Solving a Set of Truncated Dyson-Schwinger Equations with a Globally Converging Method

A globally converging numerical method to solve coupled sets of non-linear integral equations is presented. Such systems occur e.g. in the study of Dyson-Schwinger equations of Yang-Mills theory and QCD. The method is based on the knowledge of the qualitative properties of the solution functions in the far infrared and ultraviolet. Using this input, the full solutions are constructed using a globally convergent modified Newton iteration. Two different systems will be treated as examples: The Dyson-Schwinger equations of 3-dimensional Yang-Mills-Higgs theory provide a system of finite integrals, while those of 4-dimensional Yang-Mills theory at high temperatures are only finite after renormalization.Comment: 23 pages, 3 figures, 4 tables, submitted to Comput. Phys. Commun; one subsection expanded with additional technical details, a few other minor modifications and updates, version to appear in Comput. Phys. Commu

On the status of expansion by regions

We discuss the status of expansion by regions, i.e. a well-known strategy to obtain an expansion of a given multiloop Feynman integral in a given limit where some kinematic invariants and/or masses have certain scaling measured in powers of a given small parameter. Using the Lee-Pomeransky parametric representation, we formulate the corresponding prescriptions in a simple geometrical language and make a conjecture that they hold even in a much more general case. We prove this conjecture in some partial cases and illustrate them in a simple example.Comment: Published version: presentation improved, Section 7 delete

Steady state analysis of Chua's circuit with RLCG transmission line

In this paper we present a new technique to compute the steady state response of nonlinear autonomous circuits with RLCG transmission lines. Using multipoint Pade approximants, instead of the commonly used expansions around s=0 or s/spl rarr//spl infin/ accurate, low-order lumped equivalent circuits of the characteristic impedance and the exponential propagation function are obtained in an explicit way. Then, with the temporal discretization of the equations that describe the transformed circuit, we obtain a nonlinear algebraic formulation where the unknowns to be determined are the samples of the variables directly in the steady state, along with the oscillation period, the main unknown in autonomous circuits. An efficient scheme to build the Jacobian matrix with exact partial derivatives with respect to the oscillation period and with respect to the samples of the unknowns is obtained. Steady state solutions of the Chua's circuit with RLCG transmission line are computed for selected circuit parameters.Peer ReviewedPostprint (published version

Semidefinite relaxations for semi-infinite polynomial programming

This paper studies how to solve semi-infinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.Comment: 23 pages, 4 figure

On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Mor´e and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems