Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties

In this paper we introduce and discuss numerical schemes for the approximation of kinetic equations for flocking behavior with phase transitions that incorporate uncertain quantities. This class of schemes here considered make use of a Monte Carlo approach in the phase space coupled with a stochastic Galerkin expansion in the random space. The proposed methods naturally preserve the positivity of the statistical moments of the solution and are capable to achieve high accuracy in the random space. Several tests on a kinetic alignment model with self propulsion validate the proposed methods both in the homogeneous and inhomogeneous setting, shading light on the influence of uncertainties in phase transition phenomena driven by noise such as their smoothing and confidence band

Reconstruction, forecasting, and stability of chaotic dynamics from partial data

The forecasting and computation of the stability of chaotic systems from partial observations are tasks for which traditional equation-based methods may not be suitable. In this computational paper, we propose data-driven methods to (i) infer the dynamics of unobserved (hidden) chaotic variables (full-state reconstruction); (ii) time forecast the evolution of the full state; and (iii) infer the stability properties of the full state. The tasks are performed with long short-term memory (LSTM) networks, which are trained with observations (data) limited to only part of the state: (i) the low-to-high resolution LSTM (LH-LSTM), which takes partial observations as training input, and requires access to the full system state when computing the loss; and (ii) the physics-informed LSTM (PI-LSTM), which is designed to combine partial observations with the integral formulation of the dynamical system's evolution equations. First, we derive the Jacobian of the LSTMs. Second, we analyse a chaotic partial differential equation, the Kuramoto-Sivashinsky (KS), and the Lorenz-96 system. We show that the proposed networks can forecast the hidden variables, both time-accurately and statistically. The Lyapunov exponents and covariant Lyapunov vectors, which characterize the stability of the chaotic attractors, are correctly inferred from partial observations. Third, the PI-LSTM outperforms the LH-LSTM by successfully reconstructing the hidden chaotic dynamics when the input dimension is smaller or similar to the Kaplan-Yorke dimension of the attractor. This work opens new opportunities for reconstructing the full state, inferring hidden variables, and computing the stability of chaotic systems from partial data

Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks

We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.Comment: 31 page

임의성이 있는 쿠커-스메일 모형에 대한 정량적 해석에 관하여

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 하승열.In this thesis, we introduce random elements into the Cucker-Smale(C-S) model and provide quantitative analyses for those uncertainties. In real applications of the Cucker-Smale dynamics, we can expect that the C-S model contains some intrinsic uncertainties in itself and misses some extrinsic factors that might affect the dynamics of particles. Thus, to provide a better description for the dynamics of a C-S ensemble, one needs to incorporate such uncertain factors to the model and evaluate their effects on the dynamics or stability of the C-S system. To fulfill this, we first consider the macroscopic version of the Cucker-Smale model. Namely, we introduce random inputs from communication weights and initial data into the hydrodynamic Cucker-Smale (HCS) model to yield the random HCS model. Furthermore, we address extrinsic uncertainties in the microscopic and mesoscopic level, respectively. For a microscopic model, we introduce a randomly switching network structure to the Cucker-Smale model and investigate sufficient conditions for the emergence of flocking. As a mesoscopic model, we consider the kinetic Cucker-Smale equation perturbed by multiplicative white noise and study the well-posedness and asymptotic dynamics of solutions.본 학위 논문에서는, 쿠커-스메일 모형에 임의적 요소를 도입하여 그러한 불확실성에 대한 정량적 해석을 제시한다. 쿠커-스메일 모형의 다이나믹스를 실제로 응용함에 있어 우리는 쿠커-스메일 모형 자체가 몇몇 내적 불확실성을 포함하고 있으며 입자들의 다이나믹스에 영항을 줄 수 있는 몇 가지 외부적 요인을 놓치고 있음을 예상할 수 있다. 그러므로 쿠커-스메일 총체의 다이나믹스를 더 잘 서술하기 위해, 이러한 불확실성이 있는 요소를 모형에 도입하여 그것들이 쿠커-스메일 계의 다이나믹스나 안정성에 주는 영향을 평가할 필요가 있다. 이를 달성하기 위해, 우리는 우선 쿠커-스메일 모형의 거시적인 형태를 고려한다. 즉, 우리는 통신 가중치 함수와 초기값에서 오는 임의적 입력치를 유체역학 쿠커-스메일 모형에 포함시켜 임의적 유체역학 쿠커-스메일 모형을 유도한다. 더 나아가 미시적 그리고 중간보기적 단계에서 외적 불확실성에 대해 다룬다. 미시적 모형에 대해서, 쿠커-스메일 모형에 임의로 변하는 네트워크 구조를 도입하여 플로킹의 창발에 대한 충분조건을 알아본다. 중간보기적 단계의 모형으로서, 우리는 곱셈 백색 소음으로 동요된 쿠커-스메일 운동방정식을 고려하고 해의 존재성 및 유일성과 점근적 다이나믹스를 공부한다.1 Introduction 1 2 Preliminaries 9 2.1 Notation 9 2.2 Previous results 10 3 A local sensitivity analysis for the hydrodynamic Cucker-Smale model with random inputs 15 3.1 Pathwise well-posedness of z-variations 16 3.1.1 First-order z-variations 18 3.1.2 Higher-order z-variations 26 3.2 The local sensitivity analysis for stability estimates 32 3.2.1 Higher-order L2-stability 32 3.2.2 L2-stability estimates for z-variations 37 3.3 A local sensitivity analysis for flocking estimate 41 4 On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies 48 4.1 Preliminaries 49 4.1.1 Pathwise dissipative structure 49 4.1.2 A directed graph 52 4.1.3 A scrambling matrix 53 4.1.4 A state transition matrix 54 4.1.5 Previous results 55 4.2 A description of main result 57 4.2.1 Standing assumptions 57 4.2.2 Main result 58 4.3 Emergent behavior of the randomly switching system 61 4.3.1 A matrix formulation 61 4.3.2 Pathwise flocking under a priori assumptions 62 4.3.3 Emergence of stochastic flocking 70 5 Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communication 74 5.1 Preliminaries 75 5.1.1 Derivation of the SPDE 75 5.1.2 Presentation of main results 78 5.1.3 Elementary lemmas 80 5.2 A priori estimates for classical solutions 82 5.2.1 Quantitative estimates for classical solutions 86 5.3 Global well-posedness and asymptotic dynamics of strong solutions 92 5.3.1 Construction of approximate solutions 94 5.3.2 Estimates on approximate solutions 95 5.3.3 Proof of Theorem 5.1.3 103 6 Conclusion and future works 110 Appendix A Detailed proof of Chapter 3 112 A.1 Proof of Lemma 3.1.2 112 A.2 Proof of Lemma 3.1.5 115 A.3 Proof of Lemma 3.2.4 119 A.4 Proof of Theorem 3.3.2 121 Appendix B Detailed proof of Chapter 5 124 B.1 A proof of Theorem 5.2.1 124 B.2 A proof of Proposition 5.3.3 129 Bibliography 133Docto

Efficient sensitivity analysis of chaotic systems and applications to control and data assimilation

Sensitivity analysis is indispensable for aeronautical engineering applications that require optimisation, such as flow control and aircraft design. The adjoint method is the standard approach for sensitivity analysis, but it cannot be used for chaotic systems. This is due to the high sensitivity of the system trajectory to input perturbations; a characteristic of many turbulent systems. Although the instantaneous outputs are sensitive to input perturbations, the sensitivities of time-averaged outputs are well-defined for uniformly hyperbolic systems, but existing methods to compute them cannot be used. Recently, a set of alternative approaches based on the shadowing property of dynamical systems was proposed to compute sensitivities. These approaches are computationally expensive, however. In this thesis, the Multiple Shooting Shadowing (MSS) [1] approach is used, and the main aim is to develop computational tools to allow for the implementation of MSS to large systems. The major contributor to the cost of MSS is the solution of a linear matrix system. The matrix has a large condition number, and this leads to very slow convergence rates for existing iterative solvers. A preconditioner was derived to suppress the condition number, thereby accelerating the convergence rate. It was demonstrated that for the chaotic 1D Kuramoto Sivashinsky equation (KSE), the rate of convergence was almost independent of the #DOF and the trajectory length. Most importantly, the developed solution method relies only on matrix-vector products. The adjoint version of the preconditioned MSS algorithm was then coupled with a gradient descent method to compute a feedback control matrix for the KSE. The adopted formulation allowed all matrix elements to be computed simultaneously. Within a single iteration, a stabilising matrix was computed. Comparisons with standard linear quadratic theory (LQR) showed remarkable similarities (but also some differences) in the computed feedback control kernels. A preconditioned data assimilation algorithm was then derived for state estimation purposes. The preconditioner was again shown to accelerate the rate of convergence significantly. Accurate state estimations were computed for the Lorenz system.Open Acces

Coupling of morphology to surface transport in ion-beam irradiated surfaces. I. Oblique incidence

We propose and study a continuum model for the dynamics of amorphizable surfaces undergoing ion-beam sputtering (IBS) at intermediate energies and oblique incidence. After considering the current limitations of more standard descriptions in which a single evolution equation is posed for the surface height, we overcome (some of) them by explicitly formulating the dynamics of the species that transport along the surface, and by coupling it to that of the surface height proper. In this we follow recent proposals inspired by ``hydrodynamic'' descriptions of pattern formation in aeolian sand dunes and ion-sputtered systems. From this enlarged model, and by exploiting the time-scale separation among various dynamical processes in the system, we derive a single height equation in which coefficients can be related to experimental parameters. This equation generalizes those obtained by previous continuum models and is able to account for many experimental features of pattern formation by IBS at oblique incidence, such as the evolution of the irradiation-induced amorphous layer, transverse ripple motion with non-uniform velocity, ripple coarsening, onset of kinetic roughening and other. Additionally, the dynamics of the full two-field model is compared with that of the effective interface equation.Comment: 23 pages, 14 figures. Movies of figures 6, 7, and 8 available at http://gisc.uc3m.es/~javier/Movies

Identification and control of dynamical systems

Practical methods, based upon linear systems theory, are explored for applications to nonlinear phenomena and are extended to a larger class of problems. An algorithm for stabilizing, characterizing, and tracking unstable steady states and periodic orbits in multidimensional dynamical systems is developed and applied to stabilize and characterize an unstable four-cell flame front of the Kuramoto-Sivashinsky equation with six unstable degrees of freedom. A new method is presented for probing chemical reaction mechanisms experimentally with perturbations and measurements of the response. Time series analysis and the methods of linear control theory are used to determine the Jacobian matrix of a reaction at a stable stationary state subjected to random perturbations. The method is demonstrated with time series of a model system, and its performance in the presence of noise is examined. A new theory based on the construction of a multitude of linear models, each serving to represent one small region of the phase space, is presented together. Details of its implementation are presented in predicting chaotic Kuramoto-Sivashinsky wave fronts, demonstrating how it overcomes some of the problems associated with high dimensionality phase spaces. Motivated by the relationship between nonlinear prediction methods and the capabilities of neural systems, we demonstrate the possible role of nonlinear phenomena in the morphogenesis of neural tracts

Mean-Field-Type Games in Engineering

A mean-field-type game is a game in which the instantaneous payoffs and/or the state dynamics functions involve not only the state and the action profile but also the joint distributions of state-action pairs. This article presents some engineering applications of mean-field-type games including road traffic networks, multi-level building evacuation, millimeter wave wireless communications, distributed power networks, virus spread over networks, virtual machine resource management in cloud networks, synchronization of oscillators, energy-efficient buildings, online meeting and mobile crowdsensing.Comment: 84 pages, 24 figures, 183 references. to appear in AIMS 201