Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies

The front matter includes the title page, series page, copyright page, TOC, and preface.

We assume that you are here with a computer simulation of a complicated physical model that includes several input parameters. You want to perform some sort of parameter study—such as optimization, uncertainty quantification, or sensitivity analysis—with the simulation's predictions as a function of its inputs. But the methods you know for such studies are not practical with all of your parameters because of your limited computational budget. In other words, the methods need too much computation in high dimensions—more computation than you have available.

Today's models of our world are wonderfully complex thanks to fast and powerful computers. Modern simulations couple interacting physical phenomena to model rich, interconnected systems. Researchers on the frontier of computational science exploit high-performance computing to bridge the gap between micro- and macroscales. Next-generation simulations will combine physical models with high-resolution images and vast stores of measurement data to calibrate, validate, and improve predictions.

Before we can discover the active subspace, we need to know what to look for. We first characterize a class of functions that covers many quantities of interest from parameterized simulations. We then define the active subspace, which is derived from a given function. Once we know what to look for, we propose and analyze a random sampling method to find it.

If we have discovered an active subspace for a simulation's f (x), then we want to exploit it to perform the parameter studies—optimization, averaging, inversion, or response surface construction—more efficiently. The general strategy is to focus a method's efforts on the active variables and pay less attention to the inactive variables. But the details differ for each parameter study. In this chapter we set up a framework for thinking about the dimension reduction, and we address each parameter study individually. The techniques for working with response surfaces are the most mature; these can be found in our prior work [28]. The remaining parameter studies have several open questions. This chapter is full of emerging ideas, and we look forward to their continued development.

This chapter presents three engineering applications where active subspaces are helping to answer questions about the relationship between a complex simulation's many inputs and its output.

The active subspace is defined by a set of directions in a multivariate function's input space. Perturbing the inputs along these directions changes the function's output more, on average, than perturbing the inputs in orthogonal directions. The active subspace is derived from the function's gradient, so it is a property of the function. We have presented and analyzed a technique based on randomly sampling the gradient to test whether a function admits an active subspace. When an active subspace is present, one can exploit it to reduce the dimension for computational studies that seek to characterize the relationship between the function's inputs and outputs, such as optimization, uncertainty quantification, and sensitivity analysis. This is especially useful for complex engineering simulations with more than a handful of inputs.

The back matter includes bibliography and index.