The Experts below are selected from a list of 231 Experts worldwide ranked by ideXlab platform
Christoph Zechner - One of the best experts on this subject based on the ideXlab platform.
-
Sensitivity estimation for stochastic models of biochemical reaction networks in the presence of extrinsic variability
Journal of Chemical Physics, 2017Co-Authors: Jakob Ruess, Heinz Koeppl, Christoph ZechnerAbstract:Determining the sensitivity of certain system states or outputs to variations in parameters facilitates our understanding of the inner working of that system and is an essential design tool for the de-novo construction of robust systems. In cell biology, the output of interest is often the response of a certain reaction network to some input (e.g. stressors or nutrients) and one aims to quantify the sensitivity of this response in the presence of parameter heterogeneity. We argue that for such applications, parametric sensitivities in their standard form do not paint a complete picture of a system's robustness since one assumes that all cells in the population have the same parameters and are perturbed in the same way. Here, we consider stochastic reaction networks in which the parameters are randomly distributed over the population and propose a new sensitivity index that captures the robustness of system outputs upon changes in the characteristics of the parameter distribution, rather than the parameters themselves. Subsequently, we make use of Girsanov's likelihood ratio method to construct a Monte Carlo Estimator of this sensitivity index. However, it turns out that this Estimator has an exceedingly large variance. To overcome this problem, we propose a novel estimation algorithm that makes use of a marginalization of the path distribution of stochastic reaction networks and leads to Rao-Blackwellized Estimators with reduced variance.
Anis Al Gerbi - One of the best experts on this subject based on the ideXlab platform.
-
Ninomiya-Victoir scheme : strong convergence, asymptotics for the normalized error and multilevel Monte Carlo methods
2016Co-Authors: Anis Al GerbiAbstract:This thesis is dedicated to the study of the strong convergence properties of the Ninomiya-Victoir scheme, which is based on the resolution of $d+1$ ordinary differential equations (ODEs) at each time step, to approximate the solution to a stochastic differential equation (SDE), where $d$ is the dimension of the Brownian. This study is aimed at analysing the use of this scheme in a multilevel Monte Carlo Estimator. Indeed, the optimal complexity of this method is driven by the order of convergence to zero of the variance between the two schemes used on the coarse and fine grids at each level, which is related to the strong convergence order between the two schemes. In the second chapter, we prove strong convergence with order $1/2$ of the Ninomiya-Victoir scheme $X^{NV,eta}$, with time step $T/N$, to the solution $X$ of the limiting SDE. Recently, Giles and Szpruch proposed a modified Milstein scheme and its antithetic version, based on the swapping of each successive pair of Brownian increments in the scheme, permitting to construct a multilevel Monte Carlo Estimator achieving the optimal complexity $Oleft(epsilon^{-2}right)$ for the precision $epsilon$, as in a simple Monte Carlo method with independent and identically distributed unbiased random variables. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order $1$ to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo Estimator. This idea is inspired by Debrabant and R"ossler who suggest to use a scheme with high order of weak convergence on the finest grid at the finest level of the multilevel Monte Carlo method. As the optimal number of discretization levels is related to the weak order of the scheme used in the finest grid at the finest level, Debrabant and R"ossler manage to reduce the computational time, by decreasing the number of discretization levels. The coupling with the Giles-Szpruch scheme allows us to combine both ideas. By this way, we preserve the optimal complexity $Oleft(epsilon^{-2}right)$ and we reduce the computational time, since the Ninomiya-Victoir scheme is known to exhibit weak convergence with order 2. In the third chapter, we check that the normalized error defined by $sqrt{N}left(X - X^{NV,eta}right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. This result ensures that the strong convergence rate is actually $1/2$ when at least two Brownian vector fields do not commute. To link this result to the multilevel Monte Carlo Estimator, it can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central limit theorem of Lindeberg Feller type, derived recently by Ben Alaya and Kebaier for the multilevel Monte Carlo Estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. We then prove strong convergence with order $1$ in this case. The fourth chapter deals with the convergence of the normalized error process $Nleft(X - X^{NV}right)$, where $X^{NV}$ is the Ninomiya-Victoir in the commutative case. We prove its stable convergence in law to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually $1$ when the Brownian vector fields commute, but at least one of them does not commute with the Stratonovich drift vector field
-
Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel Estimators
Monte Carlo Methods and Applications, 2016Co-Authors: Anis Al Gerbi, Benjamin Jourdain, Emmanuelle ClémentAbstract:In this paper, we are interested in the strong convergence properties of the Ninomiya–Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2${1/2}$. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo Estimator: indeed, the variance of a multilevel Monte Carlo Estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo Estimator achieving the optimal complexity O(ϵ-2)${O(\epsilon^{-2})}$ for the precision ϵ. In the same spirit, we propose a modified Ninomiya–Victoir scheme, which may be strongly coupled with order 1 to the Giles–Szpruch scheme at the finest level of a multilevel Monte Carlo Estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya–Victoir scheme permits to reduce the number of discretisation levels.
-
Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel Estimators
arXiv: Computational Finance, 2015Co-Authors: Anis Al Gerbi, Benjamin Jourdain, Emmanuelle ClémentAbstract:In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order $1/2$. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo Estimator: indeed, the variance of a multilevel Monte Carlo Estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo Estimator achieving the optimal complexity $O\left(\epsilon^{-2}\right)$ for the precision $\epsilon$. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order $1$ to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo Estimator. Numerical experiments show that this choice improves the efficiency, since the order $2$ of weak convergence of the Ninomiya-Victoir scheme permits to reduce the number of discretization levels.
Emmanuelle Clément - One of the best experts on this subject based on the ideXlab platform.
-
Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel Estimators
Monte Carlo Methods and Applications, 2016Co-Authors: Anis Al Gerbi, Benjamin Jourdain, Emmanuelle ClémentAbstract:In this paper, we are interested in the strong convergence properties of the Ninomiya–Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2${1/2}$. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo Estimator: indeed, the variance of a multilevel Monte Carlo Estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo Estimator achieving the optimal complexity O(ϵ-2)${O(\epsilon^{-2})}$ for the precision ϵ. In the same spirit, we propose a modified Ninomiya–Victoir scheme, which may be strongly coupled with order 1 to the Giles–Szpruch scheme at the finest level of a multilevel Monte Carlo Estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya–Victoir scheme permits to reduce the number of discretisation levels.
-
Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel Estimators
arXiv: Computational Finance, 2015Co-Authors: Anis Al Gerbi, Benjamin Jourdain, Emmanuelle ClémentAbstract:In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order $1/2$. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo Estimator: indeed, the variance of a multilevel Monte Carlo Estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo Estimator achieving the optimal complexity $O\left(\epsilon^{-2}\right)$ for the precision $\epsilon$. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order $1$ to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo Estimator. Numerical experiments show that this choice improves the efficiency, since the order $2$ of weak convergence of the Ninomiya-Victoir scheme permits to reduce the number of discretization levels.
Don Mcleish - One of the best experts on this subject based on the ideXlab platform.
-
a general method for debiasing a Monte Carlo Estimator
Monte Carlo Methods and Applications, 2011Co-Authors: Don McleishAbstract:Consider a process, stochastic or deterministic, obtained by using a numerical integration scheme, or from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation. We will assume that we can sample from the distribution of the process from time 0 to finite time n. We propose a scheme for unbiased estimation of the limiting value of the process, together with estimates of standard error and apply this to examples including numerical integrals, root-finding and option pricing in a Heston Stochastic Volatility model. This results in unbiased Estimators in place of biased ones i nmany potential applications.
Jakob Ruess - One of the best experts on this subject based on the ideXlab platform.
-
Sensitivity estimation for stochastic models of biochemical reaction networks in the presence of extrinsic variability
Journal of Chemical Physics, 2017Co-Authors: Jakob Ruess, Heinz Koeppl, Christoph ZechnerAbstract:Determining the sensitivity of certain system states or outputs to variations in parameters facilitates our understanding of the inner working of that system and is an essential design tool for the de-novo construction of robust systems. In cell biology, the output of interest is often the response of a certain reaction network to some input (e.g. stressors or nutrients) and one aims to quantify the sensitivity of this response in the presence of parameter heterogeneity. We argue that for such applications, parametric sensitivities in their standard form do not paint a complete picture of a system's robustness since one assumes that all cells in the population have the same parameters and are perturbed in the same way. Here, we consider stochastic reaction networks in which the parameters are randomly distributed over the population and propose a new sensitivity index that captures the robustness of system outputs upon changes in the characteristics of the parameter distribution, rather than the parameters themselves. Subsequently, we make use of Girsanov's likelihood ratio method to construct a Monte Carlo Estimator of this sensitivity index. However, it turns out that this Estimator has an exceedingly large variance. To overcome this problem, we propose a novel estimation algorithm that makes use of a marginalization of the path distribution of stochastic reaction networks and leads to Rao-Blackwellized Estimators with reduced variance.