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Reconstruction of object or scene surfaces has tremendous applications in computer vision, computer graphics, and robotics. In this paper, we study a fundamental problem in this context about recovering a surface mesh from an implicit field…
We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts. The method is an extension of the multiresolution finite volume scheme used…
We generalize the technique of [Solving Dirichlet boundary-value problems on curved domains by extensions from subdomains, SIAM J. Sci. Comput. 34, pp. A497--A519 (2012)] to elliptic problems with mixed boundary conditions and elliptic…
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…
Immersed finite element (IFE) methods are a group of long-existing numerical methods for solving interface problems on unfitted meshes. A core argument of the methods is to avoid mesh regeneration procedure when solving moving interface…
In this paper, we propose a meshfree approximation method for the implicit filter developed in [2], which is a novel numerical algorithm for nonlinear filtering problems. The implicit filter approximates conditional distributions in the…
This work introduces a novel, fully robust and highly-scalable, $h$-adaptive aggregated unfitted finite element method for large-scale interface elliptic problems. The new method is based on a recent distributed-memory implementation of the…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach.…
In this article we consider two-grid finite element methods for solving semilinear interface problems in d space dimensions, for d=2 or d=3. We first describe in some detail the target problem class with discontinuous diffusion…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
Stochastic sampling methods are arguably the most direct and least intrusive means of incorporating parametric uncertainty into numerical simulations of partial differential equations with random inputs. However, to achieve an overall error…
This work proposes an $r$-adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as…
The solution $u$ of an elliptic interface problem in a domain $\Omega$ is often smooth away from the interface $\Gamma\subset \Omega$, but its gradient is discontinuous across $\Gamma$, resulting in low regularity; in particular, $u \notin…
The generation of triangle meshes from point clouds, i.e. meshing, is a core task in computer graphics and computer vision. Traditional techniques directly construct a surface mesh using local decision heuristics, while some recent methods…
Elliptic variational inequalities (EVIs) present significant challenges in numerical computation due to their inherent non-smoothness, nonlinearity, and inequality formulations. Traditional mesh-based methods often struggle with complex…
A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic…
Recent advances in implicit neural representations show great promise when it comes to generating numerical solutions to partial differential equations. Compared to conventional alternatives, such representations employ parameterized neural…
We present a locally adapted parametric finite element method for interface problems. For this adapted finite element method we show optimal convergence for elliptic interface problems with a discontinuous diffusion parameter. The method is…
This paper focuses on proposing a deep learning initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an…