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A variational framework, initially developed for high-order mesh optimisation, is being extended for r-adaptation. The method is based on the minimisation of a functional of the mesh deformation. To achieve adaptation, elements of the…
The efficiency of Gr\"obner basis computation, the standard engine for solving systems of polynomial equations, depends on the choice of monomial ordering. Despite a near-continuum of possible monomial orders, most implementations rely on…
We present a fast, direct and adaptive Poisson solver for complex two-dimensional geometries based on potential theory and fast multipole acceleration. More precisely, the solver relies on the standard decomposition of the solution as the…
This paper presents a "two-dimensional Fourier Continuation" method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly…
An adaptation of Response Surface Methodology (RSM) when the covariate is of high or infinite dimensional is proposed, providing a tool for black-box optimization in this context. We combine dimension reduction techniques with classical…
The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the…
In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated solution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity…
Numerical solution of the Poisson equation in metallic enclosures, open at one or more ends, is important in many practical situations such as High Power Microwave (HPM) or photo-cathode devices. It requires imposition of a suitable…
We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of…
Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology,…
We develop a computational approach that significantly improves the efficiency of Bayesian optimal experimental design (BOED) using local radial basis functions (RBFs). The presented RBF--BOED method uses the intrinsic ability of RBFs to…
This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform…
We target time-dependent partial differential equations (PDEs) with heterogeneous coefficients in space and time. To tackle these problems, we construct reduced basis/ multiscale ansatz functions defined in space that can be combined with…
The Hermite-Taylor method evolves all the variables and their derivatives through order $m$ in time to achieve a $2m+1$ order rate of convergence. The data required at each node of the staggered Cartesian meshes used by this method makes…
Traditional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from severe limitations when dealing with nonlinear time-dependent parametrized PDEs,…
We introduce a robust first order accurate meshfree method to numerically solve time-dependent nonlinear conservation laws. The main contribution of this work is the meshfree construction of first order consistent summation by parts…
The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily…
The shifted boundary method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods. It has proven to be quite efficient in handling problems with complex geometries,…
Mesh-free methods have significant potential for simulations in complex geometries, as the time consuming process of mesh-generation is avoided. Smoothed Particle Hydrodynamics (SPH) is the most widely used mesh-free method, but suffers…
We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The…