Predictor Corrector Methods

Since New Year’s eve of last year we’ve been managing  a minor, local cataclysm in the high-rise in which we live.  A high pressure fire control line near the top of the building froze as a result of the sub-zero temperatures and sustained gale force winds which created a wind chill of -23 F.  I walked into the stairwell a few minutes after it occurred and found it transformed into a raining, freezing environment that looked more like one of Bradley Garrett's explorations than the placid, quiet home we’re used to.  Roughly, a third of the buildings units were affected in minutes.

It felt like a scene in an old disaster movie.  I probably imagined myself as a heroic but doomed submarine captain on a sinking boat for a few seconds but didn’t find that particularly helpful.  More importantly, I found myself doing what you do in those situations:  continually prioritizing a large list of issues and attempting to respond appropriately.  As a result of continuous, pragmatic hard work since then by our resident manager and his wife, Lynn and the rest of us on the HOA board, a responsive disaster control company and our insurance agent the situation remains under control and we’re taking the right steps to mediate and repair the damage.

What makes it interesting and worth writing about is in the detail  of our small community’s response.  Not only are we doing everything required to restore the building, we’re also taking formal action, to study the building and take appropriate action to ensure we’re proof against another extreme weather event in the future.  In essence, we’re using the situation to predict a possible future event and then correct our existing infrastructure wherever required.

It’s a simple idea.  1900 saw the rise of what have come to be called “Predictor-Corrector” methods to solve difficult or even technically impossible problems in the mathematics of differential equations.  Russell and Whitehead, among others at about the same time, were showing that there were a vast set of mathematical problems that couldn’t be solved – exactly.

However, a very interesting and important set of those can be addressed by predicting an approximation of the answer, correcting it with a second step and then using that information to predict again.  In that way it’s possible to come as close to the answer to some “unsolvable” problems as is practically necessary.  Predictive-corrective methods are now part of the fabric of our technology.  It’s how our computers perform some of the most basic calculations and how a robotic craft can land on a comet.

We live in an age and cultures of systems and complexity.  I find it curious that some institutions, particularly political ones, have such slight interest in predictive-corrective strategies to solve problems when it’s probable they could be applied successfully to many of even our most controversial problems, even those of a moral character.