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Applied Asymptotic Methods
(Introduction to Boundary Function Method)
Lecture Set #1
Leonid V. Kalachev
Department of Mathematical Sciences
University of Montana
Based of the book The Boundary Function Method for Singular Perturbation
Problems by A.B. Vasil’eva, V.F. Butuzov and L.V. Kalachev, SIAM, 1995
(with additional material included)
WHY Applied Asymptotic Methods?
Asymptotic Model Reduction: parameters in a complex model are known; some parameters are small or large; reduction may be performed to produce a simpler model that is easier to understand and interpret.
Complex model:
fast and slow
processes
Small and large
parameters Simplified model:
fewer equations
and parameters
Leonid V. Kalachev 2017 UM
Reduction algorithms and their justification
is the topic of Applied Perturbation (Asymptotic) Analysis!
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What types of problems are we going to consider? Dynamic Models: in general terms, any models formulated in terms of time dependent differential (or difference) equations.
Modeling any process that may be schematically represented as a reaction scheme: chemical and bio‐chemical kinetics, population dynamics, aggregation and growth/decay processes, etc. Homogeneous models: species are well mixed, variables are
only time dependent; no spatial dependence.
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Heterogeneous models: variables are time dependent and also
spatially dependent. BEFORE COMPUTERS: asymptotic methods were widely used as a powerful
computational tool. WHAT ABOUT NOW? 1. Reduction of models (number of equations/parameters) for better understanding of underlying processes. 2. Reduction of models (number of equations/parameters) for reliable model identification, i.e.,. identification of model parameters and making reliable model predictions. Leonid V. Kalachev 2017 UM
3. Construction of asymptotic approximations to the solutions of perturbed
problems for numerical/computational purposes: (a) simple model cases;
(b) initial approximations for solution of nonlinear equations using Newton
type methods. 2
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IMPORTANT: WHY Reduction and Identification?
More on reduction and identification of models: given the data and a hypothetical model; model parameters must be estimated by fitting model solution to the data. DATA
MODEL
PARAMETERS
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Where do the models come from?
IMPORTANT: WHY Reduction and Identification?
WE WANT TO KNOW: How many meaningful models fit the data? What is the minimal number of parameters that needs to be included in a model? An “optimal model” is derived from the basic principles (physics, chemistry, etc.) and contains the smallest possible number of parameters that still allows one to describe the experimental data. How to get the “optimal model”? Via repeated application of model reduction techniques together with analysis of reliability regions for model parameter values. Leonid V. Kalachev 2017 UM
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Historically: three types of problems related to practical applications of asymptotic reduction procedures Type 1: Model equations/systems are assumed to be known; model parameters are known; small parameters are identified. Need to construct asymptotic solution as part of computational procedure with the goal of finding some approximation of the original model solution (time “before computers”). Sample applications: general physics, mechanics, semiconductor device modeling, simple chemical and biological kinetics. Models of phenomena
where mechanisms of processes are well known. Leonid V. Kalachev 2017 UM
Historically: three types of problems related to practical applications of asymptotic reduction procedures Type 2: Model equations are assumed to be known; parameters are known; small parameters are not initially identified. The model is solved
numerically, then small parameters are identified and model reduction
performed with the goal of better understanding the underlying “slow” dynamics of the original model system. Why difficulties with identification
of small parameters? Often originally unknown solution values enter as rescaling parameters that determine whether a particular non‐
dimensional model parameter is small, moderate, or large. Leonid V. Kalachev 2017 UM
Sample applications: tropospheric chemistry, complex mechanics, more complex chemical and biological kinetics. Models of phenomena
where mechanisms of processes are known, but very complex. 4
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Historically: three types of problems related to practical applications of asymptotic reduction procedures Type 3: A set of possible alternative model equations/systems is assumed to be known; model parameters are not initially known
(they must be estimated from available experimental data); small parameters are not initially identified. The goal is to choose from the set of originally specified models an optimal one, with respect to the given data, using consecutive application of reduction and optimization techniques. Leonid V. Kalachev 2017 UM
Sample applications: complex chemical and biological kinetics, ecological models, bio‐medical and bio‐engineering applications. In general, the models of phenomena where major details of process mechanisms are not originally known and need to be clarified. PRACTICAL CONSIDERATIONS: Assume that we have some model that fits the data. What is an indication of the need to use model reduction? Combination of STATISTICS / APPLIED MATH APPROACHES:
BOOTSTRAPING & MCMC (Markov Chain Monte‐Carlo)
P2
Parameter space
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P1
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PRACTICAL CONSIDERATIONS: Assume that we have some model that fits the data. What is an indication of the need to use model reduction? Combination of STATISTICS / APPLIED MATH APPROACHES:
BOOTSTRAPING & MCMC (Markov Chain Monte‐Carlo)
P2
Fitted parameters’ values
P1*, P2*
*
Parameter space
Leonid V. Kalachev 2017 UM
P1
PRACTICAL CONSIDERATIONS: Assume that we have some model that fits the data. What is an indication of the need to use model reduction? Combination of STATISTICS / APPLIED MATH APPROACHES:
BOOTSTRAPING & MCMC (Markov Chain Monte‐Carlo)
P2
..
. . . ... .....
. .*.
New parameter values
produces by fitting re‐sampled data or
by Markov Chain !
Parameter space
Leonid V. Kalachev 2017 UM
P1
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PRACTICAL CONSIDERATIONS: Assume that we have some model that fits the data. What is an indication of the need to use model reduction? Combination of STATISTICS / APPLIED MATH APPROACHES:
BOOTSTRAPING & MCMC (Markov Chain Monte‐Carlo)
P2
..
. . . ... .....
. .*.
Reliability region
estimated, e.g.,
using smoothed
histogram! Parameter space
Leonid V. Kalachev 2017 UM
P1
PRACTICAL CONSIDERATIONS: Assume that we have some model that fits the data. What is an indication of the need to use model reduction? Combination of STATISTICS / APPLIED MATH APPROACHES:
BOOTSTRAPING & MCMC (Markov Chain Monte‐Carlo)
P2
The shape of reliability
region may indicate
problems with data fit!
*
Correlated parameters;
too many parameters, etc.
Parameter space
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P1
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Simple illustrative example: Math models in chemical kinetics are usually formulated in terms of differential equations. What are the expected results of model reduction? We start with the simplest reaction mechanism (e.g., generic receptor):
K1
A B  C
K2
C  A B
Leonid V. Kalachev 2017 UM
Also assume that reactions take place in a well mixed environment
(well stirred tank reactor, small size tissue sample, etc.).
Using the Law of Mass Action, we may write a system of ordinary differential equations (ODEs) describing behavior of concentrations
[A], [B], and [C] of species A, B, and C, respectively: d [ A]
  K1[ A]  [ B]  K 2 [C ],
dt
d [ B]
  K1[ A]  [ B]  K 2 [C ],
dt
d [C ]
  K1[ A]  [ B]  K 2 [C ],
dt
0  t  T.
[ A]  A*, [ B]  B*  A*, [C ]  0.
Leonid V. Kalachev 2017 UM
Initial conditions, e.g., at time t = 0:
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Non‐dimensionalization: u  [ A] / A*, v  [ B] / A*, w  [C ] / A *
k1  K1  T  A*, k 2  K 2  T , v*  B * / A*,   t / T
The scaled model has the form (now all the parameters are non‐dimensional):
u (0)  1,
v(0)  v*,
w(0)  0.
0    1.
Leonid V. Kalachev 2017 UM
du
 k1  u  v  k 2 w,
d
dv
 k1  u  v  k 2 w,
d
dw
  k1  u  v  k 2 w,
d
Now non‐dimensional characteristic reaction times 1/ k1 and 1/ k2 may be compared with the time interval of interest [0,1]. Short characteristic times (compared to [0,1]) correspond to fast processes and long characteristic times correspond to slow processes. We may have several possibilities:
(a) Forward and reverse reactions are moderate (of order O(1)). k1  O(1), k 2  O(1)
(b) Forward and reverse reactions are slow. k1  1, k 2  1
(c) Forward and reverse reactions are fast. (d) Other: k1  1, k 2  1; k1  1, k 2  1,
etc.
Leonid V. Kalachev 2017 UM
k1  1, k 2  1
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(a) Moderate forward and reverse reactions: k1  O(1), k 2  O(1)
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(b) Slow forward and reverse reactions:
k1  1, k 2  1
For the time interval of
interest [0,1] a simpler
model may be used to describe concentrations’
dependence on time!
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In this case we may re‐scale the coefficients to obtain:
~
du ε ~
 (k1  u  v  k 2 w),
d
~
~
dv
 ε (k1  u  v  k 2 w),
d
~
~
dw
 ε ( k1  u  v  k 2 w),
d
u (0)  1,
v(0)  v*,
w(0)  0.
0    1.
Here 0< ε << 1 is a small parameter!
Leonid V. Kalachev 2017 UM
~
~
k1  k1 /   O(1), k 2  k 2 /   O(1)
In the limit as small parameter tends to zero, we obtain: du
 0,
d
dv
 0,
d
dw
 0.
d
u ( )  1,
u (0)  1,
v(0)  v*,
w(0)  0.
So,
v( )  v*,
w( )  0.
REGULARLY PERTURBED PROBLEM!
Leonid V. Kalachev 2017 UM
Approximation satisfies the initial conditions; close to the original solution in the interval of interest: 11
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(c) Fast forward and reverse reactions:
k1  1, k 2  1
Leonid V. Kalachev 2017 UM
For the interior of time interval of interest [0,1] a simpler model may be used to describe concentrations’
dependence on time!
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In this case we may re‐scale the coefficients to obtain:
~
~
du
  k1  u  v  k 2 w,
d
~
~
dv
ε
  k1  u  v  k 2 w,
d
~
~
dw
ε
  k1  u  v  k 2 w,
d
ε
u (0)  1,
v(0)  v*,
w(0)  0.
0    1.
Here 0< ε << 1 is a small parameter!
Leonid V. Kalachev 2017 UM
~
~
k1  k1    O(1), k 2  k 2    O(1)
In the limit as small parameter tends to zero, we obtain: ~
~
0  k1  u  v  k 2 w,
0    1.
! ! !
Note: initial conditions are NOT satisfied!
The reduced model consists of the above equation and
two more equations (conservation of mass): ~ ~
0  u  v  (k 2 / k1 ) w,
u  v  1  v*,
u  w  1.
! ! !
SINGULARLY PERTURBED PROBLEM!
Leonid V. Kalachev 2017 UM
Approximation does not satisfy the initial conditions; it is close to the original solution in the interior of interval of interest: 13
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(c) Some other cases: e.g., fast forward and slow reverse reactions
k1  1, k 2  1
Leonid V. Kalachev 2017 UM
This model turns out to be singularly perturbed
and may also be reduced
using one of the asymptotic
methods: Boundary Function Method, Matching Technique, etc.
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Simple illustrative example (continuation): Identification of chemical kinetics models formulated in terms of differential equations. Reliability
regions for model parameters (k1 , k2 , v(0)): comparison of moderate and fast reactions cases.
(a)
k1  O(1), k 2  O(1)
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Shapes of reliability regions projections:
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(b)
k1  1, k 2  1
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Shapes of reliability regions projections:
Two parameters are correlated;
only the ratio may be estimated:
k  k 2 / k1
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Model reduction for more general (SINGULARLY) PERTURBED
systems describing complex chemical kinetics.
dx
 f ( x, y, z, t ),
dt
dy
   g ( x, y, z , t ),
dt
dz
   h( x, y, z, t ),
dt
0  t  1.
Fast reactions
Slow reactions x(0)  x*,
y (0)  y*,
z (0)  z * .
Leonid V. Kalachev 2017 UM
Initial conditions: Moderate reactions
When, after non‐dimensionalization, small parameters appear in the original model, can we always just set small parameters zero to obtain
a reduced model? The answer is NO! Certain conditions must be satisfied!
These conditions are formulated in Tikhonov’s theorem.
Mathematics is crucial: conditions must be checked before the reduction can be performed! Leonid V. Kalachev 2017 UM
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After setting small parameter to zero, we obtain: dx
 f ( x, y, z , t ),
dt
0  g ( x, y, z, t ),
x(0)  x*,
dz
 0,
dt
z (0)  z * .
y (0)  y*,
Initial conditions for y in
general are not satisfied!
z (t )  z *
0  t  1.
After substitution, we arrive at the system:
x(0)  x*,
with initial
conditions
y (0)  y * .
Leonid V. Kalachev 2017 UM
dx
 f ( x, y, z*, t ),
dt
0  g ( x, y, z*, t ),
To apply the reduction procedure we must check that:
1. The system is solvable with respect to fast
0  g ( x, y, z*, t )
variable y (there may be more than one solution):
y  G ( x, z*, t )
INTEGRAL MANIFOLD(s)
Solution on the manifold is described by the equation:
dx
 f ( x, G ( x, z*, t ), z*, t ),
dt
x(0)  x *
J

g ( x(t ), G ( x(t ), z*, t ), z*, t )
y
must have negative real parts.
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2. The lower dimensional Integral Manifold(s) must be stable: i.e., the eigenvalues of the Jacobian matrix 19
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If there are several stable manifolds satisfying conditions 1 and 2, which one to choose?
3. To correctly choose the Integral Manifold one must check that the
initial condition y(0) = y* belongs to the domain of attraction of the
stationary solution of auxiliary system:
yˆ  G ( x*, z*, t )
dyˆ
 g ( x*, yˆ , z*, t ),
d
In which t is considered to be a parameter, and τ is a stretched variable: Conditions 1, 2, and 3 allow one to construct a UNIQUE reduced model!
Leonid V. Kalachev 2017 UM
  t /
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