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622 Plenary session
New correlative randomized algorithms for statistical modeling of radiation transfer in stochastic medium
G. A. Mikhailov1,2, I. N. Medvedev1,2
1
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2
Novosibirsk State University
Email: gam@sscc.ru, min@osmf.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-85
For solving the problems of the particle transport through a stochastic medium is often used the method
of maximum cross-section (delta Woodcock tracking), in which the random distribution density of the medium
is realized only at the end points of the free paths. A simple independent choice of the density at these points
leads to the simulation of the transport process in the averaged medium. Therefore, in this talk, new correlation
randomized algorithms for modeling the transfer process are presented. In the first of them, at the end of
the next free path, the previous value of the density is retained if the free path length does not exceed the
correlation length (correlation radius) of the medium. In the second, functional correlated algorithm, the density
value is stored with a probability equal to the corresponding value of the correlation function. The second
algorithm is more accurate but requires much more information about the density field of the medium. The
limits of applicability of the formulated algorithms are studied in detail on the basis of a test problem with extremely
anisotropic scattering and using an unbiased double randomization algorithm. It is shown that the
new algorithms make it possible to solve problems with small-scale stochasticity, for which the implementation
of unbiased transport estimators is practically impossible due to the modeling of the density field as a
whole.
This work was carried out under state contract with ICMMG SB RAS 0251-2021-0002.
Random walk on spheres based algorithm for transients of exciton transport in semiconductors with near
singular behavior of recombination rates around the dislocations
K. K. Sabelfeld1,2, A. E. Kireeva1,2
1
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2
Novosibirsk State University
E-mails: karl@osmf.sscc.ru, kireeva@ssd.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-88
A Random Walk on Spheres (RWS) based stochastic algorithm is developed in this study to simulate a non-
stationary transport of excitons in a semiconductor (SC) with a set of threading dislocations which are randomly
distributed in SC. The simulation algorithm is based on the RWS method suggested for solving the transient
drift-diffusion-reaction problems in [1]. The excitons are generated in SC uniformly over the x, y coordinates,
and exponentially decaying over the z-coordinate. The functions calculated for practical use are the fluxes to
the dislocation and substrate plane, and the cathodoluminescence intensity. The cathodoluminescence intensity
is computed as a fraction of radiatively recombined excitons. The cathodoluminescence method is employed
for the analysis of a material structure. 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 semicylinder
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 the dislocation
central line. The major challenge encountered in the present study is related to the multiscale character of
the problem: the diffusion length of excitons is assumed to be several thousands of nanometers (nm), hence
PLenary session 23
the characteristic size of the simulated semiconductor should be taken at least tens of thousands nanometers
while the dislocation diameter is 6 nm, and the piezofield is varying in the cylindrical region of radius 100 nm
around the dislocation. This means if a finite-difference method would be applied to solve this problem we
would have to introduce a mesh with at least several billions of nodes which is unrealistic. The RWS algorithm
developed is meshfree and calculates directly the fluxes and cathodoluminesence intensity in a semiconductor
which includes hundreds of dislocations with the piezoelectric fields around them.
The support of the Russian Science Foundation under grant � 19-11-00019 is gratefully acknowledged.
References
1. Sabelfeld K.K. Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems // Monte
Carlo Methods Appl. 2017. V. 23 (3), P. 189�212.
2. 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. V. 12, 054038.
3. Sabelfeld Karl K. and Kireeva A. Supercomputer Simulation of Cathodoluminescence Transients in the Vicinity of
Threading Dislocations // PCT 2018, CCIS. 2018. V. 910, P. 1�14.
Sensitivity of functionals in variational data assimilation
V. Shutyaev1, E. Parmuzin1, I. Gejadze2
1
Marchuk Institute of Numerical Mathematics, RAS
2
French National Research Institute for Agriculture, Food, and Environment, Montpellier, France
Email: victor.shutyaev@mail.ru
DOI 10.24412/cl-35065-2021-1-01-53
The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal
control problem to find the initial state and the unknown parameters of the model. A response function is
considered as a functional of the optimal solution after assimilation. The sensitivity of the response function to
the observation data is studied. The gradient of the response function with respect to observations is related
to the solution of a non-standard problem involving the coupled system of direct and adjoint equations. Based
on the Hessian of the original cost function, the solvability of the non-standard problem is studied. Algorithms
to compute the gradient of the response function with respect to observation data are formulated and justified.
Numerical examples are presented for variational data assimilation problem for the Black Sea thermodynamics
model.
This work was carried out within the Russian Science Foundation project 20-11-20057.
References
1. I. Gejadze, F.-X. Le Dimet and V. Shutyaev. On analysis error covariances in variational data assimilation. SIAM J.
Sci. Comput., 30(4), 1847-1874 (2008).
2. I. Gejadze and V.Shutyaev. On gauss-verifiability of optimal solutions in variational data assimilation problems with
nonlinear dynamics. J. Comp. Phys., 280, 439-456 (2015).
3. I. Gejadze, P. -O. Malaterre and V. Shutyaev. On the use of derivatives in the polynomial chaos based global
sensitivity and uncertainty analysis applied to the distributed parameter models. J. Comp. Phys., 381, 218-245 (2019).
4. V.P. Shutyaev, F.-X. Le Dimet. Sensitivity of functionals of variational data assimilation problems. Doklady
Mathematics, 99(3), 295-298 (2019).
5. V. Shutyaev, F.-X. Le Dimet and E. Parmuzin. Sensitivity of response functions in variational data assimilation for
joint parameter and initial state estimation. J. Comp. Appl. Math., 373 (112368), 1-14 (2020).
