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Euler's elastica model has been extensively studied and applied to image processing tasks. However, due to the high nonlinearity and nonconvexity of the involved curvature term, conventional algorithms suffer from slow convergence and high…
Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple…
This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed…
Numerical methods of approximate solution of the Cauchy problem for coupled systems of evolution equations are considered. Separating simpler subproblems for individual components of the solution achieves simplification of the problem at a…
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is…
In this paper, we propose a novel overlapping domain decomposition method that can be applied to various problems in variational imaging such as total variation minimization. Most of recent domain decomposition methods for total variation…
Image resizing is a basic tool in image processing and in literature we have many methods, based on different approaches, which are often specialized in only upscaling or downscaling. In this paper, independently of the (reduced or…
This dissertation explores block decomposable methods for large-scale optimization problems. It focuses on alternating direction method of multipliers (ADMM) schemes and block coordinate descent (BCD) methods. Specifically, it introduces a…
This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is…
In present paper we propose seemingly new method for finding solutions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that…
We revisit the operator splitting schemes proposed in a recent work of [Some extensions of the operator splitting schemes based on Lagrangian and primal-dual: A unified proximal point analysis, Feng Xue, Optimization, 2022, doi:…
In this paper we describe a systematic procedure to analyze the convergence of degenerate preconditioned proximal point algorithms. We establish weak convergence results under mild assumptions that can be easily employed in the context of…
The (efficient and parsimonious) decomposition of higher-order tensors is a fundamental problem with numerous applications in a variety of fields. Several methods have been proposed in the literature to that end, with the Tucker and PARAFAC…
We study the proximal sampler of Lee, Shen, and Tian (2021) and obtain new convergence guarantees under weaker assumptions than strong log-concavity: namely, our results hold for (1) weakly log-concave targets, and (2) targets satisfying…
This paper develops a parameter-free adaptive proximal bundle method with two important features: 1) adaptive choice of variable prox stepsizes that "closely fits" the instance under consideration; and 2) adaptive criterion for making the…
The augmented Lagrangian method (ALM) is a benchmark for convex programming problems with linear constraints; ALM and its variants for linearly equality-constrained convex minimization models have been well studied in the literature.…
The nonlinear conjugate gradient methods are known to be an effective approach for standard unconstrained optimization problems especially for large-scale problems. This paper proposes a proximal nonlinear conjugate gradient method, which…
In this contribution, we are concerned with parameter optimization problems that are constrained by multiscale PDE state equations. As an efficient numerical solution approach for such problems, we introduce and analyze a new relaxed and…
Unsupervised feature selection has drawn wide attention in the era of big data since it is a primary technique for dimensionality reduction. However, many existing unsupervised feature selection models and solution methods were presented…
We develop a new inexact interior-point Lagrangian decomposition method to solve a wide range class of constrained composite convex optimization problems. Our method relies on four techniques: Lagrangian dual decomposition, self-concordant…