Optimization Algorithms on Matrix Manifolds

This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms.

CLADP: THE CAMBRIDGE LINEAR ANALYSIS AND DESIGN PROGRAMS.

An interactive software facility for designing multivariable control systems is described. The paper discusses the desirable characteristics of such a facility, the particular capabilities of CLADP and the numerical algorithms which lie behind them.

The changing face and role of CACSD

Computer Aided Control Engineering involves three parallel streams: Simulation and modelling, Control system design (off-line), and Controller implementation. In industry the bottleneck problem has always been modelling, and this remains the case - that is where control (and other) engineers put most of their technical effort. Although great advances in software tools have been made, the cost of modelling remains very high - too high for some sectors. Object-oriented modelling, enabling truly re-usable models, seems to be the key enabling technology here. Software tools to support control systems design have two aspects to them: aiding and managing the work-flow in particular projects (whether of a single engineer or of a team), and provision of numerical algorithms to support control-theoretic and systems-theoretic analysis and design. The numerical problems associated with linear systems have been largely overcome, so that most problems can be tackled routinely without difficulty - though problems remain with (some) systems of extremely large dimensions. Recent emphasis on control of hybrid and/or constrained systems is leading to the emerging importance of geometric algorithms (ellipsoidal approximation, polytope projection, etc). Constantly increasing computational power is leading to renewed interest in design by optimisation, an example of which is MPC. The explosion of embedded control systems has highlighted the importance of autocode generation, directly from modelling/simulation products to target processors. This is the 'new kid on the block', and again much of the focus of commercial tools is on this part of the control engineer's job. Here the control engineer can no longer ignore computer science (at least, for the time being). © 2006 IEEE.

The changing face and role of CACSD

Computer Aided Control Engineering involves three parallel streams: Simulation and modelling, Control system design (off-line), and Controller implementation. In industry the bottleneck problem has always been modelling, and this remains the case - that is where control (and other) engineers put most of their technical effort. Although great advances in software tools have been made, the cost of modelling remains very high - too high for some sectors. Object-oriented modelling, enabling truly re-usable models, seems to be the key enabling technology here. Software tools to support control systems design have two aspects to them: aiding and managing the work-flow in particular projects (whether of a single engineer or of a team), and provision of numerical algorithms to support control-theoretic and systems-theoretic analysis and design. The numerical problems associated with linear systems have been largely overcome, so that most problems can be tackled routinely without difficulty - though problems remain with (some) systems of extremely large dimensions. Recent emphasis on control of hybrid and/or constrained systems is leading to the emerging importance of geometric algorithms (ellipsoidal approximation, polytope projection, etc). Constantly increasing computational power is leading to renewed interest in design by optimisation, an example of which is MPC. The explosion of embedded control systems has highlighted the importance of autocode generation, directly from modelling/simulation products to target processors. This is the 'new kid on the block', and again much of the focus of commercial tools is on this part of the control engineer's job. Here the control engineer can no longer ignore computer science (at least, for the time being). ©2006 IEEE.

Optimization on manifolds: Methods and applications

This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however,we choose to emphasise the intuition of the concepts and the structures that are important in generating practical numerical algorithms rather than the technical details of the formulation. There are a number of algorithms that can be applied to solve such problems and we discuss the steepest descent and Newton's method in some detail as well as referencing the more important of the other approaches.There are a wide range of potential applications that we are aware of, and we briefly discuss these applications, as well as explaining one or two in more detail. © 2010 Springer -Verlag Berlin Heidelberg.

Generalized phenomenological model for the viscoelasticity of idealized asphalts

A generalized theory for the viscoelastic behavior of idealized bituminous mixtures (asphalts) is presented. The mathematical model incorporates strain rate and temperature dependency as well as nonmonotonic loading and unloading with shape recovery. The stiffening effect of the aggregate is included. The model is of phenomenological nature. It can be calibrated using a relatively limited set of experimental parameters, obtainable by uniaxial tests. It is shown that the mathematical model can be represented as a special nonlinear form of the Burgers model. This facilitates the derivation of numerical algorithms for solving the constitutive equations. A numerical scheme is implemented in a user material subroutine (UMAT) in the finite-element analysis (FEA) code ABAQUS. Simulation results are compared with uniaxial and indentation tests on an idealized asphalt mix. © 2014 American Society of Civil Engineers.

Manopt, a matlab toolbox for optimization on manifolds

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design effcient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. By dealing internally with most of the differential geometry, the package aims particularly at lowering the entrance barrier. © 2014 Nicolas Boumal.

Adjoint algorithms for the Navier-Stokes equations in the low Mach number limit

This paper describes a derivation of the adjoint low Mach number equations and their implementation and validation within a global mode solver. The advantage of using the low Mach number equations and their adjoints is that they are appropriate for flows with variable density, such as flames, but do not require resolution of acoustic waves. Two versions of the adjoint are implemented and assessed: a discrete-adjoint and a continuous-adjoint. The most unstable global mode calculated with the discrete-adjoint has exactly the same eigenvalue as the corresponding direct global mode but contains numerical artifacts near the inlet. The most unstable global mode calculated with the continuous-adjoint has no numerical artifacts but a slightly different eigenvalue. The eigenvalues converge, however, as the timestep reduces. Apart from the numerical artifacts, the mode shapes are very similar, which supports the expectation that they are otherwise equivalent. The continuous-adjoint requires less resolution and usually converges more quickly than the discrete-adjoint but is more challenging to implement. Finally, the direct and adjoint global modes are combined in order to calculate the wavemaker region of a low density jet. © 2011 Elsevier Inc.

A matrix-calculation-based algorithm for numerical change propagation analysis

Engineering changes (ECs) are raised throughout the lifecycle of engineering products. A single change to one component produces knock-on effects on others necessitating additional changes. This change propagation significantly affects the development time and cost and determines the product's success. Predicting and managing such ECs is, thus, essential to companies. Some prediction tools model change propagation by algorithms, whereof a subgroup is numerical. Current numerical change propagation algorithms either do not account for the exclusion of cyclic propagation paths or are based on exhaustive searching methods. This paper presents a new matrix-calculation-based algorithm which can be applied directly to a numerical product model to analyze change propagation and support change prediction. The algorithm applies matrix multiplications on mutations of a given design structure matrix accounting for the exclusion of self-dependences and cyclic propagation paths and delivers the same results as the exhaustive search-based Trail Counting algorithm. Despite its factorial time complexity, the algorithm proves advantageous because of its straightforward matrix-based calculations which avoid exhaustive searching. Thereby, the algorithm can be implemented in established numerical programs such as Microsoft Excel which promise a wider application of the tools within and across companies along with better familiarity, usability, practicality, security, and robustness. © 1988-2012 IEEE.

Numerical simulations supporting the process design of ring rolling processes

In conventional Finite Element Analysis (FEA) of radial-axial ring rolling (RAR) the motions of all tools are usually defined prior to simulation in the preprocessing step. However, the real process holds up to 8 degrees of freedom (DOF) that are controlled by industrial control systems according to actual sensor values and preselected control strategies. Since the histories of the motions are unknown before the experiment and are dependent on sensor data, the conventional FEA cannot represent the process before experiment. In order to enable the usage of FEA in the process design stage, this approach integrates the industrially applied control algorithms of the real process including all relevant sensors and actuators into the FE model of ring rolling. Additionally, the process design of a novel process 'the axial profiling', in which a profiled roll is used for rolling axially profiled rings, is supported by FEA. Using this approach suitable control strategies can be tested in virtual environment before processing. © 2013 AIP Publishing LLC.