Related papers: Operator-difference approximations on two-dimensio…
Problems of the numerical solution of the Cauchy problem for a first-order differential-operator equation are discussed. A fundamental feature of the problem under study is that the equation includes a fractional power of the self-adjoint…
In this paper we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the…
Computing offsets of curves on parametric surfaces is a fundamental yet challenging operation in computer aided design and manufacturing. Traditional analytical approaches suffer from time-consuming geodesic distance queries and complex…
Probing the structure of complex astrophysical objects requires effective three-dimensional (3D) numerical simulation of the relevant radiative transfer (RT) processes. As with any numerical simulation code, the choice of an appropriate…
Almost commuting operators were introduced in 1985 by George Wilson to present generalizations of the Korteweg-de Vries hierarchy, nowadays known as Gelfand-Dickey (GD) hierarchies. In this paper, we review the formal construction of the…
We consider the construction of a polyhedral Delaunay partition as a limit of the sequence of power diagrams (radical partitions). The dual Voronoi diagram is obtained as a limit of the sequence of weighted Delaunay partitions. The problem…
The main objective of this paper is twofold. One is to classify and construct $SL(3,\mathbb{R})$-intertwining differential operators between vector bundles over the real projective space $\mathbb{RP}^2$. It turns out that two kinds of…
In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to a measure supported on the nonnegative integers. We show that the dual families are closely related to certain difference operators…
Machine learning, especially deep learning is gaining much attention due to the breakthrough performance in various cognitive applications. Recently, neural networks (NN) have been intensively explored to model partial differential…
Effective transport properties of heterogeneous structures are predicted by geometric microstructural parameters, but these can be difficult to calculate. Here, a boundary element code with a recurrent series method accurately and…
To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to two-dimensional…
We study the algebra of invariant differential operators on a certain homogeneous vector bundle over a Riemannian symmetric space of type $A_2$. We computed radial parts of its generators explicitly to obtain matrix-valued commuting…
Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a…
We discuss conformally covariant differential operators, which under local rescalings of the metric, \delta_\sigma g^{\mu\nu} = 2 \sigma g^{\mu\nu}, transform according to \delta_\sigma \Delta = r \Delta \sigma + (s-r) \sigma \Delta for…
Using neural networks to solve variational problems, and other scientific machine learning tasks, has been limited by a lack of consistency and an inability to exactly integrate expressions involving neural network architectures. We address…
Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of…
Symmetric orthogonalization via SVD, and closely related procedures, are well-known techniques for projecting matrices onto $O(n)$ or $SO(n)$. These tools have long been used for applications in computer vision, for example optimal 3D…
Most current results on coverage control using mobile sensors require that one partitioned cell is the sole responsibility of one sensor. In this paper, we consider a class of generalized Voronoi coverage control problems by using higher…
Of the two main objectives we pursue in this paper, the first one consists in the studying operators of the form $(\partial_t-i\triangle_{\Gamma})^{\alpha},\,\,\alpha=1/2,-1/2,-1,\ldots,$ where $\triangle_{\Gamma}$ is the Laplace-Beltrami…
We consider the problem of constructing spatial finite difference approximations on a fixed, arbitrary grid, which have analogues of any number of integrals of the partial differential equation and of some of its symmetries. A basis for the…