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Tohru Ozaki - One of the best experts on this subject based on the ideXlab platform.
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Model driven EEG/fMRI fusion of brain oscillations
Human Brain Mapping, 2009Co-Authors: Pedro A. Valdes-sosa, Jose Miguel Sanchez-bornot, Roberto Carlos Sotero, Yasser Iturria-medina, Yasser Aleman-gomez, Jorge Bosch-bayard, Felix Carbonell, Tohru OzakiAbstract:This article reviews progress and challenges in model driven EEG/fMRI fusion with a focus on brain oscillations. Fusion is the combination of both imaging modalities based on a cascade of forward models from ensemble of post-synaptic potentials (ePSP) to net primary current densities (nPCD) to EEG; and from ePSP to vasomotor feed forward signal (VFFSS) to BOLD. In absence of a model, data driven fusion creates maps of correlations between EEG and BOLD or between estimates of nPCD and VFFS. A consistent finding has been that of positive correlations between EEG alpha power and BOLD in both frontal cortices and thalamus and of negative ones for the occipital region. For model driven fusion we formulate a neural mass EEG/fMRI model coupled to a metabolic hemodynamic model. For exploratory simulations we show that the Local Linearization (LL) method for integrating stochastic differential equations is appropriate for highly nonlinear dynamics. It has been successfully applied to small and medium sized networks, reproducing the described EEG/BOLD correlations. A new LL-algebraic method allows simulations with hundreds of thousands of neural populations, with connectivities and conduction delays estimated from diffusion weighted MRI. For parameter and state estimation, Kalman filtering combined with the LL method estimates the innovations or prediction errors. From these the likelihood of models given data are obtained. The LL-innovation estimation method has been already applied to small and medium scale models. With improved Bayesian computations the practical estimation of very large scale EEG/fMRI models shall soon be possible.
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Local Linearization runge kutta llrk methods for solving ordinary differential equations
International Conference on Computational Science, 2006Co-Authors: H De La Cruz, Felix Carbonell, J C Jimenez, Rolando J Biscay, Tohru OzakiAbstract:A new class of stable methods for solving ordinary differential equations (ODEs) is introduced. This is based on combining the Local Linearization (LL) integrator with other extant discretization methods. For this, an auxiliary ODE is solved to determine a correction term that is added to the LL approximation. In particular, combining the LL method with (explicit) Runge Kutta integrators yields what we call LLRK methods. This permits to improve the order of convergence of the LL method without loss of its stability properties. The performance of the proposed integrators is illustrated through computer simulations.
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simulation of stochastic differential equations through the Local Linearization method a comparative study
Journal of Statistical Physics, 1999Co-Authors: J C Jimenez, Isao Shoji, Tohru OzakiAbstract:A new Local Linearization (LL) scheme for the numerical integration of nonautonomous multidimensional stochastic differential equations (SDEs) with additive noise is introduced. The numerical scheme is based on the Local Linearization of the SDE's drift coefficient by means of a truncated Ito–Taylor expansion. A comparative study with the other LL schemes is presented which shows some advantanges of the new scheme over other ones.
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estimation for nonlinear stochastic differential equations by a Local Linearization method 1
Stochastic Analysis and Applications, 1998Co-Authors: Isao Shoji, Tohru OzakiAbstract:This paper proposes a new Local Linearization method which approximates a nonlinear stochastic differential equation by a linear stochastic differential equation. Using this method, we can estimate parameters of the nonlinear stochastic differential equation from discrete observations by the maximum likelihood technique. We conduct the numerical experiments to evaluate the finite sample performance of identification of the new method, and compare it with the two known methods: the original Local Linearization method and the Euler methods. From the results of experiments, the new method shows much better performance than the other two methods particularly when the sampling interval is large
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the Local Linearization filter with application to nonlinear system identifications
1994Co-Authors: Tohru OzakiAbstract:Many dynamic phenomena in scientific fields are modeled by a continuous time stochastic dynamical system , $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\dot z} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} |\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } ) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w} (t),$$ where w(t) is a Gaussian white noise. The process z(t) defined by the model is a Markov diffusion proce ss. When we use such mathernati cal model s in physics or engineering sciences we often have a rough idea, from some physical understanding of the process, of the size of the parameters θ and the variance-covariance matrix Σ of the Gaussian white noise w(t) in the model. However when prediction or control of the process is concerned we sometime need more accurat e estimates of the parameters. In some cases, we don’t have much physical information helping us to guess the size of the parameters in the model. In such situations we often try to estimate parameters from observation data of the phenomenon as accurately as possible
J C Jimenez - One of the best experts on this subject based on the ideXlab platform.
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a weak Local Linearization scheme for stochastic differential equations with multiplicative noise
Journal of Computational and Applied Mathematics, 2017Co-Authors: J C Jimenez, Carlos M Mora, M SelvaAbstract:Abstract In this paper, a weak Local Linearization scheme for Stochastic Differential Equations (SDEs) with multiplicative noise is introduced. First, for a time discretization, the solution of the SDE is Locally approximated by the solution of the piecewise linear SDE that results from the Local Linearization strategy. The weak numerical scheme is then defined as a sequence of random vectors whose first moments coincide with those of the piecewise linear SDE on the time discretization. The scheme is explicit, preserves the first two moments of the solution of SDEs with linear drift and diffusion coefficients in state and time, and inherits the mean-square stability or instability that such solution may have. The rate of convergence is derived and numerical simulations are presented for illustrating the performance of the scheme.
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multiple shooting Local Linearization method for the identification of dynamical systems
Communications in Nonlinear Science and Numerical Simulation, 2016Co-Authors: Felix Carbonell, Yasser Iturriamedina, J C JimenezAbstract:Abstract The combination of the multiple shooting strategy with the generalized Gauss–Newton algorithm turns out in a recognized method for estimating parameters in ordinary differential equations (ODEs) from noisy discrete observations. A key issue for an efficient implementation of this method is the accurate integration of the ODE and the evaluation of the derivatives involved in the optimization algorithm. In this paper, we study the feasibility of the Local Linearization (LL) approach for the simultaneous numerical integration of the ODE and the evaluation of such derivatives. This integration approach results in a stable method for the accurate approximation of the derivatives with no more computational cost than that involved in the integration of the ODE. The numerical simulations show that the proposed Multiple Shooting-Local Linearization method recovers the true parameters value under different scenarios of noisy data.
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a weak Local Linearization scheme for stochastic differential equations with multiplicative noise
arXiv: Numerical Analysis, 2015Co-Authors: J C Jimenez, Carlos M Mora, M SelvaAbstract:In this paper, a weak Local Linearization scheme for Stochastic Differential Equations (SDEs) with multiplicative noise is introduced. First, for a time discretization, the solution of the SDE is Locally approximated by the solution of the piecewise linear SDE that results from the Local Linearization strategy. The weak numerical scheme is then defined as a sequence of random vectors whose first moments coincide with those of the piecewise linear SDE on the time discretization. The rate of convergence is derived and numerical simulations are presented for illustrating the performance of the scheme.
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strong Local Linearization methods for the numerical integration of stochastic differential equations with additive noise an overview
2010Co-Authors: J C JimenezAbstract:Strong Local Linearization (LL) methods conform a class of one-step explicit integrators for SDEs with additive noise derived from the following primary and common strategy: the drift coe!cient of the di"erential equation is Locally (piecewise) approximated through a linear ItoTaylor expansion at each time step, thus obtaining successive linear equations that are explicitly integrated. Hereafter, the LL approach may include some additional strategies to improve that basic a!ne approximation. Theoretical and practical results have shown that the LL integrators have a number of convenient properties. These include arbitrary order of convergence, Astability, preservation of the dynamic properties of the linear systems, low computational cost, and others. Remarkably, for nonlinear equations in general, these integrators show a stability similar to that of implicit schemes, but with much lower computational cost (comparable to conventional explicit schemes). In this paper, a review of the LL methods and their properties is presented.
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rate of convergence of Local Linearization schemes for random differential equations
Bit Numerical Mathematics, 2009Co-Authors: J C Jimenez, Felix CarbonellAbstract:Recently, two Local Linearization (LL) schemes for the numerical integration of random differential equation have been proposed, which differ with respect to the algorithm that is used for the numerical implementation of the Local Linear discretization. However, in contrast with the Local Linear discretization, the order of convergence of the LL schemes have not been studied so far. In this paper, a general theorem about this matter is presented and, on that base, additional results are derived for each particular scheme.
Felix Carbonell - One of the best experts on this subject based on the ideXlab platform.
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multiple shooting Local Linearization method for the identification of dynamical systems
Communications in Nonlinear Science and Numerical Simulation, 2016Co-Authors: Felix Carbonell, Yasser Iturriamedina, J C JimenezAbstract:Abstract The combination of the multiple shooting strategy with the generalized Gauss–Newton algorithm turns out in a recognized method for estimating parameters in ordinary differential equations (ODEs) from noisy discrete observations. A key issue for an efficient implementation of this method is the accurate integration of the ODE and the evaluation of the derivatives involved in the optimization algorithm. In this paper, we study the feasibility of the Local Linearization (LL) approach for the simultaneous numerical integration of the ODE and the evaluation of such derivatives. This integration approach results in a stable method for the accurate approximation of the derivatives with no more computational cost than that involved in the integration of the ODE. The numerical simulations show that the proposed Multiple Shooting-Local Linearization method recovers the true parameters value under different scenarios of noisy data.
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Local Linearization runge kutta methods a class of a stable explicit integrators for dynamical systems
arXiv: Numerical Analysis, 2012Co-Authors: H De La Cruz, Rolando J Biscay, J C Jimenez, Felix CarbonellAbstract:A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of the original equation in two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary ODE. The first one is solved by a Local Linearization scheme in such a way that A-stability is ensured, while the second one can be approximated by any extant scheme, preferably a high order explicit Runge-Kutta scheme. Results on the convergence and dynamical properties of this new class of schemes are given, as well as some hints for their efficient numerical implementation. An specific scheme of this new class is derived in detail, and its performance is compared with some Matlab codes in the integration of a variety of ODEs representing different types of dynamics.
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high order Local Linearization methods an approach for constructing a stable explicit schemes for stochastic differential equations with additive noise
Bit Numerical Mathematics, 2010Co-Authors: H De La Cruz Cancino, Felix Carbonell, J C Jimenez, Rolando J Biscay, T OzakiAbstract:An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.
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rate of convergence of Local Linearization schemes for random differential equations
Bit Numerical Mathematics, 2009Co-Authors: J C Jimenez, Felix CarbonellAbstract:Recently, two Local Linearization (LL) schemes for the numerical integration of random differential equation have been proposed, which differ with respect to the algorithm that is used for the numerical implementation of the Local Linear discretization. However, in contrast with the Local Linear discretization, the order of convergence of the LL schemes have not been studied so far. In this paper, a general theorem about this matter is presented and, on that base, additional results are derived for each particular scheme.
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Model driven EEG/fMRI fusion of brain oscillations
Human Brain Mapping, 2009Co-Authors: Pedro A. Valdes-sosa, Jose Miguel Sanchez-bornot, Roberto Carlos Sotero, Yasser Iturria-medina, Yasser Aleman-gomez, Jorge Bosch-bayard, Felix Carbonell, Tohru OzakiAbstract:This article reviews progress and challenges in model driven EEG/fMRI fusion with a focus on brain oscillations. Fusion is the combination of both imaging modalities based on a cascade of forward models from ensemble of post-synaptic potentials (ePSP) to net primary current densities (nPCD) to EEG; and from ePSP to vasomotor feed forward signal (VFFSS) to BOLD. In absence of a model, data driven fusion creates maps of correlations between EEG and BOLD or between estimates of nPCD and VFFS. A consistent finding has been that of positive correlations between EEG alpha power and BOLD in both frontal cortices and thalamus and of negative ones for the occipital region. For model driven fusion we formulate a neural mass EEG/fMRI model coupled to a metabolic hemodynamic model. For exploratory simulations we show that the Local Linearization (LL) method for integrating stochastic differential equations is appropriate for highly nonlinear dynamics. It has been successfully applied to small and medium sized networks, reproducing the described EEG/BOLD correlations. A new LL-algebraic method allows simulations with hundreds of thousands of neural populations, with connectivities and conduction delays estimated from diffusion weighted MRI. For parameter and state estimation, Kalman filtering combined with the LL method estimates the innovations or prediction errors. From these the likelihood of models given data are obtained. The LL-innovation estimation method has been already applied to small and medium scale models. With improved Bayesian computations the practical estimation of very large scale EEG/fMRI models shall soon be possible.
J C Jimenez - One of the best experts on this subject based on the ideXlab platform.
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convergence rate of weak Local Linearization schemes for stochastic differential equations with additive noise
Journal of Computational and Applied Mathematics, 2015Co-Authors: J C Jimenez, F. CarbonellAbstract:There exists a diversity of weak Local Linearization (LL) schemes for the integration of stochastic differential equations with additive noise, which differ in the algorithms employed for the numerical implementation of the weak Local Linear discretizations. Despite convergence results for these discretizations have been already developed, the convergence of the weak LL schemes has not been considered up to date. In this work, a general result concerning the convergence rate of the weak LL schemes is presented, as well as specificities for a number of particular schemes. As an application, the convergence of weak LL schemes for equations driven by Poisson processes is presented in addition.
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convergence rate of weak Local Linearization schemes for stochastic differential equations with additive noise
arXiv: Numerical Analysis, 2013Co-Authors: J C Jimenez, F. CarbonellAbstract:There exists a diversity of weak Local Linearization (LL) schemes for the integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is employed in the numerical implementation of the weak Local Linear discretizations. On the contrary to the Local Linear discretization, the rate of convergence of the LL schemes has not been considered up to now. In this work, a general theorem about this issue is derived and further is applied to a number of specific schemes. As application, the convergence rate of weak LL schemes for equations with jumps is also presented.
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Local Linearization runge kutta methods a class of a stable explicit integrators for dynamical systems
arXiv: Numerical Analysis, 2012Co-Authors: H De La Cruz, Rolando J Biscay, J C Jimenez, Felix CarbonellAbstract:A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of the original equation in two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary ODE. The first one is solved by a Local Linearization scheme in such a way that A-stability is ensured, while the second one can be approximated by any extant scheme, preferably a high order explicit Runge-Kutta scheme. Results on the convergence and dynamical properties of this new class of schemes are given, as well as some hints for their efficient numerical implementation. An specific scheme of this new class is derived in detail, and its performance is compared with some Matlab codes in the integration of a variety of ODEs representing different types of dynamics.
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convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise
Bit Numerical Mathematics, 2012Co-Authors: J C Jimenez, H De La Cruz CancinoAbstract:There is a variety of strong Local Linearization (LL) schemes for the numerical integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is used in the numerical implementation of the strong Local Linear discretization. However, in contrast with the Local Linear discretization, the convergence rate of the LL schemes has not been studied so far. In this paper, two general theorems about this matter are presented and, with their support, additional results are derived for some particular schemes. As a direct application, the convergence rate of some strong LL schemes for SDEs with jumps is briefly expounded as well.
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High order Local Linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise
BIT Numerical Mathematics, 2010Co-Authors: H. Cruz cancino, J C Jimenez, R. Biscay, F. Carbonell, T OzakiAbstract:An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.
T Ozaki - One of the best experts on this subject based on the ideXlab platform.
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High order Local Linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise
BIT Numerical Mathematics, 2010Co-Authors: H. Cruz cancino, J C Jimenez, R. Biscay, F. Carbonell, T OzakiAbstract:An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.
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high order Local Linearization methods an approach for constructing a stable explicit schemes for stochastic differential equations with additive noise
Bit Numerical Mathematics, 2010Co-Authors: H De La Cruz Cancino, Felix Carbonell, J C Jimenez, Rolando J Biscay, T OzakiAbstract:An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.
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a higher order Local Linearization method for solving ordinary differential equations
Applied Mathematics and Computation, 2007Co-Authors: H De La Cruz, Felix Carbonell, T Ozaki, Rolando J Biscay, J C JimenezAbstract:The Local Linearization (LL) method for the integration of ordinary differential equations is an explicit one-step method that has a number of suitable dynamical properties. However, a major drawback of the LL integrator is that its order of convergence is only two. The present paper overcomes this limitation by introducing a new class of numerical integrators, called the LLT method, that is based on the addition of a correction term to the LL approximation. In this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the LL method. In particular, it is proved that the LLT method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence. The performance of the introduced method is further illustrated through computer simulations.
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Local Linearization filters for non linear continuous discrete state space models with multiplicative noise
International Journal of Control, 2003Co-Authors: J C Jimenez, T OzakiAbstract:In this paper, the Local Linearization method for the approximate computation of the prediction and filtering estimates of continuous-discrete state space models is extended to the general case of non-linear non-autonomous models with multiplicative noise. The approximate prediction and filter estimates are obtained by applying the optimal linear filter to the piecewise linear state space model that emerges from a Local Linearization of both the non-linear state equation and the non-linear measurement equation. In addition, the solutions of the differential equations that describe the evolution of the first two conditional moments between observations are obtained, and an algorithm for their numerical computation is also given. The performance of the LL filters is illustrated by mean of numerical experiments.