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A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated…
We investigate new developments of the combined Reduced-Basis and Empirical Interpolation Methods (RB-EIM) for parametrized nonlinear parabolic problems. In many situations, the cost of the EIM in the offline stage turns out to be…
In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman-Kac formula for the Robin problem which employs…
In this work, we combine Beyn's method and the recently developed recursive integral method (RIM) to propose a contour integral-based, region partitioning eigensolver for nonlinear eigenvalue problems. A new partitioning criterion is…
Regularisation is commonly used in iterative methods for solving imaging inverse problems. Many algorithms involve the evaluation of the proximal operator of the regularisation term in every iteration, leading to a significant computational…
Physics-informed neural networks (PINNs) have recently emerged as a popular approach for solving forward and inverse problems involving partial differential equations (PDEs). Compared to fully connected neural networks, PINNs based on…
This study investigates the iterative refinement method applied to the solution of linear discrete inverse problems by considering its application to the Tikhonov problem in mixed precision. Previous works on mixed precision iterative…
In this work, we propose a model order reduction framework to deal with inverse problems in a non-intrusive setting. Inverse problems, especially in a partial differential equation context, require a huge computational load due to the…
An iterative solution method for fully nonlinear boundary value problems governing self-similar flows with a free boundary is presented. Specifically, the method is developed for application to water entry problems, which can be studied…
The conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting…
We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically,…
We present a new approach for boundary integral equations for the wave equation with zero initial conditions. Unlike previous attempts, our mathematical formulation allows us to prove that the associated boundary integral operators are…
Many nanostructures today are low-dimensional and flimsy, and therefore get easily distorted. Distortion-induced symmetry-breaking makes conventional, translation-periodic simulations invalid, which has triggered developments for new…
Recently, reduced order modeling methods have been applied to solving inverse boundary value problems arising in frequency domain scattering theory. A key step in projection-based reduced order model methods is the use of a sesquilinear…
In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss…
We apply physics-informed neural networks (PINNs) to first-order two-scale periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp…
The boundary control problem is a non-convex optimization and control problem in many scientific domains, including fluid mechanics, structural engineering, and heat transfer optimization. The aim is to find the optimal values for the…
In this paper, two numerical schemes for a nonlinear integral equation of Fredholm type with weakly singular kernel are proposed. These numerical methods combine sinc-collocation and sinc-convolution approximations with Newton and steepest…
The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform…
Bayesian statistical inverse problems are often solved with Markov chain Monte Carlo (MCMC)-type schemes. When the problems are governed by large-scale discrete nonlinear partial differential equations (PDEs), they are computationally…