4,340 research outputs found
Algorithms and complexity for approximately counting hypergraph colourings and related problems
The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows.
• When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard.
• When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist
Quantum-Classical hybrid systems and their quasifree transformations
The focus of this work is the description of a framework for quantum-classical hybrid systems.
The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case.
Here, we are going to solve two main tasks:
The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure.
Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined.
We start with a general introduction to operator algebras and algebraic quantum theory.
Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems.
Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism.
The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system.
Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz:
We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system.
Then, we present solutions for our initial tasks.
We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem.
After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra.
While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose.
They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture.
The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity.
We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations.
All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Provably Efficient Generalized Lagrangian Policy Optimization for Safe Multi-Agent Reinforcement Learning
We examine online safe multi-agent reinforcement learning using constrained
Markov games in which agents compete by maximizing their expected total rewards
under a constraint on expected total utilities. Our focus is confined to an
episodic two-player zero-sum constrained Markov game with independent
transition functions that are unknown to agents, adversarial reward functions,
and stochastic utility functions. For such a Markov game, we employ an approach
based on the occupancy measure to formulate it as an online constrained
saddle-point problem with an explicit constraint. We extend the Lagrange
multiplier method in constrained optimization to handle the constraint by
creating a generalized Lagrangian with minimax decision primal variables and a
dual variable. Next, we develop an upper confidence reinforcement learning
algorithm to solve this Lagrangian problem while balancing exploration and
exploitation. Our algorithm updates the minimax decision primal variables via
online mirror descent and the dual variable via projected gradient step and we
prove that it enjoys sublinear rate for
both regret and constraint violation after playing episodes of the game.
Here, is the horizon of each episode, and are the
state/action space sizes of the min-player and the max-player, respectively. To
the best of our knowledge, we provide the first provably efficient online safe
reinforcement learning algorithm in constrained Markov games.Comment: 59 pages, a full version of the main paper in the 5th Annual
Conference on Learning for Dynamics and Contro
Phenomenological analysis of simple ion channel block in large populations of uncoupled cardiomyocytes
Current understanding of arrhythmia mechanisms and design of anti-arrhythmic drug therapies hinges on the assumption that myocytes from the same region of a single heart have similar, if not identical, action potential waveforms and drug responses. On the contrary, recent experiments reveal significant heterogeneity in uncoupled healthy myocytes both from different hearts as well as from identical regions within a single heart. In this work, a methodology is developed for quantifying the individual electrophysiological properties of large numbers of uncoupled cardiomyocytes under ion channel block in terms of the parameters values of a conceptual fast-slow model of electrical excitability. The approach is applied to a population of nearly 500 rabbit ventricular myocytes for which action potential duration (APD) before and after the application of the drug nifedipine was experimentally measured (Lachaud et al., 2022, Cardiovasc. Res.). To this end, drug action is represented by a multiplicative factor to an effective ion conductance, a closed form asymptotic expression for APD is derived and inverted to determine model parameters as functions of APD and ΔAPD (drug-induced change in APD) for each myocyte. Two free protocol-related quantities are calibrated to experiment using an adaptive-domain procedure based on an original assumption of optimal excitability. The explicit APD expression and the resulting set of model parameter values allow (a) direct evaluation of conditions necessary to maintain fixed APD or ΔAPD, (b) predictions of the proportion of cells remaining excitable after drug application, (c) predictions of stimulus period dependency and (d) predictions of dose-response curves, the latter being in agreement with additional experimental data
A mixed-mode dependent interface and phase-field damage model for solids with inhomogeneities
The developed computational approach is capable of initiating and propagating
cracks inside materials and along material interfaces of general multi-domain
structures under quasi-static conditions. Special attention is paid to
particular situation of a solid with inhomogeneities. Description of the
fracture processes are based on the theory of material damage. It introduces
two independent damage parameters to distinguish between interface and internal
cracks. The parameter responsible for interface cracks is defined in a thin
adhesive layer of the interface and renders relation between stress and strain
quantities in fashion of cohesive zone models.The second parameter is defined
inside material domains and it is founded on the theory of phase-field fracture
guaranteeing the material damage to occur in a thin material strip introducing
a regularised model of internal cracks. Additional property of both interface
and phase-field damage is their capability to distinguish between fracture
modes which is useful if the structures is subjected to combined loading. The
solution methodology is based on a variational approach which allows
implementation of non-linear programming optimisation into standard methods of
finite-element discretisation and time stepping method.Computational
implementation is prepared in MATLAB whose numerical data validate developed
formulation for analysis of problems of fracture in multi-domain elements of
structures.Comment: 24 pages, 19 figures, to be published in Theoretical and Applied
Fracture Mechanic
Optimal speed trajectory and energy management control for connected and automated vehicles
Connected and automated vehicles (CAVs) emerge as a promising solution to improve urban mobility, safety, energy efficiency, and passenger comfort with the development of communication technologies, such as vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I). This thesis proposes several control approaches for CAVs with electric powertrains, including hybrid electric vehicles (HEVs) and battery electric vehicles (BEVs), with the main objective to improve energy efficiency by optimising vehicle speed trajectory and energy management system. By types of vehicle control, these methods can be categorised into three main scenarios, optimal energy management for a single CAV (single-vehicle), energy-optimal strategy for the vehicle following scenario (two-vehicle), and optimal autonomous intersection management for CAVs (multiple-vehicle).
The first part of this thesis is devoted to the optimal energy management for a single automated series HEV with consideration of engine start-stop system (SSS) under battery charge sustaining operation. A heuristic hysteresis power threshold strategy (HPTS) is proposed to optimise the fuel economy of an HEV with SSS and extra penalty fuel for engine restarts. By a systematic tuning process, the overall control performance of HPTS can be fully optimised for different vehicle parameters and driving cycles.
In the second part, two energy-optimal control strategies via a model predictive control (MPC) framework are proposed for the vehicle following problem. To forecast the behaviour of the preceding vehicle, a neural network predictor is utilised and incorporated into a nonlinear MPC method, of which the fuel and computational efficiencies are verified to be effective through comparisons of numerical examples between a practical adaptive cruise control strategy and an impractical optimal control method. A robust MPC (RMPC) via linear matrix inequality (LMI) is also utilised to deal with the uncertainties existing in V2V communication and modelling errors. By conservative relaxation and approximation, the RMPC problem is formulated as a convex semi-definite program, and the simulation results prove the robustness of the RMPC and the rapid computational efficiency resorting to the convex optimisation.
The final part focuses on the centralised and decentralised control frameworks at signal-free intersections, where the energy consumption and the crossing time of a group of CAVs are minimised. Their crossing order and velocity trajectories are optimised by convex second-order cone programs in a hierarchical scheme subject to safety constraints. It is shown that the centralised strategy with consideration of turning manoeuvres is effective and outperforms a benchmark solution invoking the widely used first-in-first-out policy. On the other hand, the decentralised method is proposed to further improve computational efficiency and enhance the system robustness via a tube-based RMPC. The numerical examples of both frameworks highlight the importance of examining the trade-off between energy consumption and travel time, as small compromises in travel time could produce significant energy savings.Open Acces
Learning and Control of Dynamical Systems
Despite the remarkable success of machine learning in various domains in recent years, our understanding of its fundamental limitations remains incomplete. This knowledge gap poses a grand challenge when deploying machine learning methods in critical decision-making tasks, where incorrect decisions can have catastrophic consequences. To effectively utilize these learning-based methods in such contexts, it is crucial to explicitly characterize their performance. Over the years, significant research efforts have been dedicated to learning and control of dynamical systems where the underlying dynamics are unknown or only partially known a priori, and must be inferred from collected data. However, much of these classical results have focused on asymptotic guarantees, providing limited insights into the amount of data required to achieve desired control performance while satisfying operational constraints such as safety and stability, especially in the presence of statistical noise.
In this thesis, we study the statistical complexity of learning and control of unknown dynamical systems. By utilizing recent advances in statistical learning theory, high-dimensional statistics, and control theoretic tools, we aim to establish a fundamental understanding of the number of samples required to achieve desired (i) accuracy in learning the unknown dynamics, (ii) performance in the control of the underlying system, and (iii) satisfaction of the operational constraints such as safety and stability. We provide finite-sample guarantees for these objectives and propose efficient learning and control algorithms that achieve the desired performance at these statistical limits in various dynamical systems. Our investigation covers a broad range of dynamical systems, starting from fully observable linear dynamical systems to partially observable linear dynamical systems, and ultimately, nonlinear systems.
We deploy our learning and control algorithms in various adaptive control tasks in real-world control systems and demonstrate their strong empirical performance along with their learning, robustness, and stability guarantees. In particular, we implement one of our proposed methods, Fourier Adaptive Learning and Control (FALCON), on an experimental aerodynamic testbed under extreme turbulent flow dynamics in a wind tunnel. The results show that FALCON achieves state-of-the-art stabilization performance and consistently outperforms conventional and other learning-based methods by at least 37%, despite using 8 times less data. The superior performance of FALCON arises from its physically and theoretically accurate modeling of the underlying nonlinear turbulent dynamics, which yields rigorous finite-sample learning and performance guarantees. These findings underscore the importance of characterizing the statistical complexity of learning and control of unknown dynamical systems.</p
Dynamical analysis of mushroom bifurcations: deterministic and stochastic approaches
Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Àlex Haro i Josep Sardanyés i Cayuela[en] Bifurcation theory has found contemporary applications in synthetic biology, particularly in the field of biosensors [43]. The aim of this thesis is to expand upon the framework presented in the referenced paper, which introduces a model depicting the behavior of mushroom bifurcations. The mushroom bifurcation diagram exhibits four saddle-node bifurcations and involves bistability. Our goal is to develop a comprehensive mathematical formalism that can effectively describe this behavior, both deterministically and stochastically. By doing so, we seek to uncover additional properties regarding the transients exhibited by these biosensors, specifically focusing on optimizing their timer-effect, memory properties, and signaling capabilities.
We will introduce stochastic dynamics by considering intrinsic noise in the molecular processes, allowing us to investigate the slowing-down effects in the vicinity of the saddle-nodes and transcritical bifurcations. To conduct this study, we will use three fundamental mathematical tools, which can be regarded as the backbone of our analysis. These mathematical vertebrae include the Lemma of Morse, the Weierstrass Preparation Theorem and, most notably, the Implicit Function Theorem. Through this rigorous analysis, we aim to enhance our understanding of the underlying dynamics of these biosensors and facilitate their further improvement and utilization in various applications
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