Spatial dynamics in interacting systems with discontinuous coefficients and their continuum limits

1noWe consider a discrete model in which particles are characterized by two quantities X and Y ; both quantities evolve in time according to stochastic dynamics and the equation that governs the evolution of Y is also influenced by mean-field interaction between the particles. We allow for discontinuous coefficients and random initial condition and, under suitable assumptions, we prove that in the limit as the number of particles grows to infinity the dynamics of the system is described by the solution of a Fokker–Planck partial differential equation. We provide the existence and uniqueness of a solution to the latter and show that such solution arises as the limit in probability of the empirical measures of the system.openembargoed_20200606Giovanni Alessandro ZancoZanco, Giovanni Alessandr

High-dimensional data driven parameterized macromodeling

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Infinite-dimensional methods for path-dependent stochastic differential equations

This thesis addresses the problem of extending results of stochastic analysis from the classical Markovian setting to the path-dependent setting. The two main aspects considered are: the relation between path-dependent stochastic differential equations and partial differential equations and change of variable formulae of Ito type. It also provides a comparison between the methods developed and the Functional Ito calculus recently introduced and studied by other authors. The main results obtained in the thesis are: existence and uniqueness of classical solutions to an infinite-dimensional Kolmogorov equation associated to a path-dependent stochastic differential equation; existence of Ito formulae in Hilbert and Banach spaces in situations where some of the terms appearing are ill-defined; a comparison result between the infinite-dimensional method presented and the Functional Ito calculus; a proof of existence and uniqueness of classical solutions to the path-dependent Kolmogorov equation based on the previous results. The methods used are based on a reformulation of the problems in infinite-dimensional Hilbert and Banach spaces; in particular, tools of stochastic analysis in space of continuous and càdlàg functions are investigated. The results obtained about Komogorov equations exhibit significative differences from their analogue in the classical case, both from the point of view of necessary conditions and of regularising properties of the equations. The Ito formulae obtained are suitable to be applied to different contexts and are therefore investigated in a general abstract setting. Beside their application to path-dependent equations, an application to stochastic equations with group generators is presented

Enforcing passivity of parameterized LTI macromodels via Hamiltonian-driven multivariate adaptive sampling

We present an algorithm for passivity verification and enforcement of multivariate macromodels whose state-space matrices depend in closed form on a set of external or design parameters. Uniform passivity throughout the parameter space is a fundamental requirement of parameterized macromodels of physically passive structures, that must be guaranteed during model generation. Otherwise, numerical instabilities may occur, due to the ability of non-passive models to generate energy. In this work, we propose the first available algorithm that, starting from a generic parameter-depedent state-space model, identifies the regions in the frequency-parameter space where the model behaves locally as a non-passive system. The approach we pursue is based on an adaptive sampling scheme in the parameter space, which iteratively constructs and perturbs the eigenvalue spectrum of suitable Skew-Hamiltonian/Hamiltonian (SHH) pencils, with the objective of identifying the regions where some of these eigenvalues become purely imaginary, thus pinpointing local passivity violations. The proposed scheme is able to detect all relevant violations. An outer iterative perturbation method is then applied to the model coefficients in order to remove such violations and achieve uniform passivity. Although a formal proof of global convergence is not available, the effectiveness of the proposed implementation of the passivity verification and enforcement schemes is demonstrated on several examples

Multivariate macromodeling with stability and passivity constraints

We present a general framework for the construction of guaranteed stable and passive multivariate macromodels from sampled frequency responses. The obtained macromodels embed in closed form the dependence on external parameters, through a data-driven approximation of input data samples based on orthogonal polynomial bases. The key novel contribution of this work is an extension to the multivariate and possibly high-dimensional case of Hamiltonian-based passivity check and enforcement algorithms, which can be applied to enforce both uniform stability and uniform passivity of the models. The modeling flow is demonstrated on a representative interconnect example

Uniformly Stable Parameterized Macromodeling through Positive Definite Basis Funtions

Reduced-order models are widely used to reduce the computational cost required by the numerical assessment of electrical performance during the design cycle of electronic circuits and systems. Although standard macromodeling algorithms can be considered to be well consolidated, the generation of macromodels that embed in a closed form some dependence on the design variables still presents considerable margins for improvement. One of these aspects is enforcement of uniform stability throughout the parameter space of interest. This paper proposes a novel parameterized macromodeling strategy, which enforces by construction that all macromodel poles are stable for any combination of possibly several independent design variables. The key enabling factor is adoption of positive definite multivariate basis functions for the representation of model variations induced by the parameters. This representation leads to robust model generation from tabulated frequency responses, at a computational cost that is dramatically reduced with respect to competing approaches. This result arises from a number of algebraic constraints for stability enforcement that depends on the model complexity (number of basis functions) and not on the model behavior as a function of the parameters. As a byproduct, the proposed strategy lends itself to much improved scaling with the dimension of parameter space, allowing to circumvent the curse of dimensionality that may occur when the number of independent parameters grows beyond few units. To this end, we exploit representations based on positive definite radial basis functions. The benefits of the proposed approach are demonstrated through an extensive experimental campaign applied to both passive and active devices and components, comparing the performance of different model parameterizations in terms of accuracy, time requirements and model compactness

An Adaptive Sampling Process for Automated Multivariate Macromodeling Based on Hamiltonian-Based Passivity Metrics

This paper introduces a fully automated greedy algorithm for the construction of parameterized behavioral models of electromagnetic structures, targeting at the same time uniform model stability and passivity. The proposed algorithm is able to determine a small set of parameter configurations for which an external solver provides on the fly the sampled scattering parameters of the structure over a predetermined frequency band. These samples are subjected to a multivariate rational/polynomial fitting process, which iteratively leads to a parameterized descriptor realization of the model. The main novel contribution in this work is the adoption of a model-based approach for the adaptive augmentation of an initially small set of frequency responses, each corresponding to a randomly-selected parameter configuration. In particular, the locations of the in-band passivity violations of intermediate macromodels constructed at each iteration are used as a proxy for the model-data error in those regions where input data are not available. This physics-based consistency check, which is enabled by recent developments in multivariate passivity characterization based on Skew-Hamiltonian-Hamiltonian (SHH) spectra, is combined with standard space exploration metrics to obtain a small-size and automatically-determined distribution of points in the parameter space, leading to the construction of an accurate macromodel with a very limited number of external field solver runs. The embedded passivity check and enforcement process guarantees that either the final model is passive throughout the parameter space, or the residual violations, if present, are negligible for practical purposes. Several examples validate the proposed approach for up to three concurrent parameters

Mixed Proper Orthogonal Decomposition with Harmonic Approximation for Parameterized Order Reduction of Electromagnetic Models

This paper presents some preliminary investigations on a hybrid Model Order Reduction approach for parameter-dependent electromagnetic systems. Starting from an integral equation formulation of the field problem, we introduce a first level of compression based on the well-established Proper Orthogonal Decomposition (POD). The result is a small-scale approximation of the full-order discrete field formulation, which retains an explicit dependence on the set of free parameters defining the geometry. The evaluation of the reduced model for arbitrary parameter configurations remains very expensive, as it requires the construction of the full system equations before its projection onto a lower-dimensional space. This problem is solved by constructing a surrogate macromodel of the parameterized reduced-order system through a multivariate Fourier approximation. Numerical results applied to a moving coil over a finite ground plane show model compression above 99% while preserving accuracy on currents and fields within 1%

Infinite dimensional calculus under weak spatial regularity of the processes

Two generalizations of It\uf4 formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an It\uf4 formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent It\uf4 calculus