Successful Applications of Interval Arithmetic

[David G. Hough at validgh]

George Corliss provided the following examples to the reliable_computing
mailing list from his own experience.   
There's a paper to describe the first few in more detail,
which I didn't include, but which should be available from him:

Date: Fri, 22 Dec 95 15:33 CST
From: uunet!boris.mscs.mu.edu!georgec (Dr. George F. Corliss  MU MSCS)
Subject: Applications of intervals (long)

Here are some stories I know.  Interval techniques are no "silver
bullet", but they CAN provide insight.


1.  arctan

Some years back, a major oil company was running a portion of an
oil reservoir simulation code.  They were running identical codes
on identical IBM 3090 computers with identical software configurations
at two different sites.  One machine consistently gave non-physical
negative concentrations.  After MUCH investication, the difference
was traced to different arctan routines in the vendor-supplied
libraries.  The engineers, of course, standardized on the library
that led to positive concentrations, ALTHOUGH THAT WAS THE ARCTAN
THAT YIELDED _WRONG_ RESULTS.  The engineers went away satisfied,
but the computer support people were MOST dismayed, and called me
as an interval consultant.  When I arrived, though, they preferred
that I look at different problems.  I do not know the outcome of
the engineers continuing intentionally to use the incorrect arctan.


2.  sqrt (negative)

The same oil company DID ask me to look at a crack propagation 
submodel in their reservoir simulation model.  The crack propagation
model was a nonlinear PDE, which they solve by time stepping.  At
each time step, they solved a nonlinear system by Newton's method.
Computations proceeded smoothly for about 100 time steps, when 
suddenly, computation halted with a sqrt (negative).  We used
interval techniques to validate the existing approximating
computations and to discover that the problem was a need for a more
accurate starting guess for the Newton iteration.  That modification
was made to the approximate code, and the simulation ran for more
than 200 time steps.


3.  Singular linear systems.

The same oil company gave me a linear system they were trying to solve
by a conjugate gradient algorithm.  The algorithm was not converging
as they expected.  I coded it up using the accurate linear solver
in ACRITH, and ACRITH computed 1 ULP enclosures of an answer on an
IBM 3090 while we ate lunch.  They hadn't solved the problem, but I
had.  Great!  Actually, the long accumulator had more to do with the
success than interval arithmetic, but the interval techniques DID
allow us to validate our results.  Next, I wanted to explore the
sensitivity of the result, so I inclated their matrix coefficients
by a relative width of about 10^{-3}.  ACRITH said the interval
matrix contained a singular matrix.  After further experiments,
when I inflated ANY element of their original point-valued matrix
to an interval 1 UPL wide, ACRITH again said that the matrix was
singular.  So I went to the engineers and triumphantly proclaimed,
"The reason you were having trouble is that you are trying to solve
a singular system, and interval techniques have validated that!"
One engineer looked puzzled.  "Oh, didn't we tell you?  ALL our
matrices are singular."  So I learned something, even if they knew
it already.


4.  Bounded solutions of linear systems.

A biomodeler cam to me with a problem of blood flow in a capillary
bed.  He had a pressure in a single incoming artery and the pressure
in a single outgoing vein.  In between was a complicated network
flow problem.  He had formulated the pressure model as a linear 
system, which he had solved, but he was concerned about the accuracy 
and robustness of his results.  After all, the problem was very 
sensitive, since a restriction in one vessel in the bed causes
BIG changes in the flows of nearby vesslels.  We set up an 
interval-valued system to reflect the uncertainties in his data.
This time, when we passed the system to an interval solver, we
were surprised to find everything perfectly well behaved.  There
was no evidence (in the pressures) of any sensitivity at all.
My client thought for a moment and exclaimed, "Of course!  The
pressures _must_ be bounded by the input and the output pressures,
so we have no fear of unbounded solutions!"  The application of 
interval techniques allowed him to concentrate on his model, without
fears of numerical difficulties.



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