Why parallel computing matters (and what makes it hard)

1. Why parallel computing matters (and what makes it hard) • Challenges and opportunities for parallel computing • Moore’s Law • Amdahl’s Law • The Von Neumann Bottleneck • The Speed of Light • The Heat Wall • The 3rd Pillar of Science • Simulation • Technical challenges in parallel computing • Task decomposition • Task decomposition • Data decomposition • Load Balancing • Applied to Sharks ‘n Fishes, solving systems of PDEs

2. Moore’s Law Moore’s Law: the number of transistors per processor chip by doubles every 18 months.

3. Moore’s Law • Gordon Moore (co-founder of Intel) predicted in 1965 that the transistor density of semiconductor chips would double roughly every 18 months. • Moore’s law has had a decidedly mixed impact, creating new opportunities to tap into exponentially increasing computing power while raising fundamental challenges as to how to harness it effectively. • The Red Queen Syndrome • It takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!' • Things Moore never said: • “computers double in speed every 18 months”  • “cost of computing is halved every 18 months”  • “cpu utilization is halved every 18 months” 

4. Snavely’s Top500 Laptop? • Among other startling implications is the fact that the peak performance of the typical laptop would have placed it as one of the 500 fastest computers in the world as recently as 1995. • Shouldn’t we all just go find another job now? • No because Moore’s Law has several more subtle implications and these have raised a series of challenges to utilizing the apparently ever-increasing availability of compute power; these implications must be understood to see where we are today in HPC.

5. Real Work (the fly in the ointment) • Scientific calculations involve operations upon large amounts of data, and it is in moving data around within the computer that the trouble begins. As a very simple pedagogical example consider the expression. A + B = C • This generic expression takes two values (arguments), and performs an arithmetic operation to produce a third. Implicit in such a computation is that A and B must to be loaded from where they are stored and the value C stored after it has been calculated. In this example, there are three memory operations and one floating-point operation (Flop).

6. Real Work cont. • Obviously, if the rate at which the CPU can perform the one Flop in this example increases, the time required to load and store the three values in memory had better decrease by the same proportion if the overall rate of the computation is to continue to track Moore’s Law. As has been noted, the rates of improvement in memory bandwidth have not in fact been keeping up with Moore’s Law for some time [FoSC]. • Amdahl’s Law Speedup(P) = Time(1)/Time(P) <= 1/(s +(1-s)/P) < 1/s, where P be a number of processors and s is the fraction of work done sequentially.

7. Amdahl’s Law • The law of diminishing returns • When a task has multiple parts, after you speed up one part a lot, the other parts come to dominate the total time • An example from cycling: • On a hilly closed-loop course you cannot ever average more than 2x your uphill speed even if you go downhill at the speed of light! • For supercomputers this means even though processors get faster the overall time to solution is limited by memory and interconnect speeds (moving the data around)

8. The Problem of Latency • Since memory bandwidth is also increasing faster than is memory latency, there a slower, but important overall trend towards an increase in the number of words of memory bandwidth that equate to memory latency. For a processor to avoid stalling and to realize peak bandwidths under these conditions, it must be designed to operate on multiple outstanding memory requests. The number of outstanding requests that need to be processed in order to saturate memory bandwidth is now approaching 100 to 1000 [FoSC]. • “Einstein’s Law • It is clear that latency tolerance mechanisms are an essential component of future system design.

9. Hierarchical systems • Modern supercomputers are composed of a hierarchy of subsystems: • CPU and registers • Cache (multiple levels) • Main (local) memory • System interconnect to other CPUs and memory • Disk • Archival • Latencies span 11 orders of magnitude! Nanoseconds to Minutes. This is a solar-system scale. It is 794 million miles to Saturn. • Bandwidths taper • Cache bandwidths generally are at least four times larger than local memory bandwidth, and this in turn exceeds interconnect bandwidth by a comparable amount, and so on outward to the disk and archive system.

10. “Red Shift” • While the absolute speed of all computer subcomponents have been changing rapidly, they have not all been changing at the same rate. For example, the analysis in [FoSC] indicates that, while peak processor speeds have been increasing at 59% between 1988 and 2004, memory speeds of commodity processors have been increasing at only 23% per year since 1995, and DRAM latency has only been improving at a pitiful rate of 5.5% per year. This “red shift” in the latency distance between various components has produced a crisis of sorts for computer architects, and a variety of complex latency-hiding mechanisms currently drive the design of computers as a result.

11. 3 ways of science • Experiment • Theory • Simulation

12. Parallel Models and Machines • Steps in creating a parallel program • decomposition • assignment • orchestration/coordination • mapping • Performance in parallel programs • try to minimize performance loss from • load imbalance • communication • synchronization • extra work

13. Some examples • Simulation models • A model problem: sharks and fish • Discrete event systems • Particle systems • Lumped systems (Ordinary Differential Equations, ODEs) • (Next time: Partial Different Equations, PDEs)

14. Sources of Parallelism and Locality in Simulation • Real world problems have parallelism and locality • Many objects do not depend on other objects • Objects often depend more on nearby than distant objects • Dependence on distant objects often “simplifies” • Scientific models may introduce more parallelism • When continuous problem is discretized, may limit effects to timesteps • Far-field effects may be ignored or approximated, if they have little effect • Many problems exhibit parallelism at multiple levels • e.g., circuits can be simulated at many levels and within each there may be parallelism within and between subcircuits

15. Basic Kinds of Simulation • Discrete event systems • e.g.,”Game of Life”, timing level simulation for circuits • Particle systems • e.g., billiard balls, semiconductor device simulation, galaxies • Lumped variables depending on continuous parameters • ODEs, e.g., circuit simulation (Spice), structural mechanics, chemical kinetics • Continuous variables depending on continuous parameters • PDEs, e.g., heat, elasticity, electrostatics • A given phenomenon can be modeled at multiple levels • Many simulations combine these modeling techniques

16. A Model Problem: Sharks and Fish • Illustration of parallel programming • Original version (discrete event only) proposed by Geoffrey Fox • Called WATOR • Basic idea: sharks and fish living in an ocean • rules for movement (discrete and continuous) • breeding, eating, and death • forces in the ocean • forces between sea creatures • 6 problems (S&F1 - S&F6) • Different sets of rule, to illustrate different phenomena

17. http://www.cs.berkeley.edu/~demmel/cs267/Sharks_and_Fish/ • Sharks and Fish 1. Fish alone move continuously subject to an external current and Newton's laws. • Sharks and Fish 2. Fish alone move continuously subject to gravitational attraction and Newton's laws. • Sharks and Fish 3. Fish alone play the "Game of Life" on a square grid. • Sharks and Fish 4. Fish alone move randomly on a square grid, with at most one fish per grid point. • Sharks and Fish 5. Sharks and Fish both move randomly on a square grid, with at most one fish or shark per grid point, including rules for fish attracting sharks, eating, breeding and dying. • Sharks and Fish 6. Like Sharks and Fish 5, but continuous, subject to Newton's laws.

18. Discrete Event Systems

19. Discrete Event Systems • Systems are represented as • finite set of variables • each variable can take on a finite number of values • the set of all variable values at a given time is called the state • each variable is updated by computing a transition function depending on the other variables • System may be • synchronous: at each discrete timestep evaluate all transition functions; also called a finite state machine • asynchronous: transition functions are evaluated only if the inputs change, based on an “event” from another part of the system; also called event driven simulation • E.g., functional level circuit simulation

20. Sharks and Fish as Discrete Event System • Ocean modeled as a 2D toroidal grid • Each cell occupied by at most one sea creature

21. The Game of Life (Sharks and Fish 3) • Fish only, no sharks • An new fish is born if • a cell is empty • exactly 3 (of 8) neighbors contain fish • A fish dies (of overcrowding) if • cell contains a fish • 4 or more neighboring cells are full • A fish dies (of loneliness) if • cell contains a fish • less than 2 neighboring cells are full • Other configurations are stable

22. Parallelism in Sharks and Fishes • The simulation is synchronous • use two copies of the grid (old and new) • the value of each new grid cell depends only on 9 cells (itself plus 8 neighbors) in old grid • simulation proceeds in timesteps, where each cell is updated at every timestep • Easy to parallelize using domain decomposition • Locality is achieved by using large patches of the ocean • boundary values from neighboring patches are needed • Load balancing is more difficult. The activities in this system are discrete events P4 Repeat compute locally to update local system barrier() exchange state info with neighbors until done simulating P1 P2 P3 P5 P6 P7 P8 P9

23. edge crossings = 6 edge crossings = 10 Parallelism in Circuit Simulation • Circuit is a graph made up of subcircuits connected by wires • component simulations need to interact if they share a wire • data structure is irregular (graph) • parallel algorithm is synchronous • compute subcircuit outputs • propagate outputs to other circuits • Graph partitioning assigns subgraphs to processors • Determines parallelism and locality • Want even distribution of nodes (load balance) • With minimum edge crossing (minimize communication) • Nodes and edges may both be weighted by cost • NP-complete to partition optimally, but many good heuristics (later lectures)

24. Parallelism in Asynchronous Circuit Simulation • Synchronous simulations may waste time • simulate even when the inputs do not change, little internal activity • activity varies across circuit • Asynchronous simulations update only when an event arrives from another component • no global timesteps, but events contain time stamp • Ex: Circuit simulation with delays (events are gates changing) • Ex: Traffic simulation (events are cars changing lanes, etc.)

25. Scheduling Asynch. Circuit Simulation • Conservative: • Only simulate up to (and including) the minimum time stamp of inputs • May need deadlock detection if there are cycles in graph, or else “null messages” • Speculative: • Assume no new inputs will arrive and keep simulating, instead of waiting • May need to backup if assumption wrong • Ex: Parswec circuit simulator of Yelick/Wen • Ex: Standard technique for CPUs to execute instructions • Optimizing load balance and locality is difficult • Locality means putting tightly coupled subcircuit on one processor since “active” part of circuit likely to be in a tightly coupled subcircuit, this may be bad for load balance

26. Particle Systems

27. Particle Systems • A particle system has • a finite number of particles • moving in space according to Newton’s Laws (i.e. F = ma) • time is continuous • Examples • stars in space with laws of gravity • electron beam semiconductor manufacturing • atoms in a molecule with electrostatic forces • neutrons in a fission reactor • cars on a freeway with Newton’s laws plus model of driver and engine • Reminder: many simulations combine techniques such as particle simulations with some discrete events (Ex Sharks and Fish)

28. Forces in Particle Systems • Force on each particle can be subdivided force = external_force + nearby_force + far_field_force • External force • ocean current to sharks and fish world (S&F 1) • externally imposed electric field in electron beam • Nearby force • sharks attracted to eat nearby fish (S&F 5) • balls on a billiard table bounce off of each other • Van der Wals forces in fluid (1/r^6) • Far-field force • fish attract other fish by gravity-like (1/r^2 ) force (S&F 2) • gravity, electrostatics, radiosity • forces governed by elliptic PDE

29. Parallelism in External Forces • These are the simplest • The force on each particle is independent • Called “embarrassingly parallel” • Evenly distribute particles on processors • Any distribution works • Locality is not an issue, no communication • For each particle on processor, apply the external force

30. Need to check for collisions between regions Parallelism in Nearby Forces • Nearby forces require interaction => communication • Force may depend on other nearby particles • Ex: collisions • simplest algorithm is O(n^2): look at all pairs to see if they collide • Usual parallel model is domain decomposition of physical domain • Challenge 1: interactions of particles near processor boundary • need to communicate particles near boundary to neighboring processors • surface to volume effect means low communication • Which communicates less: squares (as below) or slabs? • Challenge 2: load imbalance, if particles cluster • galaxies, electrons hitting a device wall

31. Example: each square contains at most 3 particles Load balance via Tree Decomposition • To reduce load imbalance, divide space unevenly • Each region contains roughly equal number of particles • Quad tree in 2D, Oct-tree in 3D

32. Parallelism in Far-Field Forces • Far-field forces involve all-to-all interaction => communication • Force depends on all other particles • Ex: gravity • Simplest algorithm is O(n^2) as in S&F 2, 4, 5 • Just decomposing space does not help since every particle apparently needs to “visit” every other particle • Use more clever algorithms to beat O(n^2)

33. Far-field forces: Particle-Mesh Methods • Superimpose a regular mesh • “Move” particles to nearest grid point • Exploit fact that far-field satisfies a PDE that is easy to solve on a regular mesh • FFT, Multigrid • Wait for next lecture • Accuracy depends on how fine the grid is and uniformity of particles

34. Far-field forces: Tree Decomposition • Based on approximation • O(n log n) or O(n) instead of O(n^2) • Forces from group of far-away particles “simplifies” • They resemble a single larger particle • Use tree; each node contains an approximation of descendents • Several Algorithms • Barnes-Hut • Fast Multipole Method (FMM) of Greengard/Rohklin • Anderson • Later lectures

35. Lumped SystemsODEs

36. System of Lumped Variables • Many systems approximated by • System of “lumped” variables • Each depends on continuous parameter (usually time) • Example: circuit • approximate as graph • wires are edges • nodes are connections between 2 or more wires • each edge has resistor, capacitor, inductor or voltage source • system is “lumped” because we are not computing the voltage/current at every point in space along a wire • Variables related by Ohm’s Law, Kirchoff’s Laws, etc. • Form a system of Ordinary Differential Equations, ODEs • We will refresh more on ODEs and PDEs • See http://www.sosmath.com/diffeq/system/introduction/intro.html

37. Circuit physics • The sum of all currents is 0 (Kirchoff Current Law) • The sum of all voltages around the loop is 0 • V= IR (Ohm’s Law)

38. Circuit Example • State of the system is represented by • v_n(t) node voltages • i_b(t) branch currents all at time t • v_b(t) branch voltages • Eqns. Include: Kirchoff’s current Kirchoff’s voltage Ohm’s law Capacitance Inductance • Write as single large system of ODEs (possibly with constraints) 0 A 0 v_n 0 A’ 0 -I * i_b = S 0 R -I v_b 0 0 -I C*d/dt 0 0 L*d/dt I 0

39. Systems of Lumped Variables • Another example is structural analysis in Civil Eng. • Variables are displacement of points in a building • Newton’s and Hook’s (spring) laws apply • Static modeling: exert force and determine displacement • Dynamic modeling: apply continuous force (earthquake) • The system in these case (and many) will be sparse • i.e., most array elements are 0 • neither store nor compute on these 0’s

40. Solving ODEs • Explicit methods to compute solution(t) • Ex: Euler’s method • Simple algorithm: sparse matrix vector multiply • May need to take very small timesteps, especially if system is stiff (i.e. can change rapidly) • Implicit methods to compute solution(t) • Ex: Backward Euler’s Method • Larger timesteps, especially for stiff problems • More difficult algorithm: solve a sparse linear system • Computing modes of vibration • Finding eigenvalues and eigenvectors • Ex: do resonant modes of building match earthquakes? • All these reduce to sparse matrix problems • Explicit: sparse matrix-vector multiplication • Implicit: solve a sparse linear system • direct solvers (Gaussian elimination) • iterative solvers (use sparse matrix-vector multiplication) • Eigenvalue/vector algorithms may also be explicit or implicit

41. Parallelism in Sparse Matrix-vector multiplication • y = A*x, where A is sparse and n x n • Questions • which processors store • y[i], x[i], and A[i,j] • which processors compute • y[i] = sum from 1 to n of A[i,j] * x[j] • Graph partitioning • Partition index set {1,…,n} = N1 u N2 u … u Np • for all i in Nk, store y[i], x[i], and row i of A on processor k • Processor k computes its own y[i] • Constraints • balance load • balance storage • minimize communication

42. 3 2 4 1 5 6 Graph Partitioning and Sparse Matrices • Relationship between matrix and graph 1 2 3 4 5 6 1 1 1 1 2 1 1 1 1 3 1 1 1 4 1 1 1 1 5 1 1 1 1 6 1 1 1 1 • A “good” partition of the graph has • equal number of (weighted) nodes in each part (load balance) • minimum number of edges crossing between • Can reorder the rows/columns of the matrix by putting all the nodes in one partition together

43. More on Matrix Reordering via Graph Partitioning • Goal is to reorder rows and columns to • improve load balance • decrease communication • “Ideal” matrix structure for parallelism: (nearly) block diagonal • p (number of processors) blocks • few non-zeros outside these blocks, since these require communication P0 P1 P2 P3 P4 = *

44. Summary • “Red Shift” means the promise implies by Moore’s Law is largely unrealized for scientific simulation that by necessity operates on large data • Consider “The Butterfly Effect” • Computer Architecture is a hot field again • Large centralized, specialized compute engines are vital • Grids, utility programing, SETI@home etc. do not meet all the needs of largescale scientific simulation for reason that should now be obvious • Consider a galactic scale