Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equation Q=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method

Large-Scale Multi-Agent Transport: Theory, Algorithms and Analysis

The problem of transport of multi-agent systems has received much attention in a wide range of engineering and biological contexts, such as spatial coverage optimization, collective migration, estimation and mapping of unknown environments. In particular, the emphasis has been on the search for scalable decentralized algorithms that are applicable to large-scale multi-agent systems.For large multi-agent collectives, it is appropriate to describe the configuration of the collective and its evolution using macroscopic quantities, while actuation rests at the microscopic scale at the level of individual agents. Moreover, the control problem faces a multitude of information constraints imposed by the multi-agent setting, such as limitations in sensing, communication and localization. Viewed in this way, the problem naturally extends across scales and this motivates a search for algorithms that respect information constraints at the microscopic level while guaranteeing performance at the macroscopic level.We address the above concerns in this dissertation on three fronts: theory, algorithms and analysis. We begin with the development of a multiscale theory of gradient descent-based multi-agent transport that bridges the microscopic and macroscopic perspectives and sets out a general framework for the design and analysis of decentralized algorithms for transport. We then consider the problem of optimal transport of multi-agent systems, wherein the objective is the minimization of the net cost of transport under constraints of distributed computation. This is followed by a treatment of multi-agent transport under constraints on sensing and communication, in the absence of location information, where we study the problem of self-organization in swarms of agents. Motivated by the problem of multi-agent navigation and tracking of moving targets, we then present a study of moving-horizon estimation of nonlinear systems viewed as a transport of probability measures. Finally, we investigate the robustness of multi-agent networks to agent failure, via the problem of identifying critical nodes in large-scale networks

Optimization on manifolds: A symplectic approach

There has been great interest in using tools from dynamical systems and numerical analysis of differential equations to understand and construct new optimization methods. In particular, recently a new paradigm has emerged that applies ideas from mechanics and geometric integration to obtain accelerated optimization methods on Euclidean spaces. This has important consequences given that accelerated methods are the workhorses behind many machine learning applications. In this paper we build upon these advances and propose a framework for dissipative and constrained Hamiltonian systems that is suitable for solving optimization problems on arbitrary smooth manifolds. Importantly, this allows us to leverage the well-established theory of symplectic integration to derive "rate-matching" dissipative integrators. This brings a new perspective to optimization on manifolds whereby convergence guarantees follow by construction from classical arguments in symplectic geometry and backward error analysis. Moreover, we construct two dissipative generalizations of leapfrog that are straightforward to implement: one for Lie groups and homogeneous spaces, that relies on the tractable geodesic flow or a retraction thereof, and the other for constrained submanifolds that is based on a dissipative generalization of the famous RATTLE integrator

Variational Methods for Evolution (hybrid meeting)

Variational principles for evolutionary systems take advantage of the rich toolbox provided by the theory of the calculus of variations. Such principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, but also time-incremental minimization techniques for more general evolutionary problems. The new challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), new structures (systems with nonlocal transport, gradient flows on graphs, kinetic equations, systems of equations) thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, in multi-agent systems, and in data science. This workshop brought together a broad spectrum of researchers from calculus of variations, partial differential equations, metric geometry, and stochastics, as well as applied and computational scientists to discuss and exchange ideas. It focused on variational tools such as minimizing movement schemes, optimal transport, gradient flows, and large-deviation principles for time-continuous Markov processes, $\Gamma$-convergence and homogenization

Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems

Oscillator models are central to the study of system properties such as entrainment or synchronization. Due to their nonlinear nature, few system-theoretic tools exist to analyze those models. The paper develops a sensitivity analysis for phase-response curves, a fundamental one-dimensional phase reduction of oscillator models. The proposed theoretical and numerical analysis tools are illustrated on several system-theoretic questions and models arising in the biology of cellular rhythms

Towards Understanding the Dynamics of Gaussian-Stein Variational Gradient Descent

Stein Variational Gradient Descent (SVGD) is a nonparametric particle-based deterministic sampling algorithm. Despite its wide usage, understanding the theoretical properties of SVGD has remained a challenging problem. For sampling from a Gaussian target, the SVGD dynamics with a bilinear kernel will remain Gaussian as long as the initializer is Gaussian. Inspired by this fact, we undertake a detailed theoretical study of the Gaussian-SVGD, i.e., SVGD projected to the family of Gaussian distributions via the bilinear kernel, or equivalently Gaussian variational inference (GVI) with SVGD. We present a complete picture by considering both the mean-field PDE and discrete particle systems. When the target is strongly log-concave, the mean-field Gaussian-SVGD dynamics is proven to converge linearly to the Gaussian distribution closest to the target in KL divergence. In the finite-particle setting, there is both uniform in time convergence to the mean-field limit and linear convergence in time to the equilibrium if the target is Gaussian. In the general case, we propose a density-based and a particle-based implementation of the Gaussian-SVGD, and show that several recent algorithms for GVI, proposed from different perspectives, emerge as special cases of our unified framework. Interestingly, one of the new particle-based instance from this framework empirically outperforms existing approaches. Our results make concrete contributions towards obtaining a deeper understanding of both SVGD and GVI.Comment: 59 pages, 7 figure

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques