Related papers: A Linear Combination of Unitaries Decomposition fo…
Many multilinear discrete operators are primed for pointwise decomposition; such decompositions give structural information but also an essentially optimal range of bounds. We study the (continuous) slicing method of Jeong and Lee -- which…
Comparison principles are developed for discrete quasilinear elliptic partial differential equations. We consider the analysis of a class of nonmonotone Leray-Lions problems featuring both nonlinear solution and gradient dependence in the…
In this work, we study a class of elliptic problems involving nonlinear superpositions of fractional operators of the form \[ A_{\mu,p}u := \int_{[0,1]} (-\Delta)_{p}^{s} u \, d\mu(s), \] where $\mu$ is a signed measure on $[0,1]$, coupled…
In this paper, we propose a domain decomposition method for multiscale second order elliptic partial differential equations with highly varying coefficients. The method is based on a discontinuous Galerkin formulation. We present both a…
We study the solvability of boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being a UMD complex Banach space. The originality of this work lies in the fact that we…
We consider different Linear Combination of Unitaries (LCU) decompositions for molecular electronic structure Hamiltonians. Using these LCU decompositions for Hamiltonian simulation on a quantum computer, the main figure of merit is the…
Solving elliptic PDEs in more than one dimension can be a computationally expensive task. For some applications characterised by a high degree of anisotropy in the coefficients of the elliptic operator, such that the term with the highest…
In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $\Omega$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the…
We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big)…
We study pairs $(U,L_0)$, where $U$ is a unitary operator in $H$ and $L_0\subset H$ is a closed subspace, such that $$ P_{L_0}U|_{L_0}:L_0\to L_0 $$ has a singular value decomposition. Abstract characterizations of this condition are given,…
The large sparse linear systems arising from the finite element or finite difference discretization of elliptic PDEs can be solved directly via, e.g., nested dissection or multifrontal methods. Such techniques reorder the nodes in the grid…
An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
The discretized Poisson equation matrix (DPEM) in 1D has been shown to require an exponentially large number of terms when decomposed in the Pauli basis when solving numerical linear algebra problems on a quantum computer. Additionally,…
In this paper, we propose a novel recovery based finite element method for the Cahn-Hilliard equation. One distinguishing feature of the method is that we discretize the fourth-order differential operator in a standard $C^0$ linear finite…
We present a new technique to obtain polynomial decay estimates for the matrix coefficients of unitary operators. Our approach, based on commutator methods, applies to nets of unitary operators, unitary representations of topological…
We adapt the alternating linearization method for proximal decomposition to structured regularization problems, in particular, to the generalized lasso problems. The method is related to two well-known operator splitting methods, the…
By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite-difference operators of orders up to nine are developed. The boundary closures are constructed within a diagonal-norm summation-by-parts (SBP)…
Domain decomposition methods are used for approximate solving boundary problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are taken into account in the most complete way in…
We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic and hyperbolic partial differential equations. This general approach incorporates known weak…