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CALCOLO SCIENTIFICO CALCOLO SCIENTIFICO (PARALLELO)(PARALLELO)

Prof. Luca F. Pavarino

Dipartimento di Matematica

Universita` di Milano

a.a. 20010-2011

luca.pavarino@unimi.it, http://www.mat.unimi.it/~pavarino

Corso di Laurea Magistrale e Dottorati in Matematica Applicata

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Struttura del corso

• Orario- Lunedi` 12.30 - 14.30 Aula 4

- Mertedi` 13.30 - 14.30 Aula 5 (compattare?)

- Mercoledi` 14.30 - 16.30 Aula 3

- Venerdi` 8.30 - 10.30 Aula 2

• 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 nuovo Cluster Linux Nemo

- Cluster Linux del Cilea (avogadro.cilea.it), ~1700 processori

- (nuovo IBM SP6 del Cineca (sp6.sp.cineca.it), ~5300 processori)

- Uso della libreria standard per “message passing” MPI

- Uso della libreria parallela di calcolo scientifico PETSc dell’Argonne National Lab., basata su MPI

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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, 2nd 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)

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Schedule of Topics1. 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

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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

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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 ...

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• 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,...)

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Not at all trivial that more processors help to achieve these goals:

• “If a man can dig a hole of 1 m3 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)

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Why we need powerful computers

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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

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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, …

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$5B World Market in Technical Computing

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1998 1999 2000 2001 2002 2003 Other

Technical Management andSupport

Simulation

Scientific Research and R&D

MechanicalDesign/Engineering Analysis

Mechanical Design andDrafting

Imaging

Geoscience and Geo-engineering

Electrical Design/EngineeringAnalysis

Economics/Financial

Digital Content Creation andDistribution

Classified Defense

Chemical Engineering

Biosciences

Source: IDC 2004, from NRC Future of Supercomputer Report

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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…Mega Mflop/s = 106 flop/sec Mbyte = 220 = 1048576 ~ 106

bytes

Giga Gflop/s = 109 flop/sec Gbyte = 230 ~ 109 bytes

Tera Tflop/s = 1012 flop/sec Tbyte = 240 ~ 1012 bytes

Peta Pflop/s = 1015 flop/sec Pbyte = 250 ~ 1015 bytes

Exa Eflop/s = 1018 flop/sec Ebyte = 260 ~ 1018 bytes

Zetta Zflop/s = 1021 flop/sec Zbyte = 270 ~ 1021 bytes

Yotta Yflop/s = 1024 flop/sec Ybyte = 280 ~ 1024 bytes

Current fastest (public) machine ~ 1.5 Pflop/s

Up-to-date lisy at www.top500.org

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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

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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.

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

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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

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3D simulations: isochrones of acti, repo, APD

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• 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

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Ex. 3: latest breakthrough

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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

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- 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

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Why powerful computers are

parallel

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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.

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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

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Impact of Device Shrinkage

• What happens when the feature size shrinks by a factor of x ?

• Clock rate goes up by x - actually less than x, because of power consumption

• Transistors per unit area goes up by x2

• Die size also tends to increase- typically another factor of ~x

• Raw computing power of the chip goes up by ~ x4 !- of which x3 is devoted either to parallelism or locality

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Microprocessor Transistors per Chip

i4004

i80286

i80386

i8080

i8086

R3000R2000

R10000

Pentium

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10,000

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10,000,000

100,000,000

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• Growth in transistors per chip

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Year

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(MH

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• Increase in clock rate

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But there are limiting forces

• Moore’s 2nd 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

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Revolution is Happening Now

• Chip density is continuing increase ~2x every 2 years

- Clock speed is not

- Number of processor cores may double instead

• There is little or no more hidden parallelism (ILP) to be found

• Parallelism must be exposed to and managed by software

Source: Intel, Microsoft (Sutter) and Stanford (Olukotun, Hammond)

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Parallelism in 2009-10?

• 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

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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 1012 times per second at the speed of light, c = 3x108 m/s. Thus r < c/1012 = 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 mm1 Tflop/s, 1 Tbyte sequential machine

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More Exotic Solutions on the Horizon

• GPUs - Graphics Processing Units (eg NVidia)

- Parallel processor attached to main processor

- Originally special purpose, getting more general

• FPGAs – Field Programmable Gate Arrays

- Inefficient use of chip area

- More efficient than multicore now, maybe not later

- Wire routing heuristics still troublesome

• Dataflow and tiled processor architectures

- Have considerable experience with dataflow from 1980’s

- Are we ready to return to functional programming languages?

• Cell

- Software controlled memory uses bandwidth efficiently

- Programming model not yet mature

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“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.

There are limitations to all of these:

to achieve high performance, the programmer needs to identify, schedule and coordinate parallel tasks and data.

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Processor-DRAM Gap (latency)

µProc60%/yr.

DRAM7%/yr.

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“Moore’s Law”

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Principles of Parallel Computing

• Parallelism and Amdahl’s Law

• Finding and exploiting granularity

• Preserving data locality

• Load balancing

• Coordination and synchronization

• Performance modeling

All of these issues makes parallel programming harder than sequential programming.

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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

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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.

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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.

ProcCache

L2 Cache

L3 Cache

Memory

Conventional Storage Hierarchy

ProcCache

L2 Cache

L3 Cache

Memory

ProcCache

L2 Cache

L3 Cache

Memory

potentialinterconnects

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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

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Measuring Performance: Real Performance?

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1,000

2000 2004T

eraf

lop

s1996

Peak Performance grows exponentially, a la Moore’s Law

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

was 40-50% on the vector supercomputers of 1990s

now as little as 5-10% on parallel supercomputers of today

Close the gap through ... Mathematical methods and algorithms that

achieve high performance on a single processor and scale to thousands of processors

More efficient programming models and tools for massively parallel supercomputers

PerformanceGap

Peak Performance

Real Performance

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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

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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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Simple example 1: sum of N numbers, P procs

jk

kjiij aA1)1(

N

iiaA

1

Also known as reduction (of the vector [a1,…,aN] to the scalar A)

- Assume N is an integer multiple of P: N = kP- Divide the sum into P partial sums:

Then

P

jjAA

1

P parallel tasks, each withk -1 additions of k = N/P data

Global sum (not parallel,communication needed)

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Simple example 2: pi

10

1

0

2 |)(4)1/(4 xarctgdxx

,)1/(41

2

N

iixh- 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)

hixi )2/1(

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)

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Simple example 3: prime number sieve

See exercise in class

Simple example 4: Jacobi method for BVP

See exercise in class