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Unconstrained optimization problems become more common in scientific computing and engineering applications with the rapid development of artificial intelligence, and numerical methods for solving them more quickly and efficiently have been…
A new reformulation of a free boundary problem for the Stokes equations governing a viscous flow with overdetermined condition on the free boundary is proposed. The idea of the method is to transform the governing equations to a boundary…
In this work we propose a novel approach to investigate boundary value problems (BVPs) for fully third order differential equations. It is based on the reduction of BVPs to operator equations for the nonlinear terms but not for the…
We propose a novel framework for the discretisation of multi-label problems on arbitrary, continuous domains. Our work bridges the gap between general FEM discretisations, and labeling problems that arise in a variety of computer vision…
This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling…
Image semantic segmentation aims at the pixel-level classification of images, which has requirements for both accuracy and speed in practical application. Existing semantic segmentation methods mainly rely on the high-resolution input to…
Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient…
This paper proposes a novel localized Fourier extension method for approximating non-periodic functions via domain segmentation. By partitioning the computational domain into subregions with uniform discretization scales, the method…
In this paper, we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions by means of spectral methods. The main features of this method are its…
We design a cut finite element method for the incompressible Stokes equations on curved domains. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion.…
This paper presents eigensolution and non-modal analyses for immersed boundary methods (IBMs) based on volume penalization for the linear advection equation. This approach is used to analyze the behavior of flux reconstruction (FR)…
This work extends our previous study from S. Shrestha et al. (2024) by introducing a new abstract framework for Variational Multiscale (VMS) methods at the discrete level. We introduce the concept of what we define as the optimal projector…
We propose a predictor-corrector adaptive method for the simulation of hyperbolic partial differential equations (PDEs) on networks under general uncertainty in parameters, initial conditions, or boundary conditions. The approach is based…
A new exact projective penalty method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing…
Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material).…
This work focuses on a class of elliptic boundary value problems with diffusive, advective and reactive terms, motivated by the study of three-dimensional heterogeneous physical systems composed of two or more media separated by a selective…
Chaotic free surface flows are challenging problems to simulate numerically, mainly due to the significant changes in geometry and frequent topological changes. Methods that track the evolution of the fluid in a Lagrangian formulation are a…
We study optimization algorithms for the finite sum problems frequently arising in machine learning applications. First, we propose novel variants of stochastic gradient descent with a variance reduction property that enables linear…
We present a 3D hybrid method which combines the Finite Element Method (FEM) and the Spectral Boundary Integral method (SBIM) to model nonlinear problems in unbounded domains. The flexibility of FEM is used to model the complex,…
In this work we are interested in the construction of numerical methods for high dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. A consensus-based optimization (CBO) approach combined with…