Regular sensitivity calculation and gradient-based optimization of chaotic dynamical systems

Chung, Seung Whan

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https://hdl.handle.net/2142/113952 Copy

Description

Title

Regular sensitivity calculation and gradient-based optimization of chaotic dynamical systems

Author(s)

Chung, Seung Whan

Issue Date

2021-09-15

Director of Research (if dissertation) or Advisor (if thesis)

Freund, Jonathan B

Doctoral Committee Chair(s)

Freund, Jonathan B

Committee Member(s)

• Bodony, Daniel J
• Goza, Andres
• Cyr, Eric C
• Bond, Stephen D

Department of Study

Mechanical Sci & Engineering

Discipline

Theoretical & Applied Mechans

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Chaos
• Turbulence
• Plasma kinetics
• Particle-in-cell
• Sensitivity
• Optimization
• Control

Abstract

A gradient of a quantity-of-interest J with respect to problem parameters can augment the utility of a predictive simulation. By itself, the gradient provides sensitivity information to parameters, which can aid uncertainty quantification. Gradient-based optimization can be used in both scientific and engineering applications, including design optimization, data-assimilated modeling and nonmodal stability analysis. However, obtaining useful gradients for chaotic systems is challenging. The extreme sensitivity to perturbations that defines chaos amplifies the gradient exponentially in time, which impedes both sensitivity analysis and gradient-based optimization. Fundamentally, any J defined in a chaotic system becomes highly non-convex in time. For such non-convex J, Taylor expansions are useful only in small neighborhoods, which restricts the utility of the gradients, even if computed exactly. Thus they do not indicate a useful parametric sensitivity or guidance toward a useful optimum. We examine this challenge and investigate routes to circumvent these challenges in two applications. The first is sensitivity computation in particle-in-cell (PIC) simulations involving plasma kinetics. PIC is attractive for representing non-equilibrium plasma distributions in the six-dimensional velocity--position phase-space. To do this, Lagrangian simulation particles represent the position and velocity distribution in a statistical sense. However, computing sensitivity for PIC methods is challenging due to the chaotic dynamics of these particles, and sensitivity techniques remain underdeveloped compared to those for Eulerian discretizations. This challenge is examined from a dual particle--continuum perspective that motivates a new sensitivity discretization. Two routes to sensitivity computation are presented and compared: a direct fully-Lagrangian particle-exact approach provides sensitivities of each particle trajectory, and a new particle-pdf discretization. The new formulation involves a continuum perspective but it is discretized by particles to take the advantages of the same type of Lagrangian particle description leveraged by PIC methods. Since the sensitivity particles in this approach are only indirectly linked to the plasma-PIC particles, they can be positioned and weighted independently for efficiency and accuracy. The corresponding numerical algorithms are presented in mathematical detail. The advantage of the particle-pdf approach in avoiding the spurious chaotic sensitivity of the particle-exact approach is demonstrated for Debye shielding and sheath configurations. In essence, the continuum perspective makes implicit the distinctness of the particles, which is irrelevant to most prediction goals. In this way it circumvents the Lyapunov instability of the N-body PIC system. The cost of the particle-pdf approach is comparable to the baseline PIC simulation. The other case considered is optimal control of turbulent flow. Evidence supports the possibility of control of turbulence in some applications, in that there seem to be useful, larger-scale components of the flow, which are less chaotic, in the midst of smaller-scale chaotic turbulence fluctuations. While there have been many attempts to extract model descriptions of such components from the full dynamics, such models are often limited in their applicability or accounting for nonlinearity of turbulence. Thus the full dynamics of turbulent flow is needed to be accurately predictive in simulations. However, in this case the sensitivity of the more chaotic turbulent fluctuations masks that of the useful component of the flow control. This challenge is illustrated with a model control problem of the Lorenz system and analyzed in two aspects: the growth of gradients and non-convexity of J. The horseshoe mapping of chaotic dynamical systems is identified as the root-cause mechanism for both aspects of the challenge, and its impact is quantitatively evaluated in various chaotic flow systems, ranging from the Kuramoto--Sivashinsky Equation to a three-dimensional turbulent Kolmogorov flow. A new optimization framework is proposed based on a penalty-based method. In essence, the simulation time is split into multiple intervals and auxiliary states are introduced at intermediate time points, at which the governing equation is not strictly constrained, thus introducing discontinuities in time. These discontinuities allows J to be more convex, thus enlarging search scale in the optimization space. They are exploited in this sense then gradually suppressed with increasingly stronger penalty. This multi-step penalty-based optimization is first demonstrated with a one-dimensional logistic map and the Lorenz example. Then its effectiveness is further demonstrated for more complex chaotic systems and ultimately for turbulent Kolmogorov flow. The proposed method finds a solution that suppresses large-scale pressure fluctuations without laminarization, which suggests its ability to target useful components of the flow in the midst of chaotic turbulence, thereby showing its potential for practical turbulent flow controls. It far outperforms a simple gradient-based search.

Graduation Semester

2021-12

Type of Resource

Thesis

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http://hdl.handle.net/2142/113952

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Copyright 2021 by Seung Whan Chung. All rights reserved.

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