1
CALCOLO SCIENTIFICO
(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
2
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
hardware
software
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
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
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
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
•
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
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
Why we need
powerful computers
10
10
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
11
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, …
12
$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 and
Support
Simulation
Scientific Research and R&D
Mechanical
Design/Engineering Analysis
Mechanical Design and
Drafting
Imaging
Geoscience and Geo
engineering
Electrical Design/Engineering
Analysis
Economics/Financial
Digital Content Creation and
Distribution
Classified Defense
Chemical Engineering
Biosciences
Source: IDC 2004, from NRC Future of Supercomputer Report
13
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 = 10
6
flop/sec
Mbyte = 2
20
= 1048576 ~ 10
6
bytes
Giga
Gflop/s = 10
9
flop/sec
Gbyte = 2
30
~ 10
9
bytes
Tera
Tflop/s = 10
12
flop/sec
Tbyte = 2
40
~ 10
12
bytes
Peta
Pflop/s = 10
15
flop/sec
Pbyte = 2
50
~ 10
15
bytes
Exa
Eflop/s = 10
18
flop/sec
Ebyte = 2
60
~ 10
18
bytes
Zetta
Zflop/s = 10
21
flop/sec
Zbyte = 2
70
~ 10
21
bytes
Yotta
Yflop/s = 10
24
flop/sec
Ybyte = 2
80
~ 10
24
bytes
Current fastest (public) machine ~ 1.5 Pflop/s
Up

to

date lisy at www.top500.org
14
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
15
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
16
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
17
3D simulations: isochrones of acti, repo, APD
18
•
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
19
Ex. 3: latest breakthrough
20
21
22
23
24
25
26
27
28
29
30
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
31

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
32
Why powerful
computers are
parallel
33
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.
34
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
35
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
x
2
•
Die size also tends to increase

typically another factor of ~
x
•
Raw computing power of the chip goes up by ~
x
4
!

of which
x
3
is devoted either to
parallelism
or
locality
36
Microprocessor Transistors per Chip
i4004
i80286
i80386
i8080
i8086
R3000
R2000
R10000
Pentium
1,000
10,000
100,000
1,000,000
10,000,000
100,000,000
1970
1975
1980
1985
1990
1995
2000
2005
Year
Transistors
•
Growth in transistors per chip
0.1
1
10
100
1000
1970
1980
1990
2000
Year
Clock Rate (MHz)
•
Increase in clock rate
37
37
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
38
38
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)
39
39
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
40
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
41
41
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
42
“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.
43
Processor

DRAM Gap (latency)
µProc
60%/yr.
DRAM
7%/yr.
1
10
100
1000
1980
1981
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
DRAM
CPU
1982
Processor

Memory
Performance Gap:
(grows 50% / year)
Performance
Time
“Moore’s Law”
44
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.
45
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
46
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.
47
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
48
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
49
Measuring Performance: Real Performance?
0.1
1
10
100
1,000
2000
2004
Teraflops
1996
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
Performance
Gap
Peak Performance
Real Performance
50
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
51
51
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)
52
53
54
55
56
57
58
59
Simple example 1: sum of N numbers, P procs
jk
k
j
i
i
j
a
A
1
)
1
(
N
i
i
a
A
1
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
j
j
A
A
1
P parallel tasks, each with
k

1 additions of k = N/P data
Global sum (not parallel,
communication needed)
60
Simple example 2: pi
1
0
1
0
2

)
(
4
)
1
/(
4
x
arctg
dx
x
,
)
1
/(
4
1
2
N
i
i
x
h

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)
h
i
x
i
)
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)
61
Simple example 3: prime number sieve
See exercise in class
Simple example 4: Jacobi method for BVP
See exercise in class
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο