1. Characteristics of SVD in ST Plasma Shape Reproduction Method Based on CCS Kazuo NAKAMURA 1, Shinji MATSUFUJI 2, Masashi TOMODA 2, Feng WANG 3, Osamu MITARAI 4, Kenichi KURIHARA 5, Yoichi KAWAMATA 5, Michiharu SUEOKA 5, Makoto HASEGAWA 1, Kazutoshi TOKUNAGA 1, Kohnosuke SATO 1, Hideki ZUSHI 1, Kazuaki HANADA 1, Mizuki SAKAMOTO 1, Hiroshi IDEI 1, Shoji KAWASAKI 1, Hisatoshi NAKASHIMA 1 and Aki HIGASHIJIMA 1 1 Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka, 816-8580 Japan, 2 Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka, 816-8580 Japan 3 Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, 230031 China 4 Tokai University, Toroku, Kumamoto, 862-8652 Japan 5 Japan Atomic Energy Agency, Mukoyama Naka-shi, Ibaraki-ken, 311-0193 Japan CUP2008, EAST, ASIPP, Mar. 2, 2009 (Mon.)

2. Abstract LSM (Least Square Method) has been used to solve the inverse problem from flux loop measurement to the boundary values on the CCS (Cauchy Condition Surface). When the CCS method is applied to real experimental data, noise superposition is inevitable. By introduction of SVD (Singular Value Decomposition) and truncation of the least SV components, we can expect the shape reconstruction robustness against the noise. [1] K. Kurihara: Fusion Eng. Design, Vol. 51-52 (2000) 1049-1057. [2] F. Wang, K. Nakamura, O. Mitarai, K. Kurihara, Y. Kawamata, M. Sueoka, K. N. Sato, H. Zushi, K. Hanada, M. Sakamoto, H. Idei, M. Hasegawa, et al.: Engineering Sciences Reports, Kyushu University, Vol. 29, No. 1 (2007) 7-12. Flux (known) Boundary values on CCS (unknown) Noise Inverse problem Normal calculation

3. Characteristics of SVD in ST Plasma Shape Reproduction Method Based on CCS Abstract Cauchy Condition Surface method –Observation equation –Least Square Method –Singular Value Decomposition –Magnetic surface and Largest Closed Flux Surface Plasma shape reproduction by SVD Robustness by least SV truncation Summary

4. CCS （ Cauchy Condition Surface ） Method ・ Unknown: Boundary values on CCS ・ Known: Magnetic sensor (flux loop, magnetic probe), PFC current ・ Solve the observation equation ・ Calculate vacuum magnetic surfaces ・ Identify LCFS as plasma boundary observation equation LSM Error ⇒ ( Condition number ) 2 SVD Error ⇒ ( Condition number ) 1 Condition number = max SV / min SV [1] K. Kurihara: A new shape reproduction method based on the Cauchy-condition surface for real time tokamak reactor control, Fusion Eng. Design 51-52 (2000) 1049. Vacuum Plasma We consider “vacuum region”. We think “plasma” is outside of the “vacuum region”. “Flux value” in the “vacuum region” is calculated from the boundary values on “CCS”.

5. Flux values at flux loop: Magnetic field at magnetic probe: Boundary values on CCS: Outside of plasma: Magnetic surface → LCFS → Plasma boundary Boundary values on CCS (unknown) Eddy current (unknown)

6. Least Square Method and Singular Value Decomposition ⊿ : Diagonal component is SV Observation euation Normal equation Multiply the transposed matrix Multiply the inverse matrix

7. Observation equation Minimize the error LSM Error ⇒ (Condition number) 2 Minimize the error and the norm LSM Error ⇒ (Condition number) 2 SVD Error ⇒ (Condition number) 1 Singular Value Decomposition Eigen (Characteristic) Value Decomposition Definition of eigen value and eigen vector Eigen (Characteristic) equation must hold good. In order that this equation has non-trivial solution, Larger eigen-value component has larger information. Larger singular-value component has larger information.

8. Least Square Method and Singular Value Decomposition ⊿ : Diagonal component is SV Matrix is multiplied only one time. Matrices are multiplied two times.

9. Observation equation Minimize the error LSM Error propagation ⇒ (Condition number) 2 Minimize the error and the norm LSM Error propagation ⇒ (Condition number) 2 SVD Error propagation ⇒ (Condition number) 1 Condition number = max SV / min SV

10. At flux loop: At magnetic probe: On CCS: Outside of plasma: Magnetic surface → LCFS → Plasma boundary Cauchy condition (unknown) Eddy current (unknown) (Largest Closed Flux Surface)

11. CCS （ Cauchy Condition Surface ） Method ・ Unknown: CC on CCS ・ Known: Magnetic sensor (flux loop, magnetic probe), PFC current ・ Solve the observation equation ・ Calculate vacuum magnetic surfaces ・ Identify LCFS as plasma boundary observation equation LSM Error ⇒ ( Condition number ) 2 SVD Error ⇒ ( Condition number ) 1 Condition number = max SV / min SV [1] K. Kurihara: A new shape reproduction method based on the Cauchy-condition surface for real time tokamak reactor control, Fusion Eng. Design 51-52 (2000) 1049. (Largest Closed Flux Surface)

12. SV of coefficient matrix in observation equation Flux loop value and lower modes Flux loop value and higher modes Even Odd Larger singular-value component has larger information.

13. Flux loop value and even modes SV of coefficient matrix in observation equation Flux loop value and odd modes Even Odd Larger singular-value component has larger information.

14. Error and contribution fraction  (singular value)^2 Flux loop value and even-mode accumulation W: Diagonal component is SV. SVD

15. Flux value and eigen vectors of even modes on CCS out ← up ← inin ← down ← out

16. 1 only 1 - 21 - 3 1 - 41 - 51 - 6

17. Observation equation Minimize the error LSM Error ⇒ (Condition number) 2 Minimize the error and the norm LSM Error ⇒ (Condition number) 2 SVD Error ⇒ (Condition number) 1

18. Least Square Method and Singular Value Decomposition ⊿ : Diagonal component is SV

19. Shape difference increases in proportional to noise of flux value Noise of flux value is simulated by random noise from ±1% to ±5% and shape difference is defined as the maximum difference of plasma shape.

23. Summary: Characteristics of SVD in ST Plasma Shape Reproduction Method Based on CCS Ideal equilibrium magnetic surface by equilibrium code –CCS (Cauchy-Condition Surface) method is a numerical approach to reproduce plasma shape, which has good precision in conventional tokamak and also in spherical tokamak. –When the CCS method is applied to real experimental data, noise superposition is inevitable. –SVD may be effective, since error is proportional to condition number, though the error is proportional to the square in LSM –By introduction of SVD (Singular Value Decomposition) and truncation of the least SV components, we can expect the shape reconstruction robustness against the noise. –It is expected that noise effect decreases though detailed magnetic surface information is lost, if smaller SV is neglected. –The truncation simulation of the less SV component shows shape reproduction error less than no truncation, when the measurement error exceeds a certain value. –In this configuration, the ST plasma is symmetrical with respect to the equatorial plane and the number of free parameters on CCS is effectively half of 6. –When unsymmetrical-shape plasma is considered or the degree of freedom is increased, the certain value for inversion may decrease.