Numerical Analysis of Processes

1
Numerical Analysis of Processes
NAP2
Physical models (transport equations, energy and
entropy principles). Empirical models (neural
networks and regression models). Models that use
analytical solutions (diffusion). Identification
of models based on the best agreement with
experiment (regression analysis, optimization)
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
2
Mathematical MODELs
NAP2
Mathematical model mathematical description
(program) of system characteristics (distribution
of temperature, velocity, pressure,
concentration, performance, ...), as a function
of time t, space and operating parameters
(geometry system, material properties, flow, ...).
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
3
Mathematical MODELs
NAP2
General classification of models

• Black box - a model based solely on a comparison
  with experiment. Three basic types are
• Neural networks (analogous neurons in the brain -
  the method of artificial intelligence)
• Regression models (e.g. in the form of
  correlation functions of the power law type Nu
  cRenPrm)
• Identification of the transfer function of a
  system E(t) from the measured time history of
  input x(t) and output y (t) of the system.
• White box - the model is based entirely on sound
  physical principles
• the balance of mass, momentum, energy (Fourier
  and Fick's equation, Navier Stokes equations)
• on energy principles, the desired solution
  minimizes the total energy of the system
  (elasticity),
• on simulations of random movements and
  interactions of fictitious particles (Monte
  Carlo, lattice Boltzmann ...).-
• Grey box - models, which is located between the
  above extremes. Examples are compartment models
  of flow systems, respecting the physical
  principle of conservation of mass, but other
  principles (eg balance of momentum), they are
  replaced by empirical correlations or data.

Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
4
Physical Principles
NAP2
Continuum models (white boxes)
Engineering models based on physical principles
are often based on three balances (mass, momentum
and energy). In differential form Moss
conservation Momentum conservation Energy
conservation
Cauchy equilibrium equations. Valid for
structural as well as fluid flow analysis,
compressible and incompressible.
The total energy, the sum of internal, kinetic
and potential energy
power of inner forces
heat flux
Most of the equations, you know, can be derived
from these conservation equations Bernoulli,
Euler, Navier Stokes, Fourier Kirchhoff.
5
Physical Principles
NAP2
Continuum models (white boxes)
The principles of energy are preferred especially
in the mechanics of solid phase.In the theory of
elasticity it is the Lagrange principle of
minimum of potential energy
strains and stresses can be expressed in term of
displacement u
deformation energy of internal forces (the
product of the stress and strain tensor
work of external forces (volumetric, surface,
concentrated)
at equilibrium the energy (WiWe) is minimum
Requirement of zero variation of energy
functional ?W/?u0 is the basis of many finite
elements design. This approach is represented in
dynamics by Hamilton principle of zero variation
of functional with kinetic energy
"Energy" principles are not so general each of
the variational principle can derive an
equivalent Euler Lagrange differential equation,
but the converse is not true. Especially with
flow the variational principles can be applied
only in some cases (eg creeping flow).
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Physical principles
NAP2
Models determine the type of engineers who are
engaged in addressing these issuesFUNCTIONALIST
S imagine that each system can have infinitely
many forms, and a number, functional can be
assigned to each of them. Then they plot these
numbers and look for some special points (minima,
maxima, inflection). A special mystical
significance is assigned to these characteristic
points, perhaps, the state of balance, loss of
stability, etc. Functionalists work especially in
mechanics of solids and are easily identified by
the fact that they appretiate only the finite
element method (through gritted teeth perhaps
Boundary Elements). They use words as Cauchy
Green tensor, Kirchhoff - Piola stresses, the
seventh-invariant, etc.DIFFERENTIALs believe
that the law of conservation of mass, momentum
and energy expressed in terms of differential
equations can describe the state of a system in
the infinitesimal neighborhood of any point (and
consequently the whole system too). When choosing
numerical methods the differentials are not too
picky, but generally choose the method of finite
volumes. Often use words like material
derivative.PARTICULARISTs do not believe in
continuum mechanics. Everything can be derived
from the mechanics of a particle. There are only
discrete quantities, such as in digital
computers. The complexity of reality is caused by
a large number of simple processes running in
parallel. Particularists can be identified by the
fact that shuns all others and are
discreet.FATALISTs do not believe that it is
possible for a human beeing to understand the
laws of God providence and therefore are limited
to the empirical description of the observed
phenomena. They love expert systems, artificial
intelligence methods, engineering correlations
and statistics.CHAOTs admit existence of some
principles, but do not believe in their
meaningful solvability. They hate the smooth
curves, and prefer to be spoiled by random number
generators. They love disasters, catastrophs and
attractors. They are attractive especially for
women fascinated by foreign words and colorful
flowers of fractals.
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Example of models Drying
NAP2
As an example we will use different mathematical
models of drying or moistening of the material,
such as grain (starch, corn, rice, coffee ...) in
the whole course. There will always be one of
the basic results time of drying (or relationship
between time, temperature and moisture of dried
material). For this purpose simple regression
models or neural networks are usually
sufficient. The distribution of moisture within
the grain must be known (calculated) as soon as
the microbial activity or healt risc is to be
determined. White or gray boxes models based on
transport equations and heat diffusion can be
used. When the grain has a simple geometry
(ball), and when there is no significant effect
of moisture on the diffusion coefficient, the
solution can be expressed in the form of infinite
series (analytical solution of Fick's equation
and FK, only few terms in these series are
sufficient). When the geometry is simple, but
diffusion and heat transfer coefficients are
strongly non-linear, the 1D numerical methods are
preferred (finite difference method or the
spectral method). The complicated shape of grains
(rice, beans) are evaluated by 3D numerical
finite element or control volume (commercial
software ANSYS, COMSOL and FLUENT are usually
used). At present, the concern is focused to
changing internal structure of a material, such
as cracking, which in turn significantly affect
the moisture (free water easily penetrates micro
grain). This means that in addition to transport
equations the deformation field and the
distribution of mechanical stress tensor (elastic
problem) must be solved. These models are
generally based on the finite element method.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
8
Neuronové síte
NAP2
Artificial neural networks are the brutal
techniques of artificial intelligence. Its
popularity stems from the fact that more and more
people know less and less about the real
processes and their nature (maybe it could be
called the law of exponential growth of
ignorance). ANN (Artificial Neural Network) is
designed specifically for those who knows nothing
about the processes they want to model. They just
have a lot of experimental data in which they are
unable to navigate. And have a MATLAB (for
example).
Schiele
9
Neural Networks
NAP2
Neuronmodule that calculates one output value
from several input. The behavior of a particular
neural network is given by the values of the
synaptic weights wi (coefficients amplification
of signals between interconnected neurons)
Hidden layer (here 4 neurons)
Input layer (2 neurons) X1,X2
Outlet layer (1 neuron) Y
Neurons response y to N-inputs xi
Wicoefficients of synaptic weights evaluate by
special algorithm learning network
The most frequently used activation functions f
(tangens hyperbolic, sigmoidal function, sign
function sgn). All implemented e.g. in MATLABu.
Nejcasteji používané aktivacní funkce f (tangens
hyperbolický, sigmoidální funkce, znaménková
funkce)
Nejcasteji používané aktivacní funkce f (tangens
hyperbolický, sigmoidální funkce, znaménková
funkce)
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Neural networks
NAP2
Modeling of wheat soaking using two artificial
neural networks (MLP and RBF)  Journal of Food
Engineering, Volume 91, Issue 4, April 2009,
Pages 602-607M. Kashaninejad, A.A. Dehghani, M.
Kashiri
In this article you will read how the experiments
on cereal grains humidification were evaluated.
Humidificated grains were in distilled water at
temperatures of 25, 35, 45, 55 and 65 0C for
about 15 hours (samples were weighed at 15 minute
intervals), the total available 154 values of
specific humidity of grain for various
temperatures and times. From these values only
the 99 data were used for training the network,
which had two neurons in the input layer (time,
temperature), 26 neurons in the hidden layer and
a single output neuron (humidity). Remaining 55
data (moisture) was used to verify that the
trained" neural network gives reasonable results
and what is about fault prediction. They used two
types of network MLP (Multi Layer Perceptron) and
RBF (Radial Base Functions) with different
activation functions of neurons (sigmoidal,
respectively. Gaussian basis function).
This is prediction of neural network (ANN).
MR-Moisture Ratio as a function of time and
temperature.
MLP is a classical neural network. Neuron
activation function (hyperbolic tangent ... see
previous film) with no adjustable parameters,
optimize only the weighting coefficients wij
connecting neuron with neuron j (and used the
same method as described in further regression
models). Radial basis function RBF neurons have
their own adjustable parameters - coordinates of
the neuron (determines the "distance" from the
neuron of the previous layer neurons) and "width"
basis functions. RBF is the Gaussian
function RBF networks have only one hidden layer
and weighting coefficients wij can be adjusted
between the hidden and output layer. Parameters
of the "radial" neurons (xc, ? c) is selected "ad
hoc" to the nature of the problem, while using
the statistical strategy of "cluster analysis".
It is more complicated than the MLP and the
result (at least for drying) tends to be worse.
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Regression models
NAP2
Delvaux
12
Regression models
NAP2
The regression model has the form of a relatively
simple function of the independent variable x,
and the parameters p1, p2,,pM, which is to be
determined so that the values of the function
best match the experimental values y. . Frankly
neural networks are almost the same. The search
parameters p1, p2,,pM are the coefficients of
synaptic weights linking neurons. The difference
is that the type of the model functions is more
or less unified in the ANN and the number of
weights is greater than the number of parameters
commonly used in regression functions. Regression
function is chosen on the basis of experience or
simplified ideas of modeled process (reasonable
and logically explainable behavior at very small
or large values of the independent variables is
to be required). It is also desirable that the
parameters p1, p2,,pM have clearly defined
physical meaning. However, it is true that if we
do not know a physical nature of the process or
if it is too complex (ie, the same situation as
in neural networks) the polynomial regression
function y p1p2xpMxM-1 or another neutral
function is used.
13
Regression models
NAP2
Let us consider a regression model with the
unknown vector p of M parameters
The parameters pi should be calculated so that
the model prediction best fits the N measured
points (xi,yi). The most frequently used
criterion of fit is chi-square (sum of squares of
deviations)
Number of data points N should be greater than
the number of calculated parameters M (NM means
interpolation)
The quantity ?i is standard deviation of measured
quantity yi (measurement errror). Regression
looks for minimum of the function ?2 in the
parametric space p1, p2,,pM (sometimes more
robust criteria of fit, for example the sum of
absolute values of deviations, are used). Quality
of a selected model is evaluated by the so called
correlation index, which should be close to unity
for good models
The worst value r0 corresponds to the case when
it would be better to use a constant as a
regression model )
14
Linear regression
NAP2
Linear regression model is a linear combination
of selected base functions, for example
polynomials gm(x)xm-1 or goniometric functions
gm(x)sin(mx)
Base functions are analogy of activation
functions of neurons and the parameters pm
correspond to synaptic weights.
Model prediction can be expressed in a matrix form
A is design matrix with N rows corresponding
to N/points and M columns for M-base functions.
NgtM, the case of square matrix NM is not
regression but interpolation.
As soon as the standard deviation is constant,
the ?2 value is proportional to the sum of
squares s2 of deviation between prediction
ypredikce and measured data y
This is scalar product of two vectors
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Linear regression
NAP2
Zero first derivatives of s2 with respect of all
model parameters exist at minimum
which is the system of M linear algebraic
equations for M unknown parameters
Transposed matrix AT having dimension MxN
multiplied by vector y Nx1 gives a vector of
M-values
Transposed matrix AT has dimension MxN and
when multiplied by A gives square matrix with
the dimension MxM CMxM
Matrix C enables estimate of reliability
interval of calculated parameters p1,,pM
?pk is standard deviation of calculated parameter
pk for k1,2,,M
?y jis standard deviation of measured data (it is
assumed that all data are measured with the same
accuracy)
Inversion of matrix C (inverted matrix is
called covariance matrix)
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Linear regression example noise filtration
NAP2
Principle of Savitzky Golay filter of noised data
is very simple each point of input data xi,yi is
associated with the window of Nw points left and
Nw points right and these 2Nw1 points is
approximated by regression polynomial of degree
k, where klt2Nw. Value of this regression
polynomial in the point xi substitutes original
value yi.
Number of data N1024, width of window Nw50,
quadratic polynomial.
The SG filtration is implemented in MATLAB as
function SGOLAYFILT(X,K,F) where X vector of
noised data, K degree of regression polynomial
and F2Nw1 is width of window.
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Regression models example-soaking
NAP2
Many empirical models are used for description of
relationship between moisture content in grains
and time, for example exponential Pages model
There are two model parameters p1k, p2n.
Independent variable t is time
Xe is equilibrium, XO initial moisture of grain
and even more frequently used Pelegs model (see
paper)
Parameter p2 characterises equilibrium moisture
The application of Peleg's equation to model
water absorption during the soaking of red kidney
beans (Phaseolus vulgaris L.)  Journal of Food
Engineering, Volume 32, Issue 4, June 1997, Pages
391-401Nissreen Abu-Ghannam, Brian McKenna
Pelegs model is used for beans, chickpeas, peas,
nuts
Page and Peleg models are nonlinear and in
practice are transformed to linear form by
logarithm (Page) or (Peleg) as tvar
This is new dependent variable y
and then linear regression can be applied for
identification of p1 a p2
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Regression models -soaking
NAP2
Linearisation of non-linear models (for example
linearisation of Page and Peleg models) means
that the optimised parameters p1, p2 minimise
some other criterion than the ?2 and also other
characteristics, like covariance matrix,
correlation index do not correspond to the
assumption of normal distribution of errors.
However, this effect is usually small and can be
neglected.
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Analytical solution diffusion and heat
NAP2
Vermeer
20
Analytical solutions
NAP2
Analytical solution exist only at linear
models Ordinary differential equations Partial
differential equations there exist two ways
any analytical functions that you know, for
example sin, cos, exp, tgh, Bessel functions,
are defined in fact as a solution of ordinary
differential equations (see handbooks, for
example Kamke E Differential Gleichungen,
Abramowitz M., Stegun I. Handbook of
Mathematical functions). General approach
consists in expression of solution in form of
infinite power series and identification of
coefficients by substitution the expansion into
the solved differential equation
Fourier method of separation of variables.
Solution F(t,x,y,z) is searched in the form of
product FT(t)X(x)Y(y)Z(z). Substituting into
partial differential equation results ordinary
differential equations for T(t), X(x), Y(y) and
Z(z). Application of integral transforms
Fourierovu, Laplaceovu, Hankelovu. Result is
algebraic equation, which is to be solved and
back-transformation must be used.
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Analytical solution diffusion (1/5)
NAP2
Distribution of moisture X (kg water/kg solid) is
described by Ficks equation
Xe equilibrium, X0 initial moisture
m0,1,2 for plate, cylinder, sphere.
Fourier method of separation of variables
Substituting to Ficks PDE results
in fact both terms must be constants independent
on t, r. The constant is called eigenvalue.
This term depends only on time t
this terms depends only on r
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Analytical solution diffusion (2/5)
NAP2
Solution of Gi(t) is exponetial function,
solution of Fi(r) is cos(?r) for plate (m0),
Bessel function J0(?r) for cylinder (m1) and for
sphere (m2, this is our case of spherical grains)
Phase equilibrium is assumed at the sphere
surface (r1, XXe). Therefore X(t,1)0 and
this condition must be satisfied by any function
Fi. This is the condition for eigenvalues
Dimensionless con centration profile is therefore
The coefficients ci must be selected so that the
initial condition (distribution of concentration
in time zero) will be satisfied. For constant
initial concentration XX0 must hold X1 for
arbitrary r, thewrefore
23
Analytical solution diffusion (3/5)
NAP2
The coefficients ci are evaluatred from
orthogonality of functions Fi(r). Functions are
orthogonal if their scalar product is zero.
Scalar product of functions is defined as integral
Proof
per partes integration.
Thbis is zero because both functions F must
satisfy boundary conditions
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Analytical solution diffusion (4/5)
NAP2
Let us apply orthogonality to previous equation
(multiplied by r2Fj and integrated)
Concetration profile or by integration across the
volume of sphere total moisture content as a
function of time
25
Analytical solution diffusion (5/5)
NAP2
Diffusion coefficient Def generaly depends upon
temperature and moisture and also the
equilibrioum moisture Xe is a function of
temperature, for example
Regression model is therefore strictly speaking
nonlinear (with parameters p1Dwa, p2Ea a p3b)
and the analytical solution with substituted Def
is only an approximation. This model used Katrin
Burmester for coffee grains
Heat and mass transfer during the coffee drying
process  Journal of Food Engineering, Volume 99,
Issue 4, August 2010, Pages 430-436 Katrin
Burmester, Rudolf Eggers
Modeling and simulation of heat and mass transfer
during drying of solids with hemispherical shell
geometry  Computers Chemical Engineering,
Volume 35, Issue 2, 9 February 2011, Pages
191-199I.I. Ruiz-López, H. Ruiz-Espinosa, M.L.
Luna-Guevara, M.A. García-Alvarado
26
Optimalisation
NAP2

• There are two basic optimisation techniques for
  calculation of parameters pi minimising ?2 of
  nonlinear models
• Without derivatives, when minimum can be
  identified only by repeated evaluation of s2 (or
  another criterion of fit between prediction and
  experiment) for arbitrary values of parameters
  pi. These methods are necessary at very
  complicated regression models, for example models
  based upon finite element methods.
• Derivative methods, making use values of all
  first (and sometimes second) derivatives of
  regression model with respect to all parameters
  p1,p2,,pM.

27
Optimisation methods with derivatives
NAP2
Gauss method of least sum of squares of deviations
Weight coefficients (representing for example
variable accuracy of measurin method)
model
data
Zero derivatives with respect all parameters
j1,2,,M
Linearisation of regression function by Taylor
expansion
?p is increment of parameters in one iteration
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Optimisation methods with derivatives
NAP2
Solution of systém of linear equations in each
iteration
The most frequently used modification of Gauss
method is the Marquardt Levenberg method
diagonal of matrix C is increased by adding a
constant ? in the case that the iteration are not
converging. For very high ? the matrix C is
almost diagonal and the Gauss method reduces to
the gradient steepest descent method (right hand
side is in fact gradient of the minimised
function s2). ? value is changing during
iterations when process converges ? decreases
and the faster Gauss method is preferred, while
if iterations diverge the ? increases (gradient
method is slower but more reliable).
29
Optimisation method without derivatives
NAP2
The simplest case is optimisation of only one
parameter (M1). First the global minimum is to
be localised (for example by random search). Then
the exaxt position of minimum is identified by
the method of bisection or by the Golden Section.
The golden section method reduces uncertainty
interval in each step in the ratio 0.618 and not
in the ratio 0.5 as in bisection. However only
one value of regression function need to be
evaluated in the golden section and not two
values necessary for bisection. See algorithm
Golden section search and the following slide
(definition of golden section)
f1
f4
f3
f2
L1
L20.618L1
L30.618L2
Example Initially two values f1 f2 in golden
sections of interval L1 are calculated. Because
f1gtf2 the minimum cannot be left, and the
interval of uncertainty reduces to L2. We need
again two values in this interval, but one value
(f2) was calculated in the previous step, so that
only ONE new value need to be calculated. And
that is just the glamor of the golden section
method and the secret of the magic ratio 0.618.
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Golden Section
NAP2
Quadratic equation for the ratio q/p
31
Optimisation method without derivatives
NAP2
Example How many steps of golden section method
and how many values of regression function must
be evaluated if thousand times reduction of
uncertainty interval is required?
Result, for 1000times increase accuracy it is
sufficent 14 steps.
An approach to determine diffusivity in hardening
concrete based on measured humidity
profiles  Advanced Cement Based Materials, Volume
2, Issue 4, July 1995, Pages 138-144 D. Xin, D.
G. Zollinger, G. D. Allen
Diffusion of concrete hardening, and
identification of diffusion coefficient by golden
section method.
32
Optimisation method without derivatives
NAP2
Principles of the one-parametric optimisation can
be applied also for M-parametric optimisation of
p1,,pM just repeating 1D search separately
(Rosenbrock). However, the most frequent is the
simplex method Nelder Mead . Principle is quite
simple
1. Simplex formed by M1 vertices is generated
(for two parameters p1 p2 it is a triangle). 2.
The vertex with the worst value of the regression
function is substituted by flipping, expansion or
contraction with respect to the gravity center of
simplex . Step 2 is repeated until the size of
simplex is decreased sufficiently
Animated gig from wikipedia.org
33
MATLAB
NAP2
34
Optimisation methods MATLAB
NAP2
Linear polynomial regression ppolyfit(x,y,m) Min
imalisation of function (without constraints,
method Nelder Mead) pfminsearch(fun,p0) where
fun is a reference (handle denoted by the
symbol _at_) to a user defined function, calculating
value which is to be minimised for specified
values of model parameters (for example sum of
squares of deviations). Vector p0 is initial
estimate.
Nonlinear regression (nonlinear regression
models) p nlinfit(x,y,modelfun,p0) ci
nlparci(p,resid,'covar',sigma) also statistical
evaluation of results (covariance matrix,
intervals of uncertainty)
35
Optimisation methods MATLAB
NAP2
Example Measured drying curve, 10 points (time
and moisture)
time X(moisture) 0 0.9406 1
0.7086 2 0.7196 3 0.5229 4
0.4657 5 0.3796 6 0.3023 7
0.1964 8 0.1545 9 0.1466
Apostroph vektor means transposition, instead
of row it will be columnwise vector
xdata0 1 2 3 4 5 6 7 8 9 ydata0.9406
0.7086
Plot od date by command plot(xdata,ydata,)
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Optimisation methods MATLAB
NAP2
Drying curve approximated by cubic polynomial
p1x3p4 ppolyfit(xdata,ydata,3)
p 0.0001 0.0040 -0.1322 0.9103
Vector of calculated coefficients
Plot polynomial with coefficients p(1)0.0001,
p(2)0.0040 Ypolyval(p,xdata) hold
on plot(xdata,ydata,'') plot(xdata,Y)
hold on enables plotting more curves into one
graph
All this could have been done by single command
plot(xdata,ydata,'',xdata,polyval(p,xdata))
37
Optimisation methods MATLAB
NAP2
The data of the drying curve can be better
approximated by diffusion model

• with A a scale coefficient, diffusion
  coefficient Def and radius of particle Def and R
  are not independent parameters because only the
  ratio Def/R2 appears in mthe model. The radius R
  can be selected (for example R0.01 m) and only
  wo regression parameters A, Def, should be
  minimised. How?
• Define xmodel(t,p,pa) as a sum of series (with
  params p1A, p2Def, pa1R, pa2n)
• Define function calculating sum of squares
  xdev(xmodel,p,pa,xdata,ydata)
• Calculate optimum using p fminsearch(xdev,p0)

38
Optimisation methods MATLAB
NAP2
Model definition
function xval xmodel(t,p,pa) Ap(1) Dp(2) Rp
a(1) nipa(2) xv0 for i1ni
pii(pii)2 xvxv6/piiexp(-piiDt/R2) e
nd xvalAxv
This text should be saved as M-file with the
filename xmodel.m
In this way can be defined arbitrary model of
drying, for example previously identified cubic
polynomial, Pelegs or Pages models, or even
models defined by differential integrations. This
last case should be discussed later.
39
Optimisation methods MATLAB
NAP2
Definition of goal function (sum of squares of
deviations)
function sums xdev(model,p,paux,xdat,ydat) sums
0 nlength(xdat) for i1n sumssums(model(xd
at(i),p,paux)-ydat(i))2 end
Define auxilliary parameters and search
regression parameters by fminsearch
R0.01 and number of terms in expansion n10
pa0.01,10 p fminsearch(_at_(p)
xdev(_at_xmodel,p,pa,xdata,ydata),.50.0003)
Initial estimate of parameters
The first parameter fminsearch is function with
only one parameter, vector p of optimised
parameter. Another non optimised parameters must
be in MATLAB specified by using anonymous
function _at_(p) expression.
40
Optimisation methods MATLAB
NAP2
Exactly the same procedure can be summarized in
the single M-file
function estimates, model fitcurve(xdata,
ydata) start_point 1 0.00005 Hledané
parametry jsou dva A,D. Pocátecní odhad. model
_at_expfun Expfun je predikce modelu a výpocet
souctu ctvercu odchylek estimates
fminsearch(model, start_point) volání
optimalizacní Nelder Mead metody function
sse, FittedCurve expfun(params) A
params(1) optimalizovaný škálovací parametr
D params(2) optimalizovaný difuzní
koeficient R0.01 polomer cástice (když
ho chcete zmenit musíte opravit funkci
fitcurve.m) ni10 pocet clenu
rady(správne nekonecno, ale 10 obvykle stací)
ndatalength(xdata) (pocet bodu namerené
krivky sušení) sse0 výsledek expfun
soucet ctvercu odchylek for
idata1ndata xv0 tady je treba
naprogramovat konkrétní model sušení
(Y(X,params)) for i1ni
pii(pii)2
xvxv6/piiexp(-piiDxdata(idata)/R2)
end FittedCurve(idata) A
xv ErrorVector(idata)
FittedCurve(idata) - ydata(idata)
ssesseErrorVector(idata)2 end
end end estimates, model fitcurve(xdata,ydata
)
41
EXAM
NAP 2
Regression models Optimisation
42
EXAM
NAP2
Balance equations
Lagrange variational principle (minimum WiWe)
Goal function (chi-quadrat)
Regression (parameters p minimising chi-quadrat)
Minimisation method with derivativatives
(Marquardt Levenberg) and without derivatives
(golden section, simplex method Nelder Mead)