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A new method to identify the equilibria compatible with the measurements using the technique of the ε-nets. JET-EFDA, Culham Science Centre, Abingdon, OX14 3DB, UK. 1 Fusion Advanced Research Group Ltd., Slovakia. 2 Department of Experimental Physics, Comenius University, Slovakia.
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A new method to identify the equilibria compatible with the measurements using the technique of the ε-nets JET-EFDA, Culham Science Centre, Abingdon, OX14 3DB, UK. 1Fusion Advanced Research Group Ltd., Slovakia. 2Department of Experimental Physics, Comenius University, Slovakia. 3Moscow State University, Dept. of Computational Mathematics & Cybernetics. Scientific Research Institute of System Development, Russian Academy of Science. 4Consorzio RFX-Associazione EURATOM ENEA per la Fusione, I-35127 Padova, Italy. *See the Appendix of F. Romanelli et al., Proceedings of the 23rd IAEA Fusion Energy Conference 2010, Daejeon, Korea. e-mail: feodor@zajcev.eu F.S. Zaitsev1-3, S. Matejcik2, A. Murari4, E.P. Suchkov3 and JET EFDA Contributors* 7th Workshop on Fusion Data Processing Validation and Analysis Invitation. Frascati 2012
Contents • Introduction. • Formulation of the problem. • Numerical method. • Software. • Examples. • Conclusions.
1. Introduction One of the central problems in fusion is reconstruction of plasma equilibrium using measurements. Many fundamental publications have been devoted to this problem, e.g. [1] Lao L.L., et al. Fusion Sci. Technol. 48 (2005) 968. [2] F.S. Zaitsev et al.. Nucl. Fusion. 51 (2011) 103044. The problem is strongly ill-posed. In particular the solution can be unstable with respect to the input data, i.e. small changes in the input data can produce large changes in the output. This means that current densities and safety factors, substantially different in magnitude or profile, can be compatible with the measurements within given experimental errors.
Various additional constraints are considered in order to get stable solutions to the original inverse problem. However no general theorems on constraints, which make the equilibria reconstruction problem stable with respect to the input data inaccuracies, are available. • Traditionally codes for equilibrium reconstruction, such as EFIT or SCoPE, search for one solution of the Grad-Shafranov equation with a set of constraints. But usually, the questions of • the efficiency of a constraint in selecting a solution, • the required accuracy of the measurements, • the existence of very different solutions, which are compatible with the measurement errors, • the detailed assessment of the reconstruction confidence intervals • are not addressed. However, for the correct interpretation of a pulse, it is important to have answers to all these questions. The talk presents a new method, which provides answers to the questions formulated above. The method is based on the ε-nets.
2. Formulation of the problem The solution of the general problem of equilibrium reconstruction can be separated in two stages: determining the plasma boundary (external problem) and then reconstructing the current density inside the found plasma boundary (internal problem).
External problem: Equation is valid in vacume and as the limit from vacuum at the plasma boundary and the internal surfaces of the chamber. - coordinates at which values ψk are measured, k=1,…, Ke. The problem is to find the plasma boundary, which is defined as the closed curve ψ=const of maximum “width”: If measured at all points of a closed curve then the problem would be for elliptic equation, describing the area between the plasma and this curve. But this is not the case. So, additional conditions are required: Ideal walls and passive elements: Given currents near the scrape off layer: Other constraints, e.g. from D-alpha, video, limiters and etc.
Internal problem: Grad-Shafranov equation in plasma. The plasma boundary is known here from the external problem, (ψ,p,F) - to be found. Basic additional conditions: the boundary poloidal field, determined in the external problem, given inaccuracy δ value takes account of the uncertainty in setting ϕ, given total current Basic conditions are insufficient for identification of the real physical process.
Special additional conditions in the internal problem: Motional Stark effect (MSE): Polarimetry: Interferometry: Other conditions can be added, e.g. parallel Ohm’s law for identification of evolution of function F. Then only (ψ,p) are to be reconstructed.
3. Numerical method • The main steps of the proposed algorithm: • Plasma boundary reconstruction in the external problem. • Reconstruction of p and F using a traditional algorithm with some additional conditions or solving direct problem for (p,F). • Setting regions around p and F, in which other possible reconstructions, which fit given accuracy, are to be searched. • Construction of the -net for p and the -net for F. • Solution of the standard direct problem for the Grad-Shafranov equation for each element of the (p,F) -net. • Selectionof elements of the (p,F)-net, which satisfy one or another combination of basic and special additional conditions. • Analyses of the selected elements: • checking the existence of very different solutions, which are compatible with the measurement errors; • studying the efficiency of a constraint in selecting a solution; • evaluating the required accuracy of the measurements; • assessment of the reconstruction confidence intervals. • The original space of functions is obtained in the limit -> 0.
The central part of the algorithm is the -net technique, founded in “A.N. Kolmogorov, V.M. Tihomirov. -entropy and -capacity of sets in function spaces. Uspehi Mat. Nauk. 14 (1959) 3-86”. The basic idea of the -net approach consists of constructing a final set of functions {gk} from a given class of functions G (e.g. with a given range of values and growth and decay rates) such as, for every function g from G, one can find a gk such that || g - gk || ≤ ε. The paper “F.S. Zaitsev, D.P. Kostomarov, E.P. Suchkov, V.V. Drozdov, E.R. Solano, A. Murari, S. Matejcik, N.C. Hawkes. Nucl. Fusion. 51 (2011) 103044” gives details for construction of -nets of polynomials and splines and solving the internal problem. • Two methods are proposed for polynomial -net construction: • Consideration of special sets of polynomial coefficients. • Building a grid in such a way that the polynomials, going through these points of the grid in a special way, form the -net. For splines the -net on the grid is proposed.
In the external problem coefficients of ψexpansion in polynomial series are searched (similar to code XLOC). Best fit of the plasma boundary parameterization (R(ξ),Z(ξ)) is searched from the ε-net of cubic splines. The number of elements in the net is reduced by additional conditions: closed curve, smooth at ξ = -π, convex enclosed domain, etc. Example of the -net of splines for the plasma boundary:
4. Software The numerical algorithms are implemented in the code SDSS (Substantially Different Solutions Searcher) in Fortran 2003. The code has a special graphic interface, written in Java, to help setting up the input data, constructing -nets, solving the inverse problem, visualizing and analyzing the results. Depending on the size of the -net, the required computing power is from a PC to a super-computer. The algorithm is fully parallelizable. Each process requires ~100Mb RAM. Total HD space necessary is about 500 Mb. Computational time depends on the number of elements in the -net: hours on PC for ~104 elements and on super-computer for ~106-108 elements.
5. Examples External problem: plasma boundary reconstruction. ITER: R~6.5 м, B~5.3T, I~9 MA Inaccuracy in norm C: 13% - 3 parametric, 2% - with spline -net.
Internal problem: (ψ,p,F) reconstruction. MAST: R~0.7 м,B~0.52T, I~560 kA. Forδ ≈ 2.5% very different reconstructions are present. Inclusion of the MSE in the problem with δMSE < 1.5oeliminates substantially different solutions.
Internal problem: (ψ,p,F) reconstruction. JET: R~3.05 м,B~2.4T, I~2.37 MA. Forδ ≈ 7% very different reconstructions are present. Inclusion of the MSE in the problem with δMSE < 0.3o eliminates substantially different solutions.
Internal problem: (ψ,p,F) reconstruction. ITER: R~6.5 м,B~5.3T, I~9 MA. Forδ≈ 8% very different reconstructions are present. Inclusion of the MSE in the problem with δMSE < 0.3o eliminates substantially different solutions.
Internal problem: (ψ,p,F) reconstruction. ITER: R~6.5 м,B~5.3T, I~9 MA. Forδ ≈ 8% very different reconstructions are present. Inclusion of the MSE in the problem with δMSE< 0.3o eliminates substantially different solutions.
Studying the efficiency of diagnostics in selecting a solution appropriate to the real physical process in JET-like plasmas. The threshold values for inaccuracies of MSE, polarimetry and interferometry, below which the constraint efficiently identifies one solution among several substantially different ones, have been determined: The most efficient for reconstruction are MSE and polarimetry. It was found that polarimetry and interferometry role increases with a steeper electron density profile. In general it is reasonable to include data of all three diagnostics as constraints for the reconstruction, since these diagnostics are not equivalent and can supplement each other in particular cases.
Evaluation of the confidence interval. Elements, left in the -net after application of all of the constraints, form the confidence interval of the reconstruction. The boundary of the confidence interval consists of minimum and maximum values of these -net elements. The width of the confidence interval for toroidal current density in all considered MAST, JET and ITER cases is of the order of the inaccuracies of the poloidal field at the plasma boundary δ and δMSE. The dependence of the confidence intervals for p and F on inaccuracies is more complicated and requires separate study for each pulse. Constraint with parallel Ohm’s law for identification of evolution of function F can help much in separation of p and F. With this constraint the inverse problem becomes for ψand p only at the expense of the more complicated model.
6. Conclusions • The radically new method for equilibrium reconstruction is proposed. The method is based on the -net technique. It allows solving a variety of problems which were very hard to address before: • validate the existence or absence of very different solutions, which are compatible with the same measurement errors; • studying the efficiency of a constraint in selecting a solution; • evaluating the required accuracy of the measurements; • calculating the reconstruction confidence intervals. Magnetic diagnostics alone are not sufficient for finding one reconstruction, appropriate to the real physical process. MSE and polarimetry are efficient additional constraints for selecting the correct reconstruction. The technique should be installed as a routine tool supplementing the existing methods (e.g. EFIT), since no theorems guarantee for a constraint in a traditional method to select the correct reconstruction. The technique becomes more and more attractive for different ill-posed problems due to the fast growth of computing power.
Acknowledgements. The FARG Ltd. and Comenius University work were partly funded by EURATOM, EFDA priority support No. WP11-DIA-04-01-01/CU. The MSU work was funded by the Russian Foundation for Basic Research, grants No. 11-07-00567 and 10-07-00207.
