The coupled-cluster formalism – a mathematical perspective

ABSTRACT

The Coupled-Cluster (CC) theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions – Gårding type inequalities – for the local uniqueness of the CC solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended CC theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a CC solution. Furthermore, the use of an Aubin–Nitsche duality type method in different CC approaches is discussed and contrasted with the bivariational principle.     

Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems

In this paper, we consider a class of split mixed vector quasivariational inequality problems in real Hilbert spaces and establish new gap functions by using the method of the nonlinear scalarization function. Further, we obtain some error bounds for the underlying split mixed vector quasivariational inequality problems in terms of regularized gap functions. Finally, we give some examples to illustrate our results. The results obtained in this paper are new.     

Monotonicity in high‐order curvilinear finite element arbitrary Lagrangian–Eulerian remap

The remap phase in arbitrary Lagrangian–Eulerian (ALE) hydrodynamics involves the transfer of field quantities defined on a post‐Lagrangian mesh to some new mesh, usually generated by a mesh optimization algorithm. This problem is often posed in terms of transporting (or advecting) some state variable from the old mesh to the new mesh over a fictitious time interval. It is imperative that this remap process be monotonic, that is, not generate any new extrema in the field variables. It is well known that the only linear methods that are guaranteed to be monotonic for such problems are first‐order accurate; however, much work has been performed in developing non‐linear methods, which blend both high and low (first) order solutions to achieve monotonicity and preserve high‐order accuracy when the field is sufficiently smooth. In this paper, we present a set of methods for enforcing monotonicity targeting high‐order discontinuous Galerkin methods for advection equations in the context of high‐order curvilinear ALE hydrodynamics. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.