arXiv subjects

Anand Srinivasan

Publications and source records attributed to Anand Srinivasan.

8 recordsLinked to original sources

arXiv2026

Thermodynamic structure of the Sinkhorn flow

Entropy-regularized optimal transport, which has strong links to the Schrödinger bridge problem in statistical mechanics, enjoys a variety of applications from trajectory inference to generative modeling. A major driver of renewed interest in this problem is the recent development of fast matrix-scaling algorithms\textemdash known as iterative proportional fitting or the Sinkhorn algorithm\textemdash for entropic optimal transport, which have favorable complexity over traditional approaches to the unregularized problem. Here, we take a perspective on this algorithm rooted in the thermodynamic origins of Schrödinger's problem and inspired by the modern geometric theory of diffusion: is the Sinkhorn flow (viewed in continuous-time as a mirror descent by recent results) the gradient flow of entropy in a formal Riemannian geometry? We answer this question affirmatively, finding a nonlocal Wasserstein gradient structure in the dynamics of its free marginal. This offers a physical interpretation of the Sinkhorn flow as the stochastic dynamics of a particle with law evolving by the nonlocal diffusion of a chemical potential. Simultaneously, it brings a standard suite of functional inequalities characterizing Markov diffusion processes to bear upon its geometry and convergence. We prove an entropy-energy (de Bruijn) identity, a Poincaré inequality, and a Bakry-Émery-type condition under which a logarithmic Sobolev inequality (LSI) holds and implies exponential convergence of the Sinkhorn flow in entropy. We lastly discuss computational applications such as stopping heuristics and latent-space design criteria leveraging the LSI and, returning to the physical interpretation, the possibility of natural systems whose relaxation to equilibrium inherently solves entropic optimal transport or Schrödinger bridge problems.

stat.ML↗

arXiv2024

Spatiotemporal distribution of the glycoprotein pherophorin II reveals stochastic geometry of the growing ECM of $Volvox~carteri$

The evolution of multicellularity involved the transformation of a simple cell wall of unicellular ancestors into a complex, multifunctional extracellular matrix (ECM). A suitable model organism to study the formation and expansion of an ECM during ontogenesis is the multicellular green alga $Volvox~carteri$, which, along with the related volvocine algae, produces a complex, self-organized ECM composed of multiple substructures. These self-assembled ECMs primarily consist of hydroxyproline-rich glycoproteins, a major component of which is pherophorins. To investigate the geometry of the growing ECM, we fused the $yfp$ gene with the gene for pherophorin II (PhII) in $V.~carteri$. Confocal microscopy reveals PhII:YFP localization at key structures within the ECM, including the boundaries of compartments surrounding each somatic cell and the outer surface of the organism. Image analysis during the life cycle allows the stochastic geometry of those growing compartments to be quantified. We find that their areas and aspect ratios exhibit robust gamma distributions and exhibit a transition from a tight polygonal to a looser acircular packing geometry with stable eccentricity over time, evoking parallels and distinctions with the behavior of hydrated foams. These results provide a quantitative benchmark for addressing a general, open question in biology: How do cells produce structures external to themselves in a robust and accurate manner?

cond-mat.soft↗

arXiv2024

Stochastic Voronoi Tessellations as Models for Cellular Neighborhoods in Simple Multicellular Organisms

Recent work on distinct multicellular organisms has revealed a hitherto unknown type of biological noise; rather than a regular arrangement, cellular neighborhood volumes, obtained by Voronoi tessellations of the cell locations, are broadly distributed and consistent with gamma distributions. We propose an explanation for those observations in the case of the alga $Volvox$, whose somatic cells are embedded in an extracellular matrix (ECM) they export. Both a solvable one-dimensional model of ECM growth derived from bursty transcriptional activity and a two-dimensional "Voronoi liquid" model are shown to provide one-parameter families that smoothly interpolate between the empirically-observed near-maximum-entropy gamma distributions and the crystalline limit of Gaussian distributions governed by the central limit theorem. These results highlight a universal consequence of intrinsic biological noise on the architecture of certain tissues.

cond-mat.soft↗

arXiv2023

High Order Mimetic Symplectic Methods For Hamiltonian Systems

Hamiltonian systems are known to conserve the Hamiltonian function, which describes the energy evolution over time. Obtaining a numerical spatio-temporal scheme that accurately preserves the discretized Hamiltonian function is often a challenge. In this paper, the use of high order mimetic spatial schemes is investigated for the numerical solution of Hamiltonian equations. The mimetic operators are based on developing high order discrete analogs of the vector calculus quantities divergence and gradient. The resulting high order operators preserve the properties of their continuum ones, and are therefore said to mimic properties of conservation laws and symmetries. Symplectic fourth order schemes are implemented in this paper for the time integration of Hamiltonian systems. A theoretical framework for the energy preserving nature of the resulting schemes is also presented, followed by numerical examples.

math.NA↗

arXiv2022

Contracting differential equations in weighted Banach spaces

Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, such analysis is generally restricted to norm-contracting systems. We develop a generalization of geodesic contraction rates to Banach spaces using a smoothly-weighted semi-inner product structure on tangent spaces. We show that negative contraction rates in bijectively weighted spaces imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach spaces to establish the existence of fixed points. We show that contraction in surjectively weighted spaces verify non-equilibrium asymptotic properties, such as convergence to finite- and infinite-dimensional subspaces, submanifolds, limit cycles, and phase-locking phenomena. We use contraction rates in weighted Sobolev spaces to establish existence and continuous data dependence in nonlinear PDEs, and pose a method for constructing weak solutions using vanishing one-sided Lipschitz approximations. We discuss applications to control and order reduction of PDEs.

math.DS↗

arXiv2022

Contracting dynamical systems in Banach spaces

Contraction rates of time-varying maps induced by dynamical systems illuminate a wide range of asymptotic properties with applications in stability analysis and control theory. In finite-dimensional smoothly varying inner-product spaces such as $\mathbb{R}^n$ and $\mathbb{C}^n$ with Riemannian metrics, contraction rates can be estimated by upper-bounding the real numerical range of the vector field's Jacobian. However, vector spaces with norms other than $L^2$ commonly arise in the stability analysis of infinite-dimensional systems such as those arising from partial differential equations and continuum mechanics. To this end, we present a unified approach to contraction analysis in Banach spaces using the theory of weighted semi-inner products. We generalize contraction in a geodesic distance to asymptotic stability of perturbations in smoothly varying semi-inner products, and show that the latter is a dynamical invariant similar to the coordinate-invariance of Lyapunov exponents. We show that contraction in particular weighted spaces verifies asymptotic convergence to subspaces and submanifolds, present applications to limit-cycle analysis and phase-locking phenomena, and pose general conditions for inheritance of contraction properties within coupled systems. We discuss contraction rates in Sobolev spaces for retention of regularity in partial differential equations, and suggest a type of weak solution defined by a vanishing contractive term. Lastly, we present an application to machine learning, using weighted semi-inner products to derive stability conditions for functional gradient descent in a Banach space.

math.DS↗

arXiv2021

Strong optical coupling in metallo-dielectric hybrid metasurfaces

Metasurfaces consisting of hybrid metal/dielectric nanostructures carry advantages of both material platforms. The hybrid structures can not only tightly confine subwavelength electromagnetic fields but also lower the absorption losses. Such optical characteristics are difficult to realize in metamaterials with only metal or dielectric structures. Hybrid designs also expand the scope of material choices and the types of optical modes that can be excited in a metasurface, thereby allowing novel light matter interactions. Here we present a metallo-dielectric hybrid metasurface design consisting of a high-index dielectric (silicon) nanodisk array on top of a metal layer (aluminium) separated by a buffer oxide (silica) layer. The dimensions of the nanodisks are tuned to support anapole states and the period of the nanodisk array is designed to excite surface plasmon polariton (SPP) at the metal-buffer oxide interface. The physical dimensions of the Si nanodisk and the array periods are optimized to excite the anapole and the SPP at normal incidence of light in the visible-NIR (400-900 nm) wavelength range. Finite difference time domain (FDTD) simulations show that, when the nanodisk grating is placed at a specific height (~200 nm) from the metal surface, the two modes strongly couple at zero detuning of the resonances. The strong coupling is evident from the avoided crossing of the modes observed in the reflectance spectra and in the spectral profile of light absorption inside the Si nanodisk. A vacuum Rabi splitting of up to ~ 129 meV is achievable by optimizing the diameters of Si nanodisk and the nanodisk array grating period. The proposed metasurface design is promising to realize open cavity strongly coupled optical systems operating at room temperatures.

physics.optics↗

arXiv2020

An MCMC Method for Uncertainty Set Generation via Operator-Theoretic Metrics

Model uncertainty sets are required in many robust optimization problems, such as robust control and prediction with uncertainty, but there is no definite methodology to generate uncertainty sets for nonlinear dynamical systems. In this paper, we propose a method for model uncertainty set generation via Markov chain Monte Carlo. The proposed method samples from distributions over dynamical systems via metrics over transfer operators and is applicable to general nonlinear systems. We adapt Hamiltonian Monte Carlo for sampling high-dimensional transfer operators in a computationally efficient manner. We present numerical examples to validate the proposed method for uncertainty set generation.

math.DS↗