A null-space approach for large-scale symmetric saddle point systems with a small and non zero (2, 2) block

Null-space methods have long been used to solve large sparse n × n symmetric saddle point systems of equations in which the (2, 2) block is zero. This paper focuses on the case where the (1, 1) block is ill conditioned or rank deficient and the k × k (2, 2) block is non zero and small (k ≪ n). Additionally, the (2, 1) block may be rank deficient. Such systems arise in a range of practical applications. A novel null-space approach is proposed that transforms the system matrix into a nicer symmetric saddle point matrix of order n that has a non zero (2, 2) block of order at most 2k and, importantly, the (1, 1) block is symmetric positive definite. Success of any null-space approach depends on constructing a suitable null-space basis. We propose methods for wide matrices having far fewer rows than columns with the aim of balancing stability of the transformed saddle point matrix with preserving sparsity in the (1, 1) block. Linear least squares problems that contain a small number of dense rows are an important motivation and are used to illustrate our ideas and to explore their potential for solving large-scale systems.

Hybrid inflation and waterfall field in string theory from D7-branes

We present an explicit string realisation of a cosmological inflationary scenario we proposed recently within the framework of type IIB flux compactifications in the presence of three magnetised D7-brane stacks. Inflation takes place around a metastable de Sitter vacuum. The inflaton is identified with the volume modulus and has a potential with a very shallow minimum near the maximum. Inflation ends due to the presence of “waterfall” fields that drive the evolution of the Universe from a nearby saddle point towards a global minimum with tuneable vacuum energy describing the present state of our Universe.

Accelerated Stochastic Algorithms for Convex-Concave Saddle-Point Problems

We develop stochastic first-order primal-dual algorithms to solve a class of convex-concave saddle-point problems. When the saddle function is strongly convex in the primal variable, we develop the first stochastic restart scheme for this problem. When the gradient noises obey sub-Gaussian distributions, the oracle complexity of our restart scheme is strictly better than any of the existing methods, even in the deterministic case. Furthermore, for each problem parameter of interest, whenever the lower bound exists, the oracle complexity of our restart scheme is either optimal or nearly optimal (up to a log factor). The subroutine used in this scheme is itself a new stochastic algorithm developed for the problem where the saddle function is nonstrongly convex in the primal variable. This new algorithm, which is based on the primal-dual hybrid gradient framework, achieves the state-of-the-art oracle complexity and may be of independent interest.

Semi-classical analysis of the string theory cigar

We study the semi-classical limit of the reflection coefficient for the SL(2, ℝ)k/U(1) CFT. For large k, the CFT describes a string in a Euclidean black hole of 2-dimensional dilaton-gravity, whose target space is a cigar with an asymptotically linear dilaton. This sigma-model description is weakly coupled in the large k limit, and we investigate the saddle-point expansion of the functional integral that computes the reflection coefficient. As in the semi-classical limit of Liouville CFT studied in [1], we find that one must complexify the functional integral and sum over complex saddles to reproduce the limit of the exact reflection coefficient. Unlike Liouville, the SL(2, ℝ)k/U(1) CFT admits bound states that manifest as poles of the reflection coefficient. To reproduce them in the semi-classical limit, we find that one must sum over configurations that hit the black hole singularity, but nevertheless contribute to the saddle-point expansion with finite action.

General finite-volume framework for saddle-point problems of various physics

This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.