Model Input and Output Dimension Reduction Using Karhunen–Loève Expansions With Application to Biotransport

Abstract

We consider biotransport in tumors with uncertain heterogeneous material properties. Specifically, we focus on the elliptic partial differential equation (PDE) modeling the pressure field inside the tumor. The permeability field is modeled as a log-Gaussian random field with a prespecified covariance function. We numerically explore dimension reduction of the input parameter and model output. Truncated Karhunen-Loève (KL) expansions are used to decompose the log-permeability field, as well as the resulting random pressure field. We find that although very high-dimensional representations are needed to accurately represent the permeability field, especially in presence of small correlation lengths, the pressure field is not very sensitive to high-order KL terms of the input parameter. Moreover, we find that the pressure field itself can be represented accurately using a KL expansion with a small number of terms. These observations are used to guide a reduced-order modeling approach to accelerate computational studies of biotransport in tumors.

Structure exploiting methods for fast uncertainty quantification in multiphase flow through heterogeneous media

“…. , N p are the eigenpairs of the covariance operator of Z(x, ω); see e.g., [2,6,24] for details about the use of KL expansions for representing random fields in mathematical models. For the present problem, we let N p = 100, which enables capturing over 96 percent of the average variance of the process.…”

“…The quantity r k represents the fraction of the average variance of f captured by the first k eigenvalues. The steps for computing the truncated KLE of f are included in Algorithm 1, which is adapted from [2]. Note that evaluating the truncated KLE of f requires computing the KL modes, which in turn requires a model evaluation.…”

A distributed active subspace method for scalable surrogate modeling of function valued outputs

“…These model evaluations will be used to compute the KL expansion of the model f (x, ξ). For this, we use Algorithm 1 in [2], that uses Nyström's method to compute λ k (C f ) and the corresponding eigenvectors φ k (•), k = 1, . .…”

“…In models governed by PDEs the output field often exhibits favorable regularity properties and can be represented by a KL expansion with a relatively small number of KL terms. Examples of this appear for instance in our recent works [7,2] for models governed by elliptic PDEs.…”