| 1. |
- Cullhed, Josef, et al.
(författare)
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The URDME Manual version 1.0
- 2008
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We have developed URDME, a general software for simulation of stochastic reaction-diffusion processes on unstructured meshes. This allows for a more flexible handling of complicated geometries and curved boundaries compared to simulations on structured, cartesian meshes. The underlying algorithm is the next subvolume method (NSM), extended to unstructured meshes by obtaining jump coefficients from the finite element formulation of the corresponding macroscopic equation. In this manual, we describe how to use the software together with COMSOL Multiphysics 3.4 and Matlab to set up simulations. We provide a detailed account of the code structure and of the available interfaces. This makes modifications and extensions of the code possible. We also give two detailed examples, in which we describe the process of simulating and visualizing two models from the systems biology literature in a step-by-step manner.
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| 2. |
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| 3. |
- Engblom, Stefan
(författare)
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A discrete spectral method for the chemical master equation
- 2008
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- As an equivalent formulation of the Markov-assumption of stochastic processes, the master equation of chemical reactions is an accurate description of general systems in chemistry. For D reacting species this is a differential-difference equation in D dimensions, exactly soluble for very simple systems only.We present and analyze a novel solution strategy based upon a Galerkin spectral method with an inherent natural adaptivity and a very favorable choice of basis functions.The method is demonstrated by the numerical solution of two model problems followed by two more realistic systems taken from molecular biology. It is shown that the method remains effective and accurate, providing a viable alternative to other solution methods when the dimensionality is not too high.
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| 4. |
- Engblom, Stefan
(författare)
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Computing the moments of high dimensional solutions of the master equation
- 2005
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Derived from the Markov character only, the master equation of chemical reactions is an accurate stochastic description of quite general systems in chemistry. Exact solutions of this equation are rare and the most frequently used approximative solution method is to write down the corresponding set of reaction rate equations. In many cases this approximation is not valid, or only partially so, as stochastic effects caused by the natural noise present in the full description of the problem are poorly captured. In this paper it is shown how a certain set of higher order equations can be derived. It is shown by theory and example that stochastic effects are better captured using this technique while still maintaining the computational advantages of the reaction rate approach.
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| 6. |
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| 7. |
- Engblom, Stefan
(författare)
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Gaussian quadratures with respect to discrete measures
- 2006
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In analogy to the subject of Gaussian integration formulas we present an overview of some Gaussian summation formulas. The derivation involve polynomials that are orthogonal under discrete inner products and the resulting formulas are useful as a numerical device for summing fairly general series. Several illuminating examples are provided in order to present various aspects of this not very well-known technique.
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| 8. |
- Engblom, Stefan
(författare)
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Numerical methods for the chemical master equation
- 2006
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The numerical solution of chemical reactions described at the meso-scale is the topic of this thesis. This description, the master equation of chemical reactions, is an accurate model of reactions where stochastic effects are crucial for explaining certain effects observed in real life. In particular, this general equation is needed when studying processes inside living cells where other macro-scale models fail to reproduce the actual behavior of the system considered.The main contribution of the thesis is the numerical investigation of two different methods for obtaining numerical solutions of the master equation.The first method produces statistical quantities of the solution and is a generalization of a frequently used macro-scale description. It is shown that the method is efficient while still being able to preserve stochastic effects.By contrast, the other method obtains the full solution of the master equation and gains efficiency by an accurate representation of the state space.The thesis contains necessary background material as well as directions for intended future research. An important conclusion of the thesis is that, depending on the setup of the problem, methods of highly different character are needed.
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| 9. |
- Engblom, Stefan, 1976-
(författare)
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Numerical Solution Methods in Stochastic Chemical Kinetics
- 2008
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This study is concerned with the numerical solution of certain stochastic models of chemical reactions. Such descriptions have been shown to be useful tools when studying biochemical processes inside living cells where classical deterministic rate equations fail to reproduce actual behavior. The main contribution of this thesis lies in its theoretical and practical investigation of different methods for obtaining numerical solutions to such descriptions. In a preliminary study, a simple but often quite effective approach to the moment closure problem is examined. A more advanced program is then developed for obtaining a consistent representation of the high dimensional probability density of the solution. The proposed method gains efficiency by utilizing a rapidly converging representation of certain functions defined over the semi-infinite integer lattice. Another contribution of this study, where the focus instead is on the spatially distributed case, is a suggestion for how to obtain a consistent stochastic reaction-diffusion model over an unstructured grid. Here it is also shown how to efficiently collect samples from the resulting model by making use of a hybrid method. In a final study, a time-parallel stochastic simulation algorithm is suggested and analyzed. Efficiency is here achieved by moving parts of the solution phase into the deterministic regime given that a parallel architecture is available. Necessary background material is developed in three chapters in this summary. An introductory chapter on an accessible level motivates the purpose of considering stochastic models in applied physics. In a second chapter the actual stochastic models considered are developed in a multi-faceted way. Finally, the current state-of-the-art in numerical solution methods is summarized and commented upon.
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| 10. |
- Engblom, Stefan
(författare)
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PARALLEL IN TIME SIMULATION OF MULTISCALE STOCHASTIC CHEMICAL KINETICS
- 2009
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Ingår i: Multiscale Modeling & simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 8:1, s. 46-68
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Tidskriftsartikel (refereegranskat)abstract
- A version of the time-parallel algorithm parareal is analyzed and applied to stochastic models in chemical kinetics. A fast predictor at the macroscopic scale (evaluated in serial) is available in the form of the usual reaction rate equation. A stochastic simulation algorithm is used to obtain an exact realization of the process at the mesoscopic scale (in parallel). The underlying stochastic description is a jump process driven by the Poisson measure. A convergence result in this arguably difficult setting is established, suggesting that a homogenization of the solution is advantageous. We devise a simple but highly general such technique. Three numerical experiments on models representative to the field of computational systems biology illustrate the method. For nonstiff problems, it is shown that the method is able to quickly converge even when stochastic effects are present. For stiff problems, we are instead able to obtain fast convergence to a homogenized solution. Overall, the method builds an attractive bridge between, on the one hand, macroscopic deterministic scales and, on the other hand, mesoscopic stochastic ones. This construction is clearly possible to apply also to stochastic models within other fields.
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