Transcript Water quality planning. - Computing Center of the Russian

Approximation and Visualization
of Interactive Decision Maps
Short course of lectures
Alexander V. Lotov
Dorodnicyn Computing Center of Russian
Academy of Sciences and
Lomonosov Moscow State University
Lecture 7. Non-linear Feasible Goals
Method and its applications
Plan of the lecture
1. Approximation for visualization of the feasible objective set
2. Application of the FGNL for conceptual design of future
aircrafts
3. Identification of economic systems by visualization of Y=f(X)
4. Approximation for visualization of the EPH
5. Hybrid methods for approximating the EPH in the nonconvex case
6. Statistical tests
7. A simple hybrid method: combination of two-phase, threephase and plastering methods
8. Study of a cooling equipment in continuous casting of steel
9. Parallel computing
The main problems
that arise in the non-linear case:
1. non-convexity of the set Y=f(X); and
2. time-consuming algorithms for global
scalar optimization.
Approximation for visualization of
the feasible objective set
Approximation of the feasible objective set f(X) may be
needed at least in two cases:
1) decision maker does not want to maximize or
minimize the performance indicators;
2) identification problem is considered.
We approximate the set f(X) by using simulation of
random decisions, filtering their outputs and
approximating f(X) by a system of cubes.
Then, on-line visualization of the feasible objective set is
possible. Thus, we apply simulation-based multi-criteria
optimization. Such an approach can be applied in the case
of models given by computational modules, too.
Example model: the well-known Peak Function
f ( x1, x2 )  u1 ( x1, x2 )  u2 ( x1, x2 )  u3 ( x1, x2 )  10
u1(x1 ,x 2 )  3( 1  x1 ) e
2
 x 2 (x2 1 ) 2
1
,
1
 x2  x2
3
5
u 2 ( x1 , x 2 )  10( x1  x1  x 2 ) e 1 2 ,
4
1 ( x11)2  x22
u3 ( x1 , x2 )   e
3
where
x1  [4.9; 3.2], x2  [3.5; 6]
Let us consider five criterion functions
fi ( x1, x2 )  f ( x1  i , x2  i )
which are subject of maximization.
Let us consider an example. Imagine that we
want to locate the monitoring station at the point
where maximal pollution occurs. Let f i ( x1 , x2 ) be
the pollution level forecasted by the i-th expert.
Different values of criteria characterize difference
in knowledge of experts concerning pollution
distribution.
Let us consider several three-criterion
graphs
Software demonstration
Application of the FGNL for
conceptual design of future aircrafts
Four construction parameters were considered
1. Draught of the engine per weight (P0);
2. Overall drag coefficient of the aircraft (Cx0);
3. Inductive drag coefficient of the aircraft (A);
4. Lift coefficient of the aircraft (CY).
The aircraft was described by flight characteristics:
1. Rotation speed at a given height (W_max5000);
2. Time of elevation to a given height (TimeH);
3. Time of acceleration to a given speed (TimeV).
Exploration of the decision space:
Squeezed variety of feasible values of draft of the
engine (P0), frontal resistance (СX0) and elevating
force (CY).
Identification of economic systems
by visualization of Y=f(X)
Approximation for visualization of the EPH
The EPH is approximated by the set T* that is
the union of the non-negative cones
y
m
R
with apexes in a finite number of points y of the
set Y=f(X). The set of such points y is called
the approximation base and is denoted by T.
Multiple slices of such an approximation can be
computed and displayed fairly fast.
Visualization example for 8 criteria
Goal identification
Demonstration of
Pareto Front Viewer
Hybrid methods for
approximating the EPH
in the non-convex case
The models under study
Computing of the objective functions (the model)
can be given by a computational module (black
box) provided by user and unknown for the
researcher. Thus, a very broad scope of non-linear
models can be studied.
Our methods provide inputs which depend on the
method; the module computes outputs of these
inputs (or by using simulation, or by solving
boundary problem, or in some other way).
Due to it we can even use simulation-based local
optimization of random decisions or genetic
optimization.
The scheme of the methods
We apply hybrid methods that include:
1) global random search;
2) adaptive local optimization;
3) importance sampling;
4) genetic algorithm.
Statistical tests of the approximation quality
play the leading role in approximation process.
Statistical tests
Quality of an approximation T* is studied
by using the concept of completeness
hT = Pr {f(x)  T* : x  X }.
We estimate Pr { hT > h* }   for a given
reliability  by using a random sample
HN = {x1, … , xN}.
Let hT(N)= n/N, where n=|f(xi)  T*|.
Then, hT(N) is a non-biased estimate of hT .
Moreover, hT(N) – (– ln (1 – ) / (2N) )1/2
describes the confidence interval.
Completeness function
Let (T*)ε be the ε–neighborhood of T*.
Then, hT (ε)= Pr {f(x)  (T*)ε : x  X } is the
completeness function.
Important characteristics of the function hT(N)(ε)
is the value εmax=δ(f(HN), T*).
Two optimization-based completeness functions
for different iterations (1 and 7)
The optimization-based completeness
In the problems of high dimension of decision variable
can happen that the sample completeness is equal to 1,
but the approximation is bad. In this case
optimization-based completeness function is used
hT (ε) = Pr{f(Φ(x0)) (T*)ε : x  X }
where Φ:X → X is the “improvement” mapping,
which is usually based on local optimization of a
scalar function of criteria. The mapping moves the
point f(Φ(x0)) in the direction of the Pareto frontier. As
usually, a random sample HN {x1, … , xN} is generated
and hT(N)(ε)=n(ε)/N, where n(ε)=|f(Φ(xi0))  (T*)ε| is
computed.
One-phase method
An iteration. A current approximation base T must be
given.
1. Testing the base T. Generate a random sample
HN X , compute hT(N)(ε). If hT(N)(ε) (or some
values as hT(N)(0) and εmax=δ(f(HN),T*) in
automatic testing) satisfy the requirements, stop.
2. Forming new base. Form a list that includes
points of T and sample points that not belong to
T*, exclude dominated points. By this a new
approximation base is found. Start next iteration.
Two-phase method
An iteration.
A current approximation base T must be given.
1. Testing the base T. Generate a random sample
HN X , compute Φ(HN). Construct hT(N)(ε). If the
function hT(N)(ε) (or some values as hT(N)(0) and
εmax=δ(f(Φ(HN)),T*) in the case of automatic
testing) satisfy the requirements, stop.
2. Forming new base. As usually.
Start next iteration.
Three-phase method
An iteration. Current base T and a neighborhood B
of decisions which images constitute T must be
given
1. Testing the base T. Generate two random
samples H1X and H2 B, compute Φ(H1) and
Φ(H2). Construct hT(N)(ε). If hT(N)(ε) satisfies
the requirements, stop.
2. Forming new base. As usually.
3. Forming new neighborhood B using
statistics of extreme values.
Start next iteration.
Three-phase method
An iteration. Current base T and a neighborhood B
of decisions which images constitute T must be
given
1. Testing the base T. Generate two random
samples H1X and H2 B, compute Φ(H1) and
Φ(H2). Construct hT(N)(ε). If hT(N)(ε) satisfies
the requirements, stop.
2. Forming new base. As usually.
3. Forming new neighborhood B using
statistics of extreme values.
Start next iteration.
Forming new neighborhood B
The neighborhood B is the constituted of the balls in
decision space with centers in current Pareto-optimal
decisions. They have the same radius k . The value of
the radius is computed using the statistics of extreme
values. Namely, we consider the distances of new
Pareto-optimal decisions the old Pareto-optimal
decisions. Then, we order the distances in accordance
to their growth d(N), d(N-l),… where d(N) is the most
distanced point. Then, k = d(N) + , where
θ = r(l, χ)( d(N) – d(N-l)),
while , is the reliability, 0<<1. Here
r(l, χ) = {[1-(1-χ) 1/l](-1/a) – 1}(-1) and
a = (ln l) / ln[(d(N) – d(N-l)) / (d(N) – d(N-1))],
l << N (we took l=10).
Plastering method
“Plastering” method that has some properties of genetic algorithms
(as cross-over) is used at the very end of the approximation
process.
An iteration. Current approximation base T and numbers q, 1, 2
must be given.
1. Testing the base T. Let H be the set of inputs that result in points
of the approximation base T. Select N random pairs (hi, hj) that
satisfy 1 ≤ d(f(hi), f(hj)) ≤ 2 from the set H. Select q random
points on the segment connecting the points hi and hj, and denote
them by Hl, l=1,…,N. Compute objective points for the points x 
Hl, l=1,…,N. Construct hT(N)(ε). If hT(N)(ε) satisfies the
requirements, stop.
2. Forming new base.
3. Filtering if needed
Start next iteration.
A simple hybrid method: combination of
two-phase, three-phase and plastering
methods
• Iterations of two-phase method are carried out until
hT(N)(0) and εmax=δ(f(Φ(HN)),T*) are close to zero.
• Iterations of three-phase method are carried out until
hT(N)(0) and εmax=δ(f(Φ(HN)), T*) for it are close to
zero.
• Iterations the genetic method carried out until
hT(N)(0) and εmax=δ(f(Φ(HN)), T*) for it satisfy some
requirements.
Study of a cooling equipment in
continuous casting of steel.
The research was carried out jointly
with
Dr. Kaisa Miettinen, Finland,
at the University of Jyvaskyla,
Finland.
Cooling in the continuous casting process
Criteria
J1 is the original single optimization criterion:
deviation from the desired surface temperature of the
steel strand must be minimized.
J2 to J5 are the penalty criteria introduced to
describe violation of constraints imposed on :
J2 – surface temperature;
J3 – gradient of surface temperature along the strand;
J4 – on the temperature after point z3; and
J5 – on the temperature at point z5.
J2 to J5 were considered in this study.
Description of the module
FEM/FDM module was developed in Finland,
by researchers from University of Jyvaskyla.
Properties of the model: 325 control variables that
describe intensity of water application.
Properties of local simulation-based optimization:
one local optimization required about 11-12
calculations of the gradient and about 1000-2000
additional calculations of the value of f(x).
Next pictures demonstrate the approximation
Parallel computing
Parallel computing
(processor clusters and grid-computing)
The method has the form that can be used
in parallel computing.
Thus, it can be easily implemented at
parallel platforms –
it is sufficient to separate data generation
and data analysis
(Research in the framework of contract
with Russian Federal Agency for Science).
Important property of our hybrid
methods
Since our methods are based on random
sampling, partial loss of the results is not
dangerous. It influences the reliability of
the results but does not destroy the
process. Due to it, application in GRID
network is possible.
Two platform implementation is
needed
Example of scenario template for
hybrid method
Scenario editor-1
Scenario editor - 2