Related papers: A multiscale method for semi-linear elliptic equat…
Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the…
While multilevel Monte Carlo (MLMC) methods for the numerical approximation of partial differential equations with random coefficients enjoy great popularity, combinations with spatial adaptivity seem to be rare. We present an adaptive MLMC…
In this paper, we propose and analyze a fast two-point gradient algorithm for solving nonlinear ill-posed problems, which is based on the sequential subspace optimization method. A complete convergence analysis is provided under the…
A sequential quadratic programming method is designed for solving general smooth nonlinear stochastic optimization problems subject to expectation equality constraints. We consider the setting where the objective and constraint function…
Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate…
In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of…
In this paper we investigate the application of pseudo-transient-continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual…
We consider chance-constrained problems with discrete random distribution. We aim for problems with a large number of scenarios. We propose a novel method based on the stochastic gradient descent method which performs updates of the…
We consider a randomised implementation of the finite element method (FEM) for elliptic partial differential equations on high-dimensional models. This is motivated by applications where model predictions are essential for real-time process…
We generalize our earlier results concerning meshfree collocation methods for semilinear elliptic second order problems to the quasilinear case. The stability question, however, is treated differently, namely by extending a paper on…
In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…
Non-linear least squares solvers are used across a broad range of offline and real-time model fitting problems. Most improvements of the basic Gauss-Newton algorithm tackle convergence guarantees or leverage the sparsity of the underlying…
In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems.…
In this paper, a class of optimization problems with nonlinear inequality constraints is discussed. Based on the ideas of sequential quadratic programming algorithm and the method of strongly sub-feasible directions, a new superlinearly…
We present a new adaptive collocation scheme for solving partial differential equations based on Local Coupled Multiquadrics (LCMQs) within a covers-and-nodes framework. The method, referred to as the Adaptive Ch Method, automatically…
Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…
Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis…