1. 1 METODI NUMERICI PER EDP3 (CALCOLO PARALLELO) Prof. Luca F. Pavarino Dipartimento di Matematica Universita` di Milano a.a. 2013-2014 luca.pavarino@unimi.it, http://www.mat.unimi.it/~pavarino Corso di Laurea Magistrale e Dottorati in Matematica Applicata

2. 2 Struttura del corso Orario -Lunedi` 10.30 - 12.30 Aula 3 -Mercoledi` 13.30 - 16.30 Aula 2 (Lab) -Giovedi` 10.30 - 12.30 Aula 9 12 - 13 settimane, 9 cfu (6 lezione, 3 laboratorio) Laboratorio in Aula 2 o LIR o LID: esercitazioni con - Nostro Cluster Linux (ulisse.mat.unimi.it), 104 processori - Nostro Cluster Linux Nemo - nuovo IBM BG/Q del Cineca (fermi.cineca.it), ~160K processori) -Uso della libreria standard per “message passing” MPI -Uso della libreria parallela di calcolo scientifico PETSc dell’Argonne National Lab., basata su MPI hardware software

3. 3 Materiale e Testi Slides in inglese basate su corsi di calcolo parallelo tenuti a Univ. Illinois da Michael Heath, UC Berkeley da Jim Demmel, (+ MIT da Alan Edelmann) Possibili testi: - A. Grama, A. Gupta, G. Karipys, V. Kumar, Introduction to parallel computing, 2 nd ed., Addison Wesley, 2003 - L. R. Scott, T. Clark, B. Bagheri, Scientific Parallel Computing, Princeton University Press, 2005 Molto materiale on-line, e.g. : - www-unix.mcs.anl.gov/dbpp/ (Ian Foster’s book) -www.cs.berkeley.edu/~demmel/ (Demmel’s course) -www-math.mit.edu/~edelman/ (Edelman’s course) -www.cse.uiuc.edu/~heath/ (Heath’s course) -www.cs.rit.edu/~ncs/parallel.html (Nan’s ref page)

4. 4 Schedule of Topics 1. Introduction 2. Parallel architectures 3. Networks 4. Interprocessor communications: point-to-point, collective 5. Parallel algorithm design 6. Parallel programming, MPI: message passing interface 7. Parallel performance 8. Vector and matrix products 9. LU factorization 10. Cholesky factorization 11. PETSc parallel library 12. Iterative methods for linear systems 13. Nonlinear equations and ODEs 14. Partial Differential Equations 15. Domain Decomposition Methods 16. QR factorization 17. Eigenvalues

5. 5 1) Introduction What is parallel computing Large important problems require powerful computers Why powerful computers must be parallel processors Why writing (fast) parallel programs is hard Principles of parallel computing performance

6. 6 What is parallel computing It is an example of parallel processing: -division of task (process) into smaller tasks (processes) -assign smaller tasks to multiple processing units that work simultaneously -coordinate, control and monitor the units Many examples from nature: -human brain consists of ~10^11 neurons -complex living organisms consist of many cells (although monocellular organism are estimated to be ½ of the earth biomass) -leafs of trees... Many examples from daily life: -highways tollbooths, supermarket cashiers, bank tellers, … -elections, races, competitions, … -building construction -written exams...

7. 7 Parallel computing is the use of multiple processors to execute different parts of the same program (task) simultaneously Main goals of parallel computing are: -Increase the size of problems that can be solved -bigger problem would not be solvable on a serial computer in a reasonable amount of time  decompose it into smaller problems -bigger problem might not fit in the memory of a serial computer  distribute it over the memory of many computer nodes -Reduce the “wall-clock” time to solve a problem  Solve (much) bigger problems (much) faster Subgoal: save money using cheapest available resources (clusters, beowulf, grid computing,...)

8. 8 Not at all trivial that more processors help to achieve these goals: “If a man can dig a hole of 1 m 3 in 1 hour, can 60 men dig the same hole in 1 minute (!) ? Can 3600 men do it in 1 second (!!) ?” “I know how to make 4 horses pull a cart, but I do not know how to make 1024 chickens do it” (Enrico Clementi) “ What happens if the mean-time to failure for nodes on the Tflops machine is shorter than the boot time ? (Courtenay Vaughan)

9. 9 Units of Measure in HPC High Performance Computing (HPC) units are: -Flops: floating point operations -Flops/s: floating point operations per second -Bytes: size of data (a double precision floating point number is 8) Typical sizes are millions, billions, trillions… MegaMflop/s = 10 6 flop/secMbyte = 2 20 = 1048576 ~ 10 6 bytes GigaGflop/s = 10 9 flop/secGbyte = 2 30 ~ 10 9 bytes TeraTflop/s = 10 12 flop/secTbyte = 2 40 ~ 10 12 bytes PetaPflop/s = 10 15 flop/secPbyte = 2 50 ~ 10 15 bytes ExaEflop/s = 10 18 flop/secEbyte = 2 60 ~ 10 18 bytes ZettaZflop/s = 10 21 flop/secZbyte = 2 70 ~ 10 21 bytes YottaYflop/s = 10 24 flop/secYbyte = 2 80 ~ 10 24 bytes Current fastest (public) machine ~ 1.5 Pflop/s Up-to-date lisy at www.top500.org

10. 10 Why we need powerful computers

11. 11 Simulation: The Third Pillar of Science Traditional scientific and engineering method: (1) Do theory or paper design (2) Perform experiments or build system Limitations: –Too difficult—build large wind tunnels –Too expensive—build a throw-away passenger jet –Too slow—wait for climate or galactic evolution –Too dangerous—weapons, drug design, climate experimentation Computational science and engineering paradigm: (3) Use high performance computer systems to simulate and analyze the phenomenon -Based on known physical laws and efficient numerical methods -Analyze simulation results with computational tools and methods beyond what is used traditionally for experimental data analysis Simulation Theory Experiment

12. Data Driven Science Scientific data sets are growing exponentially -Ability to generate data is exceeding our ability to store and analyze -Simulation systems and some observational devices grow in capability with Moore’s Law Petabyte (PB) data sets will soon be common: -Climate modeling: estimates of the next IPCC data is in 10s of petabytes -Genome: JGI alone will have.5 petabyte of data this year and double each year -Particle physics: LHC is projected to produce 16 petabytes of data per year -Astrophysics: LSST and others will produce 5 petabytes/year (via 3.2 Gigapixel camera) Create scientific communities with “Science Gateways” to data 12

13. 13 Some Particularly Challenging Computations Science -Global climate modeling, weather forecasts -Astrophysical modeling -Biology: Genome analysis; protein folding (drug design) -Medicine: cardiac modeling, physiology, neurosciences Engineering -Airplane design -Crash simulation -Semiconductor design -Earthquake and structural modeling Business -Financial and economic modeling -Transaction processing, web services and search engines Defense -Nuclear weapons (ASCI), cryptography, …

14. 14 Economic Impact of HPC Airlines: -System-wide logistics optimization systems on parallel systems. -Savings: approx. $100 million per airline per year. Automotive design: -Major automotive companies use large systems (500+ CPUs) for: -CAD-CAM, crash testing, structural integrity and aerodynamics. -One company has 500+ CPU parallel system. -Savings: approx. $1 billion per company per year. Semiconductor industry: -Semiconductor firms use large systems (500+ CPUs) for -device electronics simulation and logic validation -Savings: approx. $1 billion per company per year. Energy -Computational modeling improved performance of current nuclear power plants, equivalent to building two new power plants.

15. 15 $5B World Market in Technical Computing Source: IDC 2004, from NRC Future of Supercomputer Report

16. Which commercial applications require parallelism? Analyzed in detail in “Berkeley View” report www.eecs.berkeley.edu/Pubs/ TechRpts/2006/EECS-2006- 183.html

17. Motif/Dwarf: Common Computational Methods (Red Hot  Blue Cool) What do commercial and CSE applications have in common?

18. 18 Ex. 1: Global Climate Modeling Problem Problem is to compute: f(latitude, longitude, elevation, time)  temperature, pressure, humidity, wind velocity Atmospheric model: equation of fluid dynamics  Navier-Stokes system of nonlinear partial differential equations Approach: -Discretize the domain, e.g., a measurement point every 1km -Devise an algorithm to predict weather at time t+1 given t Uses: -Predict major events, e.g., El Nino -Use in setting air emissions standards Source: http://www.epm.ornl.gov/chammp/chammp.html

19. 19 Global Climate Modeling Computation One piece is modeling the fluid flow in the atmosphere -Solve Navier-Stokes equations -Roughly 100 Flops per grid point with 1 minute timestep Computational requirements: -To match real-time, need 5 x 10 11 flops in 60 seconds = 8 Gflop/s -Weather prediction (7 days in 24 hours)  56 Gflop/s -Climate prediction (50 years in 30 days)  4.8 Tflop/s -To use in policy negotiations (50 years in 12 hours)  288 Tflop/s To double the grid resolution, computation is 8x to 16x State of the art models require integration of atmosphere, clouds, ocean, sea-ice, land models, plus possibly carbon cycle, geochemistry and more Current models are coarser than this

20. 20 Climate Modeling on the Earth Simulator System  Development of ES started in 1997 in order to make a comprehensive understanding of global environmental changes such as global warming.  26.58Tflops was obtained by a global atmospheric circulation code. 26.58Tflops was obtained by a global atmospheric circulation code.  35.86Tflops (87.5% of the peak performance) is achieved in the Linpack benchmark.  Its construction was completed at the end of February, 2002 and the practical operation started from March 1, 2002

21. 21 Ex. 2: Cardiac simulation Very difficult problem spanning many disciplines: -Electrophysiology (spreading of electrical excitation front) -Structural Mechanics (large deformation of incompressible biomaterial) -Fluid Dynamics (flow of blood inside the heart) Large-scale simulations in computational electrophysiology ( joint work with P. Colli-Franzone and S. Scacchi ) -Bidomain model (system of 2 reaction-diffusion equations) coupled with Luo-Rudy 1 gating (system of 7 ODEs) in 3D -Q1 finite elements in space + adaptive semi-implicit method in time -Parallel solver based on PETSc library -Linear systems up to 36 M unknowns each time-step (128 procs of Cineca SP4) solved in seconds or minutes -Simulation of full heartbeat (4 M unknowns in space, thousands of time-steps) took more than 6 days on 25 procs of Cilea HP Superdome, then about 50 hours on 36 procs of our cluster, now 6.5 hours using multilevel preconditioner

22. 22 3D simulations: isochrones of acti, repo, APD

23. 23 Hemodynamics in circulatory system ( work in Quarteroni’s group at MOX, Polimi ) Blood flow in the heart ( Peskin’s group, CIMS, NYU ) -Modeled as an elastic structure in an incompressible fluid. -The “immersed boundary method” due to Peskin and McQueen. -20 years of development in model -Many applications other than the heart: blood clotting, inner ear, paper making, embryo growth, and others -Use a regularly spaced mesh (set of points) for evaluating the fluid - Uses -Current model can be used to design artificial heart valves -Can help in understand effects of disease (leaky valves) -Related projects look at the behavior of the heart during a heart attack -Ultimately: real-time clinical work

24. 24 Ex. 3: latest breakthrough

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35. 35 Ex. 4: Parallel Computing in Data Analysis Web search: -Functional parallelism: crawling, indexing, sorting -Parallelism between queries: multiple users -Finding information amidst junk -Preprocessing of the web data set to help find information Google physical structure (2004 estimate, check current status on e.g. wikipedia): -about 63.272 nodes (126,544 cpus) -126.544 GB RAM -5,062 TB hard drive space ( This would make Google server farm one of the most powerful supercomputer in the world ) Google index size (June 2005 estimate): -about 8 billion web pages, 1 billion images

36. 36 - Note that the total Surface Web ( = publically indexable, i.e. reachable by web crawlers) has been estimated (Jan. 2005) at over 11.5 billion web pages. -Invisible (or Deep) Web ( = not indexed by search engines; it consists of dynamic web pages, subscription sites, searchable databases) has been estimated (2001) at over 550 billion documents. -Invisible Web not to be confused with Dark Web consisting of machines or network segments not connected to the Internet Data collected and stored at enormous speeds (Gbyte/hour) -remote sensor on a satellite -telescope scanning the skies -microarrays generating gene expression data -scientific simulations generating terabytes of data -NSA analysis of telecommunications

37. 37 Why powerful computers are parallel

38. 38 Tunnel Vision by Experts “I think there is a world market for maybe five computers.” -Thomas Watson, chairman of IBM, 1943. “There is no reason for any individual to have a computer in their home” -Ken Olson, president and founder of Digital Equipment Corporation, 1977. “640K [of memory] ought to be enough for anybody.” -Bill Gates, chairman of Microsoft,1981. Slide source: Warfield et al.

39. 39 Technology Trends: Microprocessor Capacity 2X transistors/Chip Every 1.5 - 2 years Called “Moore’s Law” Moore’s Law Microprocessors have become smaller, denser, and more powerful. Gordon Moore (co-founder of Intel) predicted in 1965 that the transistor density of semiconductor chips would double roughly every 18 months. Slide source: Jack Dongarra

40. 40 Microprocessor Transistors / Clock (1970-2000)

41. 41 Impact of Device Shrinkage What happens when the feature size (transistor size) shrinks by a factor of x ? Clock rate goes up by x because wires are shorter -actually less than x, because of power consumption Transistors per unit area goes up by x 2 Die size also tends to increase -typically another factor of ~x Raw computing power of the chip goes up by ~ x 4 ! -typically x 3 is devoted to either on-chip -parallelism: hidden parallelism such as ILP -locality: caches So most programs x 3 times faster, without changing them

42. 42 But there are limiting forces Moore’s 2 nd law (Rock’s law): costs go up Demo of 0.06 micron CMOS Source: Forbes Magazine Yield -What percentage of the chips are usable? -E.g., Cell processor (PS3) is sold with 7 out of 8 “on” to improve yield Manufacturing costs and yield problems limit use of density

43. 43 Physical limits: how fast can a serial computer be? Consider the 1 Tflop/s sequential machine : -Data must travel some distance, r, to get from memory to CPU. -Go get 1 data element per cycle, this means 10 12 times per second at the speed of light, c = 3x10 8 m/s. Thus r < c/10 12 = 0.3 mm. Now put 1 Tbyte of storage in a 0.3 mm 0.3 mm area : (in fact 0.3^2 mm^2/10^12 = 9 10^(-2) 10^(-6) m^2/10^12 = 9 10^(-20) m^2 = (3 10^(-10))^2 m^2 = 3^2 A^2  -Each byte occupies less than 3 square Angstroms, or the size of a small atom! (1 Angstrom = 10^(-10) m = 0.1 nanometer) No choice but parallelism r = 0.3 mm 1 Tflop/s, 1 Tbyte sequential machine

44. Power Density Limits Serial Performance Scaling clock speed (business as usual) will not work High performance serial processors waste power -Speculation, dynamic dependence checking, etc. burn power -Implicit parallelism discovery More transistors, but not faster serial processors Concurrent systems are more power efficient –Dynamic power is proportional to V 2 fC –Increasing frequency (f) also increases supply voltage (V)  cubic effect –Increasing cores increases capacitance (C) but only linearly –Save power by lowering clock speed

45. 45 Revolution in Processors Chip density is continuing increase ~2x every 2 years Clock speed is not Number of processor cores may double instead Power is under control, no longer growing

46. 46 Parallelism in 2013? These arguments are no longer theoretical All major processor vendors are producing multicore chips -Every machine will soon be a parallel machine -To keep doubling performance, parallelism must double Which (commercial) applications can use this parallelism? -Do they have to be rewritten from scratch? Will all programmers have to be parallel programmers? -New software model needed -Try to hide complexity from most programmers – eventually -In the meantime, need to understand it Computer industry betting on this big change, but does not have all the answers -Berkeley ParLab established to work on this

47. Memory is Not Keeping Pace Technology trends against a constant or increasing memory per core Memory density is doubling every three years; processor logic is every two Storage costs (dollars/Mbyte) are dropping gradually compared to logic costs Source: David Turek, IBM Cost of Computation vs. Memory Question: Can you double concurrency without doubling memory? Strong scaling: fixed problem size, increase number of processors Weak scaling: grow problem size proportionally to number of processors Source: IBM

48. Listing the 500 most powerful computers in the world Yardstick: Rmax of Linpack -Solve Ax=b, dense problem, matrix is random -Dominated by dense matrix-matrix multiply Update twice a year: -ISC’xy in June in Germany -SCxy in November in the U.S. All information available from the TOP500 web site at: www.top500.org The TOP500 Project

49. The TOP10 in November 2012 Ran k Site Manufacture r ComputerCountryCores Rmax [Pflops] Power [MW] 1Oak Ridge National LaboratoryCray Titan Cray XK7, Opteron 16C 2.2GHz, Gemini, NVIDIA K20x USA560,640 17.59 8.21 2 Lawrence Livermore National Laboratory IBM Sequoia BlueGene/Q, Power BQC 16C 1.6GHz, Custom USA1,572,864 16.32 7.89 3 RIKEN Advanced Institute for Computational Science Fujitsu K computer SPARC64 VIIIfx 2.0GHz, Tofu Interconnect Japan705,024 10.51 12.66 4Argonne National LaboratoryIBM Mira BlueGene/Q, Power BQC 16C 1.6GHz, Custom USA786,432 8.16 3.95 5Forschungszentrum Juelich (FZJ)IBM JUQUEEN BlueGene/Q, Power BQC 16C 1.6GHz, Custom Germany393,216 4.14 1.97 6Leibniz RechenzentrumIBM SuperMUC iDataPlex DX360M4, Xeon E5 8C 2.7GHz, Infiniband FDR Germany147,456 2.90 3.42 7 Texas Advanced Computing Center/UT Dell Stampede PowerEdge C8220, Xeon E5 8C 2.7GHz, Intel Xeon Phi USA204,900 2.66 8 National SuperComputer Center in Tianjin NUDT Tianhe-1A NUDT TH MPP, Xeon 6C, NVidia, FT-1000 8C China186,368 2.57 4.04 9CINECAIBM Fermi BlueGene/Q, Power BQC 16C 1.6GHz, Custom Italy163,840 1.73.82 10IBM DARPA Trial Subset Power 775, Power7 8C 3.84GHz, Custom USA63,360 1.52 3.58

50. 36 th List: The TOP10 RankSiteManufacturerComputerCountryCores Rmax [Tflops] Power [MW] 1 National SuperComputer Center in Tianjin NUDT Tianhe-1A NUDT TH MPP, Xeon 6C, NVidia, FT-1000 8C China186,368 2,566 4.04 2 Oak Ridge National Laboratory Cray Jaguar Cray XT5, HC 2.6 GHz USA224,162 1,759 6.95 3 National Supercomputing Centre in Shenzhen Dawning Nebulae TC3600 Blade, Intel X5650, NVidia Tesla C2050 GPU China120,640 1,271 2.58 4 GSIC, Tokyo Institute of Technology NEC/HP TSUBAME-2 HP ProLiant, Xeon 6C, NVidia, Linux/Windows Japan73,278 1,192 1.40 5 DOE/SC/ LBNL/NERSC Cray Hopper Cray XE6, 6C 2.1 GHz USA153,408 1.054 2.91 6 Commissariat a l'Energie Atomique (CEA) Bull Tera 100 Bull bullx super-node S6010/S6030 France138.368 1,050 4.59 7DOE/NNSA/LANLIBM Roadrunner BladeCenter QS22/LS21 USA122,400 1,042 2.34 8 University of Tennessee Cray Kraken Cray XT5 HC 2.36GHz USA98,928 831.7 3.09 9 Forschungszentrum Juelich (FZJ) IBM Jugene Blue Gene/P Solution Germany294,912 825.5 2.26 10 DOE/NNSA/ LANL/SNL Cray Cielo Cray XE6, 6C 2.4 GHz USA107,152 816.6 2.95

51. Performance Development (Nov 2012) 59.7 GFlop/s 400 MFlop/s 1.17 TFlop/s 17.6 PFlop/s 76.5 TFlop/s 162 PFlop/s SUM N=1 N=500 1 Gflop/s 1 Tflop/s 100 Mflop/s 100 Gflop/s 100 Tflop/s 10 Gflop/s 10 Tflop/s 1 Pflop/s 100 Pflop/s 10 Pflop/s 1 Eflop/s

52. Projected Performance Development (Nov 2012) SUM N=1 N=500 1 Gflop/s 1 Tflop/s 100 Mflop/s 100 Gflop/s 100 Tflop/s 10 Gflop/s 10 Tflop/s 1 Pflop/s 100 Pflop/s 10 Pflop/s 1 Eflop/s

53. Core Count

54. Moore’s Law reinterpreted Number of cores per chip can double every two years Clock speed will not increase (possibly decrease) Need to deal with systems with millions of concurrent threads Need to deal with inter-chip parallelism as well as intra-chip parallelism

55. 55 Principles of Parallel Computing Finding enough parallelism (Amdahl’s Law) Granularity – how big should each parallel task be Locality – moving data costs more than arithmetic Load balance – don’t want 1K processors to wait for one slow one Coordination and synchronization – sharing data safely Performance modeling/debugging/tuning All of these things makes parallel programming even harder than sequential programming.

56. 56 “Automatic” Parallelism in Modern Machines Bit level parallelism -within floating point operations, etc. Instruction level parallelism (ILP) -multiple instructions execute per clock cycle Memory system parallelism -overlap of memory operations with computation OS parallelism -multiple jobs run in parallel on commodity SMPs Limits to all of these -- for very high performance, need user to identify, schedule and coordinate parallel tasks

57. 57 Amdahl’s law: Finding Enough Parallelism Suppose only part of an application seems parallel Amdahl’s law -Let s be the fraction of work done sequentially, so (1-s) is fraction parallelizable. -P = number of processors. Speedup(P) = Time(1)/Time(P) <= 1/(s + (1-s)/P) <= 1/s Even if the parallel part speeds up perfectly, we may be limited by the sequential portion of code. Ex: if only s = 1%, then speedup <= 100  not worth it using more than p = 100 processors

58. 58 Overhead of Parallelism Given enough parallel work, this is the most significant barrier to getting desired speedup. Parallelism overheads include: -cost of starting a thread or process -cost of communicating shared data -cost of synchronizing -extra (redundant) computation Each of these can be in the range of milliseconds (= millions of flops) on some systems Tradeoff: Algorithm needs sufficiently large units of work to run fast in parallel (i.e. large granularity), but not so large that there is not enough parallel work.

59. 59 Locality and Parallelism Large memories are slow, fast memories are small. Storage hierarchies are large and fast on average. Parallel processors, collectively, have large, fast memories -- the slow accesses to “remote” data we call “communication”. Algorithm should do most work on local data. Proc Cache L2 Cache L3 Cache Memory Conventional Storage Hierarchy Proc Cache L2 Cache L3 Cache Memory Proc Cache L2 Cache L3 Cache Memory potential interconnects

60. 60 Load Imbalance Load imbalance is the time that some processors in the system are idle due to -insufficient parallelism (during that phase). -unequal size tasks. Examples of the latter -adapting to “interesting parts of a domain”. -tree-structured computations. -fundamentally unstructured problems -Adaptive numerical methods in PDE (adaptivity and parallelism seem to conflict). Algorithm needs to balance load -but techniques to balance load often reduce locality -Sometimes can determine work load, divide up evenly, before starting -“Static Load Balancing” -Sometimes work load changes dynamically, need to rebalance dynamically -“Dynamic Load Balancing”

61. 61 Measuring Performance: Real Performance? 0.1 1 10 100 1,000 20002004 Teraflops 1996 Peak Performance grows exponentially, a la Moore’s Law l In 1990’s, peak performance increased 100x; in 2000’s, it will increase 1000x But efficiency (the performance relative to the hardware peak) has declined l was 40-50% on the vector supercomputers of 1990s l now as little as 5-10% on parallel supercomputers of today Close the gap through... l Mathematical methods and algorithms that achieve high performance on a single processor and scale to thousands of processors l More efficient programming models and tools for massively parallel supercomputers Performance Gap Peak Performance Real Performance

62. 62 Performance Levels Peak advertised performance (PAP) -You can’t possibly compute faster than this speed LINPACK -The “hello world” program for parallel computing -Solve Ax=b using Gaussian Elimination, highly tuned Gordon Bell Prize winning applications performance -The right application/algorithm/platform combination plus years of work Average sustained applications performance -What one reasonable can expect for standard applications When reporting performance results, these levels are often confused, even in reviewed publications

63. 63 Performance Levels (for example on NERSC-5) Peak advertised performance (PAP): 100 Tflop/s LINPACK (TPP): 84 Tflop/s Best climate application: 14 Tflop/s -WRF code benchmarked in December 2007 Average sustained applications performance: ? Tflop/s -Probably less than 10% peak! We will study performance -Hardware and software tools to measure it -Identifying bottlenecks -Practical performance tuning (Matlab demo)

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71. 71 Simple example 1: sum of N numbers, P procs Also known as reduction (of the vector [a 1,…,a N ] to the scalar A) - Assume N is an integer multiple of P: N = kP - Divide the sum into P partial sums: Then P parallel tasks, each with k -1 additions of k = N/P data Global sum (not parallel, communication needed)

72. 72 Simple example 2: pi - Use composite midpoints quadrature rule: where h = 1/N and -Decompose sum into P parallel partial sums + 1 global sum, (as before or with stride P) On processor myid = 0,…,P-1, (P = numprocs) compute: sum = 0; for I = myid + 1:numprocs:N, x = h*(I – 0.5); sum = sum + 4/(1+x*x); end; mypi = h*sum; global sum the local mypi into glob_pi (reduction)

73. 73 Simple example 3: prime number sieve See exercise in class Simple example 4: Jacobi method for BVP See exercise in class