An Approach to the Precision Parametric Reduction of Mathematical Models
The paper considers and analyzes an approach to simplification of mathematical models in order to substantiate possibility of accounting for limited computational resources needed to implement the models. Practical methods are proposed for application and evaluation of possible techniques of models’...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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| Zitieren: | An Approach to the Precision Parametric Reduction of Mathematical Models / A.A. Verlan // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 26-35. — Бібліогр.: 7 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1337502025-02-09T22:36:05Z An Approach to the Precision Parametric Reduction of Mathematical Models Verlan, A.A. The paper considers and analyzes an approach to simplification of mathematical models in order to substantiate possibility of accounting for limited computational resources needed to implement the models. Practical methods are proposed for application and evaluation of possible techniques of models’ parametric reduction for considered problem with obtainment of applicability criterion of the given approach. Розглядається і аналізується підхід до спрощення математичних моделей з метою обґрунтування можливості обліку обмеженості ресурсів комп'ютерних засобів, що реалізують моделі. Запропоновано конструктивні способи використання і оцінки можливих прийомів параметричної редукції моделей в розглянутій задачі з отриманням критерію застосовності даного підходу. 2016 Article An Approach to the Precision Parametric Reduction of Mathematical Models / A.A. Verlan // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 26-35. — Бібліогр.: 7 назв. — англ. 2308-5916 https://nasplib.isofts.kiev.ua/handle/123456789/133750 004.942, 681.518 en Математичне та комп'ютерне моделювання. Серія: Технічні науки application/pdf Інститут кібернетики ім. В.М. Глушкова НАН України |
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The paper considers and analyzes an approach to simplification of mathematical models in order to substantiate possibility of accounting for limited computational resources needed to implement the models. Practical methods are proposed for application and evaluation of possible techniques of models’ parametric reduction for considered problem with obtainment of applicability criterion of the given approach. |
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Verlan, A.A. An Approach to the Precision Parametric Reduction of Mathematical Models Математичне та комп'ютерне моделювання. Серія: Технічні науки |
| author_facet |
Verlan, A.A. |
| author_sort |
Verlan, A.A. |
| title |
An Approach to the Precision Parametric Reduction of Mathematical Models |
| title_short |
An Approach to the Precision Parametric Reduction of Mathematical Models |
| title_full |
An Approach to the Precision Parametric Reduction of Mathematical Models |
| title_fullStr |
An Approach to the Precision Parametric Reduction of Mathematical Models |
| title_full_unstemmed |
An Approach to the Precision Parametric Reduction of Mathematical Models |
| title_sort |
approach to the precision parametric reduction of mathematical models |
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Інститут кібернетики ім. В.М. Глушкова НАН України |
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2016 |
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| citation_txt |
An Approach to the Precision Parametric Reduction of Mathematical Models / A.A. Verlan // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 26-35. — Бібліогр.: 7 назв. — англ. |
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Математичне та комп'ютерне моделювання. Серія: Технічні науки |
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AT verlanaa anapproachtotheprecisionparametricreductionofmathematicalmodels AT verlanaa approachtotheprecisionparametricreductionofmathematicalmodels |
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| fulltext |
Математичне та комп’ютерне моделювання
26
4. Федорчук В. А. Дослідження динаміки нестаціонарних теплових процесів із
симетричними граничними умовами методом перерізів / В. А. Федорчук,
О. І. Махович // Математичне та комп’ютерне моделювання. Серія : технічні
науки : зб. наук. праць / Ін-т кібернетики ім. В. М. Глушкова НАН України,
Кам’янець-Подільський нац. ун-т ім. Івана Огієнка ; [редкол. : Ю. Г. Криво-
нос (відп. ред.) та ін.]. — Кам’янець-Подільський : Кам’янець-Подільський
нац. ун-т ім. Івана Огієнка, 2014. — Вип. 10. — С. 182–191.
The efficiency of numerical realization of mathematical models by the
method of supporting cuts in comparison to the widely used traditional
method of complete differences is being probed in calculable experiments.
The time indexes of the processing of the use of different methods of find-
ing of decisions are being analyzed and the recommendations to the effec-
tive use of the method of supporting cuts are formulated.
Key words: efficiency of numerical realization, time of processing, dif-
ference scheme, finite difference method.
Отримано: 16.09.2016
UDС 004.942, 681.518
A. A. Verlan, Ph. D., Associate Professor
Norwegian University of Science and Technology, Gjovik, Norway
AN APPROACH TO THE PRECISION PARAMETRIC
REDUCTION OF MATHEMATICAL MODELS
The paper considers and analyzes an approach to simplification
of mathematical models in order to substantiate possibility of ac-
counting for limited computational resources needed to implement
the models. Practical methods are proposed for application and
evaluation of possible techniques of models’ parametric reduction
for considered problem with obtainment of applicability criterion
of the given approach.
Key words: parametric reduction, evaluation of approxima-
tions, criterion of applicability.
Introduction. The more complex the objects and phenomena being
studied are, the higher is computational complexity of the models used.
Degree of complexity of mathematical descriptions of dynamics of the
objects in question goes hand in hand with requirements for computational
resources needed to run the models. Basic approach to assurance of ac-
counting for limitation of these resources consists in use of simpler
mathematical models instead of complex ones, yet preserving informative
value of the results of modeling, i.e., in certain sense preserving adequacy
© A. A. Verlan, 2016
Серія: Технічні науки. Випуск 14
27
to the object being modeled. Just the meaning of conformity of the simpli-
fied model content with the primary one characterizes great variety of ap-
proaches to simplification of complex models.
There is a long history of the question of simplification or reduction
of mathematical descriptions of objects or phenomena being studied, and it
is closely related to the identification problem [1–3], so that various tech-
niques used for identification have been successfully applied to develop-
ment of simplified descriptions of dynamic objects starting with given
complex one. The most important of these techniques are: reduction of the
equation system order, separation of motion into «fast» and «slow», re-
placement of operators, reducing the number of independent variables,
linearization of original non-linear equations, neglecting the influence of
certain factors.
Setting up the problem of models’ precision parametric reduction
of. Most commonly the problem of models’ simplification is posed as a
problem of minimizing a certain measure of deviation of the primary and
simplified models’ output for given simplification pattern. At the same time
and as a rule, it is assumed that parameters of the primary model are known
with certainty. It is clear, that in practice such condition does not exist, there-
fore the issue of mathematical model simplification can be posed as a ques-
tion of matching model type with precise characteristics of the input data.
Intuitively it is obvious that it makes no sense to use complex structure
models if their parameters are known with comparatively great inaccuracy.
This circumstance preconditions the need for adjustment of such characteris-
tics of model as complexity and accuracy of the input data. Despite the fact,
that this statement of the problem of model simplification has long been
known while it is derived from approximate calculations technique, defini-
tion of practical approaches to its solution is of great importance for study of
complex objects and, in particular, for development of algorithms and soft-
ware for various control and computation systems [4, 5].
It is common to consider modeling accuracy as a basic quality indica-
tor of computer modeling systems. Another indicator, especially for real-
time systems, is processing speed required. It is clear, that simplification
of models of objects being studied makes it possible to reduce required
processing speed, often at the cost of degraded simulation fidelity. Resolu-
tion of the contradiction between tendencies of increasing modeling fidel-
ity and decrease in required processing speed can be achieved just on the
base of matching mathematical description of given object with accuracy
characteristics of input data.
Sufficiently generalized and practical posing of the problem of
mathematical model simplification based on the concept of matching
model type with accuracy of input data was proposed by A. N. Tik-
Математичне та комп’ютерне моделювання
28
honov [6, 3], who defined it as a problem of minimizing complexity func-
tional of the class of models, formally comparable in accuracy with obser-
vations (input data). The class of formally comparable models is defined
by constraints on inequation type for precision measure.
Within this general statement of the problem of mathematical model
simplification, series of specific statements can be defined, one of which,
namely, precision parametric reduction of models, and approaches to its
solution are discussed herein.
Suppose there is some mathematical description (model) M of the
simulated phenomena, and the input data (usually numerical parameter
values) are given with certain inaccuracy, a measure of which is . Ob-
viously, if , then original mathematical description M may be
replaced with another description M ' (even with simpler one) provided
, where yp is a measure of error of output variables of the pri-
mary model, caused by inaccuracy of the input data, i.e. the presence of
. Let's assume that 0 , then replacing the original description with
another (including simpler) one will originate error 0ym . In this case,
the question of applicability of the simplified model should be resolved on
the basis of analysis of requirements for accuracy of simulation of the
phenomenon being studied. In other words, use of simplified model is ac-
ceptable if yM з , where з is a given value of modeling error meas-
ure, and vice versa. If inequation yM з is strong enough, it suggests
that primary model can be simplified even more.
In the general case, applicability of mathematical model M or M
is determined by condition
yM yp з . (1)
Here yM is either a measure of error for description of the phenomenon be-
ing studied with model M , or a measure of error conditioned by replacement
of given model M with another one, i.e. M (including a simpler one).
Since this paper addresses the issue of mathematical model simplifi-
cation, hereinafter yM will stand just for measure of error due to re-
placement of the initially given model with a simpler one.
As a characteristic property of consistency with error in the input data of
the mathematical model М', that replaces the model M , we shall accept
/yM yp (2)
which we will call concordance coefficient. Herewith applicability condi-
tion (1) of the model М' takes the form
Серія: Технічні науки. Випуск 14
29
3( 1) yp . (3)
Definition 1. Mathematical model М' is called - concordant with
error of the input data p of the model M , if yM yp and 0 .
The problem of concordance of the model M with accuracy of the
input data means obtaining model M with a certain concordance coeffi-
cient. Whereby =0.01 can be accepted as a lower limit, since further
reduction of concordance coefficient, meaning the use of models, which
differ little from the primary one, would not lead to significant gain in the
accuracy of modeling. On the other hand, if 10 , the overall modeling
error will be almost entirely determined by yM , in which case it is practi-
cally impossible to obtain reasonable values of yM while modeling com-
plex objects. Thus, it is practical to constrain maximum concordance coef-
ficient of mathematical models with the interval (0.01 10).
While addressing specific problems, maximum value of concordance
coefficient is determined by the applicability condition (3) as follows
max 3 / 1.yp (4)
Now let MN be computational complexity measure of mathematical
model M (for instance, for dynamic objects described by ordinary differ-
ential equations in normal form, number of operations reduced to the same
type and required to compute the right side can be accepted as MN ), and
be a set of mathematical models, concordance coefficient of which
with input data error does not exceed .
If max , then all the models from are equivalent in terms of
applicability, while applicability condition for any model is fulfilled (3). At
the same time, models from may not be equivalent in terms of computa-
tional complexity. Just under this condition it is possible to simplify given
model M by replacing it with model M , for which M MN N .
Definition 2. Mathematical model 0M is called optimally simpli-
fied, if
0
min MM M
N N
(5)
Obviously, the measure of computational complexity MN is a de-
creasing function of concordance coefficient. Consequently, concordance
coefficient of optimally simplified model 0M exceeds .
Development of -concordant, as well as optimally simplified mod-
els pursues dual aim. On the one hand, it is a substantiation of certain as-
Математичне та комп’ютерне моделювання
30
sumptions allowing us to simplify the model, on the other hand, it is re-
duction of requirements to processing speed of computer modeling sys-
tems. Yet the real-world solution of the problem in hand faces consider-
able difficulties, conditioned mainly by complexity of the problem of es-
timation of values yM , yp , as well as by difficulties in formal charac-
terization of the set .
Parametric approach to reduction of models. The realizable way
to overcome the difficulties is to use parametric approach to development
of simplified models. Its essence consists in replacement of certain pa-
rameter values of the original model M with such other values, which
reduce the measure of computational complexity. Usually it is replacement
of given values of some parameters of the initial model with zero values
(both for additive and multiplicative parameters) and/or with single values
(for multiplicative parameters). Such parametric simplification is helpful if
mathematical models of the processes or phenomena being studied contain
sufficiently great number of parameters, values of which are determined
from experimental data. These models may include models in the form of
linear or nonlinear algebraic equation systems of higher dimension, differ-
ential equation systems (both ordinary and partial) with coefficients which
are composite functions of input and output variables, integral equations
with kernels, form of which is determined by actual characteristics of the
objects being studied etc.
Theoretical substantiation of possibility of parametric simplification
of mathematical models for development of -concordant and optimized
simplified models is based on the results of the precision theory in relation
to inherent error analysis.
Let ( , , )M y V be mathematical model; , ,y V — vectors of output,
input variables and parameters, respectively, that belong to normed spaces
with metrics , ,y V p , that are determined by relevant norms; pN is
computational complexity measure [7].
Let 0 be an unknown vector of precise parameter values of the
model and 3 be a vector of given parameter values. It is known that
3 0( , )p p . (6)
Condition (6) determines set p , and from p follows
3( , )p p . Thus we defined the set of possible models
3( , , ) : , ( , )p pM y V ,
and model ( , , )M y V appears to be primary mathematical description.
Серія: Технічні науки. Випуск 14
31
Suppose further that 3( , )y is measure of error in the space of
output variables, conditioned by inaccuracy of assignment of the vector
3 . 3( , )y is measure of error, conditioned by replacement of given
model 3( , , )M y V with another one, particularly with a simpler model
( , , )rM y V ; 3( , )y — measure of error that characterizes distinction
between the exact model 0( , , )M y V and the model ( , , )rM y V .
Following estimates can be determined for the above errors
3( , ) / , ,y ypV (7)
3( , ) / ,y r yMV v (8)
0( , ) / .y r yrV v (9)
( v is the range of possible values of input variables), and the triangle
inequality defines their relationship
( 1)yr yM yp yp . (10)
yr yM If r , then yM yr and, consequently, the set of
possible models is a set of - concordant models for =1, i.e. 1 .
However, application of parametric approach to the search for simplified
models in the set has limited possibilities. Indeed, usually the set
does not contain points that belong to coordinate axes, and therefore
with this model simplification one can only count on reduction in the
number of significant digits in representation of given parameters. Note
that such simplification finds use in the art of approximate calculations
and can be used also in organization of computational processes in model-
ing systems on the step of computing tools selection.
Generally speaking, if r , then yM yr , and therefore con-
cordance coefficient of appropriate model will be greater than 1. Neverthe-
less, there is a possibility in principle to determine the set of
-concordant models for 1 and r . This possibility is condi-
tioned by the fact that error in the output variables space, caused by inac-
curacy of vector 3 assignment, is estimated in the metric y .
Indeterminancy of values of the primary model parameter vector
(presence of ) leads to indeterminancy of values of the output vari-
ables, quantitative measure of which is error
3( , ) ( , ) / Vy y V y V V . (11)
The range y of possible values of error y has complex configu-
ration in the space of output variables, that depends on the type of ,
Математичне та комп’ютерне моделювання
32
properties of mathematical description ( , , )M y V and is defined by the
inherent error equation. Detailed characteristic of y with techniques
used to analyze inherent error that were developed in precision theory, can
be obtained only for the simplest problems. More often, however, the
range to which the error can be attributed, is characterized by a sphere S
(in the given metric y ), radius of which (in this particular case yp ) is
determined considering the upper estimate (7) for the norm corresponding
to metric y . Due to coarseness of these estimates, S y is the case.
The sphere S in the space of output variables is matched with certain re-
gion in the space of parameters, where and
follows 3( , ) ypy . Consequently, if r , then yM yp and
set of possible models that corresponds to the set is a set of con-
sistent models for = 1.
Due to inclusion of , application of parametric approach to sim-
plification of the original description in the model class has a great
potential as compared to the class . It can be attributed to the fact that
for complex problems with a great number of parameters defined with
less-than-high accuracy, the range contains points that belong to the
coordinate axes of the parameter space. Consequently, vector with some
zero components can be selected as a vector r , and thus decrease in
value of computational complexity measure prN can be achieved.
In the general case, may not meet with the coordinate axes,
however even under these conditions, as it is not hard to show, there is a
set 3( ) : ( , ) , 1y yp that includes points of the co-
ordinate axes and defines a set of models , consistent with the input
data error with consistency coefficient 1. Indeed, assume
3( ) : ( , )p p . It is clear that (1)p p , and there is
such a value 0 1 , that for 0 the sphere ( )p meets at least one
of the coordinate axes of the parameter space. Set ( )p defines a range
of possible values of the output variable errors ( )y , which is esti-
mated in the metric y by the sphere S with radius ( ) yp , i.e.
3( , ) / ( ) ( )y yp . (12)
Серія: Технічні науки. Випуск 14
33
Here, ( ) is non-negative non-decreasing function, and (1) = 1,
we assume that function ( ) is increasing and confined one. The first
assumption means that error of output variables increases if error of the
input data increases (for practical tasks we can consider this condition as
satisfied). The second condition means correctness of all the models with
parameters from ( )p , which is a limitation for applicability of the pa-
rametric approach to model simplification.
Now let 3( ) : ( , ) ( )y yp be prototype in the
space of parameters of the sphere S . While ( ) ( ) , the set
( ) also intersects with at least one of the coordinate axes, as was to
be demonstrated. Thus, the following theorem is proved.
Theorem 1. For any 3: ( , ) , 0p p p there is such
0 0 , that for 0 in the set . of -concordant with the error of
input data well-posed models ( , , )M y V , defined by a set of parameters
3( ) : ( , )y yp , there is a non-empty subset of mod-
els ( , , )M y V for which /p pN N .
This theorem is a theoretical substantiation for possibility of applica-
tion of parametric approach to development of simplified mathematical
models based on concordance of the model type with accuracy of the input
data. The fact (proven by the theorem) of existence of -concordant
models with a lower measure of computational complexity than the pri-
mary model is the main point of the principle of precision parametric re-
duction of mathematical models that can be stated as follows.
For any model with inaccurately defined parameters, provided error
of the output variables is an increasing and confined function of the pa-
rameter error, there is a simpler -concordant model that differs from the
primary model by values of some parameters that reduce computational
complexity measure.
Of course, applicability of parametrically simplified models resulting
from precision parametric reduction is defined by condition (3), hence
obtaining of simpler mathematical descriptions than original ones with
moderate value of concordance coefficient is of practical interest. That's
why it is desirable to be able to evaluate concordance coefficient of para-
metrically simplified models a priori, with fairly simple means.
Evaluation of inherent error within the linear precision theory for a broad
class of problems (mathematical descriptions of which represent models that
can be simplified) leads to a linear dependence of yp from p , i.e.
Математичне та комп’ютерне моделювання
34
yp M pC , (13)
where MC — constant determined by character of the problem (for well-posed
problems it is norm of inverse operator of inherent error linear equations).
In this case, it is easy to get a dependable upper bound for value 0
of concordance coefficient of the models, which can be obtained as a re-
sult of the precision parametric reduction.
Theorem 2. For the minimum value 0 of concordance coefficient
of the parametrically simplified model ( , , )M y V the following estimate
is valid:
3
1,
0
min ( , )p
i m
b
i
, (14)
where , 1,i i m — coordinate axes in the parameter space.
Proof. Condition (13) implies that function ( ) has the form of
( ) . Value 0 can be defined from condition that the sphere
( ) with radius 0 p touches the nearest of the coordinate axes of the
parameter space , 1,i i m , i.e.
3
1,
0
min ( , )p
i m
i
From inclusion 0 0( ( )) ( ) follows 0 0 . QED.
As a matter of fact, estimate (14) is criterion of applicability of the
precision parametric reduction of mathematical models. Thus, higher val-
ues of b (on the order of tens) demonstrate impracticality of parametric
model simplification and, on the contrary, small values of b (on the or-
der of unities) evidence feasibility of application of the precision paramet-
ric reduction for development of acceptable simplified models.
Parametric model simplification under discussion covers also the
case with linearization of complex functional relationships. Herewith the
use of linear terms means «zeroing» of coefficients for all other terms of
the expansion.
Furthermore, it is possible to use the principle of precision parametric
reduction to assess insignificance of separate fragments of the model by
introducing fictitious multiplicative parameters (nominally equal to unity)
for the fragments under consideration with subsequent evaluation of insig-
nificance of these parameters, i.e. with assessment of possibility of re-
placement of their nominal values with zero.
Серія: Технічні науки. Випуск 14
35
Conclusions. Based on the concept of mathematical model simplifi-
cation, the principle and elements of the theory of the precision parametric
reduction of mathematical descriptions of dynamic objects are exposed as
one of the ways of accounting for computer means’ speed requirements.
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Розглядається і аналізується підхід до спрощення математичних
моделей з метою обґрунтування можливості обліку обмеженості ре-
сурсів комп'ютерних засобів, що реалізують моделі. Запропоновано
конструктивні способи використання і оцінки можливих прийомів па-
раметричної редукції моделей в розглянутій задачі з отриманням кри-
терію застосовності даного підходу.
Ключові слова: параметрична редукція, оцінка наближень, кри-
терій застосовності.
Отримано: 12.08.2016
<<
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken waarmee zakelijke documenten betrouwbaar kunnen worden weergegeven en afgedrukt. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AllowImageBreaks true
/AllowTableBreaks true
/ExpandPage false
/HonorBaseURL true
/HonorRolloverEffect false
/IgnoreHTMLPageBreaks false
/IncludeHeaderFooter false
/MarginOffset [
0
0
0
0
]
/MetadataAuthor ()
/MetadataKeywords ()
/MetadataSubject ()
/MetadataTitle ()
/MetricPageSize [
0
0
]
/MetricUnit /inch
/MobileCompatible 0
/Namespace [
(Adobe)
(GoLive)
(8.0)
]
/OpenZoomToHTMLFontSize false
/PageOrientation /Portrait
/RemoveBackground false
/ShrinkContent true
/TreatColorsAs /MainMonitorColors
/UseEmbeddedProfiles false
/UseHTMLTitleAsMetadata true
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/BleedOffset [
0
0
0
0
]
/ConvertColors /ConvertToRGB
/DestinationProfileName (sRGB IEC61966-2.1)
/DestinationProfileSelector /UseName
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements true
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MarksOffset 6
/MarksWeight 0.250000
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PageMarksFile /RomanDefault
/PreserveEditing true
/UntaggedCMYKHandling /UseDocumentProfile
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [600 600]
/PageSize [419.528 595.276]
>> setpagedevice
|