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 blocktridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistorcapacitorinductor (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 socalled nitedierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM statespace representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from themore »
This content will become publicly available on January 1, 2023
Finitedifference quadrature and inverse scattering
One of classical tasks of the network synthesis is to construct ROMs realized via ladder networks
matching rational approximations of a targeted filter transfer function. The inverse scattering can
be also viewed in the network synthesis framework. The key is continuum interpretation of the
synthesized network in terms of the underlying medium properties, aka embedding. We describe
such an embedding via finitedifference quadrature rules (FDQR), that can be viewed as extension
of the concept of the Gaussian quadrature to finitedifference schemes. One of application of this
approach is the solution of earlier intractable large scale inverse scattering problems. We also discuss
an important open question in the FDQR related to Lothar’s earlier contributions, in particular, a
possibility of finitedifference GaussKronrod rules
 Award ID(s):
 2110773
 Publication Date:
 NSFPAR ID:
 10340874
 Journal Name:
 Numerical Methods for large scale problems
 Sponsoring Org:
 National Science Foundation
More Like this


The problem of continuous inverse optimal control (over finite time horizon) is to learn the unknown cost function over the sequence of continuous control variables from expert demonstrations. In this article, we study this fundamental problem in the framework of energybased model, where the observed expert trajectories are assumed to be random samples from a probability density function defined as the exponential of the negative cost function up to a normalizing constant. The parameters of the cost function are learned by maximum likelihood via an “analysis by synthesis” scheme, which iterates (1) synthesis step: sample the synthesized trajectories from the current probability density using the Langevin dynamics via backpropagation through time, and (2) analysis step: update the model parameters based on the statistical difference between the synthesized trajectories and the observed trajectories. Given the fact that an efficient optimization algorithm is usually available for an optimal control problem, we also consider a convenient approximation of the above learning method, where we replace the sampling in the synthesis step by optimization. Moreover, to make the sampling or optimization more efficient, we propose to train the energybased model simultaneously with a topdown trajectory generator via cooperative learning, where the trajectory generator ismore »

Taylor And Francis Online (Ed.)We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of nonuniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided.

We consider damped stochastic systems in a controlled (timevarying) potential and study their transition between specified Gibbsequilibria states in finite time. By the second law of thermody namics, the minimum amount of work needed to transition from one equilibrium state to another is the difference between the Helmholtz free energy of the two states and can only be achieved by a reversible (infinitely slow) process. The minimal gap between the work needed in a finitetime transition and the work during a reversible one, turns out to equal the square of the optimal mass transport (Wasserstein 2) distance between the two endpoint distributions times the inverse of the duration needed for the transition. This result, in fact, relates nonequilibrium optimal control strategies (protocols) to gradient flows of entropy functionals via the JordanKinderlehrerOtto scheme. The purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of classical regulator theory, and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting.

Electromigration (EM) becomes a major concern for VLSI circuits as the technology advances in the nanometer regime. With Korhonen equations, EM assessment for VLSI circuits remains challenged due to the increasing integrated density. VLSI multisegment interconnect trees can be naturally viewed as graphs. Based on this observation, we propose a new graph convolution network (GCN) model, which is called {\it EMGraph} considering both node and edge embedding features, to estimate the transient EM stress of interconnect trees. Compared with recently proposed generative adversarial network (GAN) based stress imagegeneration method, EMGraph model can learn more transferable knowledge to predict stress distributions on new graphs without retraining via inductive learning. Trained on the large dataset, the model shows less than 1.5% averaged error compared to the ground truth results and is orders of magnitude faster than both COMSOL and stateoftheart method. It also achieves smaller model size, 4X accuracy and 14X speedup over the GANbased method.