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Recipes for continuation. (English) Zbl 1277.65037

Computational Science & Engineering 11. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611972-56-6/pbk; 978-1-61197-257-3/ebook). xv, 584 p. (2013).
The basic goal of this monograph is to present the parameter continuation method in theoretical rigor, theoretical development and software engineering. In the ground of this method lies the fact that as a rule solutions to parameterized mathematical equation belong to solution families parameterized by the problem parameters.
The introductory Part I (Chapters 1–5) of the book contains the parameters continuation (PC), as numerical technique for computing of implicitly given manifolds, in the view of optimization problems from variation calculus, methodologies and notations, particularly to command-line interaction with the computational continuation care (COCO) framework stipulated by the needs of software engineering.
In Chapters 6,7 of Part II, for the solving of an ordinary differential equations it is suggested the process of discretization results which in a finite-dimensional family of zero functions expressed in terms of a finite number of continuation variables, i.e., the collocation zero problem is formulated depending on decisions made in the discretization process. The here considered collocation zero problem seeks to arrive at finite-dimensional problems whose solutions converge to the solution of the original infinite-dimensional problem, with desirable convergence properties as the discretization is further refined. In Chapters 8–10, the developments of toolboxes are presented for continuing solutions to two-point boundary value problems in a single independent variable, periodic orbits in autonomous dynamical systems and generally sets of constrained orbit segments of hybrid dynamic systems, quasiperiodic invariant tori in smooth dynamic systems, and also to the continuation of connecting orbits between equilibria and periodic orbits.
Part III (Chapters 11–14) is devoted to the development of atlas algorithms, at first to general theory, then to their subclasses: the minimal single-dimensional atlas illustrating the interface to continuation afforded by AtlasBase and CirveSegment, its generalization to two-dimensional solution manifolds and generally of such implementations to the multidimensional context, with several example implementations for covering two-dimensional manifolds and in conclusion their modifications in order to accommodate volume constraints on the computational domain.
The essence of PC is the detection of special points. usually bifurcation points, connected with the critical changes of solution and solution manifolds properties. This concept of “event handling” is the main subject of Part IV. Chapter 15 defines event handling in the sense of detection and location of special points along curve segments on the solution manifold at the usage of the basic syntax for command-line handling of event in COCO. This command-line interface to event detection and location in COCO of Section 15.3 is fully compatible with embedding in COCO-compatible toolboxes. Such toolbox-specific support for event detection and location is consistent with the task embedding formalism, where events associated with toolbox-specific properties of points on the solution manifold are defined be the corresponding toolbox. Thus, Chapter 16 presents the integration of event handling in COCO-compatible atlas algorithms and toolboxes. As a closed continuation problem has been constructed, COCO-compatible zero and monitor functions can be added to the continuation problem structure. Obvious candidates for monitor functions to be added to the continuation problem structure by an atlas algorithms, are those functions which depend on the current projection condition, rather than simply the locus on the solution manifold. Here the use of such monitor functions and the relevant atlas events is demonstrated for fold- and branch-point, detection and location.
In Chapter 17 at the usage of reverse communication protocol template event handlers are provided for enabling a selective treatment of bifurcation of equilibria points and periodic orbits.
Part V considers the problem of adaptive changes to the discretization of a continuation problem during PC. In Chapter 18, two examples of adaptation are presented. In the first one the adaptation strategy is presented for the case when the discretization error estimate exceeds a critical value. In the second one the discretization parameters are included among the continuation variables and solved for accordingly. Here an augmented extended continuation problem is obtained by appending additional conditions to the contination zero problem in order to allow the discretization to change concurrently with the discretized the discretized solution during continuation.
In Chapter 19, a mesh-preserving discretization method is implemented supporting adaptive changes to the discretization order without a change in the meaning of the number of continuation variables. Chapter 20 provides examples of adaptation methods where the meaning and the number of continuation variables may change during continuation. Such moving-mesh schemes are accommodated here through some modifications of the atlas algorithms and associating with each zero function which depends on the discretization.
Each chapter is equipped with exercises and implementations in the COCO framework of algorithms and toolboxes. Part IV as epilogue includes a number of proposed development projects for young investigators.

MSC:

65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65Y20 Complexity and performance of numerical algorithms
68N30 Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.)
34B15 Nonlinear boundary value problems for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65P30 Numerical bifurcation problems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems

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