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The chemical master equation is solved by a hybrid method coupling a macroscopic, deterministic description with a mesoscopic, stochastic model. The molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules. The macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation. The probability distribution is obtained by the Stochastic Simulation Algorithm due to Gillespie. The equations are coupled via a summation over the mesoscale variables. This summation is approximated by Monte Carlo and Quasi Monte Carlo methods. The error in the approximations is analyzed. The hybrid method is applied to three chemical systems from molecular cell biology.

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“…see [23]. The local temporal discretization error τ k+1 in the difference approximation is proportional to (∆t k ) 2 .…”

Section: Solution Of the Differential-summation Equationmentioning

confidence: 99%

“…The discretization error in the approximation of the time derivative is denoted by τ k+1 . Following [13] and [23], the local error in the corrector solution φ k+1 of (13) is…”

Section: The Integration Errormentioning

confidence: 99%

Hybrid method for the chemical master equation

Hellander

Lötstedt

2007

Journal of Computational Physics

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“…al. in [5]. But, where they continuously change the time-step as τ k is computed at each time-step, we store the information about τ k and choose the new time-steps after the first solve (i) in Algorithm 1 is completed.…”

Section: Time-integration and Time-adaptivitymentioning

confidence: 99%

Pricing European multi-asset options using a space–time adaptive FD-method

Persson

Sydow

2007

Comput. Visual Sci.

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In this paper we present an adaptive technique to solve the multidimensional Black-Scholes equation. The number of grid-points required for a given tolerance of the local discretization errors is reduced substantially when compared to a standard equidistant grid. Using our adaptive methods in space and time we have control of the local discretization errors and can refine the grid where needed for accuracy reasons.

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“…In adaptive methods, the discretization and solution errors are estimated and the grid is refined or coarsened based on these estimates. Examples of adaptivity for structured grids are found in [3], [8], [12]. The grids are refined in blocks with a homogeneous grid size inside a block and with jumps at the block interfaces.…”

Section: Introductionmentioning

confidence: 99%

“…Structured grids with patches of refinement have been used for time dependent and time independent problems in applications such as fluid flow [3], [8], [10], [12], electromagnetics [11], [14], [17], [21], and plasma physics [6], [18]. The missing values of the solution at the interfaces are determined by interpolation.…”

Section: Introductionmentioning

confidence: 99%

Accurate and stable grid interfaces for finite volume methods

Ferm

Lötstedt

2004

Applied Numerical Mathematics

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A convection-diffusion equation is discretized by a finite volume method in two space dimensions. The grid is partitioned into blocks with jumps in the grid size at the block interfaces. Interpolation in the cells adjacent to the interfaces is necessary to be able to apply the difference stencils. Second order accuracy is achieved and the stability of the discretizations is investigated. The interface treatment is tested in the solution of the compressible Navier-Stokes equations. The conclusions from the scalar equation are valid also for these equations.

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Cited by 29 publications

References 32 publications

Hybrid method for the chemical master equation

Hybrid method for the chemical master equation

Pricing European multi-asset options using a space–time adaptive FD-method

Accurate and stable grid interfaces for finite volume methods

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