# Embedded Ridge Approximations

## TL;DR

This paper introduces embedded ridge approximations for scalar fields in differential-equation models, enabling efficient estimation of quantities of interest by exploiting local influence and anisotropy, with theoretical and simulation validation.

## Contribution

It formalizes embedded ridge approximations for scalar fields, connecting them with active subspaces, and develops algorithms for reduced order modeling based on ridge profiles at nodes.

## Key findings

- Embedded ridge approximations effectively estimate global quantities of interest.
- Algorithms recover field quantities at nodes using limited ridge profile data.
- The approach demonstrates favorable theoretical properties and practical performance.

## Abstract

Many quantities of interest (qois) arising from differential-equation-centric models can be resolved into functions of scalar fields. Examples of such qois include the lift over an airfoil or the displacement of a loaded structure; examples of corresponding fields are the static pressure field in a computational fluid dynamics solution, and the strain field in the finite element elasticity analysis. These scalar fields are evaluated at each node within a discretised computational domain. In certain scenarios, the field at a certain node is only weakly influenced by far-field perturbations; it is likely to be strongly governed by local perturbations, which in turn can be caused by uncertainties in the geometry. One can interpret this as a strong anisotropy of the field with respect to uncertainties in prescribed inputs. We exploit this notion of localised scalar-field influence for approximating global qois, which often are integrals of certain field quantities. We formalise our ideas by assigning ridge approximations for the field at select nodes. This embedded ridge approximation has favorable theoretical properties for approximating a global qoi in terms of the reduced number of computational evaluations required. Parallels are drawn between our proposed approach, active subspaces and vector-valued dimension reduction. Additionally, we study the ridge directions of adjacent nodes and devise algorithms that can recover field quantities at selected nodes, when storing the ridge profiles at a subset of nodes---paving the way for novel reduced order modeling strategies. Our paper offers analytical and simulation-based examples that expose different facets of embedded ridge approximations.