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1. Embedding properties of network realizations of dissipative reduced order models

   Druskin, Guddati ( July 2022 , HOUSEHOLDER SYMPOSIUM XXI)

   Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and block-tridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistor-capacitor-inductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a non-symmetric Lanczos algorithm written in J-symmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nite-dierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich data-manifolds), a nite-dierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in port-Hamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible non-positivity of conductors for the non-Stieltjes case, we introduce an additional complex s-dependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on Krein-Nudelman theory [5], that results in special data-driven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the three-point second dierences: I. A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back 

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2. An efficient partial-differential-equation-based method to compute pressure boundary conditions in regional geodynamic models

   https://doi.org/10.5194/se-13-1107-2022  

   Jourdon, Anthony ; May, Dave A. ( January 2022 , Solid Earth)

   Abstract. Modelling the pressure in the Earth's interior is a common problem in Earth sciences. In this study we propose a method based on the conservation of the momentum of a fluid by using a hydrostatic scenario or a uniformly moving fluid to approximate the pressure. This results in a partial differential equation (PDE) that can be solved using classical numerical methods. In hydrostatic cases, the computed pressure is the lithostatic pressure. In non-hydrostatic cases, we show that this PDE-based approach better approximates the total pressure than the classical 1D depth-integrated approach. To illustrate the performance of this PDE-based formulation we present several hydrostatic and non-hydrostatic 2D models in which we compute the lithostatic pressure or an approximation of the total pressure, respectively. Moreover, we also present a 3D rift model that uses that approximated pressure as a time-dependent boundary condition to simulate far-field normal stresses. This model shows a high degree of non-cylindrical deformation, resulting from the stress boundary condition, that is accommodated by strike-slip shear zones. We compare the result of this numerical model with a traditional rift model employing free-slip boundary conditions to demonstrate the first-order implications of considering “open” boundary conditions in 3D thermo-mechanical rift models. 

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3. Bringing physics into the coarse‐grid selection: Approximate diffusion distance/effective resistance measures for network analysis and algebraic multigrid for graph Laplacians and systems of elliptic partial differential equations

   https://doi.org/10.1002/nla.2539  

   Lee, Barry ( November 2023 , Numerical Linear Algebra with Applications)

   Vassilevski, Panayot (Ed.)

   In a recent paper, the author examined a correlation affinity measure for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (C nodes) in systems of elliptic partial differential equations (PDEs). This measure was applied to a set of relaxed vectors, which exposed the near‐nullspace components of the PDE operator. Selecting the CDOFs using this affinity measure and constructing the interpolation operators using a least‐squares procedure, an algebraic multigrid (AMG) method was developed. However, there are several noted issues with this AMG solver. First, to capture strong anisotropies, a large number of test vectors may be needed; and second, the solver's performance can be sensitive to the initial set of random test vectors. Both issues reflect the sensitive statistical nature of the measure. In this article, we derive several other statistical measures that ameliorate these issues and lead to better AMG performance. These measures are related to a Markov process, which the PDE itself may model. Specifically, the measures are based on the diffusion distance/effective resistance for such process, and hence, these measures incorporate physics into the CDOF selection. Moreover, because the diffusion distance/effective resistance can be used to analyze graph networks, these measures also provide a very economical scheme for analyzing large‐scale networks. In this article, the derivations of these measures are given, and numerical experiments for analyzing networks and for AMG performance on weighted‐graph Laplacians and systems of elliptic boundary‐value problems are presented.

    

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4. An extended PDE‐based statistical spatio‐temporal model that suppresses the Gibbs phenomenon

   https://doi.org/10.1002/env.2831  

   Wei, Guanzhou ; Liu, Xiao ; Barton, Russell ( October 2023 , Environmetrics)

   Partial differential equation (PDE)‐based spatio‐temporal models are available in the literature for modeling spatio‐temporal processes governed by advection‐diffusion equations. The main idea is to approximate the process by a truncated Fourier series and model the temporal evolution of the spectral coefficients by a stochastic process whose parametric structure is determined by the governing PDE. However, because many spatio‐temporal processes are nonperiodic with boundary discontinuities, the truncation of Fourier series leads to the well‐known Gibbs phenomenon (GP) in the output generated by the existing PDE‐based approaches. This article shows that the existing PDE‐based approach can be extended to suppress GP. The proposed approach starts with a data flipping procedure for the process respectively along the horizontal and vertical directions, as if we were unfolding a piece of paper folded twice along the two directions. For the flipped process, this article extends the existing PDE‐based spatio‐temporal model by obtaining the new temporal dynamics of the spectral coefficients. Because the flipped process is spatially periodic and has a complete waveform without boundary discontinuities, GP is removed even if the Fourier series is truncated. Numerical investigations show that the extended approach improves the modeling and prediction accuracy. Computer code is made available on GitHub.

    

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5. Learning to Accelerate Partial Differential Equations via Latent Global Evolution

   Wu, Tailin ; Maruyama, Takashi ; Leskovec, Jure ( January 2022 , Advances in neural information processing systems)

   Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems. However, both classical solvers and recent deep learning-based surrogate models are typically extremely computationally intensive, because of their local evolution: they need to update the state of each discretized cell at each time step during inference. Here we develop Latent Evolution of PDEs (LE-PDE), a simple, fast and scalable method to accelerate the simulation and inverse optimization of PDEs. LE-PDE learns a compact, global representation of the system and efficiently evolves it fully in the latent space with learned latent evolution models. LE-PDE achieves speedup by having a much smaller latent dimension to update during long rollout as compared to updating in the input space. We introduce new learning objectives to effectively learn such latent dynamics to ensure long-term stability. We further introduce techniques for speeding-up inverse optimization of boundary conditions for PDEs via backpropagation through time in latent space, and an annealing technique to address the non-differentiability and sparse interaction of boundary conditions. We test our method in a 1D benchmark of nonlinear PDEs, 2D Navier-Stokes flows into turbulent phase and an inverse optimization of boundary conditions in 2D Navier-Stokes flow. Compared to state-of-the-art deep learning-based surrogate models and other strong baselines, we demonstrate up to 128x reduction in the dimensions to update, and up to 15x improvement in speed, while achieving competitive accuracy. 

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