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8
DOI: 10.15514/ISPRAS-2019-31(3)-18
Modeling Nonlinear Stabilization System on Clusters with Intel Xeon Phi Coprocessors
D.V. Melnichuk, ORCID: 0000-0002-6689-8904 <melnichukdv@sgu.ru> Saratov State University, 83 Astrakhanskaya Street, Saratov, 410012, Russia
Abstract. Currently, cluster systems are widely used, the nodes of which use processors with a large number of cores. Effective software implementation on such computing systems requires that the corresponding mathematical models have a significant parallelism resource. For the problems of modeling of hybrid dynamical systems (HDS) a significant resource of parallelism is typical, since in this class of mathematical models the (theoretically infinite-dimensional) phase space of control objects with space-distributed parameters is isolated. The purpose of this work is to study the effectiveness of the software implementation on parallel computing systems of the class of modeling problems of the influence of typical nonlinearities and nonstationarity on the output vector function of the HDS. As an example, a nonlinear stabilization system for a mobile control object (the rocket taking into account the elastic deformations of its body) is considered.
Keywords: hybrid dynamical systems; processors with scalable architecture
For citation: Melnichuk D.V. Modeling of Angular Stabilization System on Processors with Scalable Architecture. Trudy ISP RAN/Proc. ISP RAS, vol. 31, issue 3, 2019. pp. 229-240. DOI: 10.15514/ISPRAS-2019-31(3)-18
Моделирование нелинейной системы стабилизации на кластерах с сопроцессорами Intel Xeon Phi
Д.В. Мельничук, ORCID: 0000-0002-6689-8904 <melnichukdv@sgu.ru> Саратовский государственный университет имени Н.Г. Чернышевского, 410012, Россия, г. Саратов, ул. Астраханская, 83
Аннотация. В настоящее время широкое распространение получают кластерные системы, в узлах которых используются процессоры с большим числом ядер. Эффективная программная реализация на подобных вычислительных системах требует, чтобы соответствующие математические модели обладали значительным ресурсом параллелизма. Для задач моделирования комбинированных динамических систем (КДС) типичен значительный ресурс параллелизма, поскольку в данном классе математических моделей (теоретически бесконечномерное) фазовое пространство объектов управления с распределенными по пространству параметрами является изолированным. Целью работы является исследование эффективности программной реализации на параллельных вычислительных системах класса задач моделирования влияния типовых нелинейностей и нестационарности на выходные вектор-функции КДС. В качестве примера рассмотрена нелинейная система стабилизации подвижного объекта управления (ракеты с учетом упругих деформаций ее корпуса).
Ключевые слова: комбинированные динамические системы; процессоры с масштабируемыми архитектурами
Для цитирования: Мельничук Д.В. Моделирование системы угловой стабилизации на процессорах с масштабируемыми архитектурами. Труды ИСП РАН, том 31, вып. 3, 2019 г., стр. 229-240 (на английском языке). DOI: 10.15514/ISPRAS-2019-31(3)-18
1. Introduction
Currently, cluster systems are widely used, in the nodes of which one or several processors with a large number of cores are used. Examples include computing systems with new Intel Xeon processors with scalable architecture or computing systems with Intel Xeon Phi coprocessors that are used as virtual cluster nodes. Parallel computational architectures of this class are effective only when solving problems with a significant parallelism resource. In this case, classes of mathematical models that are effectively implemented on Intel Xeon Phi, will be effectively implemented on modern scalable Intel Xeon architectures.
Hybrid dynamical systems (HDS) [1, 2] are mathematical models of a number of technical systems containing control objects with lumped parameters and connected to them across the boundaries of control objects with distributed parameters (see fig. 1). HDS is characterized by input and output vector functions. The nonlinear system of angular stabilization of the movable control object with deformable body is the example [3, 4]. HDS are systems of ordinary differential equations (ODE) and partial differential equations (PDE) connected by means of boundary conditions (BC) and constraint's conditions (CC) under appropriate initial conditions (IC). For the problems of HDS modeling a significant resource of parallelism is typical, since the (theoretically infinite-dimensional) phase space of control objects with distributed parameters is isolated.
Fig. 1. HDS Structural scheme
2. Related work
MPI technology is standard for cluster systems with distributed memory, and both optimization of MPI itself [5] and parallel libraries [6] and algorithms based on it are relevant. Examples are problems of mathematical physics [7, 8], graph theory [9], sparse matrix factorization [10]. Optimization of parallel algorithms for cluster systems with many-core processors was considered in [11, 12]. If adaptive algorithms of numerical modeling are used or nodes of the cluster have different capacities, dynamic balancing of computational load is required [13]. For dynamic balancing of computational load on cluster systems, the MPI-MAP parallelization pattern was previously implemented [3]. The main theorems on the stability of linearized HDS are formulated and proved in [1, 2]. In work [4] adaptive algorithms of HDS parametric synthesis are offered. Modeling of systems of angular stabilization of movable control objects with deformable body was considered in [14, 3, 4].
3. Problem formulation
The purpose of this work is to study the effectiveness of the software implementation on parallel computing systems of the class of modeling problems of the influence of typical nonlinearities and nonstationarity on the output vector function of the HDS. We consider a similar [4] system of angular stabilization of the movable control object (the rocket taking into account the deformations of its body), but providing stabilization both with respect to the vertical direction and with respect to the longitudinal axis, as well as a smooth change in the time of the thrust force of the rocket engine.
4. Parallel algorithms for modeling of hybrid dynamical systems
HDS with piecewise continuous input vector function x(t), x: R ^ Rw* and continuous output vector function y(t), y: R ^ Rwy correspond to equations
y = f(x,y,h,p,j,jtt);u = F(u,x,y,y,j, ^t),r £ n G(u,y,J)|s = 0,5 = dft;h = JsH(u, j)d5 (1)
y(0) = yo,u(r,0) = Uo(r) Here r £ RWr - are independent spatial coordinates of individual points of the object with distributed parameters, n c RWr - area occupied by an object with distributed parameters, f: Rw* x R% x x Rwp x Rwm x Rwt ^ , h: R ^ , u(r, t), u: R^rxR^ Rw" -distributed output vector function, operators F: (RWr x R ^ Rw") x (r ^ Rw*) x (R ^ R%) x (R ^ Rwy) x Rwm x RWf ^ (Rwr x R ^ Rw"), G: (Rwr x R ^ Rw") x (R ^ R%) x Rwm ^
(RWr x R ^ Rws), H: (RWr x R ^ Rw") x Rwf ^ (R ^ Rw^) correspond to partial differential equations, boundary conditions, and coupling conditions; p £ Rwp - feedback parameters; H £ - the parameters of model nonlinearities; nt £ RWt - parameters characterizing the unsteadiness of the system from the point of view of the automatic control theory; the point at the top indicates the time t differentiation. When n = j = 0 HDS (1) becomes linear stationary. After parametric synthesis, numerical simulation of the effect of typical nonlinearities and unsteadiness on the output vector function of a nonlinear HDS (1) is performed. In this case, the input vector function x(t) and the initial conditions y0, u0(r) are fixed, and the components of the vectors n and nt change with a fixed step within a parallelepiped. The element-by-element transformation of sequence ( J, j ), j = 1,2,3,... into a sequence of values characterizing the maximum and standard deviations of function y(t; j, j ) from y(t; 0,0) is parallelized (J, Jt,) ^ OV^,.)7,;' = 1,2,3,...; Vi = max |y(t; j, jt) - y(t;0,0)|
1 11 0<t<tmax
re I1/2 (2)
V = [tmL J0tmax |y(t; J, Jt) - y(t; 0,0)|2dt] , tmax >> 1
The transformation (2) can be adapted to the "two-layer" MPI-OpenMP scheme, where a separate MPI-MAP executing process performs the transformation of
{(J, JtPJ = V^} ^ {(Vi,, V21)r,; = V^} (3)
by parallelizing calculation of the values on the right side (3) based on OpenMP. Numerical integration of the initial boundary value problem (1) is implemented by the Galerkin's projection method [4] and subsequent application of the BDF method to the resulting Cauchy problem for the system of ordinary differential equations.
5. Model of stabilization system
The object moves with respect to a fixed coordinate system 00x0y0z0 (see fig. 2) under the action of force P, attraction to the Earth and external disturbing horizontal force Fe = (0, Fe , Fez0)7.
Fig. 2. Structural scheme
The coordinate system Oxyz is connected to the body 1, and r1 = (x1,y1,z1)T and pL = (Pi,i,Pi,2, P13)T characterize its linear and angular displacements relative to O0x0y0z0. Linear r2 = (x2,y2, z2)T and angular = (fi21, P22,P23)T displacement of body 2 with respect to Oxyz is caused by the elastic displacement u = (ux, uy, uz)T = u(x, t) of the centerline of the hull. The
rotation angle a = (a1, a2, a3)T of the body 2 relative to O0x0y0z0 measures the gyrostabilizer,
(c) (c)
and the control moments of the forces M' , j = 1,2,3 are formed. Under the action of M2 and
M(c) body 0 rotates at angles p0 = (0, fi0j2, p0j3)T relative to Oxyz. The moment acts on the body 1 and compensates for the rotation of the movable object relative to the longitudinal axis. Let ^0 = (^0x,^0y,^0z )T, = (n.1x,n.1y,n.1z )T, Q.2 = )T be the relative and
absolute angular velocities of bodies 0, 1, 2; Q = (Ql, Q2, Q3)T, M = (Ml, M2, M3)T be the internal forces and moments acting in the cross sections of the body. Here x(t) = (Feyo(t),Fezo(t))T and
y(t) = (fii,3(t),P2,3(t),Pi,2(t),P2,2(t), Pi,i(t), P2,i(t))T are input and output vector functions, P = (Vl,P2,...,P12)T are feedback parameters. The model equations of the nonlinear stabilization system are given in Appendix A. The set of parameters n = (p-1,^2,^3)T characterizes typical nonlinearities, and the parameter nt = [n4], n4 > 0 characterizes a smooth change in the characteristic overload according to the law
ax(t) = a(™n) + (a(mx) - a^e-^, t > 0, a(™n) < ax < a(mx) (4) At n = 0, the model equations are linearized and decomposed into three independent subsets corresponding to the motion in the O0x0y0 and O0x0z0 planes (by virtue of symmetry, they pass into each other), as well as to the rotation relative to the longitudinal axis. In this case, p5+j = pj,
j = 1,5, correspond to the stabilization system in the vertical direction, and pL1, p12 correspond to the stabilization system with respect to the longitudinal axis.
6. Numerical simulation results
In the numerical simulation of the output vector functions of the nonlinear angular stabilization system, the components of the input vector function were given as Fe (t) = 1(t), Fez0(t) = 1( t) — 1(t — 1), where 1( t) is the unit jump function of Heaviside. For stabilization system with a set of parameters
/„ = 0.02, m1 = 0.3, A = 0.07, m2 = 0.2, J2 = 0.05, а = 0.166667, a(min) = 0.2, a(max) = 2, у = 0.01,/1fc = 0.1,/2fc = 0.05,yfc = 2, ^ = 0.08, = 0.15, ^ = 0.055, = 0.05 (5) The feedback parameters of the stabilization system in the direction of vertical p1 = p6 = 6.347, p2 = p7 = 13.12, p3 = p8 = 17.59, p4 = p9 = 14.03, p5 = p10 = 5.951 were chosen on the basis of an adaptive algorithm of parametric synthesis of the family of linearized models of HDS [4]. Since the stabilization of the object with respect to the longitudinal axis is intended to compensate for the slow accumulation of errors due to nonlinear effects, the feedback parameters p11 = 0.04, p12 = 1 are selected in the central part of the stability region.
Fig. 3 presents the results of numerical simulation of the components and of the output vector functions of the original nonlinear unsteady HDS (shown as a solid line) and its linear stationary analog at = = = = 0 (shown as a dashed line). The significant difference of the results is explained by the fact that the dimensionless overload аж decreases smoothly, with the decrease of аж in the considered range of overload changes in the linear stationary system, the attenuation of transients decreases, and the characteristic value of the output vector function increases. Nevertheless, parametric synthesis by the linearized model allows to stabilize the original nonlinear system in the vertical direction in the entire range of overloads. As follows from the results presented in Fig. 4, the selected values of the feedback parameters p11 and p12 allow to stabilize the movable control object with respect to the longitudinal axis, i.e. to compensate for the slow accumulation of errors due to nonlinear effects.
Fig. 3. Stabilization in the vertical direction
Fig. 4. Stabilization with respect to the longitudinal axis
Fig. 5 shows the dependences of the parameters 6 [0,0.055] and 6 [0,0.05] at fixed n1 and ;U2 maximum v1 and standard v2 deviations (see eq. (2)) of the output vector function of the nonlinear HDS on the output vector function of the linearized HDS for tmax = 250. As follows from the data presented in Fig. 5, when changing the overload according to (9) the greatest influence on the output vector function of the nonlinear HDS has parameter characterizing the unsteadiness of the system.
Fig. 5. Maximum and standard deviations
Similar data characterizing the efficiency of stabilization with respect to the vertical, longitudinal axis, as well as the influence of the parameters of nonlinearity and unsteadiness on the output functions of the stabilization system with parameters
/„ = 0.00003, = 0.0667, /1 = 0.00009728, m2 = 0.333, /2 = 0.00345, a = 0.166667, ax = 1, y = 0.01, p1 = p6 = 4.098, p2 = p7 = 9.553, p3 = p8 = 7.687, p4 = p9 = 7.714, p5 = p10 = 3.269, /1fc = 0.002, /2fc = 0.005, )2 = 2, ^ = 0.08, = 0.2, = 0.04, = 0.05, Pii = 0.05, P12 = 1 (6)
are shown in fig. 6-8. Similarly to the previously discussed non-linear stabilization system allows to compensate for the unwanted errors throughout the range of overload (see fig. 5, 6). The greatest influence on the output vector function of the nonlinear HDS has the parameter which characterizes the unsteadiness of the system (see fig. 8).
Fig. 6. Stabilization in the vertical direction
a 10 20 30 t
Fig. 7. Stabilization with respect to the longitudinal axis
• 0,05-0,1 0,1-0,15
Fig. 8. Maximum and standard deviations
7. Efficiency analysis of parallel algorithms
Consider the effectiveness of the implementation on computer systems with coprocessors Intel Xeon Phi parallel algorithm (2), (3) modeling the effect of typical nonlinearities and unsteadiness on the output functions of the HDS. The data corresponding to the modeling of a nonlinear stabilization system with parameters (5) are presented in Table 1. The calculations were performed on a cluster of faculty of Computer Science and Informational Technologies and Volga Region Center of New Information Technologies of SSU. The four-dimensional grid of change of parameters ¡i1, ¡i2, jM3 jM4 dimension 6x9x9x9 was used.
Table 1. Modeling of the impact of model nonlinearities and non-stationary, sec.
Grid 6x9x9x9
Processor, serial/parallel Test 1 Test 2 Test 3 Test 4 Test 5
Intel Xeon E5-2603 v2, serial 16411 16325 16470 16531 16314
2 processors Intel Xeon E5-2603 v2, OpenMP. 2480 2471 2501 2492 2485
Coprocessor Intel Xeon Phi 5110P, OpenMP 1667 1635 1673 1649 1655
2 coprocessors Intel Xeon Phi 5110P, MPI-MAP/OpenMP 1023 1015 1032 1008 1040
As follows from the Table 1 results, in this case, the use of a single Intel Xeon Phi coprocessor is more efficient than the use of two quad-core CPUs. The most profitable strategy of using coprocessors is parallelization based on OpenMP inside the coprocessor and parallelization based on MPI-MAP between coprocessors. Similar data for the stabilization system with parameters for the stabilization system with a set of parameters (6) are presented in Table 2. And in this case, using one Intel Xeon Phi processor is more efficient than using two quad-core CPUs. The most profitable strategy for the use of coprocessors is the parallelization based on OpenMP within the coprocessor and parallelization based on MPI-MAP between the coprocessors.
Table 2. Modeling of the impact of model nonlinearities and non-stationary, sec.
Grid 6x9x9x9
Processor, serial/parallel. Test 1 Test 2 Test 3 Test 4 Test 5
Intel Xeon E5-2603 v2, serial. 9637 9597 9645 9657 9675
2 processors Intel Xeon E5-2603 v2, OpenMP 1443 1470 1430 1412 1467
Coprocessor Intel Xeon Phi 5110P, OpenMP 1052 1042 1063 1037 1055
2 coprocessors Intel Xeon Phi 5110P, MPI-MAP/OpenMP 703 699 710 707 705
Analogical evidence of the effectiveness of the implementation of the parallel algorithm (2), (3) using a single Intel Xeon Phi coprocessor (OpenMP) and two Intel Xeon Phi coprocessors (MPI-MAP - OpenMP) for a more detailed meshes, changing parameters are given in
Table 3 for stabilization system with parameters (5) and in Table 4 for the stabilization system with parameters (6).
Table 3 - Modeling of the impact of model nonlinearities and non-stationary, sec.
Processor, serial/parallel. Test 1 Test 2 Test 3 Test 4 Test 5
Grid 6 x 16 x 16 x 16
Coprocessor Intel Xeon Phi 5110P, OpenMP 8953 9005 8934 8902 8985
2 coprocessors Intel Xeon Phi 5110P, MPI-MAP/ OpenMP 4726 4753 4715 4703 4744
Grid 12 x 16 x 16 x 16
Coprocessor Intel Xeon Phi 5110P, OpenMP 18132 18243 18025 18187 18053
2 coprocessors Intel Xeon Phi 5110P, MPI-MAP/ OpenMP 9478 9529 9435 9501 9439
Table 4 - Modeling of the impact of model nonlinearities and non-stationary, sec.
Processor, technology of parallelization Test 1 Test 2 Test 3 Test 4 Test 5
Grid 6 x 16 x 16 x 16
Coprocessor Intel Xeon Phi 5110P, OpenMP 5671 5634 5654 5754 5698
2 coprocessors Intel Xeon Phi 5110P, MPI-MAP/ OpenMP 2988 2969 2967 3031 3002
Grid 12 x 16 x 16 x 16
Coprocessor Intel Xeon Phi 5110P, OpenMP 11205 11278 11154 11174 11237
2 coprocessors Intel Xeon Phi 5110P, MPI-MAP/ OpenMP 5777 5809 5735 5741 5798
As follows from the Table 3 and 4 results, with an increase in the average number of nodes on the grid measurement, the multiplicative contribution to the acceleration of the MPI-MAP pattern quickly tends to the number of coprocessors used.
8. Conclusions
The proposed parallel algorithm is effective on cluster systems with nodes using processors with a
large number of cores. In particular, it is effective on cluster systems with Intel Xeon Phi
coprocessors.
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The rotation of the coordinate system is characterized by angles a = (a1( a2, a3)T(in order
Appendix A.
The rotation o a3,a2,a1), and
4(a) =
cosa3 -sina3 sina3 cosa3
L0
0
2
5(a) =
cos a 0
-sina 1 0
0 sina
1 0
2 0 cos a2
10 0 0 cosa1 -sina1 0 sina1 cosa1
-sina
0 cosa1 cosa2sina1 0 -sina1 cosa2cosa1. In dimensionless variables and parameters the equations of motion of the HDS have the form " = 5(^01)01,02 = 4T ("1P2)" + 5(^102)^2, = -A^sinC"^) Ooy = ^o,2Cos((U1^o,3),^oz = A>,3,ra1ii = ^(^101)Q(0,t) - Fe + +ax[(1 + rn2)i4(^101)^(0,0o) + ^(01, 0>)] ^(a, 0) = M-1(i4(^1a)i4(^10) - £)(1,0,0)T =
= (O1(a,0),O2(a,0),O3(a,0))r,f = diag{1,1,1},/(1) = diag(/1fc,A,A) /»(" + d)0) + /(1)^1 + x (/»w0 + /(1)n1) = M(0, t) - (Mc(1), 0,0)T
L1 + w0) +7 "1
1 o
-
fl1x°0z)] =
M(c) cos(^1^>,3) + M3c) sin(^1^>,3) sin(^1^o,2)
/0 [f^1z + Ooz + "1 - f1yOoJ] =
M.
(c)
cos
("1^0,2)
= 1 arcsin("1^3(02, 01)) ,/1(z) = tgz 1 Ui^(0„0i) 1
(7)
a2 = -"1 arcsin("1^3(02,
"1^2(02,01) 1 . "1^3*(02,01)
a3 = —arcsin---— ,a1 =--arcsin----—
"1 cos("1a2) "1 cos("1a2)
^2W2 = axm2[^(0,"1(0,^2,2,^2,3)r) -^*(0,"1p1)]-4("1(0,^2,2,^2,3)r)Q(1,0
- AT
4r("101)r1 + x R2 + "1("1 • R2)" - "1"2 R2 - 2"1"1 xr2 + r2
/(2)^2 +"1"2 x/(2)f2 = -4T(("1^2,1,0,0)T)M(1,t) + + (a, 0,0)T x 4T (("1^2,1,0,0)T)Q(1, t),/(2) = diag^A^} R2 = (1 + a, 0,0)T + "^^(a, 0) = "1-1(4T("1a)4T("10) - £)(1,0,0)T = = №(a,0),O2(a,0),O2(a,0))T
U'x = "-1 [(1 - "l (w'y + u'|))1/2 - 1], ¿21 = "1^'y, ¿31 =
¿11 = (1 - ¿21 - ¿31)1/2^33 = (1 - ¿31)1/2, ¿12 = -¿21/¿33,¿22 = ¿11^33 ¿32 = 0, ¿13 = -¿3^22, ¿23 = ¿31¿12, K1 = U z(¿12¿ 22 - ¿22¿ 12)
K2 = U z¿22¿ 11 U v¿23 U Z¿33,K3 = » + U ^
¿
y ¿22
+ (4T("101)ii) • (0,1,0)T - ("1^1xUZ - fl1z(x + "1UX)) + +"J(x + "1Ux)f1x + "1Uzf1z]n1y - ^(n^ + f2z)Uy + 2"1(n1xUz -f1zUtx) = ¿21(^1 + MK2Q3 - K3^2)) + ¿22(^2 - MK1Q3 - K3O1)) + +¿23 (C3 + "1(^2 - K2Q1)) - ax №2(0,01) + ((rn2 + 1 - X)u'y)']
^Z + (4T("101)r1) • (0,0,1)T + - fl1y(x + "1Ux) +
+"1 [(x + "1Ux)f1x + "1Uyn1y]n1z - "Kn^ + f^X - -
-n1y^x) = ¿31W1 + MK2Q3 - K3^2)) + ¿33W3 + "1(K1^2 - K2^1)) -ax[02(0, 01) + ((m2 + 1 - x)u'z)'] 238
+ Kl)0i = + 1 - x)(k| + Кз) + к'з@2 - к'203 +
+2K3Ç'2 - 2кз@'3 - MiKi(K2Ç2 + K3Ç3) - ((u'x)2 + (ù'y)2 + (ù'z)2) +
+ (П1я!ц + n^i + П^)2 - (П2ж + П2у + n2z) +
+2[Lii(a ù'z -Hi7ù'y) - ¿2i(nixü'z -au'j + ¿3i(nixu'y - a u'j]}
uy(0,0 = 0,u'y(0,0 = 0,uy(1,0 =У2 -Ü^2(0,ß2),u'y(1,t) = cosCUi^2,2) •^-isin(iui^2,3),uz(0,t) = 0,u'z(0,t) = 0,uz(1,t) = Z2 -аФ3(0^), u'z(1,t) = -iu-isinOui^2,2),^2 = "x(1,0 + a^i(0,ß2),
Ç'i(0, 0 + pi(К2 (0, 003 (0, 0 - к3 (0, 002 (0, 0) = = fi • ¿GuißiX1A0)r + ахф^(0, ßi) 0'i(1, 0+ Pi (К2 (1,О03 (1,0- К3 (1, 0 02(1, 0) = = ажФ^2, ßi) + Pia(n2„ + П2,) + (1,0,0)г • ^(/^М
(9)
(10)
Mi = 4 (ß,i + K&,i - J0iKidx),M2 = К2 - 7u"z,M3 = К3 +
02 = -M'3 + Pi(K2Mi - KiM2), 03 = M'2 + Pi(K3Mi - KiM3)
ßi(0) = ßi(0) = ß2(0) = ß2(0) = ^0,2(0) = ¿0,2(0) = ^0,3(0) = ^0,3(0) =
= ri (0) = fi (0) = У2 (0) = У2 (0) = Z2 (0) = ¿2 (0) = (11)
= (x, 0) = ùy (x, 0) = uz(x, 0) = ùz(x, 0) = 0
Here (7) are ordinary differential equations, (8) are partial differential equations, (9) are boundary conditions, (10) are constraint's conditions, (11) are initial conditions, ()' = ô()/ôx.
Информация об авторе / Information about author
Дмитрий Вадимович МЕЛЬНИЧУК получил степень магистра по направлению «Прикладная математика и информатика» в Саратовском национальном исследовательском государственном университете имени Н.Г. Чернышевского, Саратов, Россия. Его исследовательские интересы включают математическое моделирование, моделирование управляемых комбинированных динамических систем, параллельные алгоритмы и параллельные вычислительные технологии.
Dmitry Vadimovich MELNICHUK received a master's degree in «Applied mathematics and Informatics» at Saratov State University, Saratov, Russia. His research interests include mathematical modeling, simulation of controlled hybrid dynamic systems, parallel algorithms, and parallel computing.
