## Ode to Inverse Problems

Deep into the night and before calling it a day for today (disgusting rhyme but “lasciamo stare“) this is one of my geek-break-musing from actual work. The topic is a fascinating family of mathematical problems called “inverse problems”.  Wikipedia is a good place to get some more formal information but to make it really short here is a definition by example: 2+4=6, no doubt about it and this constitutes our “forward problem”, i.e. given a defined input we calculate a unique result; but what if some one asked to find out what you had to add up to get 6 but had no way to ask you or otherwise determine the exact combination? There are 4 different combinations to get 6 by adding two numbers and even more if you can add more than that, so how can you say? Well you can’t really and this is what makes an “inverse problem”, i.e. when you are trying to deduce the right answer from a pool of equally plausible alternatives. Welcome to the mathematical equivalent of delusion.  Now for the necessary soundtrack I will go for something very mathematical and at the same time delusional: enter sandman in the form of Gould.

In the very essence this is an area of applied mathematics but us lowly engineers sneak in, as always, to muddle elegant theory into crude and practical results. The later, the muddle bit, is in a broader sense the topic of my doctoral dissertation.  In my thesis I look into the problems arising in the stochastic reconstruction of random heterogeneous materials, like porous rocks for instance.  In simple terms I ma trying to come up with ways to get from a statistical description of 2D images to a fully 3D representation of the material that is true to the very nature of the original. Now, how can you describe statistically a 2D image? This is hard if you think in terms of a complex full colour image but if you picture  a simple black & white checkerboard this might become easier to grasp. Such a

El Tunel

pattern can easily be translated to some rule (read statistic) that simply says “every 1 step of 10cm paint a black square, immediately followed by the same in white and then repeat  until you have reached so-and-so dimensions”. This is the recipe to make a black-and-white Rubik-type of cube were each sub-cube is of alternate colour. This is the idea that is applied to complex heterogeneous materials, a nasty term that means “something like a (natural) sponge, all irregular and weird but also the same way weird no matter which way you look at it”.  Now that we have established our toolbox of statistical descriptors we can venture to the real world. This methodology suffers from a serious practical limitation: it is not computationally feasible and to an extend even theoretically possible to have a complete statistical description of the intricate morphology of a complex material just by examining 2D sections of it. As always the mathematicians did their brilliant part and then we engineers started cutting corners and trying to fit our need to the delusional nature of this endeavour. My contribution, more like a mashup really, was to pick-and-choose ways to improve getting to a realistic representation of the real material by (in effect) cheating.  The “cheat”  consist of some educated guesses that use previously know information on how nature or industry generates/constructs a material and use this information to guide our solution algorithm through the maze of multiple plausible solutions to the one like the actual material we are trying to recreate. I’ve written about 180 pages of instructions manual to practising (dis)illusionists, called it a thesis and hope to get a PhD out of it the 17-Feb-09.

To end this musing I want to draw the attention to an intriguing and very personal experience of  inverse problems, that is learning from experience. Now at this point a disclaimer is of order; I am going to misuse and abuse scientific terms a little to make this artsy-fartsy, literary ending sound pompous. Don’t buy it, it is just fluff with no substance but I can’t resist making analogies out of analogies.  Think about it every time you are trying make a rational choice based on previous experience. What you  (we)  are really trying to solve is a horribly inverse problem. We are trying to estimate a new state of future affairs based on the (arguably) incomplete evaluation of past events, a classic in parameter estimation for those versed in reaction engineering. Now how much this helps when getting into an argument with your concubitus is another story and I wish you my best of luck!