executing models in less time—some solver insight

1© 2011 The MathWorks, Inc.

Executing models in less time—some solver insight

Pieter J. Mosterman

2

What all may affect simulation performance?

� Model– Initialization (images, ML script, set_params, …)

– Execution

– Code generation

� Solver� Processing (Engine, Code Generator, Compiler)

– Optimization

– Diagnostics

� Interaction– Debugging

– Logging

– Viewing

� Simulation mode� Platform� Use scenarios

– Simstate

http://www.mathworks.com/company/events/webinars/webinarconf.html?id=54186&language=en

3

The make up of a simulation

initialize compilemodel(ctrl+d)

generatecode

compilecode

executeloop

terminate

simulation timebudget

• Integrate• Reduce step• Find zero crossing• Solve algebraic loop• …

4

Comparing performance of different simulation modes

Help Search: comparing performance

must rebuild code fromblocks (e.g., model reference blocks, MATLAB Function blocks, Stateflow charts)

No need to rebuild code from blocks

5

Agenda

� Constructing models with increasing fidelity� Execution of models explained

– Fixed step solver methods

– Variable step solver methods

– Zero crossings

� Conclusions

6

Host stack Target stack

Elaborate

Synthesize

Raises level of abstraction

Enables continuous testing

Model-Based Design

High-level design Detailed design Implementation

ExploreVerifyTest

ExploreVerifyTest

ExploreVerifyTest

ExploreVerifyTest

7


Create executable models in all phases

8


Make the computational approximation the primary design deliverable—the real thing!

9

Agenda


– A reference model



– Zero crossings

� Conclusions

10

A Simscape model—what solver to choose?

11

Using a Simscape solver to generate reference behavior

12

The corresponding Simulink model with an ideal diode

13

Agenda





– Zero crossings

� Conclusions

14

ode1: forward Euler numerical integration

14

),( txfx =�

kt

kx�

kkkke htxtxtx )()()(ˆ 1 �+=+

kx

1+kt

1+kx

1ˆ +kx

),( txφ

kkk tth −= +1

)()( 21 kk hOt =+ε

)(!2

)(

!1

)()()( 32

1 kkk

kk

kk hOhtx

htx

txtx +++=+��

Step h in time

Along instantaneous vector component

Gives the estimate

Comparison with Taylor series expansion

Gives error of the estimate

15

ode1: stability analysis

1)0(; == xkxx� ktetx =)(

kkkke htxtxtx )()()(ˆ 1 �+=+

)(ˆ)1()(ˆ)(ˆ)(ˆ)(ˆ 1 ken

nkekekeke txkhtxtxkhtxtx +=⇒+= ++

{ }11|, <+∈⇒=∈ zCzkhzCk

Re(z)

Im(z)

In MATLAB:>> clear i>> [X,Y]= meshgrid(-3:0.01:1,-3:0.01:3);>> Mu = X + i*Y;>> R = 1 + Mu;>> Rhat = abs(R);>> contour(X,Y,Rhat,[1, 1],’b’)>> grid>> hold>> plot([0 0],[-3 3],’k’)

16

Trapezoidal numerical integration

16

)()()(

)( 1k

k

kkk hO

h

txtxtx +−= + ��

�� kt 1+kt

kx1+kx

kkk

kkt htxtx

txtx2

)()()()(ˆ 1

1

�� ++= ++

)(2

)(

2

)()()()( 31

1 kkk

kk

kkkk hOhtx

htx

htxtxtx +−++= ++

��

)(ˆ 1+kt tx

Average beginning and end point

Finite difference …

… to approximate Taylor series expansion

)(!2

)(

!1

)()()( 32

1 kkk

kk

kk hOhtx

htx

txtx +++=+��

)( 1+ktε

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ode2: Heun method

� But, we do not have the value at tk+1 because that is what we are trying to compute

� So, combine trapezoidal with Euler approximation

kkk

kkt htxtx

txtx2

)()()()(ˆ 1

1

�� ++= ++

( )kkk ttxftx ),()( =�

( )11 ),()()( ++ += kkkk ttxhtxftx ��

18

ode2: (Heun) stability

In MATLAB:>> clear i>> [X,Y]= meshgrid(-3:0.01:1,-3:0.01:3);>> Mu = X + i*Y;>> R = 1 + Mu;>> Rhat = abs(R);>> contour(X,Y,Rhat,[1, 1],’b’)>> grid>> hold>> plot([0 0],[-3 3],’k’)>> R = 1 + Mu + .5*Mu.^2;>> Rhat = abs(R);>> contour(X,Y,Rhat,[1, 1],’m’)

Stability area of Runge-Kutta methods increases with order

<++∈⇒=∈ 1

2

11|, 2zzCzkhzCk

19

In general, varying order of numerical solvers

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

Two-stage Multi-stage

kt

kx

1+kt

1+kx

One-stage

For example: ode1, ode2, ode3, ode4, ode5, ode8

More accurate and stable as order increases

20

Stiff systems mix behavior at widely differing time scales

21

Fixed step and solver order must be chosen carefully—ode1, 6e-5

22


23


24


25


26


27

Step size and order determine simulation time

28


29


30

But these solvers really used an explicit formulation

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

Two-stage Multi-stageOne-stage

31

How about a truly implicit approach?

� Trapezoidal integration– Requires future points!

– Impossible?

� Can do, but requires inverting the system

kt

kx

1+kt

1+kxk

kkkkt h

txtxtxtx

2

)()()()(ˆ 1

1

�� ++= ++

kktkt

ktkt htxAtxA

txtx2

)(ˆ)(ˆ)(ˆ)(ˆ 1

1

++= ++

( ) ( ) )(ˆ2)(ˆ2 1 ktkt txAhItxAhI +=− +

( ) ( ) )(ˆ22)(ˆ 11 ktkt txAhIAhItx +−= −

+

32

� Trapezoidal is stable in entire left half plane

� But, expensive!– Linearization

– Inversion

Plusses and deltas?

In MATLAB:>> clear i>> [X,Y]= meshgrid(-3:0.01:1,-3:0.01:3);>> Mu = X + i*Y;>> R = 1 + Mu;>> Rhat = abs(R);>> contour(X,Y,Rhat,[1, 1],’b’)>> grid>> hold>> plot([0 0],[-3 3],’k’)>> R = 1 + Mu + .5*Mu.^2;>> Rhat = abs(R);>> contour(X,Y,Rhat,[1, 1],’m’)>> R = (2 + Mu)./(2-Mu);>> Rhat = abs(R);>> contour(X,Y,Rhat,[1, 1],’r’)

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Advantage: a large step size is possible

34

Still stable with very large step size

35

Agenda





– Zero crossings

� Conclusions

36

A fixed step size has to be stable for the fastest behavior anywhere in a simulation

kt

kx

1−kt kt

kx

1−kt

kt

kx

1+kt

1+kx

1−kt

From k-1 to kFast behavior, so a small step is necessary

From k to k+1Slow behavior, so a large step is possible

Is the ‘fast’ behavior holding us hostage?

37

Step size control

� Compute Euler approximation

� Compute Heun approximation

� Compare the results for the error estimate, ε

� Reduce step size from hmax till ε < tol

– For example, bisection

37

( ) kkkke htxftxtx )()()(ˆ 1 +=+

( ) ( )k

kkekkt h

txftxftxtx

2

)()(ˆ)()(ˆ 1

1

++= ++

)(!2

)()()( 1

211 +++ =≈− kek

kktke th

txtxtx ε

��

38

Tolerance consists of two components

� Relative tolerance depends on signal magnitude� Absolute tolerance is constant

0

abstol

-abstol

magnitude + reltol * magnitude

magnitude - reltol * magnitude

magnitude

39

Approximations determined by absolute and relative tolerance

� Absolute tolerance prevents infinitely accurate solution around 0

0

abstol

-abstol



magnitude

40

Approximations determined by absolute and relative tolerance

� Combined relative and absolute tolerance drive solver step size selection

0

abstol

-abstol



magnitude

41

Variable step solvers

� ode45– Compares RK methods of order 4 and 5

� ode23– Compares RK methods of order 2 and 3

� ode23tb– Trapezoidal (implicit) integration with BDF error estimate

But now we better be careful with our computational complexity?

42

A variable-step solver may actually take much less time to simulate

43

And we can do even better: multi-step solvers reuse effort

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

1−kt

Implicit multi-stage(e.g., ode23tb, ode23t)

Explicit multi-step(e.g., ode113)

1−kt kt

kx

1+kt

1+kx

1−kt

Implicit multi-step(e.g., ode15s)

44

So, yet more efficient integration

ode45: 1.055492

ode113: 0.8866972

multi-stage is ~19% slower than multi-step

45

But the reuse may turn against us

1−t

kx

0t

1+kx

2−t

We do not have initial values at t-2 and t-1

How do we build this history?• Single-step integration algorithm• Either implicit or explicit with very small step size, ε

How about t1 in case of implicit methods?• Linearize system and invert system matrix

1t

But now when we reach a discontinuity …… smoothness is violated … solver reset!

46

Many discontinuities require many solver resets

diode current

47

A single-step solver …

48

…versus a multi-step solver …

49

… and another multi-step solver …

50

But there is more to it than just the single step nature

51

Or even …

52

Note: extrapolation becomes unreliable for high order

kt

kx

1+kt

1+kx

1−kt

ktkx�

1+kt

1+kx�

1−kt

1−kx�

0th order

1st order

2nd order

Stability area of multistep methods (explicit: ode113, implicit: ode15s) decreases with order!

)(tx

)(tx�

53

Agenda




– Zero crossings

� Conclusions

54

Discontinuities

� Variable-step solvers ‘zoom in’ on zero crossings– Is solver step size reduction an efficient mechanism?

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

55

� How about we use a dedicated root-finding algorithm?– Bisection, Newton-Raphson

� Disregard the discontinuity, then search

Discontinuities

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

kt

kx

1+kt

1+kx

56

Reduced simulation time for ode23t

Zero-crossing location OFF>> tic;sim(gcs,’reltol’,’1e-4’,’Solver’,’ode23t’,’stoptime’,’0.5’);tocElapsed time is 6.255886 seconds.

Zero-crossing location ON>> tic;sim(gcs,’reltol’,’1e-4’,’Solver’,’ode23t’,’stoptime’,’0.5’);tocElapsed time is 5.634365 seconds.

57

But not for ode113 … because it resets its history anyway

Zero-crossing location OFF>> tic;sim(gcs,’reltol’,’1e-4’,’Solver’,’ode113’,’stoptime’,’0.5’);tocElapsed time is 3.947801 seconds.

Zero-crossing location ON>> tic;sim(gcs,’reltol’,’1e-4’,’Solver’,’ode113’,’stoptime’,’0.5’);tocElapsed time is 4.155692 seconds.

58

Is computational complexity the only zero crossing issue?

kt

kx

1+kt

1+kx

2+kt

2+kx

kt

kx

1+kt

1+kx

59

Chattering in our electrical circuit

60

Resolve the chattering by adaptive zero crossings

61

Or, employ a Simscape fixed-step solver with fixed cost

62

But a fixed-cost solver may be less accurate

63

But a fixed-cost solver may be less accurate

64

Agenda




– Zero crossings

� Conclusions

65

Conclusions

� Solvers selection is a trade off– Stability

– Accuracy

– Time to simulate

� Compare different solvers and solver parameters– Use a computationally expensive variable-step solver to

generate a reference solution

– If the model is sensitive, investigate why

66

Characterization

� My model– Must have a predictable

execution time (could be long)

– Must have a short simulation time

– Has widely varying time constants (stiff)

– Includes discontinuities

– Includes many discontinuities

– Exhibits chattering

– Should not be dissipative

� My solver– Fixed step vs. variable

step

– Explicit vs. implicit

– Single step vs. multi step

– Zero crossing location or not

NB: Fixed step solvers do not do root finding

67

Characterization









� My solver– Fixed-step integration

� Experiment with integration order, step size, and accuracy

68

Characterization










� If the accuracy suffices

– Variable-step integration� Multi-step methods are

preferred� Zero-crossing on may

shorten simulation time

69

Characterization









� My solver– Preferably implicit

– Fixed step� ode14x

– Variable step� ode15s, ode23s, ode23t,

ode23tb

70

Characterization









� My solver– Variable-step integration

� No integration history (at least single step) with zero crossing location

� ode23t

71

Characterization










� If the accuracy suffices� Choose the fundamental

sample time as the step size

72

Characterization










� If the accuracy suffices

– Variable-step integration� Adaptive zero crossing

location

73

Characterization









� My solver– Nondissipative integration

method� ode23t

74

And …

Do not fall in love with a model --- Jacques LeFèvre

75

®

executing models in less time—some solver insight

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