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An Adaptive delta f Monte Carlo Method
KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
2010 (English)In: IEEE Transactions on Plasma Science, ISSN 0093-3813, E-ISSN 1939-9375, Vol. 38, no 9, 2190-2197 p.Article in journal (Refereed) Published
Abstract [en]

A new adaptive delta f Monte Carlo method is presented with an application to radio frequency heating and transport in fusion plasmas. The method is suitable when an initial zeroth-order approximation of the distribution function is known. The difference between our method and earlier delta f methods is that we model the source term, obtained from the delta f ansatz, by adding particles. The rate of particles is defined by the inhomogeneous term in the Fokker-Planck equation. We develop an adaptive scheme for modifying the unperturbed part G(x) such that the number of particles used in the simulation for a fixed weight is minimized. This implicitly reduces the variance and improves computational efficiency. The method is tested on a one-dimensional Fokker-Planck model for RF-heating and compared against the analytical stationary solution.

Place, publisher, year, edition, pages
2010. Vol. 38, no 9, 2190-2197 p.
Keyword [en]
Control-variate, Monte Carlo (MC), variance reduction, delta f method
National Category
Fusion, Plasma and Space Physics
Identifiers
URN: urn:nbn:se:kth:diva-27685DOI: 10.1109/TPS.2010.2051686ISI: 000283252500015Scopus ID: 2-s2.0-77956619121OAI: oai:DiVA.org:kth-27685DiVA: diva2:380409
Note
QC 20101221Available from: 2010-12-21 Created: 2010-12-20 Last updated: 2013-04-05Bibliographically approved
In thesis
1. Variance reduction methods for numerical solution of plasma kinetic diffusion
Open this publication in new window or tab >>Variance reduction methods for numerical solution of plasma kinetic diffusion
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Performing detailed simulations of plasma kinetic diffusion is a challenging task and currently requires the largest computational facilities in the world. The reason for this is that, the physics in a confined heated plasma occur on a broad range of temporal and spatial scales. It is therefore of interest to improve the computational algorithms together with the development of more powerful computational resources. Kinetic diffusion processes in plasmas are commonly simulated with the Monte Carlo method, where a discrete set of particles are sampled from a distribution function and advanced in a Lagrangian frame according to a set of stochastic differential equations. The Monte Carlo method introduces computational error in the form of statistical random noise produced by a finite number of particles (or markers) N and the error scales as αNβ where β = 1/2 for the standard Monte Carlo method. This requires a large number of simulated particles in order to obtain a sufficiently low numerical noise level. Therefore it is essential to use techniques that reduce the numerical noise. Such methods are commonly called variance reduction methods. In this thesis, we have developed new variance reduction methods with application to plasma kinetic diffusion. The methods are suitable for simulation of RF-heating and transport, but are not limited to these types of problems. We have derived a novel variance reduction method that minimizes the number of required particles from an optimization model. This implicitly reduces the variance when calculating the expected value of the distribution, since for a fixed error the  optimization model ensures that a minimal number of particles are needed. Techniques that reduce the noise by improving the order of convergence, have also been considered. Two different methods have been tested on a neutral beam injection scenario. The methods are the scrambled Brownian bridge method and a method here called the sorting and mixing method of L´ecot and Khettabi[1999]. Both methods converge faster than the standard Monte Carlo method for modest number of time steps, but fail to converge correctly for large number of time steps, a range required for detailed plasma kinetic simulations. Different techniques are discussed that have the potential of improving the convergence to this range of time steps.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. viii, 42 p.
Series
Trita-EE, ISSN 1653-5146 ; 2012:007
Keyword
variance reduction, Monte Carlo, quasi-Monte Carlo, kinetic diffusion, stochastic differential equation
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-91332 (URN)978-91-7501-278-0 (ISBN)
Presentation
2012-03-30, Seminarierummet, Teknikringen 31, KTH, Stockholm, 12:24 (English)
Opponent
Supervisors
Note
QC 20120314Available from: 2012-03-14 Created: 2012-03-13 Last updated: 2012-03-14Bibliographically approved
2. Numerical solution of quasilinear kinetic diffusion equations in toroidal plasmas
Open this publication in new window or tab >>Numerical solution of quasilinear kinetic diffusion equations in toroidal plasmas
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

One of the main challenges for the realization of a working fusion power plant is an increased detailed understanding of kinetic phenomena in toroidal plasmas. The tokamak is a toroidal, magnetically confined plasma device and is currently the main line towards a power plant. The spatial and temporal scales in a tokamak plasma are extreme and the only tractable path for quantitative studies is to rely on computer simulations. Present day simulation codes can resolve only some of these scales. Nevertheless they still require the largest high performance computing (HPC) resources available in the world. In combination with the increase of computational performance, it is therefore necessary to improve the numerical algorithms used in the simulations.

In this thesis we have developed new numerical methods designed for Monte Carlo simulation of plasma kinetic diffusion. Examples are simulation of fast-ion thermalization and radio-frequency heating. The aim has been to reduce the statistical random noise in particle codes, produced by a finite number of particles (or markers). Traditionally the statistical noise is improved by increasing the number of particles (N) or by simulating the perturbation of the distribution (with particles) from a known distribution function. This is the well known δf method. In this thesis we have developed a new type of δf method, which minimizes the number of particles used in a simulation. The computational speedup of the new method is substantial. In this thesis, we have further benchmarked quasi-Monte Carlo techniques that improve the convergence rate from N−1/2 to N−1 for some cases.

In Monte Carlo simulations, error appears also from the time step discretization. Based on the mathematics of operator splitting, a new scheme for the pitch-angle scattering diffusion process has been developed that outperforms the standard methods. Finally this thesis also presents a new code, SELFO-light, for self-consistent simulations of ion cyclotron resonance heating, suitable for routine calculations, which couples a one dimensional Fokker-Planck model with the finite element wave solver LION.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. xiii, 54 p.
Series
Trita-EE, ISSN 1653-5146 ; 2013:012
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-120345 (URN)978-91-7501-697-9 (ISBN)
Public defence
2013-04-26, F3, Lindstedtsvägen 26, KTH, Stockholm, 13:15 (English)
Opponent
Supervisors
Note

QC 20130405

Available from: 2013-04-05 Created: 2013-04-04 Last updated: 2013-04-05Bibliographically approved

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