Nonlinear dimensionality reduction for parametric problems: A kernel proper orthogonal decomposition

Abstract

Reduced-order models are essential tools to deal with parametric problems in the context of optimization, uncertainty quantification, or control and inverse problems. The set of parametric solutions lies in a low-dimensional manifold (with dimension equal to the number of independent parameters) embedded in a large-dimensional space (dimension equal to the number of degrees of freedom of the full-order discrete model). A posteriori model reduction is based on constructing a basis from a family of snapshots (solutions of the full-order model computed offline), and then use this new basis to solve the subsequent instances online. Proper orthogonal decomposition (POD) reduces the problem into a linear subspace of lower dimension, eliminating redundancies in the family of snapshots. The strategy proposed here is to use a nonlinear dimensionality reduction technique, namely, the kernel principal component analysis (kPCA), in order to find a nonlinear manifold, with an expected much lower dimension, and to solve the problem in this low-dimensional manifold. Guided by this paradigm, the methodology devised here introduces different novel ideas, namely, 1) characterizing the nonlinear manifold using local tangent spaces, where the reduced-order problem is linear and based on the neighboring snapshots, 2) the approximation space is enriched with the cross-products of the snapshots, introducing a quadratic description, 3) the kernel for kPCA is defined ad hoc, based on physical considerations, and 4) the iterations in the reduced-dimensional space are performed using an algorithm based on a Delaunay tessellation of the cloud of snapshots in the reduced space. The resulting computational strategy is performing outstandingly in the numerical tests, alleviating many of the problems associated with POD and improving the numerical accuracy.

A multiparametric advection-diffusion reduced-order model for molecular transport in scaffolds for osteoinduction

“…The analysis for all possible combinations of parameters, along with the required nonlinear system of equations with a high number of degrees of freedom per each case, motivates the use of a ROM. The proposal presented here is to use a posteriori ROM, following the ideas in Díez et al (2021).…”

“…The number of parameters in the model (six in total) motivates the use of a ROM for multiple evaluation. We study the performance of standard POD and local POD, accounting for both the original snapshots and new, quadratically generated, ones (Díez et al 2021).…”

Operator inference for non-intrusive model reduction with quadratic manifolds

“…where ε(t) denotes the error associated with the projection of s(t) − s ref onto the linear subspace V. While many different manifold constructs are conceivable, we focus on a polynomial mapping between the highdimensional data samples and their lower-dimensional representations. Explicit nonlinear mappings with a polynomial structure were originally proposed in manifold learning [42], but similar formulations have emerged recently in model reduction [2,3,5,43,44]. Our attention will be restricted to the case of quadratic Kronecker products:…”

“…The mapping can be constructed in non-intrusive fashion using linear regression and is driven by physics-based training data. A similar concept was also explored in [5], where a kernel principal component analysis is used to find a nonlinear manifold with lower dimensionality. In that work, the approximation space is enriched with element-wise cross-products of the snapshots, thereby establishing globally curved manifolds.…”

“…In that work, the approximation space is enriched with element-wise cross-products of the snapshots, thereby establishing globally curved manifolds. Our approach differs from [5] in that we explicitly derive a data-driven basis to represent the quadratic components of the manifold.…”