“SIAM Spotlights” is a new series of short books on topics in computational mathematics and scientific computing. The topic here is active subspaces and how they can reduce the dimension of a parameter study. As described here, parameter studies begin with a simulation model with m well-defined input parameters that computes a single quantity of interest. Each parameter ranges over a finite interval about a starting point, and there are resources for running the simulation multiple times. The author further assumes that the simulation computes a Lipshitz continuous function, f(x), whose domain is the Cartesian product of intervals about the starting values. Finally, there is a probability distribution defined on that domain. An obvious example of a parameter study is establishing tolerances for the parameters of a finished design of a product such as a camera lens. Once the designers have determined nominal values of the parameters with an acceptable measure of quality, they need to find out if combinations of small errors might produce an unacceptable measure. If there are more than a few parameters, this can be a daunting question. According to this author, “active subspaces offer a more general approach to reduce the study’s dimension. They identify a set of important directions in the input space.”
Chapter 1 begins by describing the requirements for the simulation, and then defines three different algorithms for finding possible important directions. These algorithms are applied to some examples with m = 2. Chapters 2 and 3 present some theoretical motivation for the algorithms based on the directional derivative. For a fixed direction, the directional derivative of f at a random x is a random variable, and the mean square of that variable is hTCh, where h is a unit vector in the given direction and C is a covariance matrix derived from the gradient of f. The eigenvectors of C turn out to be the important directions.
The author distinguishes four types of studies depending on whether you want a global approximation of f(x) or only a dependable estimate of the probable range of values of f, and chapter 4 describes how to exploit active subspaces for each type. In general, if there are m parameters and we have approximated C and m eigenvectors and values, then we look for a group of n larger eigenvalues separated by a significant gap from the remaining m – n. This decomposes the parameter space into two orthogonal subspaces W1 and W2. We then look for a function of n parameters g(y) with y = W1Tx that can represent f(x). To get a global model for f(x), we can define g(y) as the expectation of f(W1 y + W2TW2x) conditional on y = W1T x. The author shows that the mean-square error of this model for f is bounded by a constant times the sum of the smaller eigenvalues.
The last two chapters of this short work contain three detailed examples and a discussion of possible directions for further research. The author includes a bibliography of 130 references; since the text is only 88 pages, I believe his major contribution here is the condensation of much other work to concise, well-written text.
