To the problem of the recovery of nonlinearities in equations of mathematical physics

Демидов, А. В.; Kochurov, A. S.; Popov, A. Yu.

doi:10.1007/s10958-009-9658-x

Cited by 9 publications

(7 citation statements)

References 14 publications

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“…The link with the Schiffer and Pompeiu conjectures which is clearly pointed out in this paper is particularly interesting. In [25] results of unicity for a class of affine functions or for exponential functions are given for special smooth boundaries and results of non-unicity for doublettype configurations. Finally in [26] identifiability results are given for the full Grad-Shafranov equation in a domain with a corner, with some exceptions for the angle.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time

Blum

Boulbe

Faugeras

2012

Journal of Computational Physics

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The reconstruction of the equilibrium of a plasma in a Tokamak is a free boundary problem described by the Grad-Shafranov equation in axisymmetric configuration. The right-hand side of this equation is a nonlinear source, which represents the toroidal component of the plasma current density. This paper deals with the identification of this nonlinearity source from experimental measurements in real time. The proposed method is based on a fixed point algorithm, a finite element resolution, a reduced basis method and a least-square optimization formulation. This is implemented in a software called Equinox with which several numerical experiments are conducted to explore the identification problem. It is shown that the identification of the profile of the averaged current density and of the safety factor as a function of the poloidal flux is very robust.

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Section: Introductionmentioning

confidence: 99%

Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time

Blum

Boulbe

Faugeras

2012

Journal of Computational Physics

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“…(1) construction of the optimal sets of functions dp/dρ and dF 2 /dρ (this is the key step); (2) solution of the direct problems ( 1)-( 4), (7), reduced to ( 8)-( 11 The construction of sets of functions dp/dρ and dF 2 /dρ can be done by different methods. One is based on the ε-nets of polynomials [18,20], which cover a priori defined sufficiently broad classes of functions dp(ρ(ψ))/dρ and dF 2 (ρ(ψ))/dρ with given absolute inaccuracy ε. We have improved the method of [18] for constructing the ε-net for polynomials and proposed the method for constructing the ε-net for splines.…”

Section: Numerical Methods For Construction Of Substantially Differen...mentioning

confidence: 99%

“…Up to now, the question about the uniqueness of the solution of the inverse problem ( 1)-( 5) with δ = 0 has been studied analytically only in some particular very idealized cases [18]. The answer depends on the form of the area S and the form of the right-hand side (2) in (1).…”

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

Analyses of substantially different plasma current densities and safety factors reconstructed from magnetic diagnostics data

et al. 2011

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The problem of plasma current density and safety factor reconstruction using magnetic field measurements is considered. In the traditional formulation, the problem is strongly ill-posed. In particular, substantially different current densities and safety factors can be equally well attributed to the same set of measurements. The paper presents an accurate mathematical formulation of the inverse problem and its variants. A numerical algorithm is given, which allows to find all substantially different solutions or to prove the absence of multiple solutions. Examples of very different current density and safety factor reconstructions for measurements with finite accuracy are presented. Cases of MAST, JET and ITER-like plasmas are considered. It is shown that including the Motional Stark Effect (MSE) measurements as constraints, provided the accuracy of MSE measurements is sufficient, allows identifying one solution. The approach of the paper can be used for a wide range of inverse problems in physics and help in selecting additional conditions, which can identify the most likely solution among several.

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“…Изучению обратной задачи по определению профиля тока на основе магнитных измерений посвящён целый ряд работ. Исследования проводятся как в направлении разработки эффективного вычислительного алгоритма [5-10, 12, 16], так и в теоретическом направле-нии -например, доказательства теорем единственности в приближении цилиндрической геометрии [17]. В рамках модели тороидальной геометрии таких теорем нет.…”

Section: анализ возможности определения профиля тока на основе магнитunclassified

Numerical Simulation of Electromagnetic Diagnostics System of the Tokamak T-15

Zotov¹,

Белов²,

Sychugov³

et al. 2015

PAST-TF

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В настоящее время проводится модернизация токамака Т-15. Одной из задач исследований на Т-15 является получение разря-дов с вытянутым поперечным сечением плазмы, окружённым сепаратрисой. Присутствие на сепаратрисе Х-точки, где магнит-ное поле обращается в ноль, позволяет использовать данную конфигурацию для создания дивертора -устройства для вывода из плазмы примесей и продуктов реакции. Для оптимизации дивертора и контроля плазмы предполагается в каждый момент разряда проводить реконструкцию границы плазменного шнура. Достоверность такой реконструкции зависит от влияния раз-личных факторов на сигналы электромагнитной диагностики. В работе на основе одного из возможных сценариев разряда в Т-15 проведено исследование точности реконструкции границы плазмы и профиля тока в зависимости от погрешности измере-ний и прочих факторов. Рассматривается задача определения границы плазмы на лимитерной и диверторной стадиях разряда, исследуется влияние распределения тока на сигналы в магнитных датчиках. Так как подобная задача является некорректной, изучается вопрос о выборе параметра регуляризации, в частности, когда погрешность измерений неизвестна. Входными дан-ными задачи являются число магнитных датчиков и их расположение, а также погрешность измерения ими магнитных полей. Исследуется точность восстановления границы плазмы в целом и X-точки сепаратрисы отдельно в зависимости от погрешности измерений и количества датчиков. Моделирование сценария разряда проводилось с помощью кодов TOKAMEQ и DINA. Об-ратные задачи МГД-равновесия решались на основе кода RPB (Reconstruction of Plasma Boundary).Ключевые слова: токамак Т-15, обратная задача МГД-равновесия, электромагнитная диагностика, граница плазмы, профиль тока. The T-15 Tokamak is now under modernisation. One of the objectives of research in the T-15 is obtaining a discharge with an elongated cross-section of the plasma, surrounded with a separatrix. The presence of Х-point on the separatrix, where the magnetic field vanishes, allows to create a divertor-the device for removal of plasma impurities and reaction products. In order to optimize and control the divertor plasma is expected at every moment of the discharge to carry out the reconstruction of the plasma boundary. Reliability of this reconstruction depends on the influence of various factors on the electromagnetic signals of diagnostics. In this paper on the basis of one of the possible scenarios discharge in the T-15 we are interested in the plasma boundary reconstruction accuracy and current profile according to the measurement error and other factors. The problem of determining the plasma boundary on the limiter and divertor discharge stage is studied, examines the impact of the plasma current density at the signals of the magnetic sensors. As this problem is illposed, we study choosing the regularization parameter, in particular for the case when the measurement error is unknown. The problem input data are the number of magnetic sensors and their location and the accuracy of the magnetic fields measurement. We s...

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To the problem of the recovery of nonlinearities in equations of mathematical physics

Cited by 9 publications

References 14 publications

Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time

Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time

Analyses of substantially different plasma current densities and safety factors reconstructed from magnetic diagnostics data

Numerical Simulation of Electromagnetic Diagnostics System of the Tokamak T-15

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