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Published at December 3

AAROC: Reduced Over-Collocation Method with Adaptive Time Partitioning and Adaptive Enrichment for Parametric Time-Dependent Equations

math.NA

cs.NA

physics.flu-dyn

Released Date: December 3, 2024

Authors: Lijie Ji¹, Zhichao Peng², Yanlai Chen³

Aff.: ¹Shanghai University; ²The Hong Kong University of Science and Technology; ³University of Massachusetts Dartmouth

Arxiv: http://arxiv.org/pdf/2412.02152v1

$\boldsymbol{\mu}=(\mu_{1},\dots,\mu_{p})$	Parameter in $p$ -dimensional parameter domain $\Omega_{p}\subseteq\mathbb{R}^{p}$
$\Xi_{\rm{train}}$	Parameter training set, a finite subset of $\Omega_{p}$
$u(\boldsymbol{\mu})$	Function-valued solution of a parameterized PDE on $\Omega_{\boldsymbol{x}}\subset\mathbb{R}^{d}$
$\mathcal{P}(u(\boldsymbol{\mu}),t,x;\boldsymbol{\mu})$	A (nonlinear) PDE operator
$N_{x}$	Number of finite difference intervals in $x-$ direction of the physical domain
$\mathcal{N}$	Degrees of freedom (DoF) of a high-fidelity PDE discretization, the “truth” solver
$X^{\mathcal{N}}$	A size- $\mathcal{N}$ (full) collocation grid
$N_{\rm max}$	Number of reduced basis snapshots, $N_{\rm max}\ll\mathcal{N}$
$N_{\rm tpar}$	Number of time segments
$I_{j}$	The $j$ -th time segment, $j=1,\cdots,N_{\text{tpar}}$
$\boldsymbol{\mu}^{j}$	“Snapshot” parameter values, $j=1,\ldots,N_{\rm max}$
$V_{n}$	$n$ -dimensional RB space
$\widehat{u}_{n}(\boldsymbol{\mu},t)$	Reduced basis solution in the $n$ -dimensional RB space $V_{n}$
$E_{n}(\boldsymbol{\mu})$	Relative Reduced basis solution error
$\Delta_{{N}}\left(\boldsymbol{\mu}\right)$	A residual-based error estimate
$X^{n-1}_{r,j}$	The reduced collocation grid at $n$ -th greedy loop in $j$ -th segment, a subset of $X^{\mathcal{N}}$ determined based on residuals, $j=1,\cdots,N_{\text{tpar}}$
$X^{n}_{s,j}$	An additional reduced collocation grid at $n$ -th greedy loop in $j$ -th segment, a subset of $X^{\mathcal{N}}$ determined based on the solutions, $j=1,\cdots,N_{\text{tpar}}$
$X^{n}_{j}$	The whole reduced collocation grid at $n$ -th greedy loop in $j$ -th segment, defined by $X^{n-1}_{r,j}\cup X^{n}_{s,j}$ , $j=1,\cdots,N_{\text{tpar}}$
$T$	Final time for the time-dependent problems
$\Delta t$	Time stepsize for the time dependent problems
$t_{i}$	Time level $i$ , $i=1,\ldots,\mathcal{N}_{t}$
$\mathcal{N}_{t}$	Total number of time levels, i.e. $\mathcal{N}_{t}=T/\Delta t$
${\mathcal{T}}_{f}$	The set of uniformly discretized time nodes $t_{i}$ of the entire time domain $[0,T]$
${\mathcal{T}}_{j}$	The set of uniformly discretized time nodes $t_{i}$ within $j$ -th time segment $I_{j}$ , $j=1,\cdots,N_{\text{tpar}}$
${\mathcal{T}}_{r}$	The set of selected time nodes for the RB space during the offline process
$\rho_{\Delta}^{n}$	The robustness indicator of $n-$ th greedy loop
$\gamma$	Control parameter for the robustness indicator of the AAROC method
$p_{\text{adap}}$	Percent parameter for the adaptive enrichment strategy
$n_{\text{adap}}^{\rm{incre}}$	Stepsize of the adaptive enrichment strategy
$n_{\rm{adap}}^{\max}$	The maximum number of adaptive enriched collocation points
$\epsilon_{\mathrm{tol}}$	Error estimate stopping tolerance in greedy sweep
Offline process	The pre-computation phase, during which the reduced basis and reduced collocation points in each time segment are determined through a greedy loop.
Online process	The process of solving the reduced problem to obtain the reduced order solution.