linear inverse problems is considered. The regularized model is reformulated as a saddle-point problem, and the primal and dual variables are discretized in the piecewise affine and piecewise constant finite element spaces, respectively. An accelerated primal-dual iterative scheme with an O( ¹N2 ) convergence rate is proposed for the discretized problem, where N is the iteration counter. Both the regularization and perturbation errors of the regularized model, and the finite element discretization and iteration errors of the accelerated primal-dual scheme, are estimated. Preliminary numerical results are reported to show the efficiency of the proposed iterative scheme.
Linear inverse problems with total variation regularization can be reformulated as saddle-point problems; the primal and dual variables of such a saddle-point reformulation can be discretized in piecewise affine and constant finite element spaces, respectively. Thus, the well-developed primal-dual approach (a.k.a. the inexact Uzawa method) is conceptually applicable to such a regularized and discretized model. When the primal-dual approach is applied, the resulting subproblems may be highly nontrivial and it is necessary to discuss how to tackle them and thus make the primal-dual approach implementable. In this paper, we suggest linearizing the data-fidelity quadratic term of the hard subproblems so as to obtain easier ones. A linearized primal-dual method is thus proposed. Inspired by the fact that the linearized primal-dual method can be explained as an application of the proximal point algorithm, a relaxed version of the linearized primal-dual method, which can often accelerate the convergence numerically with the same order of computation, is also proposed. The global convergence and worst-case convergence rate measured by the iteration complexity are established for the new algorithms. Their efficiency is verified by some numerical results.
