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18Численное статистическое моделирование и методы Монте-Карло
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ном фантоме [3] проведен численный анализ качества томограмм при различных угловых и временных распределениях внешнего источника излучения.
Исследование выполнено при финансовой поддержке Российского фонда фундаментальных исследований (код проекта 20-01-00173).
Список литературы
1. Аниконов Д.С., Прохоров И.В. Определение коэффициента уравнения переноса при энергетических и угловых особенностях внешнего излучения //Доклады АН. 1992. Т. 327. № 2. С. 205-207.
2. Anikonov D. S., Kovtanyuk A. E., Prokhorov I. V. Transport Equation and Tomography. Inverse and Ill-Posed Problems Series, 30, VSP, Boston - Utrecht, 2002.
3. Steiding C., Kolditz D., Kalender W. A. A quality assurance framework for the fully automated and objective evaluation of image quality in cone-beam computed tomography // Medical Physics. 2014. V. 41. № 3. 031901.
Рандомизированный проекционный метод для решения нелинейного уравнения Больцмана в трехмерном случае
С. В. Рогазинский
Новосибирский государственный университет
Институт вычислительной математики и математической геофизики СО РАН
Email: svr@osmf.sscc.ru
DOI: 10.24411/9999-017A-2020-10094
В [1] рандомизированный проекционный метод использовался для решения нелинейного уравнения Больцмана и была получена оценка нормы в L^R1) скорости сходимости проекционного ряда по функциям Эрмита.
В данной работе рассмотрен проекционный метод в трехмерном случае. Получены оценки нормы в L2(R3) скорости сходимости проекционного ряда по функциям Эрмита в зависимости от гладкости разлагаемой функции.
Численные эксперименты подтверждают полученные оценки.
Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (коды проектов 18-01-00356, 18-01-00599) и в рамках проекта НИР 0315-2019-0002.
Список литературы
1. Sergey V. Rogasinsky. Two variants of Monte Carlo projection method for numerical solution of nonlinear Boltzmann equation // Russ. J. Numer. Anal. Math. Modelling, Volume 34, Issue 3 (Jun 2019).
Probabilistic models and algorithms for PDEs in high dimensions with applications to narrow escape problems in cells and semiconductors
K. K. Sabelfeld
Institute of Computational Mathematics and Mathematical Geophysics, SB RAS
Novosibirsk State University
Email: karl@osmf.sscc.ru
DOI: 10.24411/9999-017A-2020-10381
In this study probabilistic models and stochastic simulation algorithms for solving high-dimensional PDEs with focus on drift-diffusion-recombination transport problems are presented. Application of the developed stochastic algorithms to narrow escape problems are considered. Our recent research in this application field has been published in [1]. We discuss in this talk the transport in cells and biological tissues where the mean time and flux of a molecular agent to a target are of principal interest. Another important application handled in this work is the electron-hole and exciton transport in semiconductors [3]. The presented stochastic simulation algorithms are based on the random walk on spheres method suggested for solving the transient drift-diffusion-reaction problems in [2] and a global random walk method recently developed and published in [4].
The support of the Russian Science Foundation under grant № 19-11-00019 is gratefully acknowledged.
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References
1. Sabelfeld K.K. Stochastic simulation algorithms for solving narrow escape diffusion problems by introducing a drift to the target, Journal of Computational Physics, 410 (2020), Article ID 109406.
2. Sabelfeld K.K. Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems Monte Carlo Methods Appl. 2017. Vol. 23 (3), P. 189-212.
3. Kaganer Vladimir M., Lahnemann Jonas, Pfuller Carsten, Sabelfeld Karl K., Kireeva Anastasya E., and Brandt Oliver. Determination of the Carrier Diffusion Length in GaN from Cathodoluminescence Maps Around Threading Dislocations: Fallacies and Opportunities, Physical Review Applied. 2019. Vol. 12, 054038.
4. K. Sabelfeld, A. Kireeva. A new Global Random Walk algorithm for calculation of the solution and its derivatives of elliptic equations with constant coefficients in an arbitrary set of points, Applied Mathematics Letters, vol. 107, 2020, 106466.
Monte Carlo methods for solving a system of nonlinear parabolic-elliptic equations of semiconductors
K. K. Sabelfeld1,2, A. E. Kireeva1
lInstitute of computational mathematics and mathematical geophysics, SB RAS
2Novosibirsk State University
Email: karl@osmf.sscc.ru
DOI: 10.24411/9999-017A-2020-10095
In this study we develop a Monte Carlo method for solving a system of nonlinear parabolic-elliptic equations governing the transport and recombination of electrons and holes in semiconductors. This field attracts considerable experimental and theoretical interest because the optoelectronic properties of technologically important semiconductor materials have been found to be controlled by the electron-hole recombination dynamics. A stochastic method for solving a nonlinear system of divergence free drift-diffusion-Poisson equations is developed in [1]. It is based on the global Random Walk algorithm [2] which calculates a gradient of the solution of the Poisson equation in arbitrary family of points of the domain. The drift-diffusion-Poisson system is solved by the iteration procedure including alternating simulation of the drift-diffusion processes and calculating the gradient of the solution to the Poisson equation. In the present study we extend the stochastic algorithm to solve the nonlinear system of drift-diffusion-Poisson equations in the general divergence form.
The support of the Russian Science Foundation under grant № 19-11-00019 is gratefully acknowledged. References
1. Sabelfeld K., Kireeva A. Stochastic simulation algorithms for solving a nonlinear system of drift-diffusion-Poisson equations // BIT Numerical Mathemaatics, submitted 2020.
2. Sabelfeld Karl K. A global random walk on spheres algorithm for transient heat equation and some extensions // Monte Carlo Methods and Applications. 2019. № 25(1). P. 85-96.
Stochastic simulation of transients of cathodoluminescence intensity: impact of randomly distributed dislocations
K. K. Sabelfeld^ A. E. Kireeva1
lInstitute of computational mathematics and mathematical geophysics, SB RAS
2Novosibirsk State University
Email: karl@osmf.sscc.ru
DOI: 10.24411/9999-017A-2020-10096
In this study a stochastic algorithm of simulation of exciton diffusion and drift in a semiconductor in vicinity of randomly distributed dislocations is developed. The Monte Carlo algorithm is based on the random walk on spheres method suggested for solving the transient drift-diffusion-reaction problems in [1]. The cathodoluminescence intensity is computed as a fraction of the radiatively recombined excitons. The cathodoluminescence method is employed for the analysis of a material structure. Threading dislocations are visible as dark spots in cathodoluminescence maps. The recent experiments [2] showed that the strain field in the vicinity of dislocations produces a piezoelectric field which affects the exciton life-time close to the dislocation edge and causes a drift of excitons. In our previous model [3] we simulate the threading dislocation as a semi-cylinder whose surface adsorbs excitons. In the present work, the dislocation is simulated with its piezoelectric field around which defines the life-time and the drift of excitons depending on the distance from
