Peter Aronsson

Automatic Parallelization of Simulation Code from Equation Based Simulation Languages

Peter Aronsson,

Industrial phd student, PELAB SaS IDA

Linköping University, Sweden

Based on Licentiate presentation & CPC’03 Presentation

Peter Aronsson

Outline

• Introduction

• Task Graphs

• Related work on Scheduling & Clustering

• Parallelization Tool

• Contributions

• Results

• Conclusion & Future Work

Peter Aronsson

Introduction

• Modelica– Object Oriented, Equation Based, Modeling Language

• Modelica enable modeling and simulation of large and complex multi-domain systems

• Large need for parallel computation– To decrease time of executing simulations

– To make large models possible to simulate at all.

– To meet hard real time demands in hardware-in-the-loop simulations

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Examples of large complex systems in Modelica

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Modelica Example - DCmotorR1 I1

emf

ground

stepload

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Modelica examplemodel DCMotor import Modelica.Electrical.Analog.Basic.*; import Modelica.Electrical.Sources.StepVoltage; Resistor R1(R=10); Inductor I1(L=0.1); EMF emf(k=5.4); Ground ground; StepVoltage step(V=10); Modelica.Mechanics.Rotational.Inertia load(J=2.25); equation connect(R1.n, I1.p); connect(I1.n, emf.p); connect(emf.n, ground.p); connect(emf.flange_b, load.flange_a); connect(step.p, R1.p); connect(step.n, ground.p);end DCMotor;

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Example – Flat set of EquationsR1.v = -R1.n.v+R1.p.v 0 = R1.n.i+R1.p.i R1.i = R1.p.i R1.i*R1.R = R1.v I1.v = -I1.n.v+I1.p.v 0 = I1.n.i+I1.p.i I1.i = I1.p.i I1.L*I1.der(i) = I1.v emf.v =-emf.n.v+emf.p.v 0 = emf.n.i+emf.p.i emf.i = emf.p.i emf.w = emf.flange_b.der(phi) emf.k*emf.w = emf.v emf.flange_b.tau = -emf.i*emf.k ground.p.v = 0 step.v = -step.n.v+step.p.v 0 = step.n.i+step.p.i step.i = step.p.i step.signalSource.outPort.signal[1] = (if time < step.signalSource.p_startTime[1] then 0 else step.signalSource.p_height[1])+step.signalSource.p_offset[1] step.v = step.signalSource.outPort.signal[1] load.flange_a.phi = load.phi load.flange_b.phi = load.phi load.w = load.der(phi) load.a = load.der(w) load.a*load.J = load.flange_a.tau+load.flange_b.tau R1.n.v = I1.p.v I1.p.i+R1.n.i = 0 I1.n.v = emf.p.v emf.p.i+I1.n.i = 0 emf.n.v = step.n.v step.n.v = ground.p.v emf.n.i+ground.p.i+step.n.i = 0 emf.flange_b.phi = load.flange_a.phiemf.flange_b.tau+load.flange_a.tau = 0 step.p.v = R1.p.v R1.p.i+step.p.i = 0 load.flange_b.tau = 0 step.signalSource.y = step.signalSource.outPort.signal

Peter Aronsson

0.5 1 1.5 2

1

2

3

4

5

•load.flange_a.tau

•load.w

•load.flange_a.tau

•load.w

Plot of Simulation result

Peter Aronsson

Task Graphs

• Directed Acyclic Graph (DAG)G = (V,E, ,c)V – Set of nodes, representing computational tasksE – Set of edges, representing communication of data

between tasks(v) – Execution cost for node v c(i,j) – Communication cost for edge (i,j)

• Referred to as the delay model (macro dataflow model)

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Small Task Graph Example

12

32

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41

52

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81

5 10

5 5 5

10 1010

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Task Scheduling Algorithms• Multiprocessor Scheduling Problem

– For each task, assign• Starting time• Processor assignment (P1,...PN)

– Goal: minimize execution time, given• Precedence constraints• Execution cost• Communication cost

• Algorithms in literature– List Scheduling approaches (ERT, FLB)– Critical Path scheduling approaches (TDS, MCP)

• Categories: Fixed No. of Proc, fixed c and/or , ...

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Granularity

• Granularity g = min((v))/max(c(i,j)) • Affects scheduling result

– E.g. TDS works best for high values of g, i.e. low communication cost

• Solutions:– Clustering algorithms

• IDEA: build clusters of nodes where nodes in the same cluster are executed on the same processor

– Merging algorithms• Merge tasks to increase computational cost.

Peter Aronsson

Task Clustering/Merging Algorithms

• Task Clustering Problem:– Build clusters of nodes such that parallel time decreases

– PT(n) = tlevel(n)+blevel(n)

– By zeroing edges, i.e. putting several nodes into the same cluster => zero communication cost.

• Literature:– Sarkars Internalization alg., Yangs DSC alg.

• Task Merging Problem– Transform the Task Graph by merging nodes

• Literature: E.g. Grain Packing alg.

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Clustering v.s. Merging

12

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0 0 0

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Clustered Task Graph

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mer

ging

Merged Task Graph

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3,66

2,5,64

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1010

10

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DSC algorithm

1. Initially, put each node a separate cluster.

2. Traverse Task Graph– Merge clusters as long as Parallel Time does

not increase.

• Low complexity O((n+e) log n)

• Previously used by Andersson in ObjectMath (PELAB)

Peter Aronsson

Modelica Compilation

a...

a...

a...

a...

a...

a...

inertialxy

r...

r...

r...

r...

r...

r...

b0 c...

b...

b...

b...

b...

b...

b...

b...

c...

b...

b...

l...

r...

Modelica model (.mo)

Modelica semantics

Equation system(DAE)

Opt. Rhscalculations

Flat modelica (.mof)

Numericalsolver

C code

Structure of simulation code:for t=0;t<stopTime;t+=stepSize { x_dot[t+1] = f(x_dot[t],x[t],t); x[t+1] = ODESolver(x_dot[t+1]);}

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Optimizations on equations• Simplification of equations

E.g. a=b, b=c eliminate => b

• BLT transformation, i.e. topological sorting into strongly connected components(BLT = Block Lower Triangular form)

• Index reduction, Index is how many times an equation needs to be differentiated in order to solve the equation system.

• Mixed Mode /Inline Integration, methods of optimizing equations by reducing size of equation systems

ab

cd

e

0

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Generated C Code Content• Assignment statements• Arithmetic expressions (+,-,*,/), if-expressions• Function calls

– Standard Math functions• Sin, Cos, Log

– Modelica Functions• User defined, side effect free

– External Modelica Functions• In External lib, written in Fortran or C

– Call function for solving subsystems of equations• Linear or non-linear

• Example Application– Robot simulation has 27 000 lines of generated C code

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Parallelization Tool Overview

ModelicaCompiler

C compiler

Model.mo

C code

C compiler

ParallelizerParallelizer

Parallel C codeSolver

libMPIlib

Seq exe

Parallel exe

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Parallelization Tool Internal Structure

Parser

Task Graph Builder

Symbol Table

Scheduler

Code Generator Debug & Statistics

Sequential C code

Parallel C code

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Task Graph building• First graph: corresponds to individual

arithmetic operations, assignments, function calls and variable definitions in the C code

• Second graph: Clusters of tasks from first task graph

Example:

+

- *

foo -/

+

*

a b cd

defs

+,-,* +,*

foo /,-

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Investigated Scheduling Algorithms

• Parallelization Tool – TDS (Task Duplications Scheduling Algorithm)– Pre – Clustering Method– Full Task Duplication Method

• Experimental Framework (Mathematica)– ERT– DSC– TDS– Full Task Duplication Method– Task Merging approaches (Graph Rewrite Systems)

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Method 1:Pre Clustering algorithm

– buildCluster(n:node, l:list of nodes, size:Integer)

– Adds n to a new cluster– Repeatedly adds nodes until the

size(cluster)=size – Children to n– One in-degree children to cluster– Siblings to n– Parents to n– Arbitrary nodes

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Managing cycles• When adding a node to a

cluster the resulting graph might have cycles

• Resulting graph when clustering a and b is cyclic since you can reach {a,b} from c

• Resulting graph not a DAG– Can not use standard scheduling

algorithms

a

b

c

de

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Pre Clustering Results

• Did not produce Speedup– Introduced far too many dependencies in

resulting task graph– Sequentialized schedule

• Conclusion:– For fine grained task graphs:

• Need task duplication in such algorithm to succeed

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Method 2: Full Task Duplication• For each node:n with successor(n)={}

– Put all pred(n) in one cluster• Repeat for all nodes in cluster

– Rationale: If depth of graph limited, task duplication will be kept at reasonable level and cluster size reasonable small.

– Works well when communication cost >> execution cost

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Full Task Duplication (2)

• Merging clusters1. Merge clusters with load balancing strategy,

without increasing maximum cluster size

2. Merge clusters with greatest number of common nodes

• Repeat (2) until number of processors requirement is met

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Full Task Duplication Results

• Computed measurements– Execution cost of largest cluster +

communication cost

• Measured speedup– Executed on PC Linux

cluster SCI network interface,

using SCAMPI

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Robot Example Computed Speedup

• Mixed Mode / Inline Integration

1 2 4 9# Proc

0.250.5

0.751

1.251.5

1.752

Speedup

c10

c100

c1000

1 2# Proc

0.250.5

0.751

1.251.5

1.752

Speedup

c10

c100

c1000

With MM/IIWithout MM/II

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Thermofluid pipe executed on PC Cluster

• Pressurewavedemo in Thermofluid package 50 discretization points

1 2 4 8 16# Proc

0.25

0.5

0.75

1

1.25

1.5

1.75

2Speedup

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Thermofluid pipe executed on PC Cluster

• Pressurewavedemo in Thermofluid package 100 discretization points

1 2 4 8 16# Proc

0.5

1

1.5

2

2.5

3Speedup

Peter Aronsson

Task Merging using GRS

• Idea: A set of simple rules to transform a task graph to increase its granularity (and decrease Parallel Time)

• Use top level (and bottom level) as metric:

• Parallel Time = max tlevel + max blevel

tleveln 0 , predn maxkpredntlevelk k L ck,n

B, predn

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Rule 1

• Merging a single child with only one parent.

• Motivation: The merge does not decrease amount of parallelism in the task graph. And granularity can possibly increase.

p

ctlevelj, j prednp p’

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tlevelc maxpipredctlevelpipj

pjpredc

c

Rule 2• Merge all parents of a node together with the node

itself.

• Motivation: If the top level does not increase by the merge the resulting task will increase in size, potentially increasing granularity.

p1

c c’

p2 pn

tlevelj, j prednpi, pi

…

Peter Aronsson

p L cp, ciB

, i 1..n

tlevelci tlevelpj L cpj, ciB

p i 1..n pj predcipj p

Rule 3• Duplicate parent and merge into each child node

• Motivation: As long as each child’s tlevel does not increase, duplicating p into the child will reduce the number of nodes and increase granularity.

c2

p

tlevelj, j prednpcnc1

c2’ cn’c1’…

…

Peter Aronsson

p1 p2 ... pnmax k ,

i1

k

pi maxSize, 1 k n

Rule 4• Merge siblings into a single node as long as a parameterized

maximum execution cost is not exceeded.

• Motivation: This rule can be useful if several small predecessor nodes exist and a larger predecessor node which prevents a complete merge. Does not guarantee decrease of PT.

tlevelj, j prednpp1

c

p2 pnp´

c

Pk+1 pn… …

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Results – Example

• Task graph from Modelica simulation code– Small example from the

mechanical domain.

– About 100 nodes built on expression level, originating from 84 equations & variables

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Result Task Merging example

• B=1, L=1

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Result Task Merging example

– B=1, L=10

– B=1, L=100

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Conclusions

• Pre Clustering approach did not work well for the fine grained task graphs produced by our parallelization tool

• FTD Method– Works reasonable well for some examples

• However, in general: – Need for better scheduling/clustering

algorithms for fine grained task graphs

Peter Aronsson

Conclusions (2)

• Simple delay model may not be enough– More advanced model require more complex

scheduling and clustering algorithms

• Simulation code from equation based models– Hard to extract parallelism from– Need new optimization methods on DAE:s or

ODE:s to increase parallelism

Peter Aronsson

Conclusions Task Merging using GRS

• A task merging algorithm using GRS have been proposed– Four rules with simple patterns => fast pattern matching

• Can easily be integrated in existing scheduling tools.• Successfully merges tasks considering

– Bandwidth & Latency– Task duplication– Merging criterion: decrease Parallel Time, by decreasing

tlevel (PT)

• Tested on examples from simulation code

Peter Aronsson

Future Work

• Designing and Implementing Better Scheduling and Clustering Algorithms– Support for more advanced task graph models– Work better for high granularity values

• Try larger examples• Test on different architectures

– Shared Memory machines– Dual processor machines

Peter Aronsson

Future Work (2)

• Heterogeneous multiprocessor systems– Mixed DSP processors, RISC,CISC, etc.

• Enhancing Modelica language with data parallelism– e.g. parallel loops, vector operations

• Parallelize e.g. combined PDE and ODE problems in Modelica.

• Using e.g. SCALAPACK for solving subsystems of linear equations. How to integrate into scheduling algorithms?