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What is called "numerical reproducibility" is the problem of getting the same result when the scientific computation is run several times, either on the same machine or on different machines, with different types and numbers of processing…
We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The…
The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the…
Hybrid spectral-spatial representations are introduced to rapidly calculate periodic scalar and dyadic Green's functions of the Helmholtz equation for 2D and 3D configurations with a 1D (linear) periodicity. The presented schemes work…
In this paper, we propose an incremental abstraction method for dynamically over-approximating nonlinear systems in a bounded domain by solving a sequence of linear programs, resulting in a sequence of affine upper and lower hyperplanes…
We present an approach to accelerate real-space electronic structure methods several fold, without loss of accuracy, by reducing the dimension of the discrete eigenproblem that must be solved. To accomplish this, we construct an efficient,…
Classical algorithms in numerical analysis for numerical integration (quadrature/cubature) follow the principle of approximate and integrate: the integrand is approximated by a simple function (e.g. a polynomial), which is then integrated…
Many parallel algorithms which solve basic problems in computer science use auxiliary space linear in the input to facilitate conflict-free computation. There has been significant work on improving these parallel algorithms to be in-place,…
Two fundamental difficulties are encountered in the numerical evaluation of time-dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts -…
We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…
This paper introduces a new discrete fracture model accounting for non-isothermal compositional multiphase Darcy flows and complex networks of fractures with intersecting, immersed and non immersed fractures. The so called…
The need for efficient normal integration methods is driven by several computer vision tasks such as shape-from-shading, photometric stereo, deflectometry, etc. In the first part of this survey, we select the most important properties that…
The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the…
These notes are concerned with harmonic and holomorphic functions on Euclidean spaces, using quaternions and Clifford algebras in higher dimensions. The main themes are weak solutions, the mean-value property, and subharmonicity.
The solutions to many problems in the mathematical, computational, and physical sciences often involve multidimensional integrals. A direct numerical evaluation of the integral incurs a computational cost that is exponential in the number…
Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…
Implicitly described domains are a well established tool in the simulation of time dependent problems, e.g. using level-set methods. In order to solve partial differential equations on such domains, a range of numerical methods was…
In this paper we analyze the approximation of multivariate integrals over the Euclidean plane for functions which are analytic. We show explicit upper bounds which attain the exponential rate of convergence. We use an infinite grid with…
Additional integral inequalities are obtained for integrals of the differences of subharmonic functions by Borel measures on balls in a multidimensional Euclidean space. These integrals are still estimated from above through the Nevanlinna…
It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of…