Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper `Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation' by P. van der Kamp et al., it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line $ell(x,y)=-c$, and let $B_3,B_4$ be any two distinct points on the line $ell(x,y)=c$. Set $B_0=tfrac{1}{2}(B_1+B_3)$ and $B_5=tfrac{1}{2}(B_2+B_4)$; these points lie on the line $ell(x,y)=0$. Finally, let $B_infty$ be the point at infinity on this line. Let $mathfrak E$ be the pencil of conics with the base points $B_1,B_2,B_3,B_4$. Then the composition of the $B_infty$-switch and of the $B_0$-switch on the pencil $mathfrak E$ is the Kahan discretization of a Hamiltonian vector field $f=ell(x,y)begin{pmatrix}partial H/partial y -partial H/partial x end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map $Phi_f:mathbb C P^2dashrightarrowmathbb C P^2$ has three singular points $B_0,B_2,B_4$, while the inverse map $Phi_f^{-1}$ has three singular points $B_1,B_3,B_5$.
Kahan离散化适用于任何二次向量场，并产生近似于相流位移的双象图。对于具有线性泊松张量和二次汉密尔顿函数的平面二次哈密顿向量场，该映射已知是可积的，并保留了一束圆锥曲线。在P. van der Kamp等人的论文“Kahan离散化是广义Manin变换的根的三类二次向量场”中，证明了Kahan离散化可以表示为二次曲线上的两个对合的复合。本说明可视为对该文件的评论，我们在其中表明，这一结果是可以逆转的。对于一个线性形式$ell(x,y)$，设$B_1,B_2$是直线$ell(x,y)=-c$上任意两个不同的点，设$B_3,B_4$是直线$ell(x,y)=c$上任意两个不同的点。设置$B_0=tfrac{1}{2}(B_1+B_3)$和$B_5=tfrac{1}{2}(B_2+B_4)$;这些点位于$ell(x,y)=0$直线上。最后，设$B_infty$为这条直线上无穷远处的点。设$mathfrak E$为以$B_1,B_2,B_3,B_4$为基点的曲线铅笔。然后，$B_infty$ -开关和铅笔$mathfrak E$上的$B_0$ -开关的组合是哈密顿向量场$f=ell(x,y)begin{pmatrix}partial H/partial y -partial H/partial x end{pmatrix}$与二次哈密顿函数$H(x,y)$的Kahan离散化。这个双国映射$Phi_f:mathbb C P^2dashrightarrowmathbb C P^2$有三个奇异点$B_0,B_2,B_4$，而逆映射$Phi_f^{-1}$有三个奇异点$B_1,B_3,B_5$。
{"title":"Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor","authors":"M. Petrera, Y. Suris","doi":"10.3934/jcd.2019020","DOIUrl":"https://doi.org/10.3934/jcd.2019020","url":null,"abstract":"Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper `Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation' by P. van der Kamp et al., it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line $ell(x,y)=-c$, and let $B_3,B_4$ be any two distinct points on the line $ell(x,y)=c$. Set $B_0=tfrac{1}{2}(B_1+B_3)$ and $B_5=tfrac{1}{2}(B_2+B_4)$; these points lie on the line $ell(x,y)=0$. Finally, let $B_infty$ be the point at infinity on this line. Let $mathfrak E$ be the pencil of conics with the base points $B_1,B_2,B_3,B_4$. Then the composition of the $B_infty$-switch and of the $B_0$-switch on the pencil $mathfrak E$ is the Kahan discretization of a Hamiltonian vector field $f=ell(x,y)begin{pmatrix}partial H/partial y -partial H/partial x end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map $Phi_f:mathbb C P^2dashrightarrowmathbb C P^2$ has three singular points $B_0,B_2,B_4$, while the inverse map $Phi_f^{-1}$ has three singular points $B_1,B_3,B_5$.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"5 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2018-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76367615","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}
A single queueing system with time-dependent exponentially distributed arrival processes and exponential machine processes (Kendall notation $M_t/M_t/1$) is analyzed. Modeling the time evolution for the discrete queue-length distribution by a continuous drift-diffusion process a Smoluchowski equation on the half space is derived approximating the forward Kolmogorov equations. The approximate model is analyzed and validated, showing excellent agreement for the probabilities of all queue lengths and for all queuing utilizations, including ones that are very small and some that are significantly larger than one. Having an excellent approximation for the probability of an empty queue generates an approximation of the expected outflow of the queueing system. Comparisons to several well-established approximation from the literature show significant improvements in several numerical examples.
{"title":"Continuous approximation of $ M_t/M_t/ 1 $ distributions with application to production","authors":"D. Armbruster, Simone Gottlich, S. Knapp","doi":"10.3934/jcd.2020010","DOIUrl":"https://doi.org/10.3934/jcd.2020010","url":null,"abstract":"A single queueing system with time-dependent exponentially distributed arrival processes and exponential machine processes (Kendall notation $M_t/M_t/1$) is analyzed. Modeling the time evolution for the discrete queue-length distribution by a continuous drift-diffusion process a Smoluchowski equation on the half space is derived approximating the forward Kolmogorov equations. The approximate model is analyzed and validated, showing excellent agreement for the probabilities of all queue lengths and for all queuing utilizations, including ones that are very small and some that are significantly larger than one. Having an excellent approximation for the probability of an empty queue generates an approximation of the expected outflow of the queueing system. Comparisons to several well-established approximation from the literature show significant improvements in several numerical examples.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"18 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2018-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79431836","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 the mean-median map as a dynamical system on the space of finite multisets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive family of rational points, the local minima. By constructing a simpler map which represents the dynamics in such neighbourhoods, we extend the results of Cellarosi and Munday (arXiv:1408.3454v1 [math.CO]) by two orders of magnitude. Based on these computations, we conjecture that the Hausdorff dimension of the graph of the limit function of the multiset $[0,x,1]$ is greater than 1.
{"title":"Geometrical properties of the mean-median map","authors":"Jonathan Hoseana, F. Vivaldi","doi":"10.3934/jcd.2020004","DOIUrl":"https://doi.org/10.3934/jcd.2020004","url":null,"abstract":"We study the mean-median map as a dynamical system on the space of finite multisets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive family of rational points, the local minima. By constructing a simpler map which represents the dynamics in such neighbourhoods, we extend the results of Cellarosi and Munday (arXiv:1408.3454v1 [math.CO]) by two orders of magnitude. Based on these computations, we conjecture that the Hausdorff dimension of the graph of the limit function of the multiset $[0,x,1]$ is greater than 1.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"34 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2018-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82378097","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}
Manin transformations are maps of the plane that preserve a pencil of cubic curves. We generalise to maps that preserve quadratic, and certain quartic and higher degree pencils, and show they are measure preserving. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where two double base points go to infinity. On the other hand, each generalised Manin transformation can be brought into QRT-form by a fractional affine transformation. We also classify the generalised Manin transformations which admit a root.
{"title":"Generalised Manin transformations and QRT maps","authors":"Peter H. van der Kamp, D. McLaren, G. Quispel","doi":"10.3934/JCD.2021009","DOIUrl":"https://doi.org/10.3934/JCD.2021009","url":null,"abstract":"Manin transformations are maps of the plane that preserve a pencil of cubic curves. We generalise to maps that preserve quadratic, and certain quartic and higher degree pencils, and show they are measure preserving. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where two double base points go to infinity. On the other hand, each generalised Manin transformation can be brought into QRT-form by a fractional affine transformation. We also classify the generalised Manin transformations which admit a root.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"66 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2018-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74724727","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}
Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples.
{"title":"Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations","authors":"P. Benner, T. Breiten, C. Hartmann, B. Schmidt","doi":"10.3934/jcd.2020001","DOIUrl":"https://doi.org/10.3934/jcd.2020001","url":null,"abstract":"Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2017-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84229114","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}
Hao Zhang, Scott T. M. Dawson, C. Rowley, Eric A. Deem, L. Cattafesta
Dynamic mode decomposition (DMD) gives a practical means of extracting dynamic information from data, in the form of spatial modes and their associated frequencies and growth/decay rates. DMD can be considered as a numerical approximation to the Koopman operator, an infinite-dimensional linear operator defined for (nonlinear) dynamical systems. This work proposes a new criterion to estimate the accuracy of DMD on a mode-by-mode basis, by estimating how closely each individual DMD eigenfunction approximates the corresponding Koopman eigenfunction. This approach does not require any prior knowledge of the system dynamics or the true Koopman spectral decomposition. The method may be applied to extensions of DMD (i.e., extended/kernel DMD), which are applicable to a wider range of problems. The accuracy criterion is first validated against the true error with a synthetic system for which the true Koopman spectral decomposition is known. We next demonstrate how this proposed accuracy criterion can be used to assess the performance of various choices of kernel when using the kernel method for extended DMD. Finally, we show that our proposed method successfully identifies modes of high accuracy when applying DMD to data from experiments in fluids, in particular particle image velocimetry of a cylinder wake and a canonical separated boundary layer.
{"title":"Evaluating the accuracy of the dynamic mode decomposition","authors":"Hao Zhang, Scott T. M. Dawson, C. Rowley, Eric A. Deem, L. Cattafesta","doi":"10.3934/jcd.2020002","DOIUrl":"https://doi.org/10.3934/jcd.2020002","url":null,"abstract":"Dynamic mode decomposition (DMD) gives a practical means of extracting dynamic information from data, in the form of spatial modes and their associated frequencies and growth/decay rates. DMD can be considered as a numerical approximation to the Koopman operator, an infinite-dimensional linear operator defined for (nonlinear) dynamical systems. This work proposes a new criterion to estimate the accuracy of DMD on a mode-by-mode basis, by estimating how closely each individual DMD eigenfunction approximates the corresponding Koopman eigenfunction. This approach does not require any prior knowledge of the system dynamics or the true Koopman spectral decomposition. The method may be applied to extensions of DMD (i.e., extended/kernel DMD), which are applicable to a wider range of problems. The accuracy criterion is first validated against the true error with a synthetic system for which the true Koopman spectral decomposition is known. We next demonstrate how this proposed accuracy criterion can be used to assess the performance of various choices of kernel when using the kernel method for extended DMD. Finally, we show that our proposed method successfully identifies modes of high accuracy when applying DMD to data from experiments in fluids, in particular particle image velocimetry of a cylinder wake and a canonical separated boundary layer.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"28 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86131916","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}
Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm.
{"title":"Approximation of Lyapunov functions from noisy data","authors":"P. Giesl, B. Hamzi, M. Rasmussen, K. Webster","doi":"10.3934/jcd.2020003","DOIUrl":"https://doi.org/10.3934/jcd.2020003","url":null,"abstract":"Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"20 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90297100","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}
Aromatic B-series are a generalization of B-series. Some of the operations defined for B-series can be defined analogically for aromatic B-series. This paper derives combinatorial formulas for the composition and substitution laws for aromatic B-series.
{"title":"Algebraic structure of aromatic B-series","authors":"Geir Bogfjellmo","doi":"10.3934/jcd.2019010","DOIUrl":"https://doi.org/10.3934/jcd.2019010","url":null,"abstract":"Aromatic B-series are a generalization of B-series. Some of the operations defined for B-series can be defined analogically for aromatic B-series. This paper derives combinatorial formulas for the composition and substitution laws for aromatic B-series.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"54 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2015-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91150985","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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