Introduction to linear and nonlinear programming

Digital twin can be defined as a virtual representation of a physical asset enabled through data and simulators for real-time prediction, optimization, monitoring, controlling, and improved decision making. Recent advances in computational pipelines, multiphysics solvers, artificial intelligence, big data cybernetics, data processing and management tools bring the promise of digital twins and their impact on society closer to reality. Digital twinning is now an important and emerging trend in many applications. Also referred to as a computational megamodel, device shadow, mirrored system, avatar or a synchronized virtual prototype, there can be no doubt that a digital twin plays a transformative role not only in how we design and operate cyber-physical intelligent systems, but also in how we advance the modularity of multi-disciplinary systems to tackle fundamental barriers not addressed by the current, evolutionary modeling practices. In this work, we review the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective. Our aim is to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.

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Digital Twin: Values, Challenges and Enablers From a Modeling Perspective

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Summary We propose a new method for estimating parameters in models that are defined by a system of non-linear differential equations Such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself Current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation The paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non-linear differential equations The approach is based on a modification of data smoothing methods along with a generalization of profiled estimation We derive estimates and confidence intervals, and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology The performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus

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Parameter estimation for differential equations: a generalized smoothing approach

This paper reviews the development of the ensemble Kalman filter (EnKF) for atmospheric data assimilation. Particular attention is devoted to recent advances and current challenges. The distinguishing properties of three well-established variations of the EnKF algorithm are first discussed. Given the limited size of the ensemble and the unavoidable existence of errors whose origin is unknown (i.e., system error), various approaches to localizing the impact of observations and to accounting for these errors have been proposed. However, challenges remain; for example, with regard to localization of multiscale phenomena (both in time and space). For the EnKF in general, but higher-resolution applications in particular, it is desirable to use a short assimilation window. This motivates a focus on approaches for maintaining balance during the EnKF update. Also discussed are limited-area EnKF systems, in particular with regard to the assimilation of radar data and applications to tracking severe storms ...

Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation

Dynamic data assimilation is the assessment, combination and synthesis of observational data, scientific laws and mathematical models to determine the state of a complex physical system, for instance as a preliminary step in making predictions about the system's behaviour. The topic has assumed increasing importance in fields such as numerical weather prediction where conscientious efforts are being made to extend the term of reliable weather forecasts beyond the few days that are presently feasible. This book is designed to be a basic one-stop reference for graduate students and researchers. It is based on graduate courses taught over a decade to mathematicians, scientists, and engineers, and its modular structure accommodates the various audience requirements. Thus Part I is a broad introduction to the history, development and philosophy of data assimilation, illustrated by examples; Part II considers the classical, static approaches, both linear and nonlinear; and Part III describes computational techniques. Parts IV to VII are concerned with how statistical and dynamic ideas can be incorporated into the classical framework. Key themes covered here include estimation theory, stochastic and dynamic models, and sequential filtering. The final part addresses the predictability of dynamical systems. Chapters end with a section that provides pointers to the literature, and a set of exercises with instructive hints.

Dynamic Data Assimilation: A Least Squares Approach

This survey provides a comprehensive and unified analysis of symmetry in a wide variety of Cayley graphs of permutation groups. These include the star graph, bubble-sort graph, modified bubble-sort graph, complete-transposition graph, prefix-reversal graph, alternating-group graph, binary and base-b (b ≥ 3) hypercube, cube connected cycles, bisectional graph, folded hypercube and binary orthogonal graph. In addition, we also define a variety of generalizations of the hypercube and orthogonal graphs. The types of symmetry analyzed include vertex and edge transitivity, distance regularity and distance transitivity. Since these notions of symmetry depend on the shortest paths, as a by product we also describe the shortest path routing algorithms for these graphs. We present a number of open problems related to the networks described in this paper.

Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey

Publisher Summary This chapter presents a survey on various parallel sorting algorithms. Sorting is a nontrivial problem and has widespread commercial and business applications. Serial algorithms for sorting have been available since the days of punched-card machines. At present, there is a considerable body of literature on serial sorting algorithms. Parallel algorithms for sorting are of a recent origin and came into existence over the past decade. The chapter presents a unified treatment of various parallel sorting algorithms by bringing out clearly the relation between the architecture of parallel computers and the structure of algorithms. In the design of parallel algorithms in general, and of parallel sorting algorithms in particular, two models have been widely used: (1) models based on fixed interconnection networks such as the same or single instruction on multiple data (SIMD) machine mesh-connected network and (2) models based on a global memory, which is shared by various processors. The special-purpose network-sorting algorithms are described. Algorithms for SIMD machines are given.

Parallel Sorting Algorithms

This paper introduces a new class of interconnection scheme based on the Cayley graph of the alternating group. It is shown that this class of graphs are edge symmetric and 2-transitive. We then describe an algorithm for (a) packet routing based on the shortest path analysis, (b) finding a Hamiltonian cycle, (c) ranking and unranking along the chosen Hamiltonian cycle, (d) unit expansion and dilation three embedding of a class of two-dimensional grids, (e) unit dilation embedding of a variety of cycles, and (f) algorithm for broadcasting messages. The paper concludes with a short analysis of contention resulting from a typical communication scheme. Although this class of graphs does not possess many of the symmetry properties of the binary hypercube, with respect to the one source broadcasting, these graphs perform better than does a hypercube, and with respect to the contention problem, these graphs perform better than do the star graphs and are close to the hypercube. © 1993 by John Wiley & Sons, Inc.

A new class of interconnection networks based on the alternating group

There has been a constant barrage of worms over the Internet during the recent past. Besides threatening network security, these worms cause an enormous economic burden in terms of loss of productivity at the victim hosts. In addition, these worms create unnecessary network data traffic that causes network congestion, thereby hurting all users. To develop appropriate tools for thwarting quick spread of worms, researchers are trying to understand the behavior of the worm propagation with the aid of epidemiological models. In this study, we apply the classical SIS model and a modification of SIR model to simulate worm propagation in two different network topologies. Whereas in the SIR model once a node is cured after infection it becomes permanently immune, our modification allows this immunity to be temporary, since the cured nodes may again become infected, maybe with a different strain of the same worm. The simulation study also shows that time to infect a large portion of the network vary significantly depending on where the infection begins. This information could be usefully employed to choose hosts for quarantine to delay worm propagation to the rest of the network

Measurement and analysis of worm propagation on Internet network topology

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