Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hessian-based sampling for high-dimensional model reduction   In this work we develop a Hessian-based sampling method for the construction
of goal-oriented reduced order models with high-dimensional parameter inputs.
Model reduction is known very challenging for high-dimensional parametric
problems whose solutions also live in high-dimensional manifolds. However, the
manifold of some quantity of interest (QoI) depending on the parametric
solutions may be low-dimensional. We use the Hessian of the QoI with respect to
the parameter to detect this low-dimensionality, and draw training samples by
projecting the high-dimensional parameter to a low-dimensional subspace spanned
by the eigenvectors of the Hessian corresponding to its dominating eigenvalues.
Instead of forming the full Hessian, which is computationally intractable for a
high-dimensional parameter, we employ a randomized algorithm to efficiently
compute the dominating eigenpairs of the Hessian whose cost does not depend on
the nominal dimension of the parameter but only on the intrinsic dimension of
the QoI. We demonstrate that the Hessian-based sampling leads to much smaller
errors of the reduced basis approximation for the QoI compared to a random
sampling for a diffusion equation with random input obeying either uniform or
Gaussian distributions.

20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph