Categorical Tensor Network States by Biamonte, S, Jaksch, D

“…This general framework of categorical tensor network<br/> states, where a combination of generic and algebraically defined tensors<br/> appear, enhances the theory of tensor network states.…”

Existence of Global Weak Solutions for Some Polymeric Flow Models by Barrett, J, Schwab, C, Suli, E

“…The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain for the velocity and the pressure of the fluid, with an extra-stress tensor as right-hand side in the momentum equation. …”

Existence of global weak solutions to kinetic models for dilute polymers by Barrett, J, Suli, E

“…The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. …”

Numerical approximation of corotational dumbbell models for dilute polymers by Barrett, J, Suli, E

“…The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω in R d, d=2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. …”

Sparse finite element approximation of high-dimensional transport-dominated diffusion problems by Schwab, C, Suli, E, Todor, R

“…We develop the numerical analysis of stabilized sparse tensor-product finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations, using piecewise polynomials of degree p &gt; 0. …”

Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers by Barrett, J, Suli, E

“…The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain $\Omega \subset R^d$, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. …”

Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems by Ortner, C, Suli, E

“…An optimal order bound is derived on the discretization error in each case without requiring the global Lipschitz continuity of the tensor $S$. We then further relax our hypotheses: using a broken G{\aa}rding inequality we extend our optimal error bounds to the case of quasilinear hyperbolic systems where, instead of assuming that $S$ is uniformly monotone, we only require that the fourth-order tensor $A=\nabla S$ is satisfies a Legendre--Hadamard condition. …”

Particle tracking methods for residence time calculations in incompressible flow by Glasgow, C, Parrott, A, Handscomb, D

“…Numerical methods are presented for the calculation of residence time distributions in steady incompressible fluid flow using a given set of normal fluid fluxes, defined across the cell faces of a cartesian tensor product mesh. A particle tracking approach is adopted involving the construction of a piecewise polynomial representation of the velocity distribution, and subsequent integration of this representation for the determination of individual particle trajectories.…”

Multiple knot B-spline representation of incompressible flow by Glasgow, C, Parrott, A, Handscomb, D

“…The fluxes should be defined across the face-centres of a cartesian tensor product mesh. The proposed spline representation interpolates the given fluxes exactly and also enables the normal fluid velocity to be set identically to zero across or around the surfaces of an arbitrary number of rectangular regions lying in specified planes.…”

A Green's function preconditioner for the steady-state Navier-Stokes equations by Kay, D, Loghin, D

“…The solver is an iterative method of Krylov subspace type for which we devise a preconditioner based on Green's tensor for the Oseen operator. The preconditioner supersedes existing preconditioners for the Oseen problem in that it exhibits only a mild dependence on the viscosity (inverse Reynolds number) and, most importantly, improved performance with the size of the problem. …”

Computing the common zeros of two bivariate functions via Bezout resultants by Nakatsukasa, Y, Noferini, V, Townsend, A

“…The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bezout matrices with polynomial entries. …”

A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems by Georgoulis, E, Lasis, A

“…We do not assume uniform ellipticity of the diffusion tensor. Moreover, diffusion tensors or arbitrary form are covered in the theory presented. …”

A Categorical semantics of Quantum Protocols by Abramsky, S, Coecke, B

“…It also shows the degrees of axiomatic freedom: we can show what requirements are placed on the (semi)ring of scalars C(I,I), where C is the category and I is the tensor unit, in order to perform various protocols such as teleportation. …”

Categorical Quantum Circuits by Bergholm, V, Biamonte, J

“…Our specific approach has further applications in applying category theory and related ideas to tensor network simulation.…”

Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions by Schwab, C, Suli, E

“…Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin discretizations, based on a tensorized Riesz basis, built from biorthogonal piecewise polynomial wavelet bases in time and the Hermite polynomial chaos in the Wiener-Itô decomposition of $L^{2}(H,\mu)$, are introduced and are shown to converge quasioptimally with respect to the nonlinear, best $N$-term approximation benchmark. …”