P. Giesl, S. Hafstein, Magnea Haraldsdottir, D. Thorsteinsson, C. Kawan
. We propose a subgradient algorithm for the computation of contraction metrics for systems with an exponentially stable equilibrium. We show that for sufficiently smooth systems our method is always able to compute a contraction metric on any forward-invariant compact neighbourhood of the equilibrium, which is a subset its basin of attraction. We demonstrate the applicability of our method by constructing contraction metrics for three planar and one three-dimensional systems
{"title":"Subgradient algorithm for computing contraction metrics for equilibria","authors":"P. Giesl, S. Hafstein, Magnea Haraldsdottir, D. Thorsteinsson, C. Kawan","doi":"10.3934/jcd.2022030","DOIUrl":"https://doi.org/10.3934/jcd.2022030","url":null,"abstract":". We propose a subgradient algorithm for the computation of contraction metrics for systems with an exponentially stable equilibrium. We show that for sufficiently smooth systems our method is always able to compute a contraction metric on any forward-invariant compact neighbourhood of the equilibrium, which is a subset its basin of attraction. We demonstrate the applicability of our method by constructing contraction metrics for three planar and one three-dimensional systems","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90242066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximated exponential integrators for the stochastic Manakov equation","authors":"A. Berg, David Cohen, G. Dujardin","doi":"10.3934/jcd.2023002","DOIUrl":"https://doi.org/10.3934/jcd.2023002","url":null,"abstract":"","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73251821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Karthik Elamvazhuthi, Siting Liu, Wuchen Li, S. Osher
{"title":"Dynamical optimal transport of nonlinear control-affine systems","authors":"Karthik Elamvazhuthi, Siting Liu, Wuchen Li, S. Osher","doi":"10.3934/jcd.2023006","DOIUrl":"https://doi.org/10.3934/jcd.2023006","url":null,"abstract":"","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80407603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Katerina Karoni, Benedict Leimkuhler, Gabriel Stoltz
We describe a family of descent algorithms which generalizes common existing schemes used in applications such as neural network training and more broadly for optimization of smooth functions–potentially for global optimization, or as a local optimization method to be deployed within global optimization schemes. By introducing an auxiliary degree of freedom we create a dynamical system with improved stability, reducing oscillatory modes and accelerating convergence to minima. The resulting algorithms are simple to implement, and convergence can be shown directly by Lyapunov's second method.Although this framework, which we refer to as friction-adaptive descent (FAD), is fairly general, we focus most of our attention on a specific variant: kinetic energy stabilization (which can be viewed as a zero-temperature Nosé–Hoover scheme with added dissipation in both physical and auxiliary variables), termed KFAD (kinetic FAD). To illustrate the flexibility of the FAD framework we consider several other methods. In certain asymptotic limits, these methods can be viewed as introducing cubic damping in various forms; they can be more efficient than linearly dissipated Hamiltonian dynamics (LDHD).We present details of the numerical methods and show convergence for both the continuous and discretized dynamics in the convex setting by constructing Lyapunov functions. The methods are tested using a toy model (the Rosenbrock function). We also demonstrate the methods for structural optimization for atomic clusters in Lennard–Jones and Morse potentials. The experiments show the relative efficiency and robustness of FAD in comparison to LDHD.
{"title":"Friction-adaptive descent: A family of dynamics-based optimization methods","authors":"Katerina Karoni, Benedict Leimkuhler, Gabriel Stoltz","doi":"10.3934/jcd.2023007","DOIUrl":"https://doi.org/10.3934/jcd.2023007","url":null,"abstract":"We describe a family of descent algorithms which generalizes common existing schemes used in applications such as neural network training and more broadly for optimization of smooth functions–potentially for global optimization, or as a local optimization method to be deployed within global optimization schemes. By introducing an auxiliary degree of freedom we create a dynamical system with improved stability, reducing oscillatory modes and accelerating convergence to minima. The resulting algorithms are simple to implement, and convergence can be shown directly by Lyapunov's second method.Although this framework, which we refer to as friction-adaptive descent (FAD), is fairly general, we focus most of our attention on a specific variant: kinetic energy stabilization (which can be viewed as a zero-temperature Nosé–Hoover scheme with added dissipation in both physical and auxiliary variables), termed KFAD (kinetic FAD). To illustrate the flexibility of the FAD framework we consider several other methods. In certain asymptotic limits, these methods can be viewed as introducing cubic damping in various forms; they can be more efficient than linearly dissipated Hamiltonian dynamics (LDHD).We present details of the numerical methods and show convergence for both the continuous and discretized dynamics in the convex setting by constructing Lyapunov functions. The methods are tested using a toy model (the Rosenbrock function). We also demonstrate the methods for structural optimization for atomic clusters in Lennard–Jones and Morse potentials. The experiments show the relative efficiency and robustness of FAD in comparison to LDHD.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136257203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate a matrix dynamical system related to optimal mass transport in the linear category, namely, the problem of finding an optimal invertible matrix by which two covariance matrices are congruent. We first review the differential geometric structure of the problem in terms of a principal fiber bundle. The dynamical system is a gradient flow restricted to the fibers of the bundle. We prove global existence of solutions to the flow, with convergence to the polar decomposition of the matrix given as initial data. The convergence is illustrated in a numerical example.
{"title":"Convergence of the vertical gradient flow for the Gaussian Monge problem","authors":"Erik Jansson, Klas Modin","doi":"10.3934/jcd.2023008","DOIUrl":"https://doi.org/10.3934/jcd.2023008","url":null,"abstract":"We investigate a matrix dynamical system related to optimal mass transport in the linear category, namely, the problem of finding an optimal invertible matrix by which two covariance matrices are congruent. We first review the differential geometric structure of the problem in terms of a principal fiber bundle. The dynamical system is a gradient flow restricted to the fibers of the bundle. We prove global existence of solutions to the flow, with convergence to the polar decomposition of the matrix given as initial data. The convergence is illustrated in a numerical example.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135505525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a Weiner process that is conditioned to pass through a finite set of points and consider the dynamics generated by iterating a sample path from this process. Using topological techniques we are able to characterize the global dynamics and deduce the existence, structure and approximate location of invariant sets. Most importantly, we compute the probability that this characterization is correct. This work is probabilistic in nature and intended to provide a theoretical foundation for the statistical analysis of dynamical systems which can only be queried via finite samples.
{"title":"Conditioned Wiener processes as nonlinearities: A rigorous probabilistic analysis of dynamics","authors":"K. Mischaikow, Cameron Thieme","doi":"10.3934/jcd.2023004","DOIUrl":"https://doi.org/10.3934/jcd.2023004","url":null,"abstract":"We study a Weiner process that is conditioned to pass through a finite set of points and consider the dynamics generated by iterating a sample path from this process. Using topological techniques we are able to characterize the global dynamics and deduce the existence, structure and approximate location of invariant sets. Most importantly, we compute the probability that this characterization is correct. This work is probabilistic in nature and intended to provide a theoretical foundation for the statistical analysis of dynamical systems which can only be queried via finite samples.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79597207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in begin{document}$ mathbb{R}^3 $end{document} that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary begin{document}$ n $end{document} -homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of begin{document}$ mathbb{R}^3 $end{document} with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.
We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in begin{document}$ mathbb{R}^3 $end{document} that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary begin{document}$ n $end{document} -homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of begin{document}$ mathbb{R}^3 $end{document} with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.
{"title":"Computing connecting orbits to infinity associated with a homoclinic flip bifurcation","authors":"A. Giraldo, B. Krauskopf, H. Osinga","doi":"10.3934/jcd.2020020","DOIUrl":"https://doi.org/10.3934/jcd.2020020","url":null,"abstract":"We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in begin{document}$ mathbb{R}^3 $end{document} that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary begin{document}$ n $end{document} -homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of begin{document}$ mathbb{R}^3 $end{document} with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87250152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior need to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pullback attractors, as well as forward limit sets persist and that the latter two notions even converge under perturbation. As concrete application, we study integrodifference equation under spatial discretization of collocation type.
{"title":"Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors","authors":"H. Huynh, P. Kloeden, Christian Potzsche","doi":"10.3934/jcd.2023003","DOIUrl":"https://doi.org/10.3934/jcd.2023003","url":null,"abstract":"In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior need to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pullback attractors, as well as forward limit sets persist and that the latter two notions even converge under perturbation. As concrete application, we study integrodifference equation under spatial discretization of collocation type.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91369379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit.Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy).This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics.
{"title":"Review on contraction analysis and computation of contraction metrics","authors":"P. Giesl, S. Hafstein, C. Kawan","doi":"10.3934/jcd.2022018","DOIUrl":"https://doi.org/10.3934/jcd.2022018","url":null,"abstract":"Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit.Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy).This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74668366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we propose a new numerical scheme for solving Richards' equation within Gardner's framework and accomplishing mass conservation. In order to do so, we resort to Kirchhoff transformation of Richards' equation in mixed form, so to exploit specific Gardner model features, obtaining a linear second order partial differential equation. Then, leveraging the mass balance condition, we integrate both sides of the equation over a generic grid cell and discretize integrals using trapezoidal rule. This approach provides a linear non-homogeneous initial value problem with respect to the Kirchhoff transform variable, whose solution yields the sought numerical scheme. Such a scheme is proven to be begin{document}$ l^{2} $end{document}-stable and convergent to the exact solution under suitably conditions on step-sizes, retaining the order of convergence from the underlying quadrature formula.
In this work we propose a new numerical scheme for solving Richards' equation within Gardner's framework and accomplishing mass conservation. In order to do so, we resort to Kirchhoff transformation of Richards' equation in mixed form, so to exploit specific Gardner model features, obtaining a linear second order partial differential equation. Then, leveraging the mass balance condition, we integrate both sides of the equation over a generic grid cell and discretize integrals using trapezoidal rule. This approach provides a linear non-homogeneous initial value problem with respect to the Kirchhoff transform variable, whose solution yields the sought numerical scheme. Such a scheme is proven to be begin{document}$ l^{2} $end{document}-stable and convergent to the exact solution under suitably conditions on step-sizes, retaining the order of convergence from the underlying quadrature formula.
{"title":"A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards' equation","authors":"M. Berardi, F. Difonzo","doi":"10.3934/jcd.2022001","DOIUrl":"https://doi.org/10.3934/jcd.2022001","url":null,"abstract":"In this work we propose a new numerical scheme for solving Richards' equation within Gardner's framework and accomplishing mass conservation. In order to do so, we resort to Kirchhoff transformation of Richards' equation in mixed form, so to exploit specific Gardner model features, obtaining a linear second order partial differential equation. Then, leveraging the mass balance condition, we integrate both sides of the equation over a generic grid cell and discretize integrals using trapezoidal rule. This approach provides a linear non-homogeneous initial value problem with respect to the Kirchhoff transform variable, whose solution yields the sought numerical scheme. Such a scheme is proven to be begin{document}$ l^{2} $end{document}-stable and convergent to the exact solution under suitably conditions on step-sizes, retaining the order of convergence from the underlying quadrature formula.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84102798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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