=Paper=
{{Paper
|id=Vol-2864/paper36
|storemode=property
|title=Parameters Identification for Fractional-fractal Model of Filtration-Consolidation Using GPU
|pdfUrl=https://ceur-ws.org/Vol-2864/paper36.pdf
|volume=Vol-2864
|authors=Vsevolod Bohaienko,Anatolij Gladky
|dblpUrl=https://dblp.org/rec/conf/cmis/BohaienkoG21
}}
==Parameters Identification for Fractional-fractal Model of Filtration-Consolidation Using GPU==
https://ceur-ws.org/Vol-2864/paper36.pdf
Parameters identification for fractional‐fractal model of
filtration‐consolidation using GPU
Vsevolod Bohaienkoa, Anatolij Gladkya
a
V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, Glushkov ave., 40, Kyiv, 03187, Ukraine
Abstract
The paper considers some computational problems arising in the important practical field of
the determination of safe operation conditions of engineering facilities that pollute soils and
groundwater. In the case of complex geological and hydrological conditions, such problems
are widely considered using mathematical modeling of deformation and consolidation
processes in water-saturated soils, particularly, in the foundations of hydraulic structures. To
simulate the dynamics of such processes, we use a fractional-fractal approach that allows
considering temporal non-locality of transfer processes in media of fractal structure. The used
one-dimensional differential model contains a non-local Caputo derivative with respect to the
time variable and a local fractal derivative with respect to the space variable. Some of
model’s parameters, namely the orders of fractional derivatives, can only be determined
fitting them to the measured data related to the state of a process. We propose to use particle
swarm optimization algorithm to perform an identification of fractional derivatives’ orders
and present the results of its testing on noised subsets of direct problem solutions. In this
context, we have determined that the order of space-fractal derivative is restored with a
relative error of not more than 1% while the order of time-fractional derivative is restored
with higher errors of not more than 10%. The lowest number of observation points that
ensures stable restoration of the orders was equal to 25. As high computational complexity is
combined with highly independent computational blocks while applying evolutionary
optimization algorithms to the problems of differential models’ parameters identification, we
implemented the proposed algorithm on graphical processing units (GPU) using OpenCL
framework and on multi-threaded systems using OpenMP. The results of performance testing
showed up to 4-times lower GPU execution time compared to the case of multi-threaded
execution on 6 cores of central processing unit (CPU).
Keywords
filtration-consolidation processes, fractional-fractal model, parameters identification, particle
swarm optimization, GPU, OpenCL.
1. Introduction
The determination of the conditions for safe functioning of numerous engineering facilities that
pollute soils and groundwater are among the most important and relevant in the connection with
environment protection issues. This makes urgent the development of effective and reliable methods
for mathematical modeling of deformation and consolidation dynamics in saturated soils, particularly,
in the foundations of hydraulic structures. Such methods are usually based on the solution of initial-
boundary value problems for the corresponding systems of partial differential equations (e.g. [1-3]).
Recently, in order to take into account the effects of temporal and spatial correlations, a series of
models have been developed by introducing into them local and non-local operators of fractional-
order differentiation (e.g. [4-6]).
CMIS-2021: The Fourth International Workshop on Computer Modeling and Intelligent Systems, April 27, 2021, Zaporizhzhia, Ukraine
EMAIL: sevab@ukr.net (V. Bohaienko); gladky@ukr.net (A. Gladky)
ORCID: 0000-0002-3317-9022 (V. Bohaienko); 0000-0002-3839-9550 (A. Gladky)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
In this context, we continue the studies presented in [6] on the simulation of anomalous processes
of filtration-consolidation in water-saturated soils under the influence of salt transfer. The model
developed in [6] uses the fractional-fractal approach [7-10] that enables modeling of temporal non-
locality of processes in soils of fractal structure. Comparing to the previously developed mass transfer
models within such an approach this model widens its application taking into consideration soil
compaction and chemical osmosis.
Practical usage of fractional-order differential models to predict the dynamics of mass transfer and
soil compaction requires identification of their parameters. However, the orders of fractional
derivatives usually do not have technical means of measurement. The only approach that can be used
to determine their values is to select them the way to make the model best describe available
measurements.
In general, two approaches used to identify the parameters of fractional-differential models can be
singled out. Within the first one, corresponding inverse problems are solved analytically or using
numerical-analytical methods (e.g. [11, 12]). Such an approach is usually applied to relatively simple,
mostly one-dimensional models. Despite some problems of restoration of fractional derivative orders
are solved by least-squares techniques [13] or the Tikhonov method [14], the second approach that is
based on metaheuristic optimization techniques (e.g. [15-17]) have less limitation on its usage. One of
such algorithms that were efficiently applied [17] to some models of mass transfer processes in soils
is the particle swarm optimization (PSO) algorithm [18]. Regarding most metaheuristic optimization
algorithms it should also be noted that data parallelism can be easily exploited in them.
Hence, in this paper we study the accuracy and performance of the PSO algorithm applying it to
solve parameters identification problem for the fractal-fractional model of filtration-consolidation
proposed recently in [6]. We also consider PSO algorithm’s implementation on graphical processors
(GPU) and study its performance comparing it with the case of multi-threaded implementation
on CPU.
2. Direct problem
Considering the non-local in time isothermal filtration-consolidation process in a soil of fractal
structure saturated with a salt solution, we study the following mathematical model [6]:
(1)
Dt( ) H C H C
x x
C C (2)
Dt( ) C d* kH C
x x x x
where H ( x, t ) is the water head ( m ), C ( x, t ) is the concentration of salts in the liquid phase
( kg / m3 ), k is the filtration coefficient in fractal dimension ( m 2 1 / s ), is the coefficient of
m3 2
chemical osmosis in fractal dimension ( ), d* is the coefficient of convective diffusion in
kg s
fractal dimension ( m 2 / s ), C is the consolidation coefficient in fractal dimension ( m 2 / s ),
C
t
1
, is the porosity of a medium, Dt( ) f (t ) (t ) f ( ) d f (0) is the
k (1 ) t 0
regularized fractional Caputo derivative of the order , 0 1 with respect to the variable t
[19-21], is the operator of fractal derivative [7-9], which can be represented under the known
x
d 1
conditions as f ( x) f ( x) 1 , 0.5 is the fractal dimension.
x dx x
The equations (1),(2) contains coefficients in fractal dimension in space and time. To make them
consistent with measurable values, we, following [22], introduce parameters t ( s ), x ( m ) of time
x2 2
and space dimensions and represent fractal dimensional filtration coefficient as k k where k
t1
is the measurable value of the coefficient. The same is performed for the other coefficients in fractal
dimension. While the choice of the values of dimensional parameters t , x strongly influence
simulation results, there are significant difficulties of their direct determination. Thus, we consider the
model (1),(2) as semi-empirical and set t x 1 (one of the cases studied in [22]) further focusing
on the identification of fractional derivatives' orders only.
Let us note, that from equations (1), (2) when 1 we obtain a system of equations [23] of the
corresponding fractional-differential model that does not consider fractal properties of a soil. When
, 1 we obtain the integer-order classical model [1, 2].
Using in (1), (2) the representation of the fractal derivative operator on the base of integer-order
derivative [7-9] we obtain the system of equations in the following form [6]:
2 H H 2C C (3)
Dt( ) H C s ( x) 2 r ( x)
s ( x ) r ( x )
x x x 2 x
2C C C (4)
Dt( ) C d* s ( x) 2 r ( x) s ( x)
x x x x
where
1 1
( x, t ) kH ( x, t ) C ( x, t ) , r ( x) 2 x1 2 , s ( x) 2 x 2(1 ) .
To describe the dynamics of the considered process in the domain ( x, t ) : 0 x l , t 0
with permeable boundaries we complement the equations (3), (4) with boundary conditions
H (0, t ) 0, H (l , t ) 0, H ( x, 0) H 0 (5)
C (0, t ) C0 , C (l , t ) 0, C ( x, 0) 0 (6)
where H 0 is the initial value of water head, C0 is the specific value of salts concentration at the inlet
of filtration flow.
3. Numerical method for the direct problem
We numerically solve the boundary value problem (3)-(6) using a finite-difference technique
briefly described below.
In the grid domain h ( xi , t j ) : xi ih (i 0, m 1 ), t j j ( j 0, n ) where h, are the
grid steps with respect to the spatial variable and time the considered problem can be discretized using
the linearized Crank–Nicholson scheme written using the notations from [24] as [6]
x x
t( ) C 0.5d* s Cˆ xx C xx r Cˆ 0 C 0 0.5s 0 Cˆ 0 C 0
x
x x
(7)
x
x
t( ) H 0.5C s Hˆ xx H xx r Hˆ 0 H 0 0.5 s Cˆ xx C xx r Cˆ 0 C 0 x x
(8)
( ) ( )
where t u is the discrete analogue of the fractional derivative Dt u defined as
u j 1 u j j 1
u v 1 u v 1 (9)
t( j1) u v( j ) , v( j ) ( j v 1)1 ( j v)1 ,
(2 ) v 0 (2 )
and ( z ) is the Euler's gamma function.
Substituting (9) into (7), (8) we reduce the solution of the problem (3)-(6) on the ( j 1) -th time
step to the solution of the systems of linear algebraic equations
Ai j Ci j11 Si j Ci j 1 Bi j Ci j11 Fi j (i 1, m; j 0, n) , (10)
Ai j H i j 11 Si j H i j 1 Bi j H i j 11 Fi j (i 1, m; j 0, n) , (11)
C0j 1 C0 , Cmj 11 0, Ci0 0 (i 0, m 1; j 0, n) , (12)
H 0j 1 0, H mj 11 0, H i0 H 0 (i 0, m 1; j 0, n) (13)
where
0.5 si ri si j 0.5 si ri si
Ai j *
d
h h 2 4h
j
i 1 i 1 , Bi
j
d* i 1 i 1 ,
h h 2 4h
j j
Si j Ai j Bi j
,
(2 )
j 1 C v 1 Civ Ci j 0.5d* si ri
Fi j v( j ) i i 1 i i 1 Ci 1 Ci 1
2
j j j j j
C C C
v 0 (2 ) h h 2
0.5s j
i
4h 2
i1 ij1 Ci j1 Cij1 ,
0.5C si ri j 0.5C si ri j j j 1
Ai j , Bi , Si Ai Bi ,
h h 2 h h 2 (2 )
j 1 v 1
H iv Hij 0.5C si ri
H
Fi j v( j ) i
(2 )
h h
H i
j
1 2 H i
j
H i
j
1
2
H i j1 H i j1
v 0
0.5 si ri
Ci 1 2Ci Ci 1 Ci 1 2Ci Ci 1 Ci 1 Ci 1 Ci 1 Ci 1 ,
j 1 j 1 j 1 j 1 j 1
j j j j j
h h 2
si s ( xi ) , ri r ( xi ) , i j kH i j Ci j .
Equations (10)-(13) can be effectively solved by the Thomas algorithm [24].
4. Inverse problem and particle swarm optimization algorithm
Having the direct problem and numerical method stated, we further consider the inverse problem
of identification of derivatives’ orders , in the model (3)-(6) on the base of some subset of
possibly noised measurements. The inverse problem is posed the following way:
- assume there are N known values Ci , H i , i 1,..., N of concentration and water head
measured in the moments of time Ti in the depths xi ;
- assuming that the values of other model’s parameters are known, find such values , of
orders , that minimize the following goal function:
F (ˆ , ˆ ) H ( xi , Ti , ˆ , ˆ ) H i C ( xi , Ti ,ˆ , ˆ ) Ci (14)
N 2 2
i 1
where C ( x, t ,ˆ , ˆ ) , H ( x, t ,ˆ , ˆ ) are the solutions of the direct problem (3)-(6) with ˆ , ˆ in
the point x at the time t .
Taking into account the complexity of the inverse problem and the fact that the parameters to be
identified are floating-point numbers, we propose to solve it by the particle swarm optimization (PSO)
algorithm [18], which can be briefly described as follows:
- denote S as the number of particles in the swarm; xi (ˆ , ˆ ) , vi as the coordinates and
velocity of the particle i ; pi as the coordinates of the particle i that corresponds to the best goal
function (14) value obtained by it; g as the coordinates that corresponds to the best goal function
value obtained by the swarm;
- introduce the parameters of the algorithm , p , g ;
- at the initialization stage, the coordinates of the particles are generated randomly and
velocities are set to zero. The values of the goal function are calculated for each particle along with
the initial values of pi and g .
- on the iteration j for the particle i
o generate random numbers rP , rg [0,1] ;
o modify the velocity: vi( j 1) = vi( j ) p rp ( pi( j ) xi( j ) ) g rg ( g ( j ) xi( j ) ) ;
o modify particle's coordinates: xi( j 1) = xi( j ) vi( j ) ;
o calculate goal function value and modify pi and g .
- Iteration process is finished when the given maximal number of iterations I max is reached, the
zero best value of the goal function is achieved, or the difference between the best and the worst
values of the goal function for the particles in the swarm becomes lower than the given constant.
In our case, the computation of direct problem’s solutions is the most time-consuming part when
identifying parameters values. As the changes in particles’ states are independent, the corresponding
direct problems can be efficiently solved on shared-memory parallel systems.
5. Parallel implementation
The following scheme is used to implement the solution of parameters’ identification problem
on GPU.
As the solution of direct problems while calculating goal function values forms the main part of
algorithm’s computational complexity, we choose it as the only part that is executed on GPU. We
allocate a group of threads for each particle and, correspondingly, for a set of derivatives’ orders for
which the problem (3)-(6) has to be solved.
The main share of time while solving the direct problem is formed by the calculation of sums
derived from the discretization (9) of the Caputo fractional derivative because the number of terms in
them grows linearly with the increase of time step number. These sums must be calculated on each
time step for each grid node, thus, within a thread block we use thread to grid node mapping. So, on
each time step, at the first stage all threads compute the values of sums in the right parts of (10),(11)
and then the threads with local ids 0 perform the solution of (10),(11). Solution on GPU is performed
up to the moment of time max Ti without interaction with CPU. Then the solutions for the time steps
i
j : t j Ti t j 1 are transferred into the memory of CPU and the values of H ( xi , Ti ,ˆ , ˆ ) and
C ( xi , Ti , ˆ , ˆ ) are determined using linear interpolation both in space and time.
In such a computational scheme, the amount of used GPU memory is S ( m 2) max Ti / and
i
the number of running GPU threads is S ( m 2) . Assuming that the time spent on calculations on
CPU can be neglected comparing with the time spent on GPU and recalling that the Thomas
algorithm has the complexity order O(m) , the execution time of the presented algorithm is
T ( S , m, n) k m S ( m 2) / N c n (n 1) / 2 where k is the parameter of system’s performance,
N c is the number of GPU cores, n is the number of time steps.
Further, we compare GPU implementation’s performance with the performance of multi-threaded
implementation in which for the same procedure CPU core to particle mapping is used.
6. Accuracy testing results
The described solution schemes for direct and inverse problems were implemented in C++
language. The source codes are publicly available through
https://github.com/sevaboh/cons_chem_osm. To experimentally study the accuracy of the inverse
problem’s solution algorithm, we generate testing datasets as described below.
We fix the following values of physical parameters [6]: Cv 0.34 , l 25 , 0.00095 ,
0.38 , d 0.02 , k 0.01 , 2.8 105 , C0 200 , H 0 10 . With 0.01 and fixed m, , ,
the problem (3)-(6) is solved up to the time step for which Te 1 . Then, the given per cent P of grid
nodes that simulate the number of observation points are randomly selected as xi . Setting all Ti Te ,
we further set H i H ( xi , Te ,ˆ , ˆ ) r , Ci C ( xi , Te , ˆ , ˆ ) r where r is the random variable that is
uniform in 0.5 R,0.5R , R is the given noise level. The solution to the problem (3)-(6) obtained for
Te 2 and noised values in observation points for R 2 , P 0.5 , 0.6, 0.8 are depicted in
Figure 1.
Figure 1: Solutions and noised values in observation points
Several series of computational experiments were performed for m 100 and different values of
, , P, R . In the first one with fixed 0.6, 0.8 and S 10, p g 0.5 , we tested the
influence of noise level and number of observation points on the accuracy of fractional derivatives
orders restoration. The values of , were restricted to the range 0.5,1 ; P was equal to 0.75, 0.5,
0.25, and 0.1; R was equal to 1, 2, and 3. We performed 10 runs of the algorithm with 30 iterations in
each run.
The value of was in all conducted experiments restored with the maximal relative error of
0.65%. Standard deviation within the performed 10 runs was not more than 0.29% of the obtained
average values of .
The relative errors for the case of are given in Table. 1. Here the errors increase with the
increase of noise level R . For P 0.25 (more than 25 observation points) changes in the errors are
small but its decrease to 0.1 (10 observation points) up to 3 times decrease the accuracy. Standard
deviation behave the similar way being 4% of the average values for P 0.25 and becoming up to
8% for P 0.1 .
Table 1
Relative errors of the restoration of
R P 0.75 0.5 0.25 0.1
1 4.98% 1.85% 4.36% 16.66%
2 6.78% 6.86% 7.10% 26.13%
3 9.13% 8.74% 10.16% 34.16%
To determine the influence of initial values of and on the accuracy of their restoration we
change in the range [0.55, 0.95] for the fixed 0.75 and vice versa change in the same range
for 0.75 . The obtained relative errors are given for P 0.5, R 2 in Tables 2 and 3. In these
computational experiments, was as in the previous one restored with rather high accuracy (relative
error not more than 0.6% in all cases). The error of restoration being not more than 8.43% was
higher when tends to the edges of the range.
The last series of experiments with variable Te showed that better accuracy of parameters
identification can be achieved for higher values of Te : relative error of restoration with R 2 ,
P 0.5 lowered from ~7% for 0.1 Te 1 to ~1.5% for Te 2 .
Table 2
Relative errors e , e of and restoration for the fixed 0.75
e e
0.55 6.96% 0.30%
0.6 6.67% 0.28%
0.7 3.26% 0.25%
0.8 0.94% 0.32%
0.95 8.43% 0.60%
Table 3
Relative errors e , e of and restoration for the fixed 0.75
e e
0.55 0.48% 0.18%
0.6 0.23% 0.16%
0.7 4.73% 0.22%
0.8 0.96% 0.43%
0.95 5.52% 0.55%
7. Solution time testing results
GPU implementation of the PSO algorithm was performed using the OpenCL framework while
multi-threaded implementation used OpenMP.
Time spent on parameters' identification using GPU and multi-threaded implementations was
measured solving the problem with 0.6, 0.8, P 0.5, R 2 on one node of SCIT-4 cluster of
VM Glushkov Institute of cybernetics of NAS of Ukraine (NVidia RTX 2080 Ti GPU, 2 Intel(R)
Xeon(R) Bronze 3104 CPUs, 1 CPU with 6 cores was allocated for OpenMP).
During the tests we changed the number m of nodes in the finite-difference grid and the number
S of particles in PSO swarm. Average solution time among 10 runs is given in Table 4. Regarding
the accuracy, it increased with the increase of the number of observation points as m increases with
fixed P reaching the maximal relative error of 0.24% for m 200 .
Table 4
Average solution time, ms
m 50 m 100 m 200
S GPU CPU GPU CPU GPU CPU
10 10524 11819 19944 23586 38791 46907
20 10539 23162 20010 46407 38930 92550
30 10590 29163 20082 58477 39008 116521
40 10652 40834 20089 80818 39160 162622
As can be seen from the data in Table 4, execution time of GPU algorithm does not depend on the
number of particles and linearly increases with the increase of grid size. This can be due to the non-
complete allocation of GPU’s computational resources. On the other hand, CPU execution time
linearly increases both with the increase of m and S that proves the efficiency of GPU
implementation, which in the conducted experiments gave up to 4-times decrease of solution time.
The efficiency of GPU implementation slightly increased with the increase of m : average solution
time per one grid node per particle decreased from 21.1 ms for m 50 to 19.4 ms for m 200 . The
corresponding indicator for CPU implementation did not change significantly with the change of m
but decreased from 23.6 ms for S 10 to 20.4 ms for S 40 . Such behavior reflects the general
heuristic that parallel implementation’s efficiency increase with the increase of the volume of
computations.
The series of computational experiments conducted for S 10 , N 100 with variable
Te 0.5,1, 2,3, 4 confirmed quadratic dependency of execution time on the number of time steps
(Figure 2). GPU implementation here allowed obtaining slower increase of solution time.
Figure 2: Execution time of GPU and CPU implementations subject to the ending time of simulation
8. Conclusions
In the paper we presented the computational scheme for identifying the values of fractional
derivatives orders in the fractional-fractal model of filtration-consolidation based on the available
measurements. The scheme is based on the particle swarm optimization technique and is implemented
in multi-threaded mode and for the execution on GPU.
The results of computational experiments show that the order of the local fractal derivative with
respect to the spatial variable can be restored with significantly higher accuracy than the order of non-
local time-fractional derivative. Expectedly, relative errors increase with the increase of noise level.
Considering the number of observation points, errors less than ~10% were obtained in all cases for
their number more or equal to 25.
OpenCL GPU implementation of the inverse problem’s solution algorithm performed closely to
the OpenMP CPU implementation that used 6 CPU cores on the minimal tested volume of
computations ( m 50 , S 10 ). With the increase of the volume, GPU overperformed CPU by up to
4 times.
Thus, the proposed computational scheme can be effectively applied with regard to both accuracy
and speed to adapt the fractional-fractal model to the observed dynamics of salt transfer and
compaction of water-saturated soils in complex geological and hydrological conditions.
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