Related papers: An online adaptive finite-element method for nonsm…
Sparse learning is a very important tool for mining useful information and patterns from high dimensional data. Non-convex non-smooth regularized learning problems play essential roles in sparse learning, and have drawn extensive attentions…
We formulate and analyze a goal-oriented adaptive finite element method for a symmetric linear elliptic partial differential equation (PDE) that can simultaneously deal with multiple linear goal functionals. In each step of the algorithm,…
Convex and nonconvex finite-sum minimization arises in many scientific computing and machine learning applications. Recently, first-order and second-order methods where objective functions, gradients and Hessians are approximated by…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and…
An efficient and accurate finite-element algorithm is described for the numerical solution of the incompressible Navier-Stokes (INS) equations. The new algorithm that solves the INS equations in a velocity-pressure reformulation is based on…
In this work we present TRFD, a derivative-free trust-region method based on finite differences for minimizing composite functions of the form $f(x)=h(F(x))$, where $F$ is a black-box function assumed to have a Lipschitz continuous…
We consider the unconstrained optimization problem whose objective function is composed of a smooth and a non-smooth conponents where the smooth component is the expectation a random function. This type of problem arises in some interesting…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…
An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages…
We propose a penalty-based smoothing framework for convex nonsmooth functions with a supremum structure. The regularization yields a differentiable surrogate with controlled approximation error, a single-valued dual maximizer, and explicit…
An adaptive regularization algorithm for unconstrained nonconvex optimization is proposed that is capable of handling inexact objective-function and derivative values, and also of providing approximate minimizer of arbitrary order. In…
In this paper we present a finite element method for the direct transcription of constrained non-linear optimal control problems. We prove that our method converges of high order under mild assumptions. Our analysis uses a regularized…
A novel refinement measure for non-intrusive surrogate modelling of partial differential equations (PDEs) with uncertain parameters is proposed. Our approach uses an empirical interpolation procedure, where the proposed refinement measure…
We consider adaptive finite element methods for solving a multiscale system consisting of a macroscale model comprising a system of reaction-diffusion partial differential equations coupled to a microscale model comprising a system of…
We introduce a new $hp$-adaptive strategy for self-adjoint elliptic boundary value problems that does not rely on using classical a posteriori error estimators. Instead, our approach is based on a generally applicable prediction strategy…
In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such…
In this paper, we consider a class of structured nonconvex nonsmooth optimization problems, in which the objective function is formed by the sum of a possibly nonsmooth nonconvex function and a differentiable function whose gradient is…
In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several…
We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is…