We assume that Numerical Methods are virtually
complete and that all is well and ready in this area.
It is not so. How can THAT be?!
Take a very simple problem, like solving an n-by-n
matrix (linear system) Equation Ax = b, where A and
b are known (by some method - for instance A and b
may derive from an instantaneous (know Newton's
method for finding x to f(x) = 0? - when x can be an
n-vector X, and f(x) can be another n-vector F(X),
there is STILL a Newton's Method for solving F(X) = 0
- trouble is, it often breaks down in real problems) -
linearization of some physical and chemical equations
written to try to faithfully represent a real world
problem, which in itself is not always so clear or
complete in itself!
Solving for x when A and b are known - provided 'A'
is NOT 'singular' (zero determinant) - is 'cut and dried',
a real 'piece of cake', taught in a first Linear Algebra or
Numerical Linear Algebra Numerics course - RIGHT?!
Well, a method - there are many - is, well, yes, taught,
usually the Gaussian 'Elimination and Backsubstitution' -
or its equivalent as LU (LowerUpper) factorization in A,
using parallel manipulations in 'b'. Simple?!
NOT so Simple.
For one thing, in many realistic problems, the generated
'A' matrix may be 'practically' nonsingular, or as better
stated, 'ill-conditioned'. This is usually interpreted as
the ratio of the magnitude of the Largest Eigenvalue
(remember EIGENVALUES?! These are numbers 'c'
such that Av = cv for some 'Eigenvector'. Eignevalues
can be REAL. Then 'A' doesn't change the direction of
the eigenvector 'v', it just stretches it, compresses it,
or reverses its direction, or does a combo of these.
When an eignevalue 'c' is 'imaginary' - in the complex
numbers sense - then 'A' rotates the Eigenvector 'v'
when multiplied on as in Av - rotated around SOME
Axis in n-dimensional space.
Generally Complex Eignevalues 'c' indicate rotation
along with stretching or compression.) to the magnitude
of the smallest Eigenvalue.
It turns out that as the Condition Number (let's call it CN -
okay?!) tends to get larger and larger for related structure
matrices A (say these may come from finer and finer
discretization - more and more elements or grid cells -
of an 'attempting to be realistic' PDE - partially differential
equation).
And in General, the Classical, and Modern too, Matrix
Algebra (Abstract or Numerical) Linear System Solution
Algorithms get worse and worse in producing an accurate
determination of 'x' from Ax = b, with A and b in hand and
'known', as the CN grows larger!
Classical Gaussian Elimination or LU factorization
easily starts to fail when 'too large' and 'too fine'
a discretized system of a PDE is attempted.
But NOT only accuracy fails, the method itself
may start giving first bad, then just outrageously
wrong answers. Or completely brake down, often
not 'bombing' but just making 'so much progress'
and then 'No More' towards a good approximate
solution of JUST THIS INTERMEDIATE Linear
SYSTEM. Like Gears breaking off teeth somewhere
in the depths of a '24 Jewel' fine mechanical watch.
Unfortunately, Gaussian Elimination was generally
abandoned long ago for the intermediate, iterative,
linearization stages, of Important Real WORLD
Fluid Flow Problems - SUCH AS Advanced
Aerodynamics, WEATHER and CLIMATE.
Why?! Because the work and storage required
grow outrageously fast with increases in problem
size (finer mesh, more discretization cells or
elements).
What is really bad is that the Replacement Methods,
Classical Modern and Post-Modern, are often NOT
any better behaved do the growth of Condition
Number, but often are even MORE seriously
affected. Thus for example Gauss-Sidel SOR,
which was a standby for much of the mid to later
29th Century. All sorts of arcane mathematics
was done - really wasted effort, wasted on a
lame horse! - to develop 'good ways' of picking
an instant relaxation factor 'Omega' to 'make
GS-SOR' Go! The only place it needs to go,
however, is AWAY - Far, Far Away!
At Texaco an 8-year experienced Engineering
Program with an M.Sc. in Chemical Engineering
(lots and lots of hokey numerical calculations
are routinely bit off in Chemical Engineering)
was under my supervision for writing a Black
Oil Simulator at Texaco Canada. in the mid
1980's. Of COURSE he insisted on GS-SOR,
in which he was a 'past master'. So he made
a preliminary simulator and I gave it a test.
I told him the answers looked wrong.
He insisted they were right.
Then I had him substitute the values he found
for the physical variables back into the equations
he THOUGHT he was solving. Well, they sure
didn't balance out - there were huge areas.
Many of the flows were only about 30% of
flows he found when he executed the algorithms
I insisted upon, and then checked to see that
the New Methods balanced the equations with
much much smaller errors. His GS-SOR had
simply Not Enough POWER to solve the ugly
systems which naturally evolved from the quite
ordinary 'Black Oil Equations' - a very simple
type of Oil, Water and GAs simulation regimen.
Of course, we used relatively small test problems.
On 'Larger Problems' - that is in dividing the
Reservoir into larger numbers of smaller grid
cells -
NB: Methods are always improving, but the Best tend
NOT to be over-decoupling of neighboring regions
contributions, and they tend to be highly recursive.
Unfortunately these are NOT the conditions most
favorable for efficient uses of large multi-multi-CPU
'Super Computing Machines'.
However, ONCE such a machine is Installed, its users,
and requisitioners, must Make it Look GOOD to justify
their continuing good performance rating and jobs.
It is a situation rife with opportunities and incentives for
both conscious and true-believing unwitting, or even
subconscious, cheating and other scientific corruption.
- the methods I suggested also deteriorated,
but much more slowly (with increasing
problem size). Some methods are less sensitive
to CN increase, but ALL deteriorate.
I was simply MORE experienced in the particular problems,
and am a fanatic for doubt and checking. That is all
I really had over the poor, confident, self-superior,
proud, Industrial Scientific/Engineering Programmer!
Ning, my temporary supervisee, simply had been
used to 'getting away' with GS-SOR, which he
had been taught in School, and which he had
used - successfully, he claimed - in pure water
flow, and low concentration pollutant convection
problems he had worked on in previous jobs in
Hydrogeological Simulation.
Of course, because Ning had such Confidence
in the value of GS-Sor, he deemed 'convergence'
was attained when successive iterations starting
yielding miniscule adjustments of the proposed
'numerical solution'. And he never bothered to
check how small the 'residual errors' - the "out
of balance" in the governing equations - were.
This was the basis of a SUBCONSCIOUS cheat -
failing to check, and merely presuming that a famed
algorithm, once it stopped producing serious
adjustments of the estimated solution, had simply
"completed its job handily"!
Why waste Precious Computer Resources
CHECKING on such matters, when YOU
BELIEVE the Methods you are using are
the Very Best?!
Well, the Stinking Oil Industry - its kind of
Sloppy - Right, but AERODYNAMICS!!!! ...
Ok, Ok, Great Aerodynamic Design is done
these days - Right?!
Yes, indeed, but the problem is masked by all the
many Wind Tunnel Measurements (I have made
wind tunnel measurements by the way, when I was
a graduate student in Mechanical/Wind Engineering;
and for a couple of decades in the Canadian Oil
Industry, I did and advised on Numerical Flow
Simulation of Underground and Well Flow for
Oil Field Production and Development, Before
I studied Mechanical Engineering, I took an M.Sc.
in Mathematical Hydrodynamics from the Math
Department at University of BC. Earlier, when I
was in Alaska, I had an Introduction to Physical
Oceanography. I know quite a bit about a
variety of fluid and gas flow. And as well, many
aspects of Numerical Simulation of the same.),
which allow all sorts of 'fixes' and 'tweaks' to
be worked out for the methods - not only on a
particular design project, but often in a whole
region or type of flow regime application in
Aerodynamic Simulation Software Systems.
Yes, even a Black Box with lots of screws for
adjustments can be 'tuned in' with enough
trials and error correcting adjustments when
repeated compared against physical 'reality'!
Unfortunately, we DON'T have an accurate
'Wind Tunnel' equivalent for the Earth ...
for Weather ... and for Climate.
Yet Multi-Multi-Multi-CPU Machines are
requisitioned and placed in action by NASA
and NOAA for exactly things like Climate,
Ocean Currents, and Weather.
Although there are wonderful discoveries in abstract
algorithms using 'Divide and Conquer' Algorithms for
solving large linear systems, the logistics, data &
computation 'flow' (ie, BUS & CPU's Waiting for data
which Other CPUs are generating) problems involved
in use of STRONG, usually highly Recursive Methods,
to attack instant linearizations of Non repeatable Real
World Problems, or the Immense Data and computational
Resources required to - say - adequately represent the
Earth's Fluid spheres, even in abject oversimplification,
promotes the Feckless IDEA that FASTER is Better.
Well, the Faster which 512-Cpu or 1024-CPU machines
illicit is often 'tested' on Small Discretizatons of known or
sort-of-known Problems, inspires 'decoupling' of
related variables and cell groupings, so as to distribute
'the workload' over the Big Machine and take advantage
of its promise in parallel Computing ...
A BIG Obstacle: Known Solution problems are quite
Often Linear, Nearly Linear or Drastically OverSimplified,
or when recorded by some measuring instruments, of
less than full value short time duration.
Yet the questions we want to know with Climate,
for instance, is what the average weather will be
like in 20 or 50 years, so that we can 'plan for it'!
Simulations which 'Look Good in the Shower'
over a 5 year or less period of retro-historical
'matching' may be impotent and invalid over
fifty years, and as dominating Weather cycles
and fluctuations an be 'debalanced' by a spate
of 'El Ninnos/La Ninnas' or Volcanic Eruptions,
or disturbances of the Jet Stream by Pulses of
Auroral Energy falling in the Arctic Atmosphere,
what matches for the last 5 or 10 years may
become less relevant even for the next 5 or 10!
All we need is for the Sun to enter some somewhat
infrequent regime of gross energy output or distribution
of its energy flows between Light and Particles, and
the whole applecart may be upset!
I have seen many cases in Petroleum Reservoir Engineering,
where a Simulation Set Up is well fitted and matched to the
processes in the early or mid phase of the depletion of a Field,
but somewhat irrelevant for the New effects which occur when
water or gas production suddenly starts to bypass Oil flow
and changes the distribution of produced product in later
phases of a field's production.
Unfortunately, the life of a Weather Cycle can be as long
or longer than the life of an aggressively produced Oil Field,
and when Solar, Climate or Mini-Ice Age Cycles (like the
one from 1250-1850 which froze the Vikings from Greenland
and impaired food production in much of Northern Europe,
stopping, for whatever reason, Long Before the Great
American Forest and Prairie Clearings for Agriculture or
the massive growth of Coal Consuming Industry - and,
Why did it START even?!) are involved? ... well those
Simulations can be just stuffed in the 'Circular File'!
Remember the Bogus Club of ROME Simulations in the
Late 1960's or Early 1970's?! Well, Today's Club of
ROME may be based NOT on a mistake in programming
as the CoR Simulations were, but rather upon undue
faith in the Power of More CPUs and FASTER and
FASTER 'throughput'.
By the way, what is the Point of Fast Climate Simulations?!
DOES IT MATTER if they take 5 hours or 5 weeks?!
Well, it DOES, if 'Errors and Bugs' are being repeatedly
found, and WORSE YET, when Algorithmic Deficiencies
NOT Found in the Small, or Too Linear, or OverSimple
TESTS, are discovered and have to become corrected,
"BEFORE SOMEONE FINDS OUT!" Quick, Quick!
THEY Must have SOMETHING to Prove! ...
Ultimately, as I have described, even large LINEAR systems
have problems, as the 'Resolution' gets finer and finer.
Another difficulty in Time Dependent PDEs: When spatial
resolution gets finer and finer, timesteps often have to be
taken finer and finer, also. THAT drastically increases the
work which must be done to complete a simulation.
And when 'Trouble Spots' gets really finely 'gridded'
the same applies. Problem Nonlinearity and Difficulty,
ultimately Condition Number of Linearized Iterates,
Climbs like a Rocket bent for the Nearest Star!
Yeh! Bigger, Faster, Distributed, Expensive MultiCPU
Monster Computers for Bigger FASTER Erroneous
Throughput! For SOME Problems, with Zillions of
cases, it's The Cat's Meow, but were intrinsically
integral entire global problems? ... Where'd that
Almanac Go?!
But you know that one of the BIG PROBLEMS
of Computers are DATA Buses, with Traffic Control
on Data Buses, in particular. How are these many
microscopic units to be interfaced together?!
This is still a problem for Todays Optically Imaged
Lithographic, essentially 2-D machines. It will
be even more trouble with the extra degrees of
freedom (don't forget pivoting, rotation, and flexure!)
on jumping to Actual, Full, Bottom-Up 3-D!
Connections Upon Connections!
"The Wring Experts! - WE WIRE YOU GOOD!"
"The Bionic Connection
Are we already a lot closer to a mind-machine
interface than we ever guessed?
By Jocelyn Selim
"Kevin Warwick is about to become telepathic,
luck and technology permitting. His lips are
parted expectantly as he sits blindfolded and
perched on a lab stool at Reading University
in England. One inch below his left wrist,
a pincushion array of 100 silicon electrodes,
all of them together about one-sixteenth the
size of a dime, has been surgically inserted
into his median nerve. From the electrodes,
22 wires run eight inches under the skin and
exit one inch below his elbow. There they are
soldered to a connector board where a
2-inch-by-2-inch maze of circuitry amplifies,
filters, and converts the electrochemical
impulses coming down his median nerve into
digital signals."
www.discover.com/nov_02/gthere.html?article=featbionic.html
OUCH!
I will look.
Some Things we may just Never Get Right,
so don't be a Sop for all the Overwhelming
Optimisms of the Self-Promoting Worldlies -
Y'All Hear, Now?!
GGF - The Iowan Idiot
PS: "Legions and Legions of Nanobots,
All Lined in Fine Array, Ready to HELP
The Denti$t, Find all the Tooth Decay!
©2002 Gary G. Ford
| 1. Notes |
