arXiv subjects

Amirhossein Taghvaei

Publications and source records attributed to Amirhossein Taghvaei.

At least 55 records · Page 3Linked to original sources

arXiv2022

Thermodynamic engine powered by anisotropic fluctuations

The purpose of this work is to present the concept of an autonomous Stirling-like engine powered by anisotropy of thermodynamic fluctuations. Specifically, simultaneous contact of a thermodynamic system with two heat baths along coupled degrees of freedom generates torque and circulatory currents -- an arrangement referred to as a Brownian gyrator. The embodiment that constitutes the engine includes an inertial wheel to sustain rotary motion and average out the generated fluctuating torque, ultimately delivering power to an external load. We detail an electrical model for such an engine that consists of two resistors in different temperatures and three reactive elements in the form of variable capacitors. The resistors generate Johnson-Nyquist current fluctuations that power the engine, while the capacitors generate driving forces via a coupling of their dielectric material with the inertial wheel. A proof-of-concept is established via stability analysis to ensure the existence of a stable periodic orbit generating sustained power output. We conclude by drawing a connection to the dynamics of a damped pendulum with constant torque and to those of a macroscopic Stirling engine. The sought insights aim at nano-engines and biological processes that are similarly powered by anisotropy in temperature and chemical potentials.

cond-mat.stat-mech↗

arXiv2022

Optimality vs Stability Trade-off in Ensemble Kalman Filters

This paper is concerned with optimality and stability analysis of a family of ensemble Kalman filter (EnKF) algorithms. EnKF is commonly used as an alternative to the Kalman filter for high-dimensional problems, where storing the covariance matrix is computationally expensive. The algorithm consists of an ensemble of interacting particles driven by a feedback control law. The control law is designed such that, in the linear Gaussian setting and asymptotic limit of infinitely many particles, the mean and covariance of the particles follow the exact mean and covariance of the Kalman filter. The problem of finding a control law that is exact does not have a unique solution, reminiscent of the problem of finding a transport map between two distributions. A unique control law can be identified by introducing control cost functions, that are motivated by the optimal transportation problem or Schrödinger bridge problem. The objective of this paper is to study the relationship between optimality and long-term stability of a family of exact control laws. Remarkably, the control law that is optimal in the optimal transportation sense leads to an EnKF algorithm that is not stable.

math.OC↗

arXiv2021

Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks

Wasserstein Barycenter is a principled approach to represent the weighted mean of a given set of probability distributions, utilizing the geometry induced by optimal transport. In this work, we present a novel scalable algorithm to approximate the Wasserstein Barycenters aiming at high-dimensional applications in machine learning. Our proposed algorithm is based on the Kantorovich dual formulation of the Wasserstein-2 distance as well as a recent neural network architecture, input convex neural network, that is known to parametrize convex functions. The distinguishing features of our method are: i) it only requires samples from the marginal distributions; ii) unlike the existing approaches, it represents the Barycenter with a generative model and can thus generate infinite samples from the barycenter without querying the marginal distributions; iii) it works similar to Generative Adversarial Model in one marginal case. We demonstrate the efficacy of our algorithm by comparing it with the state-of-art methods in multiple experiments.

cs.LG↗

arXiv2021

An optimal control approach to particle filtering

We present a novel particle filtering framework for continuous-time dynamical systems with continuous-time measurements. Our approach is based on the duality between estimation and optimal control, which allows reformulating the estimation problem over a fixed time window into an optimal control problem. The resulting optimal control problem has a cost function that depends on the measurements and the closed-loop dynamics under optimal control coincides with the posterior distribution over the trajectories for the corresponding estimation problem. This type of stochastic optimal control problem can be solved using a remarkable technique known as path integral control. By recursively solving these optimal control problems using path integral control as new measurements become available we obtain an optimal control-based particle filtering algorithm. A distinguishing feature of the proposed method is that it uses the measurements over a finite-length time window instead of a single measurement for the estimation at each time step, resembling the batch methods of filtering, and improving fault tolerance. The efficacy of our algorithm is illustrated with several numerical examples.

math.OC↗

arXiv2021

Energy harvesting from anisotropic fluctuations

We consider a rudimentary model for a heat engine, known as the Brownian gyrator, that consists of an overdamped system with two degrees of freedom in an anisotropic temperature field. Whereas the hallmark of the gyrator is a nonequilibrium steady-state curl-carrying probability current that can generate torque, we explore the coupling of this natural gyrating motion with a periodic actuation potential for the purpose of extracting work. We show that path-lengths traversed in the manifold of thermodynamic states, measured in a suitable Riemannian metric, represent dissipative losses, while area integrals of a work-density quantify work being extracted. Thus, the maximal amount of work that can be extracted relates to an isoperimetric problem, trading off area against length of an encircling path. We derive an isoperimetric inequality that provides a universal bound on the efficiency of all cyclic operating protocols, and a bound on how fast a closed path can be traversed before it becomes impossible to extract positive work. The analysis presented provides guiding principles for building autonomous engines that extract work from anistropic fluctuations.

cond-mat.stat-mech↗

arXiv2021

Stein particle filtering

We present a new particle filtering algorithm for nonlinear systems in the discrete-time setting. Our algorithm is based on the Stein variational gradient descent (SVGD) framework, which is a general approach to sample from a target distribution. We merge the standard two-step paradigm in particle filtering into one step so that SVGD can be used. A distinguishing feature of the proposed algorithm is that, unlike most particle filtering methods, all the particles at any time step are equally weighted and thus no update on the weights is needed. We further extended our algorithm to allow for updating previous particles within a sliding window. This strategy may improve the reliability of the algorithm with respect to unexpected disturbance in the dynamics or outlier-measurements. The efficacy of the proposed algorithms is illustrated through several numerical examples in comparison with a standard particle filtering method.

cs.CE↗

arXiv2021

On the relation between information and power in stochastic thermodynamic engines

The common saying, that information is power, takes a rigorous form in stochastic thermodynamics, where a quantitative equivalence between the two helps explain the paradox of Maxwell's demon in its ability to reduce entropy. In the present paper, we build on earlier work on the interplay between the relative cost and benefits of information in producing work in cyclic operation of thermodynamic engines (by Sandberg etal. 2014). Specifically, we study the general case of overdamped particles in a time-varying potential (control action) in feedback that utilizes continuous measurements (nonlinear filtering) of a thermodynamic ensemble, to produce suitable adaptations of the second law of thermodynamics that involve information.

cond-mat.stat-mech↗

arXiv2021

Optimal Transportation Methods in Nonlinear Filtering: The feedback particle filter

Feedback particle filter (FPF) is a Monte-Carlo (MC) algorithm to approximate the solution of a stochastic filtering problem. In contrast to conventional particle filters, the Bayesian update step in FPF is implemented via a mean-field type feedback control law. The objective for this paper is to situate the development of FPF and related controlled interacting particle system algorithms within the framework of optimal transportation theory. Starting from the simplest setting of the Bayes' update formula, a coupling viewpoint is introduced to construct particle filters. It is shown that the conventional importance sampling resampling particle filter implements an independent coupling. Design of optimal couplings is introduced first for the simple Gaussian settings and subsequently extended to derive the FPF algorithm. The final half of the paper provides a review of some of the salient aspects of the FPF algorithm including the feedback structure, algorithms for gain function design, and comparison with conventional particle filters. The comparison serves to illustrate the benefit of feedback in particle filtering.

eess.SY↗

arXiv2021

Feedback Particle Filter for Collective Inference

The purpose of this paper is to describe the feedback particle filter algorithm for problems where there are a large number ($M$) of non-interacting agents (targets) with a large number ($M$) of non-agent specific observations (measurements) that originate from these agents. In its basic form, the problem is characterized by data association uncertainty whereby the association between the observations and agents must be deduced in addition to the agent state. In this paper, the large-$M$ limit is interpreted as a problem of collective inference. This viewpoint is used to derive the equation for the empirical distribution of the hidden agent states. A feedback particle filter (FPF) algorithm for this problem is presented and illustrated via numerical simulations. Results are presented for the Euclidean and the finite state-space cases, both in continuous-time settings. The classical FPF algorithm is shown to be the special case (with $M=1$) of these more general results. The simulations help show that the algorithm well approximates the empirical distribution of the hidden states for large $M$.

math.OC↗

arXiv2021

Harvesting energy from a periodic heat bath

The context of the present paper is stochastic thermodynamics - an approach to nonequilibrium thermodynamics rooted within the broader framework of stochastic control. In contrast to the classical paradigm of Carnot engines, we herein propose to consider thermodynamic processes with periodic continuously varying temperature of a heat bath and study questions of maximal power and efficiency for two idealized cases, overdamped (first-order) and underdamped (second-order) stochastic models. We highlight properties of optimal periodic control, derive and numerically validate approximate formulae for the optimal performance (power and efficiency).

eess.SY↗

arXiv2020

Deep FPF: Gain function approximation in high-dimensional setting

In this paper, we present a novel approach to approximate the gain function of the feedback particle filter (FPF). The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian. The numerical problem is to approximate the exact gain function using only finitely many particles sampled from the probability distribution. Inspired by the recent success of the deep learning methods, we represent the gain function as a gradient of the output of a neural network. Thereupon considering a certain variational formulation of the Poisson equation, an optimization problem is posed for learning the weights of the neural network. A stochastic gradient algorithm is described for this purpose. The proposed approach has two significant properties/advantages: (i) The stochastic optimization algorithm allows one to process, in parallel, only a batch of samples (particles) ensuring good scaling properties with the number of particles; (ii) The remarkable representation power of neural networks means that the algorithm is potentially applicable and useful to solve high-dimensional problems. We numerically establish these two properties and provide extensive comparison to the existing approaches.

cs.LG↗

arXiv2020

Optimal transport mapping via input convex neural networks

In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel minimax optimization. Building upon recent advances in the field of input convex neural networks, we propose a new framework where the gradient of one convex function represents the optimal transport mapping. Numerical experiments confirm that we learn the optimal transport mapping. This approach ensures that the transport mapping we find is optimal independent of how we initialize the neural networks. Further, target distributions from a discontinuous support can be easily captured, as gradient of a convex function naturally models a {\em discontinuous} transport mapping.

cs.LG↗

arXiv2020

Lasso formulation of the shortest path problem

The shortest path problem is formulated as an $l_1$-regularized regression problem, known as lasso. Based on this formulation, a connection is established between Dijkstra's shortest path algorithm and the least angle regression (LARS) for the lasso problem. Specifically, the solution path of the lasso problem, obtained by varying the regularization parameter from infinity to zero (the regularization path), corresponds to shortest path trees that appear in the bi-directional Dijkstra algorithm. Although Dijkstra's algorithm and the LARS formulation provide exact solutions, they become impractical when the size of the graph is exceedingly large. To overcome this issue, the alternating direction method of multipliers (ADMM) is proposed to solve the lasso formulation. The resulting algorithm produces good and fast approximations of the shortest path by sacrificing exactness that may not be absolutely essential in many applications. Numerical experiments are provided to illustrate the performance of the proposed approach.

math.OC↗

arXiv2020

On the Lyapunov Foster criterion and Poincaré inequality for Reversible Markov Chains

This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincaré inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.

math.PR↗

arXiv2020

Maximal power output of a stochastic thermodynamic engine

Classical thermodynamics aimed to quantify the efficiency of thermodynamic engines by bounding the maximal amount of mechanical energy produced compared to the amount of heat required. While this was accomplished early on, by Carnot and Clausius, the more practical problem to quantify limits of power that can be delivered, remained elusive due to the fact that quasistatic processes require infinitely slow cycling, resulting in a vanishing power output. Recent insights, drawn from stochastic models, appear to bridge the gap between theory and practice in that they lead to physically meaningful expressions for the dissipation cost in operating a thermodynamic engine over a finite time window. Building on this framework of {\em stochastic thermodynamics} we derive bounds on the maximal power that can be drawn by cycling an overdamped ensemble of particles via a time-varying potential while alternating contact with heat baths of different temperature ($T_c$ cold, and $T_h$ hot). Specifically, assuming a suitable bound $M$ on the spatial gradient of the controlling potential, we show that the maximal achievable power is bounded by $\frac{M}{8}(\frac{T_h}{T_c}-1)$. Moreover, we show that this bound can be reached to within a factor of $(\frac{T_h}{T_c}-1)/(\frac{T_h}{T_c}+1)$ by operating the cyclic thermodynamic process with a quadratic potential.

math.OC↗

arXiv2019

An Optimal Transport Formulation of the Ensemble Kalman Filter

Controlled interacting particle systems such as the ensemble Kalman filter (EnKF) and the feedback particle filter (FPF) are numerical algorithms to approximate the solution of the nonlinear filtering problem in continuous time. The distinguishing feature of these algorithms is that the Bayesian update step is implemented using a feedback control law. It has been noted in the literature that the control law is not unique. This is the main problem addressed in this paper. To obtain a unique control law, the filtering problem is formulated here as an optimal transportation problem. An explicit formula for the (mean-field type) optimal control law is derived in the linear Gaussian setting. Comparisons are made with the control laws for different types of EnKF algorithms described in the literature. Via empirical approximation of the mean-field control law, a finite-$N$ controlled interacting particle algorithm is obtained. For this algorithm, the equations for empirical mean and covariance are derived and shown to be identical to the Kalman filter. This allows strong conclusions on convergence and error properties based on the classical filter stability theory for the Kalman filter. It is shown that, under certain technical conditions, the mean squared error (m.s.e.) converges to zero even with a finite number of particles. A detailed propagation of chaos analysis is carried out for the finite-$N$ algorithm. The analysis is used to prove weak convergence of the empirical distribution as $N\rightarrow\infty$. For a certain simplified filtering problem, analytical comparison of the m.s.e. with the importance sampling-based algorithms is described. The analysis helps explain the favorable scaling properties of the control-based algorithms reported in several numerical studies in recent literature.

eess.SY↗

arXiv2019

Bio-inspired Learning of Sensorimotor Control for Locomotion

This paper presents a bio-inspired central pattern generator (CPG)-type architecture for learning optimal maneuvering control of periodic locomotory gaits. The architecture is presented here with the aid of a snake robot model problem involving planar locomotion of coupled rigid body systems. The maneuver involves clockwise or counterclockwise turning from a nominally straight path. The CPG circuit is realized as a coupled oscillator feedback particle filter. The collective dynamics of the filter are used to approximate a posterior distribution that is used to construct the optimal control input for maneuvering the robot. A Q-learning algorithm is applied to learn the approximate optimal control law. The issues surrounding the parametrization of the Q-function are discussed. The theoretical results are illustrated with numerics for a 5-link snake robot system.

eess.SY↗

arXiv2019

Diffusion map-based algorithm for Gain function approximation in the Feedback Particle Filter

Feedback particle filter (FPF) is a numerical algorithm to approximate the solution of the nonlinear filtering problem in continuous-time settings. In any numerical implementation of the FPF algorithm, the main challenge is to numerically approximate the so-called gain function. A numerical algorithm for gain function approximation is the subject of this paper. The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian $Δ_ρ$. The numerical problem is to approximate this solution using {\em only} finitely many particles sampled from the probability distribution $ρ$. A diffusion map-based algorithm was proposed by the authors in a prior work to solve this problem. The algorithm is named as such because it involves, as an intermediate step, a diffusion map approximation of the exact semigroup $e^{Δ_ρ}$. The original contribution of this paper is to carry out a rigorous error analysis of the diffusion map-based algorithm. The error is shown to include two components: bias and variance. The bias results from the diffusion map approximation of the exact semigroup. The variance arises because of finite sample size. Scalings and upper bounds are derived for bias and variance. These bounds are then illustrated with numerical experiments that serve to emphasize the effects of problem dimension and sample size. The proposed algorithm is applied to two filtering examples and comparisons provided with the sequential importance resampling (SIR) particle filter.

math.OC↗