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Mori–Zwanzig Modal Decomposition

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arXiv 2023年

作者： Woodward, Michael Tian, Yifeng Lin, Yen Ting Hader, Christoph Fasel, Hermann Livescu, Daniel Graduate Interdisciplinary Program in Applied Mathematics Department of Mathematics University of Arizona TucsonAZ85721 United States Computer Computational and Statistical Sciences Division Los Alamos National Laboratory Los AlamosNM87544 United States Department of Aerospace and Mechanical Engineering University of Arizona TucsonAZ85721 United States

We introduce the Mori-Zwanzig (MZ) Modal Decomposition (MZMD), a novel technique for performing modal analysis of large scale spatio-temporal structures in complex dynamical systems, and show that it represents an efficient generalization of Dynamic Mode Decomposition (DMD). The MZ formalism provides a mathematical framework for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, incorporating the effects of unresolved dynamics through the memory kernel and orthogonal dynamics. We present a formulation and analysis of the modes and spectrum from MZMD and compare it to DMD when applied to a complex flow: a Direct Numerical Simulation (DNS) data-set of laminar-turbulent boundary-layer transition flow over a flared cone at Mach 6. We show that the addition of memory terms by MZMD improves the resolution of spatio-temporal structures within the transitional/turbulent regime, which contains features that arise due to nonlinear mechanisms, such as the generation of the so-called "hot" streaks on the surface of the flared cone. As a result, compared to DMD, MZMD improves future state prediction accuracy, while requiring nearly the same computational cost. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamic mode decomposition

Data-Driven Mori-Zwanzig: Approaching a Reduced Order Model for Hypersonic Boundary Layer Transition

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arXiv 2023年

作者： Woodward, Michael Tian, Yifeng Mohan, Arvind Lin, Yen Ting Hader, Christoph Fasel, Hermann Chertkov, Misha Livescu, Daniel Graduate Interdisciplinary Program in Applied Mathematics Department of Mathematics University of Arizona TucsonAZ85721 United States Computer Computational and Statistical Sciences Division Los Alamos National Laboratory Los AlamosNM87544 United States Department of Aerospace and Mechanical Engineering University of Arizona TucsonAZ85721 United States

In this work, we apply, for the first time to spatially inhomogeneous flows, a recently developed data-driven learning algorithm of Mori-Zwanzig (MZ) operators, which is based on a generalized Koopman’s description of dynamical systems. The MZ formalism provides a mathematically exact procedure for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. The algorithm developed in this work applies Mori’s linear projection operator and an SVD based compression to the selection of the resolved variables (equivalently, a low rank approximation of the two time covariance matrices). We show that this MZ decomposition not only identifies the same spatio-temporal structures found by DMD, but it can also be used to extract spatio-temporal structures of the hysteresis effects present in the memory kernels. We perform an analysis of these structures in the context of a laminar-turbulent boundary-layer transition flow over a flared cone at Mach 6, and show the dynamical relevance of the memory kernels. Additionally, by including these memory terms learned in our data-driven MZ approach, we show improvement in prediction accuracy over DMD at the same level of truncation and at a similar computational cost. Furthermore, an analysis of the spatio-temporal structures of the MZ operators shows identifiable structures associated with the nonlinear generation of the so-called "hot" streaks on the surface of the flared cone, which have previously been observed in experiments and direct numerical simulations. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamic mode decomposition

Computational design of anisotropic stealthy hyperuniform composites with engineered directional scattering properties

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Physical Review E 2023年第4期108卷 045306-045306页

作者： Wenlong Shi David Keeney Duyu Chen Yang Jiao Salvatore Torquato Materials Science and Engineering Arizona State University Tempe Arizona 85287 USA Materials Research Laboratory University of California Santa Barbara California 93106 USA Department of Physics Arizona State University Tempe Arizona 85287 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Disordered hyperuniform materials are an emerging class of exotic amorphous states of matter that endow them with singular physical properties, including large isotropic photonic band gaps, superior resistance to fracture, and nearly optimal electrical and thermal transport properties, to name but a few. Here we generalize the Fourier-space-based numerical construction procedure for designing and generating digital realizations of isotropic disordered hyperuniform two-phase heterogeneous materials (i.e., composites) developed by Chen and Torquato [Acta Mater. 142, 152 (2018)] to anisotropic microstructures with targeted spectral densities. Our generalized construction procedure explicitly incorporates the vector-dependent spectral density function χ̃V(k) of arbitrary form that is realizable. We demonstrate the utility of the procedure by generating a wide spectrum of anisotropic stealthy hyperuniform microstructures with χ̃V(k)=0 for k∈Ω, i.e., complete suppression of scattering in an “exclusion” region Ω around the origin in Fourier space. We show how different exclusion-region shapes with various discrete symmetries, including circular-disk, elliptical-disk, square, rectangular, butterfly-shaped, and lemniscate-shaped regions of varying size, affect the resulting statistically anisotropic microstructures as a function of the phase volume fraction. The latter two cases of Ω lead to directionally hyperuniform composites, which are stealthy hyperuniform only along certain directions and are nonhyperuniform along others. We find that while the circular-disk exclusion regions give rise to isotropic hyperuniform composite microstructures, the directional hyperuniform behaviors imposed by the shape asymmetry (or anisotropy) of certain exclusion regions give rise to distinct anisotropic structures and degree of uniformity in the distribution of the phases on intermediate and large length scales along different directions. Moreover, while the anisotropic exclusion regions

关键词： Amorphous materials Complex materials Simulated annealing

Embedded (4,5) pairs of explicit 7-stage Runge-Kutta methods with FSAL property

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arXiv 2021年

作者： Stepanov, Misha Department of Mathematics and Program in Applied Mathematics University of Arizona TucsonAZ85721 United States

The general case of embedded (4,5) pairs of explicit 7-stage Runge-Kutta methods with FSAL property (a7 j = bj, 1 ≤ j ≤ 7, c7 = 1) is considered. Besides exceptional cases, the pairs form five 4-dimensional families. The pairs within two (already known) families satisfy the simplifying assumption ∑jaijcj = c2i/2, i ≥ *** Codes 65L05, 65L06 Copyright © 2021, The Authors. All rights reserved.

关键词： Runge Kutta methods

Machine Learning for Electricity Market Clearing

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arXiv 2022年

作者： Pagnier, Laurent Ferrando, Robert Dvorkin, Yury Chertkov, Michael Program in Applied Mathematics University of Arizona TucsonAZ United States Electrical and Computer Engineering New York University New YorkNY United States

This paper seeks to design a machine learning twin of the optimal power flow (OPF) optimization, which is used in market-clearing procedures by wholesale electricity markets. The motivation for the proposed approach stems from the need to obtain the digital twin, which is much faster than the original, while also being sufficiently accurate and producing consistent generation dispatches and locational marginal prices (LMPs), which are primal and dual solutions of the OPF optimization, respectively. Availability of market-clearing tools based on this approach will enable computationally tractable evaluation of multiple dispatch scenarios under a given unit commitment. Rather than direct solution of OPF, the Karush–Kuhn–Tucker (KKT) conditions for the OPF problem in question may be written, and in parallel the LMPs of generators and loads may be expressed in terms of the OPF Lagrangian multipliers. Also, taking advantage of the practical fact that many of the Lagrangian multipliers associated with lines will be zero (thermal limits are not binding), we build and train an ML scheme which maps flexible resources (loads and renewables) to the binding lines, and supplement it with an efficient power-grid aware linear map to optimal dispatch and LMPs. The scheme is validated and illustrated on IEEE models. We also report a trade of analysis between quality of the reconstruction and number of samples needed to train the model. © 2022, CC BY.

关键词： Acoustic generators

ON THE GLOBAL WELL-POSEDNESS AND GEVREY REGULARITY OF SOME ELECTRODIFFUSION MODELS

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arXiv 2022年

作者： Abdo, Elie Lee, Fizay-Noah Wang, Weinan Department of Mathematics Temple University PhiladelphiaPA19122 United States Program In Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics University of Arizona TucsonAZ85712 United States

We consider the Nernst-Planck equations describing the nonlinear time evolution of multiple ionic concentrations in a two-dimensional incompressible fluid. The velocity of the fluid evolves according to either the Euler or Darcy’s equations, both forced nonlinearly by the electric forces generated by the presence of charged ions. We address the global well-posedness and Gevrey regularity of the resulting electrodiffusion models in the periodic setting. © 2022, CC BY.

关键词： Nonlinear equations

Langevin Equations in the Small-Mass Limit: Higher-Order Approximations

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Annales Henri Poincare 2020年第6期21卷 1765-1811页

作者： Birrell, Jeremiah Wehr, Jan Department of Mathematics and Statistics University of Massachusetts Amherst Amherst 01003 MA United States Department of Mathematics and Program in Applied Mathematics University of Arizona Tucson 8572 AZ United States

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order m^ℓ ^/ ² over compact time intervals for any ℓ∈ Z⁺. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the m→ 0 limit, which result in order m^{1 / 2} approximations. Our results cover bounded forces, for which we prove convergence in L^p norms and unbounded forces, in which case we prove convergence in probability. © 2020, Springer Nature Switzerland AG.

关键词： 60H10 82C31

Microscopic charge carrier dynamics within non-normal incidence VECSEL cavities 25

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Microscopic charge carrier dynamics within non-normal incide...

Vertical-Cavity Surface-Emitting Lasers XXV 2021

作者： McLaren, S. Kilen, I. Moloney, J.V. Program in Applied Mathematics University of Arizona 617 N. Santa Rita Ave. TucsonAZ85721 United States Arizona Center for Mathematical Sciences University of Arizona 1630 E University Blvd. TucsonAZ85721 United States Wyant College of Optical Sciences University of Arizona 1630 East University Boulevard TucsonAZ85721 United States Department of Mathematics University of Arizona 617 N. Santa Rita Ave. TucsonAZ85721 United States

ISBN: (数字)9781510642447

ISBN: (纸本)9781510642430

Mode-locked vertical external-cavity surface emitting lasers are promising compact sources for high-power, ultrafast pulses with excellent beam quality and the flexibility offered by an external cavity. Typical models of these lasers use macroscopic or quasistatic approaches based on rate or delay differential equations. Although these approaches have shown widespread success, they often require numerous experimentally tuned parameters and cannot capture the ultrafast nonequilibrium dynamics present as the field interacts with the quantum well. The Maxwell Semiconductor Bloch Equations has reduced parametrization and captures the carrier dynamics by coupling together a numerical wave propagator to a first principles of quantum mechanical description of the induced microscopic polarization within the active semiconductor quantum well. We expand on this model utilizing a reference frame transform to model modelocking within VECSEL cavities with non-normally incident semiconductor heterostructures. Specifically, we demonstrate the effect of increased pumping on the fundamental and harmonic modelocking behaviors of V-cavity VECSELs as well as transverse kinetic hole burning during colliding pulse operation as seen in modelocked ring cavities. © 2021 SPIE. © 2021 SPIE. All rights reserved.

关键词： Semiconductor quantum wells

Utilizing probabilistic entanglement between sensors in quantum networks

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arXiv 2024年

作者： Van Milligen, Emily A. Gagatsos, Christos N. Kaur, Eneet Towsley, Don Guha, Saikat Department of Physics The University of Arizona 1630 East University Boulevard TucsonAZ85721 United States Department of Electrical and Computer Engineering The University of Arizona 1630 East University Boulevard TucsonAZ85721 United States Wyant College of Optical Sciences The University of Arizona 1630 East University Boulevard TucsonAZ85721 United States Program in Applied Mathematics The University of Arizona TucsonAZ85721 United States Cisco Quantum Lab Los Angeles United States College of Computer Science University of Massachusetts AmherstMA United States

One of the most promising applications of quantum networks is entanglement assisted sensing. The field of quantum metrology exploits quantum correlations to improve the precision bound for applications such as precision timekeeping, field sensing, and biological imaging. When measuring multiple spatially distributed parameters, current literature focuses on quantum entanglement in the discrete variable case—and quantum squeezing in the continuous variable case distributed amongst all of the sensors in a given network. However, it can be difficult to ensure all sensors pre-share entanglement of sufficiently high fidelity. This work probes the space between fully entangled and fully classical sensing networks by modeling a star network with probabilistic entanglement generation that is attempting to estimate the average of local parameters. The quantum Fisher information is used to determine which protocols best utilize entanglement as a resource for different network conditions. It is shown that without entanglement distillation there is a threshold fidelity below which classical sensing is preferable. For a network with a given number of sensors and links characterized by a certain initial fidelity and probability of success, this work outlines when and how to use entanglement, when to store it, and when it needs to be distilled. © 2024, CC BY.

关键词： Quantum entanglement