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Variance reduction methods for numerical solution of plasma kinetic diffusion
KTH, School of Electrical Engineering (EES), Fusion Plasma Physics.
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 [en]
variance reduction, Monte Carlo, quasi-Monte Carlo, kinetic diffusion, stochastic differential equation
National Category
Fusion, Plasma and Space Physics
Identifiers
URN: urn:nbn:se:kth:diva-91332ISBN: 978-91-7501-278-0 (print)OAI: oai:DiVA.org:kth-91332DiVA: diva2:509600
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
List of papers
1. Adaptive delta f Monte Carlo Method for Simulation of RF-heating and Transport in Fusion Plasmas
Open this publication in new window or tab >>Adaptive delta f Monte Carlo Method for Simulation of RF-heating and Transport in Fusion Plasmas
2009 (English)In: Radio Frequency Power in Plasmas, American Institute of Physics (AIP), 2009, Vol. 1187, 589-592 p.Conference paper, Published paper (Refereed)
Abstract [en]

Essential for modeling heating and transport of fusion plasma is determining the distribution function of the plasma species. Characteristic for RF-heating is creation of particle distributions with a high energy tail. In the high energy region the deviation from a Maxwellian distribution is large while in the low energy region the distribution is close to a Maxwellian due to the velocity dependency of the collision frequency. Because of geometry and orbit topology Monte Carlo methods are frequently used. To avoid simulating the thermal part, delta f methods are beneficial. Here we present a new delta f Monte Carlo method with an adaptive scheme for reducing the total variance and sources, suitable for calculating the distribution function for RF-heating

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2009
Series
AIP Conference Proceedings, ISSN 0094-243X ; 1187
Keyword
RF-heating, delta-f method, Monte Carlo method, Variance reduction, Fokker-Planck equation
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-25566 (URN)10.1063/1.3273820 (DOI)000276069400122 ()2-s2.0-73449103937 (Scopus ID)978-0-7354-0753-4 (ISBN)
Conference
18th Topical Conference on Radio Frequence Power in Plasmas Ghent, Belgium, June 24-26, 2009
Note

QC 20101118

Available from: 2010-11-18 Created: 2010-10-27 Last updated: 2014-10-14Bibliographically approved
2. An Adaptive delta f Monte Carlo Method
Open this publication in new window or tab >>An Adaptive delta f Monte Carlo Method
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.

Keyword
Control-variate, Monte Carlo (MC), variance reduction, delta f method
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-27685 (URN)10.1109/TPS.2010.2051686 (DOI)000283252500015 ()2-s2.0-77956619121 (Scopus ID)
Note
QC 20101221Available from: 2010-12-21 Created: 2010-12-20 Last updated: 2013-04-05Bibliographically approved
3. Randomized quasi-Monte Carlo simulation of fast-ion thermalization
Open this publication in new window or tab >>Randomized quasi-Monte Carlo simulation of fast-ion thermalization
2012 (English)In: Computational Science & Discovery, ISSN 1749-4680, E-ISSN 1749-4699, Vol. 5, no 1, 014010- p.Article in journal (Refereed) Published
Abstract [en]

This work investigates the applicability of the randomized quasi-Monte Carlo method for simulation of fast-ion thermalization processes in fusion plasmas, e.g. for simulation of neutral beam injection and radio frequency heating. In contrast to the standard Monte Carlo method, the quasi-Monte Carlo method uses deterministic numbers instead of pseudo-random numbers and has a statistical weak convergence close to O(N -1), where N is the number of markers. We have compared different quasi-Monte Carlo methods for a neutral beam injection scenario, which is solved by many realizations of the associated stochastic differential equation, discretized with the Euler-Maruyama scheme. The statistical convergence of the methods is measured for time steps up to 2 14.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2012
Keyword
Monte Carlo, quasi-Monte Carlo, low discrepancy sequence, variance reduction
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-91197 (URN)10.1088/1749-4699/5/1/014010 (DOI)2-s2.0-84866342521 (Scopus ID)
Note

QC 20121114. Updated from submitted to published.

Available from: 2012-03-13 Created: 2012-03-09 Last updated: 2017-12-07Bibliographically approved

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