Novel Mathematics Inspired by Industrial Challenges

In science and engineering, simulation tasks often involve numerical time integration of differential equations. Usually, these systems contain different time constants of the involved components and/or right-hand side. This multirate behavior may be caused by coupling subsystems in multiphysics problems acting on different time scales. Such a behavior does already occur if one deals with just single-physics problems: for example, the activity level of components in electrical networks may strongly vary depending on the according functional purpose, physics or time; another example is given in lattice QCD, where the equations of motion may depend on weak and strong forces, which demand to sample these forces with different frequencies to gain the same rate of approximation.

To be efficient or to speed up simulation of highly complex coupled systems is necessary for many design and optimization work flows. To this end, numerical integration schemes have to be adapted to exploit this multirate behavior. One idea proposed by Rice in 1960 are multirate schemes, which use different step sizes adapted to the various activity levels. In the last 50 years, the methodology of numerical time integration schemes has been advanced in a constant interplay between the demands defined by the need of exploiting multirate behavior in different fields of applications and the development of tailored multirate schemes to answer these demands.

Ever since the 1960s, the semiconductor industry has heavily relied on simulation in order to analyze and verify the design of integrated circuits before actual chips are manufactured. Over the decades, the algorithms and tools of circuit simulation have evolved in order to keep up with the ever-increasing complexity of integrated circuits, and at certain points of this evolution, new simulation techniques were required. Such a pointwas reached in the early 1990s, when a newapproachwas needed to efficiently and accurately simulate the effects of the ever-increasing amount of on-chip wiring on the proper functioning of the chip. The industry’s proposed solution for this task, the AWE approach, worked well for small- to moderate-size networks of on-chip wiring, but suffered from numerical issues for larger networks. It turned out that for the special case of networks with single inputs and single outputs, these problems can be remedied by exploiting the connection between AWE and the classical Lanczos algorithm for single starting vectors. However, the general case of on-chip wiring involves networks with multiple inputs and outputs, and so a Lanczostype algorithm was needed that could handle such multiple starting vectors. Since no such extension existed, a new band Lanczos algorithm for multiple starting vectors was developed. It turned out that this new band approach can also be employed to devise extensions of other Krylov-subspace methods. In this chapter, we describe the band Lanczos algorithm and the band Arnoldi process and how their developments were driven by the need to efficiently and accurately simulate the effects of on-chip wiring of integrated circuits.

In industrial design processes, the system dynamics of complex engineering structures is modelled by a network approach that results in differentialalgebraic model equations. For this problem class, well-established modular simulation techniques like multi-rate or multi-method approaches, co-simulation or waveform relaxation may suffer from exponential instability. With a novel framework for the stability and convergence analysis of modular time integration methods for differential-algebraic systems the sources of numerical instability could be identified and eliminated. Stabilized modular methods have been developed for such diverse fields of application like multibody system dynamics and circuit simulation. This mathematical research has strongly influenced the design of an industrial interface standard for co-simulation in system dynamics.

In the 1970s of the last century, the progress in powerful simulation software for mechanical multibody systems and for electrical circuits led to a new class of models that is characterized by differential equations and algebraic constraints. These Differential-Algebraic Equations (DAEs) became soon a hot topic, and the methodology that has emerged since then represents now a general field in applied and computational mathematics, with various new applications in science and engineering. Taking a historical approach, the present article introduces a summarizing view at DAEs, with emphasis on numerical aspects and without aiming for completeness. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.

The numerical simulation of electrical machines is of crucial interest in electrical engineering, where design optimization, time-to-market and cost effectiveness have become major concerns in order to face competition in the global market. In the case of squirrel-cage induction machines, one of the challenges is to reduce the CPU time needed to reach the steady-state during the computations, which can take several days whereas engineers are only interested in the last machine revolution needing just a few minutes. Thus, being able to reduce this simulation time is crucial to get a competitive tool at the design stage. This chapter summarizes the history of how a real industrial problem led to the development of a novel numerical algorithm for the solution of the above problem, since the mathematical methods existing at that moment were not completely satisfactory, in terms of computational time, to approximate the steady-state behaviour of squirrel-cage induction machines. Starting from a brief historical background, the authors explain the main problem to solve, the limitations of the existing techniques and the novel methodology, illustrated with very promising numerical results. The method still has different aspects to be exploited that would allow not only to expand the range of applications in the field of electrical engineering, but also in other areas such as acoustics or hydrodynamics.

Radio frequency (RF) circuits are ubiquitous in daily live. However, they are extremely hard to design. Even experienced RF designers typically need several re-designs before the circuit fulfills the specifications.

In this work we present an integrated computational pipeline involving several model order reduction techniques for industrial and applied mathematics, as emerging technology for product and/or process design procedures. Its data-driven nature and its modularity allow an easy integration into existing pipelines.

Four classical systems, the Kelvin gyrostat, the Maclaurin spheroids, the Brouwer rotating saddle, and the Ziegler pendulum have directly inspired development of the theory of Pontryagin and Krein spaces with indefinite metric and singularity theory as independent mathematical topics, not to mention stability theory and nonlinear dynamics.

It has been observed since a long time that data are often carrying interesting topological and geometric structures. Characterizing such structures and providing efficient tools to infer and exploit them is a challenging problem that asks for new mathematics and that is motivated by a real need from applications.

In this chapter, different dynamic regression models designed for the prediction of variables related with energy production, mainly, with variables associated with the price and the energy demand are presented.

In this paper we provide an introduction to quantization with applications in quantitative finance.We start with a review of vector quantization (VQ), a method originally devised for lossy signal compression. The basics of VQ are presented and applied to probability distributions.