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)
Index Determination in DAEs using indexdet and ADOL-C 3

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 in DAEs using indexdet and ADOL-C 4

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 in DAEs using indexdet and ADOL-C 5

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!!
Index Determination in DAEs using indexdet and ADOL-C 6

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
Index Determination in DAEs using indexdet and ADOL-C 7

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
Index Determination in DAEs using indexdet and ADOL-C 8

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 in DAEs using indexdet and ADOL-C 9

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
Index Determination in DAEs using indexdet and ADOL-C 10

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
Index Determination in DAEs using indexdet and ADOL-C 11

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
Index Determination in DAEs using indexdet and ADOL-C 12

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_*/
Index Determination in DAEs using indexdet and ADOL-C 13

EDynamo1.h
#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_ Class name
#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_*/
Index Determination in DAEs using indexdet and ADOL-C 14

EDynamo1.h
#ifndef EDYNAMO1_H_ Nr. equations in dyn(...),
#define EDYNAMO1_H_ nr. equations in dae(...),
#include "IDExample.h" t0, and
class EDynamo1 : public IDExample
text used for printout
{
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_*/
Index Determination in DAEs using indexdet and ADOL-C 15

EDynamo1.h
#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_ Trajectory
#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_*/
Index Determination in DAEs using indexdet and ADOL-C 16

EDynamo1.h
#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_ Dynamic
#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_*/
Index Determination in DAEs using indexdet and ADOL-C 17

EDynamo1.h
#ifndef EDYNAMO1_H_
#define EDYNAMO1_H_ DAE
#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_*/
Index Determination in DAEs using indexdet and ADOL-C 18

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, 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 );
#endif /*EDYNAMO1_H_*/
}
Index Determination in DAEs using indexdet and ADOL-C 19

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
Index Determination in DAEs using indexdet and ADOL-C 20

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
Index Determination in DAEs using indexdet and ADOL-C 21

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
Index Determination in DAEs using indexdet and ADOL-C 22

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!!!
Index Determination in DAEs using indexdet and ADOL-C 23

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
Index Determination in DAEs using indexdet and ADOL-C 24

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
Index Determination in DAEs using indexdet and ADOL-C 25

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
Index Determination in DAEs using indexdet and ADOL-C 26

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
Index Determination in DAEs using indexdet and ADOL-C 27

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
Index Determination in DAEs using indexdet and ADOL-C 28

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!!!
Index Determination in DAEs using indexdet and ADOL-C 29

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
Index Determination in DAEs using indexdet and ADOL-C 30

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 indexdet and ADOL-C 31