English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We consider approximating solutions to parameterized linear systems of theform $A(\mu_1,\mu_2) x(\mu_1,\mu_2) = b$, where $(\mu_1, \mu_2) \in\mathbb{R}^2$. Here the matrix $A(\mu_1,\mu_2) \in \mathbb{R}^{n \times n}$ isnonsingular, large, and sparse and depends nonlinearly on the parameters$\mu_1$ and $\mu_2$. Specifically, the system arises from a discretization of apartial differential equation and $x(\mu_1,\mu_2) \in \mathbb{R}^n$, $b \in\mathbb{R}^n$. This work combines companion linearization with the Krylovsubspace method preconditioned bi-conjugate gradient (BiCG) and a decompositionof a tensor matrix of precomputed solutions, called snapshots. As a result, areduced order model of $x(\mu_1,\mu_2)$ is constructed, and this model can beevaluated in a cheap way for many values of the parameters. The decompositionis performed efficiently using the sparse grid based higher-order propergeneralized decomposition (HOPGD), and the snapshots are generated as onevariable functions of $\mu_1$ or of $\mu_2$. Tensor decompositions performed ona set of snapshots can fail to reach a certain level of accuracy, and it is notpossible to know a priori if the decomposition will be successful. This methodoffers a way to generate a new set of solutions on the same parameter space atlittle additional cost. An interpolation of the model is used to produceapproximations on the entire parameter space, and this method can be used tosolve a parameter estimation problem. Numerical examples of a parameterizedHelmholtz equation show the competitiveness of our approach. The simulationsare reproducible, and the software is available online.

Sparse grid based Chebyshev HOPGD for parameterized linear systems