Related papers: A regularized Newton-like method for nonlinear PDE
We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion…
We propose a regularized Hessian-free Newton-type method for minimizing smooth convex functions with Lipschitz continuous Hessians. The algorithm constructs an approximate Hessian by finite differences and selects the regularization…
The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for…
In this paper, we consider a class of systems of nonlinear equations, which arise in discretized mixed formulations of problems in solid mechanics by $hp$-finite elements. We introduce a semismooth Newton solver for this specific class and…
Newton's method for finding an unconstrained minimizer for strictly convex functions, generally speaking, does not converge from any starting point. We introduce and study the damped regularized Newton's method (DRNM). It converges globally…
This paper concerns the composite problem of minimizing the sum of a twice continuously differentiable function $f$ and a nonsmooth convex function. For this class of nonconvex and nonsmooth problems, by leveraging a practical inexactness…
We propose a novel finite element method scheme for singularly perturbed advection-diffusion-reaction problems, which combines certain quantum-assisted stabilization scheme with a classical h-adaptive approach to provide automatic error…
We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the…
We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving ERM problems with a nonsmooth regularization term. Current second-order and quasi-Newton methods for this…
In this paper we develop adaptive numerical schemes for certain nonlinear variational problems. The discretization of the variational problems is done by representing the solution as a suitable frame decomposition, i.e., a complete, stable,…
Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression…
We present iterative solvers to approximate the solution of numerical schemes for stochastic Stefan problems. After briefly talking about the convergence results, we tackle the question of efficient strategies for solving the nonlinear…
Adaptive Finite Element Method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property…
We study the numerical approximation of a class of degenerate parabolic stochastic partial differential equations on non-compact metric graphs, which naturally arise in the asymptotic analysis of Hamiltonian flows under small noise…
We investigate a level-set type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable…
This paper proposes and develops new Newton-type methods to solve structured nonconvex and nonsmooth optimization problems with justifying their fast local and global convergence by means of advanced tools of variational analysis and…
Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT…
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to…
The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by…
A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure…