global-local approximations puneet singlapsingla/teaching/mae502/class... · boundary value...

GLOBAL-LOCAL APPROXIMATIONS

Puneet Singla

MAE 502

Department of Mechanical & Aerospace EngineeringUniversity at Buffalo

http://www.acsu.buffalo.edu/ psingla/MAE502/

November 12, 2007

PUNEET SINGLA 1 / 15

http://www.acsu.buffalo.edu/~psingla/MAE502/

WHAT ARE WE LOOKING FOR?

Interpolation.

Local support

Interelement continuity/compatibility

Completeness.

Choices: FEM, MLS, Splines, RBF etc.

Most Important Characteristic: Independent LocalApproximations



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1

0

1

2

3

4

A

B

C



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1

0

1

2

3

4

Local Approximations

Resulting Curve

A

B

C


AVERAGING PROCESS

1

0

h h h h

1 1 1( ) ( ) ( ) ( ) ( )I I

I I IF X w x F X w x F X−− −= +

( )IF X1( )IF X−

1 1 I I IX X X X− +

1( )IF X+

( ) / ; 1I I Ix X X h x= − ≤

1( )Iw x +1( )Iw x + ( )Iw x

1 1 I I Ix x x x− +

11( ) ( ) ( ) ( ) ( )I I

I I IF X w x F X w x F X++= +


AVERAGING PROCESS

1 The first derivative of the weighting function must have anmth-order osculation with w(0) = 1 at the centroid of itsrespective local approximation.

w(0) = 1dkwdxk |x=0 = 0 k = 0,1, · · · ,m

(3.4)

2 The weighting function must have an (m+1)th-order zero at thecentroid of its neighboring local approximation.

w(1) = 0dkwdxk |x=1 = 0 k = 0,1, · · · ,m

(3.5)

3 The sum of two neighboring weighting functions must be unityover the entire closed interval.

w(Ix)+w(Ix−1) = 1 ∀ x, −1 ≤ x ≤ 1 (3.6)


BOUNDARY VALUE SOLUTION FOR WEIGHTING

FUNCTIONS

We assume the following particular form for weighting function,

w(x) = 1− J(x) (1)

where, J(x) is a polynomial in the independent variable whosefirst derivative is given by following expression:

dJ(x)dx

= Cxm(1− x)m (2)

NOTE: This particular form for the weighting function is inaccordance with the fact that first m partial derivatives of theweighting function vanishes at end points and w(0) = 1.

The remaining boundary conditions on the weighting function,w(x), will completely define the constants in Eq. (2).



FUNCTIONS

We have:

J(1) = C1∫

0

xm(1− x)mdx = 1 (3)

That means the appropriate value for constant, C, is given by:

C =

1∫0

xm(1− x)mdx

−1

=(2m+1)!

(m!)2 (4)

Thus, the general form for weighting function can now bewritten as:

w(x) = 1− (2m+1)!(m!)2

x∫0

xm(1− x)mdx (5)



FUNCTIONS

Further, the binomial theorem allows us to expand the aboveintegrand:

w(x) = 1− (2m+1)!(m!)2

x∫0

m

∑n=0

mCnxmxm−n(−1)ndx (6)

Now, integrating the above expression term by term yield thefollowing expression for weighting function, w(x):

w(x) = 1−Km

∑n=0

Anx2m−n+1 (7)

where, K and An are given by following expressions:

K = (2m+1)!(−1)m

(m!)2 An = (−1)n mCn2m−n+1 (8)

Finally, to obtain the expression for weighting function ininterval [−1,1] instead of [0,1] the absolute value of x is used asindependent variable than x.


WEIGHTING FUNCTIONS

TABLE: Weight functions for higher order continuity.

order ofpiecewisecontinuity

Weight Function:

w0,0,··· .0(x1,x2, · · · ,xN) =N∏i=1

w(xi)

w(x), for all x ∈ {−1 ≤ x ≤ 1}, y ∆= |x|0 w(x) = 1− y1 w(x) = 1− y2(3−2y)2 w(x) = 1− y3(10−15y+6y2)3 w(x) = 1− y4(35−84y+70y2−20y3)...

...

m w(x) = 1− ym+1{

(2m+1)!(−1)m

(m!)2

m∑

k=0

(−1)k

2m−k+1

(mk

)ym−k

}PUNEET SINGLA 7 / 15

AVERAGING PROCESS

+

+

+

(0,0)

(1,1)

y

y

y

y

11

Preliminary Weight functions: Approximations: ( , ) F x y 2 2

11

01 01 11

00

( , ) (3 2 ) (3 2 )

( , ) ( , ) (1 , )

( , )

w x y x x y y

F x y w x y w x y

F x y

= − −

= −

00 11

10 10 11

( , ) (1 ,1 )

( , ) ( , ) ( ,1 )

w x y w x y

F x y w x y w x y

= − −

= −

Weight functions are Final Approximation: a partition

1 1 1 1

0 0 0 0

of unity:

( , ) ( , ) ( , ) ( , ) 1

valid over {0 1,0 1}

ij ij iji j i j

F x y w x y F x y w x y

x y= = = =

= =

≤ ≤ ≤ ≤

∑∑ ∑∑


WEIGHTING FUNCTIONS (FOR APPROXIMATION IN 2-D)


PRELIMINARY APPROXIMATIONS

The preliminary approximations , while arbitrary, in particularcould be chosen to minimize the least square criterion:

J = 12

1∫−1

w(x)[F(X)−FI(X)]2dx (3.21)

If FI(X) is considered as a linear combination of a set of linearlyindependent basis functions:

FI(X) =n∑

i=0aiφi(x); X = IX +hx (3.22)

The least square criterion gives the optimum (minimum integralleast square fit error) coefficients as: a = M−1c




J = 12

1∫−1

w(x)[F(X)−FI(X)]2dx (3.21)


FI(X) =n∑

i=0aiφi(x); X = IX +hx (3.22)

The least square criterion gives the optimum (minimum integralleast square fit error) coefficients as: a = M−1cwith,

c j = < F(x),φ j(x) >

Mi j = < φi,φ j >




J = 12

1∫−1

w(x)[F(X)−FI(X)]2dx (3.21)


FI(X) =n∑

i=0aiφi(x); X = IX +hx (3.22)

The least square criterion gives the optimum (minimum integralleast square fit error) coefficients as: a = M−1c

If basis functions satisfy the orthogonality condition:< φi(x),φ j(x) >= kiδi j, ki

∆= µii

Then least square solution simplifies to the simple uncoupledresult to compute the Fourier coefficients:ai = <F(x),φi(x)>

ki, i = 1,2, · · · ,n (3.30)


PRELIMINARY APPROXIMATIONS(THE GENERALIZATION TO N-DIMENSIONS IS REALLY, REALLY BEAUTIFUL!)

(a) 1-D Basis Functions (b) 2-D Basis Functions

w(.) =NΠ

i=0[1− x2

i (3−2|xi|)], φi1···iN (.) =NΠ

i=0φi(xi)

ai1···iN =< φi1···iN (x1, · · · ,xN),F(X1, , · · · ,XN) >

N∏j=1

k j

(3.45)


BASIC IDEA


ALGORITHM IMPLEMENTATION

The main steps of the GLO-MAP algorithm are as follows:

Choose a set of sequential neighboring points, IX , arbitrary innumber and location. These points serve as the centroids ofvalidity for the local functional approximation, FI .

The density and location of these points depends upon numerousfactors like location and density of available measurement dataand desired degree of approximation.




Choose set of basis functions based on computational efficiencyor a priori knowledge of the nature of the given input-output datato approximate F(X) in local neighborhood of a centroid ofvalidity, IX .

One attractive choice for the basis functions is the orthogonalpolynomial basis functions as discussed earlier.




Determine the Fourier coefficients corresponding to each localapproximation.

For the approximation of a given continuous functional form orfrom dense discretely measurable functions, the Fouriercoefficients can be computed by numerically evaluating theintegral expressions using standard numerical integrationalgorithms.




In case of the GLO-MAP algorithm the numerical integrationprocedure can be summarized as below:

1 Determine Gaussian quadrature points, XG, in unit hypercube.2 For each quadrature point, determine the measurement points Xi,

for which weight function has non-zero value.3 Determine function value at quadrature point, F(XG) can be

taken as a weighted average of function value at Xi.

F(XG) =1

∑i w(Xi)∑

iF(Xi)w(Xi) (9)

4 Evaluate the numerical integrals.




It should be noted that Gaussian quadrature points are the samefor each local approximation and can be pre-computed.

However, the numerical evaluation of integral expression ofseveral variables, over regions with dimension greater than one, isnot easy.As a rule of thumb, the number of function evaluations needed tosample an N-dimensional space increases as the Nth power of thenumber needed to do a one-dimensional integral, resulting inincreased computational cost associated with numericalintegration proportion to the Nth power.Use discrete orthogonal basis functions.




The final step of the GLO-MAP algorithm is the use ofweighting functions to merge the local approximations into asingle mth order continuous functional model.

Note, this step is the most important feature of the GLO-MAPalgorithm as it reduces the systematic errors introduced due tothe neglected interaction between different local models byblending overlapping local approximations into a global one.



Input-Output Data

Choose a set of sequential points as centroidsfor local approximations Choose set of orthogonal basis functions

For each centroid point, find the measurementdata points which lies in the local neighborhood

of the centroid

Determine Fourier coefficients pertaining toeach local approximation

Merge different local approximations usingweighting functions given in Table 1

Compute measure of approximation error by computing theglobal covariance matrix by merging covariance matrix

corresponding to different local approximations.

Is approximation error lessthan pre-specified tolerance?

Stop

Refine the distance betweendifferent centroid points. Include higher order basis

functions

FIGURE: Flowchart for the GLO-MAP based multi-resolution algorithm.


SEQUENTIAL VERSION

There are many engineering application problems which needsto be solved in an iterative manner in real-time by successivelyapproximating the input-output data.

Kalman filter algorithm can be used to update a priori learnedGLO-MAP network parameters in real time whenever newmeasurements are available.

The main challenge associated with the use of the Kalman filterin the GLO-MAP algorithm is the dynamic state vector i.e. thecomponents of state vector of Kalman filter changes with everymeasurement data depending upon the location of measurementdata relative to centroids of validity of local approximations.

However, when a new measurement is available, only theneighboring 2N local approximations (associated with the 2N

vertices of the hypercube containing the new measurement)willhave non-zero contribution in final global map.


SEQUENTIAL VERSION

The main steps involved in the implementation of sequential versionof the GLO-MAP algorithm are as follows:

Choose a set of sequential neighboring points, IX , arbitrary innumber and location. These points serve as the centroids ofvalidity for the local functional approximation, FI .

The density and location of these points depends upon numerousfactors like location and density of available measurement dataand desired degree of approximation.


SEQUENTIAL VERSION


Choose a set of basis functions (preferably orthogonal functions)to approximate F(X) in the local neighborhood of a centroid ofvalidity, IX .

Initialize the coefficients of each basis functions to be zero andassociated covariance matrix (PG) to be identity times a largenumber.


SEQUENTIAL VERSION


Given a new measurement data point, (X,F(X)), find theneighboring 2N centroids such that weight function associatedwith these 2N centroids have non-zero value at the measurementpoint.

Include the coefficients of the basis functions associated withthese centroids in the local algebraic Kalman filter state vector,denoted by x. Let xG denotes the global super-set of coefficientsand M be a selection matrix consisting of zeros and ones thatsatisfies x = MxG.Extract rows and columns of PG corresponding to the coefficientsassociated with these 2N centroids and denote them byP−

k = MPGMT .


SEQUENTIAL VERSION


Use the following equations to compute the local Kalman gainand update state vector, x, and the associated covariance matrix,P.

x+k = x−k +K(F(X)−ΦΦΦ(.)x−k ) (9)

P+k = (I−KΦΦΦ(.))P−

k (10)

K = P−k ΦΦΦ(.)T (ΦΦΦ(X)P−

k ΦΦΦ(.)T +Rk)−1 (11)

Matrices ΦΦΦ and R are given by following equations:

ΦΦΦk =[

φ1(Xk) · · · φN(Xk)]

(12)

Rk = σw(Xk) (13)


SEQUENTIAL VERSION


Update rows and columns of the global covariance matrix, PG

and coefficients associated with each local approximation.

Once the value of PG is less than a pre-specified tolerance, useappropriate weighting functions to merge various localapproximations into a single mth order continuous functionalmodel.


RESULTS

(a) True Surface (b) True ContoursIn first test case, the continuous functional expression is used to obtainthe coefficients of the preliminary local approximations

In second test case a discrete measurement data is used to computepreliminary approximations.

100 random samples over the interval [0-10,0-10] for both X1 andX2, giving total 104 measurements.


RESULTS

(c) True Surface (d) True Contours

(e) Orthogonalization Error (f) Approximation Error


RESULTS

(g) True Surface (h) True Contours

(i) Error Surface (j) Error Contours


PORK-CHOP PLOT APPROXIMATION

Departure Date

Tim

e O

f Flig

ht (

days

)

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07

150

200

250

300

350

5

10

15

20

25

30

35

40

(k) True porkchop plot for de-parture ∆V∞

Departure Date

Tim

e O

f Flig

ht (

days

)

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07

150

200

250

300

350

5

10

15

20

25

30

35

40

(l) True porkchop plot for ar-rival ∆V∞

(m) True surface plot for de-parture ∆V∞

(n) True surface plot for ar-rival ∆V∞

FIGURE: True departure and arrival ∆V∞ plots for a mission to the asteroid2003-Y N107.



Departure Date

Tim

e O

f Flig

ht (

days

)

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07

150

200

250

300

350

5

10

15

20

25

30

35

40

(a) Approximated porkchopplot for departure ∆V∞

Departure Date

Tim

e O

f Flig

ht (

days

)

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07

150

200

250

300

350

5

10

15

20

25

30

35

40

(b) Approximated porkchopplot for arrival ∆V∞

(c) Approximated surfaceplot for departure ∆V∞

(d) Approximated surfaceplot for arrival ∆V∞

FIGURE: Approximated departure and arrival ∆V∞ plots for a mission to theasteroid 2003-Y N107.



Departure Date

Tim

e O

f Flig

ht (

days

) −0.0017681

0.0035897

−0.0017681

0.0035897

−0.0017681

0.0035897

−0.0017681

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07220

230

240

250

260

270

280

290

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(a) Error contour plot for220 ≤ TOF ≤ 300

Departure Date

Tim

e O

f Flig

ht (

days

) −0.069026−0.069026

−0.069026−0.069026

−0.069026−0.069026

−0.069026

−0.0690262.3774 1.3289

−4.262−1.1175

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07120

130

140

150

160

170

180

190

200

210

−10

−5

0

5

10

15

20

(b) Error contour plot forTOF < 220

Departure Date

Tim

e O

f Flig

ht (

days

)

046003

0.046003

0.0460030.046003

0.046003

0.0460030.046003

0.046003

3.68710.95627

0.0460030.046003

0.34943

0.046003

0.046003

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07300

310

320

330

340

350

360

−20

−15

−10

−5

0

5

(c) Error contour plot forTOF > 300

Departure Date

Tim

e O

f Flig

ht (

days

)

0.34592 0.34592

0.34592

−0.10591 2.15321.7014

0.34592

−0.10591−0.10591−1.0096

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07

150

200

250

300

350

−20

−15

−10

−5

0

5

10

15

20

(d) Error contour plot forwhole domain



Departure Date

Tim

e O

f Flig

ht (

days

)

0.0019424

0.0019424

0.001942

0.080756

0.0334680.080756

0.32245

0.0019424

−0.003311

0.212110.04923102/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07

220

230

240

250

260

270

280

290

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

(e) Error contour plot for220 ≤ TOF ≤ 300

Departure Date

Tim

e O

f Flig

ht (

days

)

2.4956−0.62737 1.3245

−1.4081−6.0925

0.9341

−0.237 −0.237

−0.237

−0.237 −0.237

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07120

130

140

150

160

170

180

190

200

210

−10

−5

0

5

10

15

20

25

(f) Error contour plot forTOF < 220

Departure Date

Tim

e O

f Flig

ht (

days

)

0.240060.24006

−0.065179

−0.065179

5.1238

2.3767

−2.8123−0.67565

−1.2861

1.1558

−0.065179

−0.065179

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07300

310

320

330

340

350

360

−20

−15

−10

−5

0

5

(g) Error contour plot forTOF > 300

Departure Date

Tim

e O

f Flig

ht (

days

)0.22060.2206

−0.25762 −0.25762

0.2206

0.2206

1.6553

−3.1269

−1.2141

0.2206

02/04/07 03/26/07 05/15/07 07/04/07 08/23/07 10/12/07 12/01/07

150

200

250

300

350

−20

−15

−10

−5

0

5

10

15

20

25

(h) Error contour plot forwhole domain



Original measurement points for approximation purposes:3.(365×246) = 269370 .

GLO-MAP approximation: 8.(625+2.400) = 11400 realnumbers.


global-local approximations puneet singlapsingla/teaching/mae502/class... · boundary value...

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