High order recovery of geometric interfaces from cell-average data

TL;DR

This work considers the problem of recovering multivariate characteristic functions u := χΩ from cell-average data on a coarse grid, motivated in particular by the accurate treatment of interfaces in finite volume schemes, and derives two main families of local reconstruction schemes based on nonlinear least-squares fitting.

Abstract

We consider the problem of recovering characteristic functions $u:=χ_Ω$ from cell-average data on a coarse grid, and where $Ω$ is a compact set of $\mathbb{R}^d$. This task arises in very different contexts such as image processing, inverse problems, and the accurate treatment of interfaces in finite volume schemes. While linear recovery methods are known to perform poorly, nonlinear strategies based on local reconstructions of the jump interface $Γ:=\partialΩ$ by geometrically simpler interfaces may offer significant improvements. We study two main families of local reconstruction schemes, the first one based on nonlinear least-squares fitting, the second one based on the explicit computation of a polynomial-shaped curve fitting the data, which yields simpler numerical computations and high order geometric fitting. For each of them, we derive a general theoretical framework which allows us to control the recovery error by the error of best approximation up to a fixed multiplicative constant. Numerical tests in 2d illustrate the expected approximation order of these strategies. Several extensions are discussed, in particular the treatment of piecewise smooth interfaces with corners.

Figures (26)

• Figure 1: ENO-SR in one dimension: the jump point $X_0$ is identified by matching the average on the singular cell with the piecewise polynomial reconstruction.
• Figure 2: Local approximation of a smooth interface by a line interface.
• Figure 3: Line interfaces
• Figure 4: Circular interfaces
• Figure 5: Quadratic subgraphs

Theorems & Definitions (15)

• Remark 1.1
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Theorem 3.1
• Theorem 3.2
• Corollary 3.3