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In this paper, we consider nonconvex optimization problems with nonsmooth nonconvex objective function and nonlinear equality constraints. We assume that both the objective function and the functional constraints can be separated into 2…
We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the…
Solving parabolic optimal control problems can be inherently challenging in the field of science and engineering, especially with constraints on the nonsmooth distributed control. Motivated by the extensive applicability of the alternating…
The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate…
This work proposes an adaptive sequential Monte Carlo sampling algorithm to solve Bayesian inverse problems in scenarios where likelihood evaluations are costly but can be approximated using a surrogate model built from previous evaluations…
When an inverse problem is solved by a gradient-based optimization algorithm, the corresponding forward and adjoint problems, which are introduced to compute the gradient, can be also solved iteratively. The idea of iterating at the same…
There are some types of ill-conditioned algebraic equations that have difficulty in obtaining accurate roots and coefficients that must be expressed with a multiple precision floating-point number. When all their roots are simple, the…
In this paper, we study a class of bilevel optimization problems where the lower-level problem is a convex composite optimization model, which arises in various applications, including bilevel hyperparameter selection for regularized…
Multi-agent distributed consensus optimization problems arise in many signal processing applications. Recently, the alternating direction method of multipliers (ADMM) has been used for solving this family of problems. ADMM based distributed…
Monge-Amp\`{e}re equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the $C^0$ interior penalty method to approximation the viscosity solution of the…
The alternating direction method of multipliers (ADMM) is a popular method for solving convex separable minimization problems with linear equality constraints. The generalization of the two-block ADMM to the three-block ADMM is not trivial…
This paper proposes and establishes the iteration-complexity of an inexact proximal accelerated augmented Lagrangian (IPAAL) method for solving linearly constrained smooth nonconvex composite optimization problems. Each IPAAL iteration…
Recently, a class of inexact Picard iteration method for solving the absolute value equation: $Ax-|x~|=b$ have been proposed in [Optim Lett 8:2191-2202,2014]. To further improve the performance of Picard iteration method, a new inexact…
We develop an accelerated gradient descent algorithm on the Grassmann manifold to compute the subspace spanned by a number of leading eigenvectors of a symmetric positive semi-definite matrix. This has a constant cost per iteration and a…
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…
We propose a multigrid correction scheme to solve a new Steklov eigenvalue problem in inverse scattering. With this scheme, solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems…
This paper proposes and analyzes an accelerated inexact dampened augmented Lagrangian (AIDAL) method for solving linearly-constrained nonconvex composite optimization problems. Each iteration of the AIDAL method consists of: (i) inexactly…
A new iterative method for solving large scale symmetric nonlinear eigenvalue problems is presented. We firstly derive an infinite dimensional symmetric linearization of the nonlinear eigenvalue problem, then we apply the indefinite Lanczos…
The classical Alexandrov estimate controls the oscillation of a convex function by the mass of its associated Monge-Amp\`ere measure and yields, for two convex functions of $n$ variables with the same boundary values, a sup-norm bound with…
This paper is devoted to first-order algorithms for smooth convex optimization with inexact gradients. Unlike the majority of the literature on this topic, we consider the setting of relative rather than absolute inexactness. More…