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I am solving an ill-posed inverse problem and having a difficult time researching related methodologies because I don't know the appropriate jargon/nomenclature.

I have a system with several parameters, $x_1,x_2,\dots,x_N$ (some continuous, and some discrete), where $N\approx 10$. The values of two properties, $y_1$ and $y_2$, are dependent on these parameters:

$$y_1 = f\left(x_1,x_2,\dots,x_N\right)$$ $$y_2 = f\left(x_1,x_2,\dots,x_N\right)$$

My task is to find a combination (or multiple combinations) of parameters that result in a specific value pair for $y_1$, and $y_2$, provided they fall within the parameter space (if they are not in the parameter space, then find the combination that results in the nearest match).

Many of the searches I've done for material on inverse-problems are related to continuum mathematics, which my problem is not.

Where do I start looking for material on solving problems like this?

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  • $\begingroup$ Do you have a sample of (x,y) pairs? Have you looked into some sort of regression method if so? $\endgroup$
    – Chester
    Aug 7, 2015 at 21:49
  • $\begingroup$ All I have is the target $y_1$ and $y_2$ values and the functions used to calculate them. I don't have data a priori to generate a regression. $\endgroup$
    – Neal Kruis
    Aug 7, 2015 at 22:13
  • $\begingroup$ I suppose I could generate enough combinations to train a regression, but that could be fairly substantial considering the size of the parameter space. $\endgroup$
    – Neal Kruis
    Aug 7, 2015 at 22:18

1 Answer 1

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Metaheuristics. Metaheuristics sample a set of solutions which is too large to be completely sampled. In my case, I am exploring the use of genetic algorithms and particle swarm optimization to minimize the error in my search space which I defined as:

$$\sqrt{\left(y_{1,target}-y_1\right)^2+\left(y_{2,target}-y_2\right)^2}$$

These techniques allow me to efficiently search the parameter space for global optima without prior knowledge of the result surface.

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