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E De Schutter - One of the best experts on this subject based on the ideXlab platform.
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accurate reaction Diffusion Operator splitting on tetrahedral meshes for parallel stochastic molecular simulations
Journal of Chemical Physics, 2016Co-Authors: Iain Hepburn, Weiliang Chen, E De SchutterAbstract:Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, which has led to the development of parallel methods that can take advantage of the power of modern supercomputers in recent years. We systematically test suggested components of stochastic reaction-Diffusion Operator splitting in the literature and discuss their effects on accuracy. We introduce an Operator splitting implementation for irregular meshes that enhances accuracy with minimal performance cost. We test a range of models in small-scale MPI simulations from simple Diffusion models to realistic biological models and find that multi-dimensional geometry partitioning is an important consideration for optimum performance. We demonstrate performance gains of 1-3 orders of magnitude in the parallel implementation, with peak performance strongly dependent on model specification.
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accurate reaction Diffusion Operator splitting on tetrahedral meshes for parallel stochastic molecular simulations
arXiv: Quantitative Methods, 2015Co-Authors: Iain Hepburn, Weiliang Chen, E De SchutterAbstract:Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, meaning that the serial limit has already been reached in sub-cellular models. This calls for parallel simulations that can take advantage of the power of modern supercomputers; however exact methods are known to be inherently serial. We introduce an Operator splitting implementation for irregular grids with a novel method to improve accuracy, and demonstrate potential for scalable parallel simulations in an initial MPI version. We foresee that this groundwork will enable larger scale, whole-cell stochastic simulations in the near future.
Selime Gurol - One of the best experts on this subject based on the ideXlab platform.
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modelling spatially correlated observation errors in variational data assimilation using a Diffusion Operator on an unstructured mesh
Quarterly Journal of the Royal Meteorological Society, 2019Co-Authors: Oliver Guillet, Xavier Vasseur, Yann Michel, Serge Gratton, Anthony T Weaver, Selime GurolAbstract:We propose a method for representing spatially correlated observation errors in variational data assimilation. The method is based on the numerical solution of a Diffusion equation, a technique commonly used for representing spatially correlated background errors. The discretization of the pseudo‐time derivative of the Diffusion equation is done implicitly using a backward Euler scheme. The solution of the resulting elliptic equation can be interpreted as a correlation Operator whose kernel is a correlation function from the Matern family. In order to account for the possibly heterogeneous distribution of observations, a spatial discretization technique based on the finite element method (FEM) is chosen where the observation locations are used to define the nodes of an unstructured mesh on which the Diffusion equation is solved. By construction, the method leads to a convenient Operator for the inverse of the observation‐error correlation matrix, which is an important requirement when applying it with standard minimization algorithms in variational data assimilation. Previous studies have shown that spatially correlated observation errors can also be accounted for by assimilating the observations together with their directional derivatives up to arbitrary order. In the continuous framework, we show that the two approaches are formally equivalent for certain parameter specifications. The FEM provides an appropriate framework for evaluating the derivatives numerically, especially when the observations are heterogeneously distributed. Numerical experiments are performed using a realistic data distribution from the Spinning Enhanced Visible and InfraRed Imager (SEVIRI). Correlations obtained with the FEM‐discretized Diffusion Operator are compared with those obtained using the analytical Matern correlation model. The method is shown to produce an accurate representation of the target Matern function in regions where the data are densely distributed. The presence of large gaps in the data distribution degrades the quality of the mesh and leads to numerical errors in the representation of the Matern function. Strategies to improve the accuracy of the method in the presence of such gaps are discussed.
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Modelling spatially correlated observation errors in variational data assimilation using a Diffusion Operator on an unstructured mesh
Quarterly Journal of the Royal Meteorological Society, 2019Co-Authors: Oliver Guillet, Anthony Weaver, Xavier Vasseur, Yann Michel, Serge Gratton, Selime GurolAbstract:We propose a method for representing spatially correlated observation errors in variational data assimilation. The method is based on the numerical solution of a Diffusion equation, a technique commonly used for representing spatially correlated background errors. The discretization of the pseudo‐time derivative of the Diffusion equation is done implicitly using a backward Euler scheme. The solution of the resulting elliptic equation can be interpreted as a correlation Operator whose kernel is a correlation function from the Matérn family.In order to account for the possibly heterogeneous distribution of observations, a spatial discretization technique based on the finite element method (FEM) is chosen where the observation locations are used to define the nodes of an unstructured mesh on which the Diffusion equation is solved. By construction, the method leads to a convenient Operator for the inverse of the observation‐error correlation matrix, which is an important requirement when applying it with standard minimization algorithms in variational data assimilation. Previous studies have shown that spatially correlated observation errors can also be accounted for by assimilating the observations together with their directional derivatives up to arbitrary order. In the continuous framework, we show that the two approaches are formally equivalent for certain parameter specifications. The FEM provides an appropriate framework for evaluating the derivatives numerically, especially when the observations are heterogeneously distributed.Numerical experiments are performed using a realistic data distribution from the Spinning Enhanced Visible and InfraRed Imager (SEVIRI). Correlations obtained with the FEM‐discretized Diffusion Operator are compared with those obtained using the analytical Matérn correlation model. The method is shown to produce an accurate representation of the target Matérn function in regions where the data are densely distributed. The presence of large gaps in the data distribution degrades the quality of the mesh and leads to numerical errors in the representation of the Matérn function. Strategies to improve the accuracy of the method in the presence of such gaps are discussed.
Iain Hepburn - One of the best experts on this subject based on the ideXlab platform.
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accurate reaction Diffusion Operator splitting on tetrahedral meshes for parallel stochastic molecular simulations
Journal of Chemical Physics, 2016Co-Authors: Iain Hepburn, Weiliang Chen, E De SchutterAbstract:Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, which has led to the development of parallel methods that can take advantage of the power of modern supercomputers in recent years. We systematically test suggested components of stochastic reaction-Diffusion Operator splitting in the literature and discuss their effects on accuracy. We introduce an Operator splitting implementation for irregular meshes that enhances accuracy with minimal performance cost. We test a range of models in small-scale MPI simulations from simple Diffusion models to realistic biological models and find that multi-dimensional geometry partitioning is an important consideration for optimum performance. We demonstrate performance gains of 1-3 orders of magnitude in the parallel implementation, with peak performance strongly dependent on model specification.
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accurate reaction Diffusion Operator splitting on tetrahedral meshes for parallel stochastic molecular simulations
arXiv: Quantitative Methods, 2015Co-Authors: Iain Hepburn, Weiliang Chen, E De SchutterAbstract:Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, meaning that the serial limit has already been reached in sub-cellular models. This calls for parallel simulations that can take advantage of the power of modern supercomputers; however exact methods are known to be inherently serial. We introduce an Operator splitting implementation for irregular grids with a novel method to improve accuracy, and demonstrate potential for scalable parallel simulations in an initial MPI version. We foresee that this groundwork will enable larger scale, whole-cell stochastic simulations in the near future.
Anthony T Weaver - One of the best experts on this subject based on the ideXlab platform.
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modelling spatially correlated observation errors in variational data assimilation using a Diffusion Operator on an unstructured mesh
Quarterly Journal of the Royal Meteorological Society, 2019Co-Authors: Oliver Guillet, Xavier Vasseur, Yann Michel, Serge Gratton, Anthony T Weaver, Selime GurolAbstract:We propose a method for representing spatially correlated observation errors in variational data assimilation. The method is based on the numerical solution of a Diffusion equation, a technique commonly used for representing spatially correlated background errors. The discretization of the pseudo‐time derivative of the Diffusion equation is done implicitly using a backward Euler scheme. The solution of the resulting elliptic equation can be interpreted as a correlation Operator whose kernel is a correlation function from the Matern family. In order to account for the possibly heterogeneous distribution of observations, a spatial discretization technique based on the finite element method (FEM) is chosen where the observation locations are used to define the nodes of an unstructured mesh on which the Diffusion equation is solved. By construction, the method leads to a convenient Operator for the inverse of the observation‐error correlation matrix, which is an important requirement when applying it with standard minimization algorithms in variational data assimilation. Previous studies have shown that spatially correlated observation errors can also be accounted for by assimilating the observations together with their directional derivatives up to arbitrary order. In the continuous framework, we show that the two approaches are formally equivalent for certain parameter specifications. The FEM provides an appropriate framework for evaluating the derivatives numerically, especially when the observations are heterogeneously distributed. Numerical experiments are performed using a realistic data distribution from the Spinning Enhanced Visible and InfraRed Imager (SEVIRI). Correlations obtained with the FEM‐discretized Diffusion Operator are compared with those obtained using the analytical Matern correlation model. The method is shown to produce an accurate representation of the target Matern function in regions where the data are densely distributed. The presence of large gaps in the data distribution degrades the quality of the mesh and leads to numerical errors in the representation of the Matern function. Strategies to improve the accuracy of the method in the presence of such gaps are discussed.
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representation of correlation functions in variational assimilation using an implicit Diffusion Operator
Quarterly Journal of the Royal Meteorological Society, 2010Co-Authors: Isabelle Mirouze, Anthony T WeaverAbstract:Correlation models are required in data assimilation to characterize the error structures of variables defined on a numerical grid. Previous studies have shown that the Diffusion equation can provide a flexible and computationally efficient framework for representing grid-point correlation functions for problems of large dimension such as those encountered in atmospheric or ocean variational data assimilation. In this article, an implicit formulation of the Diffusion-based correlation model is presented as an alternative to the traditional explicit formulation. The implicit formulation is analyzed in detail for the one-dimensional (1D) problem and shown to be closely related to the first-order recursive filter. Integrating a 1D implicit Diffusion equation, with constant coefficient, over M steps is shown to be equivalent to convolving the initial condition with an Mth order auto-regressive (AR) function. Expressions for both the length-scale of the AR function and the normalization factor required to generate unit-amplitude (correlation) functions are given in terms of M and the Diffusion coefficient. For a fixed length-scale the Gaussian function, which is the only function that can be represented using an explicit formulation of the constant-coefficient Diffusion equation, is the limiting case as M → ∞ of the AR functions generated by the implicit Diffusion equation. Generalizations of the Diffusion model are discussed to allow for different shapes in the correlation function and spatial variations in the length-scale. An important consequence of employing spatially varying length-scales is that the normalization factors are no longer constant. Approximate expressions for the normalization factors are evaluated in terms of their effectiveness to provide viable alternatives to estimates produced using expensive algorithms such as randomization. Boundary conditions can distort the correlation functions near the boundaries and significantly degrade the accuracy of the analytical expressions for the normalization factors. These problems can be avoided through a straightforward extension of the Diffusion model that makes the boundaries effectively transparent, although the solution comes at the expense of an extra application of the Diffusion equation. Extensions of the method to construct two- and three-dimensional correlation models are discussed. Copyright © 2010 Royal Meteorological Society
Olivier Pannekoucke - One of the best experts on this subject based on the ideXlab platform.
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heterogeneous correlation modeling based on the wavelet diagonal assumption and on the Diffusion Operator
Monthly Weather Review, 2009Co-Authors: Olivier PannekouckeAbstract:Abstract This article discusses several models for background error correlation matrices using the wavelet diagonal assumption and the Diffusion Operator. The most general properties of filtering local correlation functions, with wavelet formulations, are recalled. Two spherical wavelet transforms based on Legendre spectrum and a gridpoint spherical wavelet transform are compared. The latter belongs to the class of second-generation wavelets. In addition, a nonseparable formulation that merges the wavelets and the Diffusion Operator model is formally proposed. This hybrid formulation is illustrated in a simple two-dimensional framework. These three formulations are tested in a toy experiment on the sphere: a large ensemble of perturbed forecasts is used to simulate a true background error ensemble, which gives a reference. This ensemble is then applied to compute the required parameters for each model. A randomization method is utilized in order to diagnose these different models. In particular, their abi...
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estimation of the local Diffusion tensor and normalization for heterogeneous correlation modelling using a Diffusion equation
Quarterly Journal of the Royal Meteorological Society, 2008Co-Authors: Olivier Pannekoucke, S MassartAbstract:As the background error covariance matrix is a key component of any assimilation system, its modelling is an important step. Usually, this matrix is decomposed into correlations and variance matrices. An interesting method for modelling the correlation matrix of the background error for complex geometry, like ocean grid, consists in computing correlation functions using a Diffusion Operator. The background error correlation functions can be estimated for example from an ensemble of perturbed forecasts. The Diffusion Operator is able to represent heterogeneous correlation functions at a reasonable numerical cost. But afirst challenge resides in the determination of the local Diffusion tensor corresponding to the local correlation function. Then the second challenge resides in the determination of the normalization to make sure that the matrix modelled through the Diffusion Operator is a correlation matrix. In this article, we propose to build a background error correlation matrix using a Diffusion Operator based on a local Diffusion tensor. The estimation of this local tensor is performed using an ensemble of perturbed forecasts. A validation within a randomization method illustrates the feasibility and the accuracy of the proposed method. In particular, it is shown that the local geographical variations of diagnosed correlation functions (through an ensemble of perturbed forecast) are well represented. This is first illustrated in an analytical one-dimensional framework. In that context, the Diffusion field and the normalization field are deduced from a given correlation length-scale field. The resulting length-scales are shown to correspond to the initial length-scale when the given length-scale field spectrum is red. The approximate normalization, computed from the local length-scale, is close to the true normalization under the same condition of a red spectrum. Then, the method is illustrated in a real context using an ensemble of perturbed forecasts from the MOCAGE-PALM assimilation system. In that case, length-scale and anisotropy diagnosis reveal the complexity of the correlation of stratospheric ozone forecast errors. The local Diffusion tensor deduced from these diagnosis are shown be able to represent such an existing heterogeneity and anisotropy. As in the one-dimensional case, the approximate normalization, based on the local Diffusion tensor, appears to be a really good approximation of the true normalization.