相关论文: Approximate Linearization of Fixed Point Iteration…
We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to…
We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a…
Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based…
An adaptive regularization strategy for stabilizing Newton-like iterations on a coarse mesh is developed in the context of adaptive finite element methods for nonlinear PDE. Existence, uniqueness and approximation properties are known for…
In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the…
This paper is concerned with goal-oriented a posteriori error estimation for nonlinear functionals in the context of nonlinear variational problems solved with continuous Galerkin finite element discretizations. A two-level, or discrete,…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
We analyze inexact fixed point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed point…
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…
Interior-point methods for linear programming problems require the repeated solution of a linear system of equations. Solving these linear systems is non-trivial due to the severe ill-conditioning of the matrices towards convergence. This…
In this paper we study the behavior of finite dimensional fixed point iterations, induced by discretization of a continuous fixed point iteration defined within a Banach space setting. We show that the difference between the discrete…
We develop adaptive discretization algorithms for locally optimal experimental design of nonlinear prediction models. With these algorithms, we refine and improve a pertinent state-of-the-art algorithm in various respects. We establish…
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic…
Finite elasticity problems commonly include material and geometric nonlinearities and are solved using various numerical methods. However, for highly nonlinear problems, achieving convergence is relatively difficult and requires small load…
In recent years, it has been convincingly shown that weather forecasting models can be run in single-precision arithmetic. Several models or components thereof have been tested with even lower precision than this. This previous work has…
Nonlinear convex problems arise in various areas of applied mathematics and engineering. Classical techniques such as the relaxed proximal point algorithm (PPA) and the prediction correction (PC) method were proposed for linearly…
This work proposes a general strategy for solving possibly nonlinear problems arising from implicit time discretizations as a sequence of explicit solutions. The resulting sequence may exhibit instabilities similar to those of the base…
Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations…
In this work we present an adaptive Newton-type method to solve nonlinear constrained optimization problems in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive…
The impact of different linearisation and iterative solution strategies for fully-coupled pressure-based algorithms for compressible flows at all speeds is studied, with the aim of elucidating their impact on the performance of the…