1. 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

2. Peter Aronsson Outline Introduction Task Graphs Related work on Scheduling & Clustering Parallelization Tool Contributions Results Conclusion & Future Work

3. 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

4. Peter Aronsson Examples of large complex systems in Modelica

5. Peter Aronsson Modelica Example - DCmotor

6. Peter Aronsson Modelica example model 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;

7. Peter Aronsson Example – Flat set of Equations R1.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.phi emf.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

8. Peter Aronsson load.flange_a.tau load.w load.flange_a.tau load.w Plot of Simulation result

9. Peter Aronsson Task Graphs Directed Acyclic Graph (DAG) G = (V,E, ,c) V – Set of nodes, representing computational tasks E – 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)

10. Peter Aronsson Small Task Graph Example 1212 3232 2121 4141 5252 6262 7171 8181 5 10 55 5

11. Peter Aronsson Task Scheduling Algorithms Multiprocessor Scheduling Problem –For each task, assign Starting time Processor assignment (P 1,...P N ) –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 ,...

12. Peter Aronsson 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.

13. 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.

14. Peter Aronsson Clustering v.s. Merging 1212 3232 2121 4141 5252 6262 7171 8181 50 00 0 00 10 Clustered Task Graph 1212 3232 2121 4141 5252 6262 7171 8181 5 10 5 5 5 merging Merged Task Graph 1212 3,6 6 2,5,6 4 7171 8181 5 10

15. Peter Aronsson 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)

16. Peter Aronsson Modelica Compilation Modelica model (.mo) Modelica semantics Equation system (DAE) Opt. Rhs calculations Flat modelica (.mof) Numerical solver 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]); }

17. Peter Aronsson 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 a b c d e 0

18. Peter Aronsson 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

19. Peter Aronsson Parallelization Tool Overview Modelica Compiler C compiler Model.mo C code C compiler Parallelizer Parallel C code Solver lib MPI lib Seq exe Parallel exe

20. Peter Aronsson Parallelization Tool Internal Structure Parser Task Graph Builder Symbol Table Scheduler Code Generator Debug & Statistics Sequential C code Parallel C code

21. Peter Aronsson 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 - / + * abc d defs +,-,* +,* foo/,-

22. Peter Aronsson 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)

23. Peter Aronsson 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

24. Peter Aronsson 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 d e

25. Peter Aronsson 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

26. Peter Aronsson 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

27. Peter Aronsson Full Task Duplication (2) Merging clusters 1.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

28. Peter Aronsson 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

29. Peter Aronsson Robot Example Computed Speedup Mixed Mode / Inline Integration With MM/II Without MM/II

30. Peter Aronsson Thermofluid pipe executed on PC Cluster Pressurewavedemo in Thermofluid package 50 discretization points

31. Peter Aronsson Thermofluid pipe executed on PC Cluster Pressurewavedemo in Thermofluid package 100 discretization points

32. 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

33. Peter Aronsson 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 c p’

34. Peter Aronsson 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. p1p1 c c’ p2p2 pnpn …

35. Peter Aronsson 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. c2c2 p cncn c1c1 c2’c2’ cn’cn’c1’c1’ … …

36. Peter Aronsson 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. p1p1 c p2p2 pnpn p´ c P k+1 pnpn … …

37. Peter Aronsson 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

38. Peter Aronsson Result Task Merging example B=1, L=1

39. Peter Aronsson Result Task Merging example –B=1, L=10 –B=1, L=100

40. Peter Aronsson 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

41. 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

42. 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

43. 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

44. 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?