Index Determination in DAEs using the Library indexdet and the ADOL-C Package for Algorithmic Differentiation

AD 2008
The Fifth International Conference on Automatic Differentiation
August 11-15, 2008, Bonn, Germany

Index Determination in DAEs using the
Library indexdet and the ADOL-C Package
for Algorithmic Differentiation

Dagmar Monett
René Lamour
Andreas Griewank

DFG Research Center Matheon
Mathematics for key technologies

Motivation
Some applications
 Circuit simulation (focus of original project)
 Electromechanical problems (dynamo, plate condenser)
 Multiple pendulum
 Robotic arm

Criteria for
index 3 or higher:
CV-loops
&
LI-cutsets

Index Determination in DAEs using indexdet and ADOL-C 2

Background

Project D7
Numerical simulation of integrated circuits
for future chip generations

• MATHEON - Mathematics for key technologies: Modeling,
simulation and optimization of real-world processes
• MATHEON is a Research Centre funded by the DFG, Deutsche
Forschung Gemeinschaft (German Research Foundation)


Background

D7 history
 First funding period
 Circuit-device coupled simulations
 Determination of suitable discretizations
 Second funding period, first stage (Tischendorf)
 Incorporation of a system integrator
 Discretization of new semiconductor device models
 Inclusion of 2D models by applying the Scharfetter-Gummel approach

Occurring challenges
 Determination of the tractability index treated in
 Computation of consistent initial values second stage


Index determination

DAEs given by the general equation:

f ((d ( x(t ))' , x(t ))  0, t  I z (t )  d ( x (t ))'

“Random” path x (t ) yields linearized system with coefficients:

f
A(t )  ( z (t ), x (t )),
z
f
B (t )  ( z (t ), x (t )),
z
d
D(t )  ( x (t ))
x


Index determination

Continuous matrix function sequence:

( if det( Gi 1 )  0
return index = i + 1 )

[R.Lamour. Index Determination and Calculation of
Consistent Initial Values for DAEs. Computers and
Mathematics with Applications, 50:1125-1140, 2005]

Originally: differentiation approximated by finite differences!!!

Repeated Differentiations  Taylor series arithmetic
+
Shift op. on original specification!!


Results

“Algorithmic Differentiation” yields the matrices of polynomials
A(t ), B (t ), D(t )
No truncation errors

Term Q1,13 rel.err.
(t - 1)0 0.0 Quadratic complexity in degree
(t - 1)1 5.000e-16 Elapsed running time by the program varying the number of Taylor
coefficients (averages over 20 independent runs)
(t - 1)2 2.000e-16
(t - 1)3 1.478e-15 0,7

0,6
(t - 1)4 6.153e-15
Elapsed running time (sec)
0,5
Term Q2,13 rel.err. 0,4

(t - 1)0 6.790e-14 0,3

(t - 1)1 1.404e-13 0,2

(t - 1)2 2.094e-13 0,1

(t - 1)3 2.785e-13 0
1 2 3 4 5 10 30 50 70 90 110 130 160 190 230 270 310

(t - 1)4 – Nr. of Taylor coefficients


Use of Algorithmic Differentiation techniques
problem dimensions,
ADOL-C
t0 , x, d ( x, t ), f ( z , x, t )
active and passive variables,
User class ‘asectra‘ initial Taylor coefficients of t
active section and tag 0
daeIndexDet.cpp forward mode
calculation of the
Global trajectory, x (t )
parameters Taylor coefficients of independent (i.e., t)
and dependent variables (i.e. x (t ) )
E.g. print out parameters,
degree of Taylor coefficients

‘asecdyn‘ active and passive variables,
active section and initial Taylor coefficients of x (t ) and t
calculation of the tag 1
dynamic, d ( x (t ), t ) forward and
and its derivative reverse modes
z (t )  d ( x(t ), t )
using shift operator
Taylor coefficients of independent
(i.e., t and x (t ) ) and dependent variables
(i.e. d ( x (t ), t ) ) and adjoints

active and passive variables,
initial Taylor coefficients of d( x (t ), t ),
‘asecdae‘ x (t ) and t
active section and tag 2
forward and
calculation of the
reverse modes
DAE, f ( z (t ), x (t ), t )

Taylor coefficients of independent
(i.e., d( x (t ), t ) and x (t )) and dependent
variables (i.e. f ( z (t ), x (t ), t ) ) and adjoints

Construct
matrices
A, B, and D


Workflow overview
User class
‘asecdyn‘
Calculate
‘asectra‘ ‘asecdae‘ Construct ‘hqrfcp‘
dynamic d ( x (t ), t )
daeIndexDet.cpp Calculate Calculate matrices Householder QR
and its derivative
trajectory x (t ) DAE f ( z (t ), x (t ), t ) A, B, and D of matrix A
z (t )  d ( x(t ), t )
using shift operator
rA
Global
parameters Init. + AD ‘hqrfcp‘
Householder QR
ADOL-C of matrix D

Matrix sequence loop rD
Not properly
Initial computations stated rA  rD
leading term YES
Terminate
NO
YES YES
Initialize matrices r0 ‘hqrfcp‘
Compute Compute NO
2 dim  m and variables rA  r0 Householder QR
D =G0 * A V0, U01, G0
i = 1, dim = r0 of matrix G0=A*D

NO

NO NO NO when i = 0,
i++
Qi2  Qi  0 Qi * Qi1  0 Gi * Qi  0
Compute Pi, Wi, Qi P0  D * D

YES YES NO
YES YES
Error


Index determination and AD

daeIndexDet:
A program for the index determination in DAEs
using the indexdet library and the ADOL-C package for
Automatic Differentiation

indexdet.lib
adolc.lib
matrixseq.h

EDynamo1.h daeIndexDet.cpp daeIndexDet.exe



daeIndexDet:

indexdet.lib
adolc.lib
matrixseq.h



EDynamo1.h

#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_
#include "IDExample.h"
class EDynamo1 : public IDExample
{
public:
EDynamo1(void) : IDExample(4, 5, 0.1, "EDynamo1-01")
{}
void tra(adouble t, adouble *ptra)
{}
void dyn(adouble *px, adouble t, adouble *pd)
{}
void dae(adouble *py, adouble *px,
adouble t, adouble *pf)
{}
};
#endif /*EDYNAMO1_H_*/


EDynamo1.h

#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_ Class name
{
public:
{}
{}
{}
{}
};


EDynamo1.h

#ifndef EDYNAMO1_H_ Nr. equations in dyn(...),
#define EDYNAMO1_H_ nr. equations in dae(...),
#include "IDExample.h" t0, and
text used for printout
{
public:
{}
{}
{}
{}
};


EDynamo1.h

#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_ Trajectory
{
public:
{}
{/*...*/}
{/*...*/}
{/*...*/}
};


EDynamo1.h

#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_ Dynamic
{
public:
{}
{/*...*/}
{/*...*/}
{/*...*/}
};


EDynamo1.h

#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_ DAE
{
public:
{}
{/*...*/}
{/*...*/}
{/*...*/}
};


EDynamo1.h

#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_
{
public:
{}
{/*...*/}
{
{/*...*/} /*...*/
void dae(adouble *py, dp - v;
pf[0] = adouble *px,
adouble t, = dv - f( *pf)
pf[1]
adouble v, jL, t, k0, k, m, alpha );
pf[2] = C*de + G*e - jL;
{/*...*/} pf[3] = dfi - e;
}; pf[4] = fi - fiL( v, t, k0, k, r );
}



daeIndexDet:

indexdet.lib
adolc.lib
matrixseq.h



Classes and supporting files
38 114

IDException.h IDException.cpp

110 965 53 517

TPolyn.h TPolyn.cpp ioutil.h ioutil.cpp
33
164 3107
defaults.h
adolc.lib Matrix.h Matrix.cpp

42 33 76 2780

IDOptions.h defs.h linalg.h linalg.cpp

195 79 66 3280

IDOptions.cpp IDExample.h matrixseq.h matrixseq.cpp

207 193

EDynamo1.h daeIndexDet.cpp


Example: Bike Dynamo

(In cooperation with project MATHEON Project D13)

Electromechanical problem

p : position of the mass point

v : velocity

e : voltage
  : magnetic flow
jL : conductance
m, k , k0 , C , G are known
L R C


Example: Bike Dynamo
(Index 4)

pv
~
  f ( v, j L , t )  
v
0  p  z (t )

Ce  Ge  jL  0

 e  0
  L (v, t ).
with
~ f 2
f (v, jL , t )  mech  k0 jL sin(2kvt )
m mk
L (v, t )  k0 r cos(2kvt )
f mech  gm tan( )
E.g. t  0 Singular point

The new method correctly computes the index even at problem singularities!!!


Example: Bipolar ring oscillator with inductor
Circuit simulation (Index 2)
G (e1  e11 )  jC1  j B 3  j1, 3  0
G (e2  e11 )  jC 2  j B 4  j2, 3  0
jE 3  j E 4  i  j1,3  j2, 3  0
G (e4  e11 )  jC 3  j B 5  j4, 6  0
G (e5  e11 )  jC 4  j B 6  j5, 6  0
jE 5  jE 6  i  j4, 6  j5, 6  0
v G (e7  e11 )  jC 5  jB1  j7 ,9  0
ja ,b 
 q (ea  eb ) q (v)  4 10 11 vB 1 
vB G (e8  e11 )  jC 6  jB 2  j8,9  0
jB  jBE  jBC
jE1  j E 2  i  j7 ,9  j8,9  0
v BE v BC
IS vT IS vT jv  3i  0
 (e  1)  (e  1)
 r u10  v  0
jC  jBE  (1   r ) jBC Lj 'L  e11  0
L,  ,  r , v, vT , vB , G, I S , i are known G (6e11  e1  e2  e4  e5  e7  e8 )  jL  0


Achievements

Main achievements

 Exact differentiations without explicit specification of derivatives
expressions
 High accurate results (e.g. checking eps-conditions: equal to 0 up
to machine precision). Accurate calculation of the index /
consistent initial values (for the linear case)
 New matrix-algebra operations to deal with AD (e.g.
implementation of special matrix-matrix multiplications, QR
decomposition of matrices of Taylor polynomials, etc.) with
operator overloading in C++
 New program and library for the index determination and the
consistent initialization


Software

daeIndexDet: Program for the index determination in DAEs
indexdet: Corresponding library
Using Algorithmic Differentiation techniques (ADOL-C package for AD)
http://www.mathematik.hu-berlin.de/~monett/indexdet/indexdet.html


Consistent initialization

• Lots of variables are used from previous computations and structure
• Classes and supporting files are already available

Algorithm
Taylor coefficients (ADOL-C) of
x, d, f and matrixes A, B, D
Index computation,
projectors and other variables
Calculate variables vi
and conform solution x
Calculate
xi+1 = xi + λ zi
Verify convergence


Further work

1. Computation of consistent initial values
 Nonlinear case
2. Sparse implementation
 Sparse LU-based implementation using Taylor arithmetic
3. Structural analysis
 Exploration of extension of [Pantelides/Pryce] to computational graphs

4. If 3. promising

Development and implementation of method


Topic

DAE Example f  x, h ( x, y )   0
 
Corresponding
g  y , h ( x, y )   0
  computational graph
x , y , h, f , g 
x
f
Jacobian 
x h
 f 
  
y
 ( f , g )  h   h h  g
    y
 ( x, y )  g   x
    y 

 
 h 
has rank  1

 
Transformation to ODE for ( x, y ) impossible!
Structural Analysis à la Pantelides fails!!!


Plans for 2008/2009 and beyond

Solution of introductory example based on graph

Expanded system Maximal Degree of Variables
 
z  h ( x, y )  0 x y z
1 1 0
f ( x, z )  0
0 – 0
g ( y, z )  0
– 0 0


Plans for 2008/2009 and beyond

Solution of introductory example based on graph

Expanded system Maximal Degree of Variables
 
z  h ( x, y )  0 x y z
1 1 0
f ( x, z )  0
0 – 0
g ( y, z )  0
– 0 0

Occurrence of derivatives determines
structural index via maximal transversal

Combinatorial subtasks:
 Identification of replicated subgraphs
 Maximal Weighted Matching on DAG
 Linear Assignment Problem on Matrix

Index Determination in DAEs using the Library indexdet and the ADOL-C Package for Algorithmic Differentiation

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