Detection and prediction of equilibrium states in kinetic plasma simulations via mode tracking using reduced-order dynamic mode decomposition
Introduction
Kinetic plasma simulations are important for a wide range of applications, including but not limited to the design and analysis of high-power microwave sources, particle accelerators, laser ignited devices, and ionosphere and magnetosphere problems [1], [2], [3], [4], [5], [6], [7]. Electromagnetic particle-in-cell (EMPIC) algorithms are typically used for simulating kinetic collisionless plasmas governed by Maxwell-Vlasov equations. EMPIC algorithms compute the electromagnetic field on the spatial mesh based on a discretized form of Maxwell's equations while simultaneously updating, via a kinetic model based on the Lorentz force equation, the velocity and position of computational superparticles that effect a coarse-graining of the phase space of charged particles in the plasma [8], [9], [10], [11], [12]. The inherent nonlinearity and multi-scale nature of the problem make the interpretation of the underlying physics often difficult and serve as one of the motivations for a reduced-order model that can characterize, with sufficient accuracy, the plasma system using a small number of degrees of freedom. Reduced-order models may also facilitate the possible use of model-based control methods such as model predictive control (MPC) [13], [14]. Several recent studies [15], [16], [17], [18] in the plasma physics community have indicated the practicality of adopting a lower dimensional feature space that can model the system through a small set of spatio-temporal coherent structures. A variety of model-order reduction techniques, such as proper orthogonal decomposition (POD) [18], [19], [20], bi-orthogonal decomposition (BOD) [21], [22], principal component analysis (PCA) [23] have been proposed in the past. These methods are invariably limited in their ability to resolve the time dynamical properties using low rank modeling. Dynamic mode decomposition (DMD) [24], [25], [26] helps to overcome this difficulty. In particular, it was recently shown in [27], [28], [29] that DMD can efficiently extract the underlying characteristic features of (fluid-model) magnetohydrodynamics based plasma simulations with reasonable accuracy. Our preliminary study [30] shows promise of DMD in reconstructing self electric fields from (kinetic-model) EMPIC plasma simulations. However, a detailed analysis regarding ability of DMD to capture relevant plasma dynamics from the particle-in-cell simulation is yet to be explored.
Another challenge of particle-in-cell (PIC) based algorithms is the large computational load [31]. Several improvements have been proposed in the literature to speed up PIC simulations, ranging from computational architecture to the underlying algorithmic structure, e.g. see [32], [33], [34]. Here, we address the issue also from a reduced-order model perspective. In order to minimize the computational cost, ideally one would like to perform reduced-order modeling such as DMD using data from high-fidelity simulations based on relatively short time windows and extrapolate the results in future time. However, as is shown in this work, accurate prediction of the equilibrium dynamics using data-driven methods such as DMD requires sufficient data harvesting near equilibrium. As a result, a related important question to be addressed is how to leverage DMD to optimally predict the equilibrium state. The question becomes particularly crucial for timely termination of the high-fidelity simulations such as those based on EMPIC algorithms.
In order to exploit the time extrapolation ability of DMD for reducing computation cost of high-fidelity simulations, it is important to identify the transition from transient to equilibrium state of a dynamical system in an in-line fashion. Several past works [35], [36], [37], [38] deal with identification of state transition in high-dimensional physical systems. Some recent publications [39], [40] highlight the importance of DMD in identifying such regime transitions. The authors in [39] rely on the DMD reconstruction error difference between transient and equilibrium states of a dynamical system to identify such transitions. However, one of the key assumptions in [39] is the fast relaxation of the dynamical system in transience, i.e. a faster time scale of the transient dynamics compared to equilibrium dynamics. The present work does not rely on the fast relaxation assumption since the transience is characterized by temporal variations in the amplitude and changing frequency content. Rather, we compute the residue based on the relative position of dominant DMD eigenvalues with respect to the unit circle. While Ref. [40] also performs identification of regime transition, it does so by observing the variation of a DMD-based least-squares residual term as the DMD window is gradually increased to span the spatial domain. In contrast, the residual term in this work is based on the loci of DMD eigenvalues in the complex plane. We keep track of the residual term as a fixed-width DMD window is moved forward in time. Finally, in [40], the change in the slope of the residual term is detected by fitting two straight lines. This work employs instead a rolling average to detect non-negative slopes that is suitable for in-line application. This work addresses all these issues in the context of kinetic plasma simulations from a modal analysis perspective. The main contributions of the present work can be summarized as follows:
- 1.
As mentioned above, while DMD has been recently applied to fluid-based plasma simulations, it has not yet been studied for kinetic plasma simulations. In this work we study the performance of DMD in reconstructing the self electric fields and its effect on the superparticle dynamics, for several test cases.
- 2.
We propose an algorithm for in-line detection of the onset of the equilibrium state of a dynamical system using a sliding-window DMD approach. This advancement has the potential to speed up EMPIC simulations for long term predictions when combined with the time-extrapolation ability of DMD. We propose a sliding window approach that tracks the position of DMD eigenvalues relative to the unit circle on the complex plane for detecting the equilibrium state. We analyze the prediction error in self-field pattern, as well as the superparticle dynamics, produced by the reduced-order model extrapolated solution.
- 3.
We perform a first-of-its-kind analysis to investigate the convergence in DMD mode shapes and shifting of DMD eigenvalues as the DMD window slides from transient to the equilibrium state. We do so by in-line tracking of the DMD modes and eigenvalues as a part of equilibrium detection algorithm. Such analysis can provide insight on how hidden features in the transient state can manifests itself as the system approaches equilibrium.
Section snippets
EMPIC algorithm
The EMPIC algorithm [11], [41], [42], [43], [44], [45], [46] generates the high-fidelity data for the DMD reduced-order model. It executes a marching-on-time procedure in four stages (Fig. 1) during each timestep: field-update, gather, particle-pusher and scatter.
For the field update, time-dependent Maxwell's equations are discretized on simplicial (triangular or tetrahedral) meshes using finite elements based on discrete exterior calculus [47], [48], [49], [50], [51], [52], [53], [54]. The
Equilibrium state identification
Detection of onset of the “equilibrium state” is motivated by the need to identify the ideal data-harvesting window for data-driven reduced-order methods, as well as for control applications. Accurate long term prediction of equilibrium behavior requires the DMD harvesting region to include the equilibrium region. One may terminate high-fidelity simulations once the system has reached equilibrium, ensuring enough quality data for the DMD to work with. Therefore, in-line detection (i.e.
Results
In this section we apply the tracking and equilibrium detection algorithm for several plasma examples. The effectiveness of DMD in the modeling and prediction of self electric fields as well as its effect on the particle dynamics is demonstrated. For application of the proposed equilibrium detection algorithm to a classic textbook use-case, the reader is referred to Appendix A, wherein the well-known Lorenz'96 oscillator is studied. This section presents three test cases. The first two examples
Computational complexity
The timestep complexity (runtime computational complexity to evolve through one timestep) in our explicit particle-in-cell algorithm is [34] where is the number of particles and N represents aggregate mesh dimension. More typically, for implicit field solvers the timestep complexity1
Concluding remarks
This work introduced a DMD approach for the reduced-order modeling of kinetic plasmas. Data is harvested from high-fidelity EMPIC simulations and used to extract key (low-dimensional) features as well as to predict/extrapolate the problem dynamics to later times. Extraction of key features/modes is shown to be instrumental in providing physical insight into the problem and can facilitate the application of model predictive control methods. Accurate prediction of nonlinear limit-cycle behavior
CRediT authorship contribution statement
Indranil Nayak: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft. Mrinal Kumar: Conceptualization, Methodology, Supervision, Writing – review & editing. Fernando L. Teixeira: Conceptualization, Funding acquisition, Project administration, Resources, Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was partially supported by the Defense Threat Reduction Agency under Grant HDTRA1-18-1-0050, the Air Force Office of Scientific Research under Grant No. FA9550-20-1-0083 and the Ohio Supercomputer Center under Grant PAS-0061.
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