Quadratic Weighted Histopolation on Tetrahedral Meshes with Probabilistic Degrees of Freedom
Allal Guessab, Federico Nudo
TL;DR
This work tackles accurate 3D function reconstruction on tetrahedra from integral data by enlarging the classical linear histopolation with three quadratic, probabilistically weighted enrichment schemes: face–volume, purely volumetric, and edge–face. Each scheme builds a set of quadratic degrees of freedom from densities and orthogonal polynomials, and a unisolvence theorem guarantees well-posedness through a rank condition on the quadratic moment matrix; explicit Dirichlet and Beta-type density families are analyzed to ensure positive definiteness and practical computability. An explicit reconstruction operator is derived for each enrichment, accompanied by an adaptive parameter-selection procedure that tunes density parameters (e.g., $\alpha,\beta$) via grid search. Numerical experiments demonstrate substantial accuracy gains over classical linear histopolation, with the volumetric enrichment often offering the smallest errors and the other enrichments providing directional benefits, thereby enabling more accurate three-dimensional reconstructions in applications like tomography and PDE simulations.
Abstract
In this paper we introduce three complementary three-dimensional weighted quadratic enrichment strategies to improve the accuracy of local histopolation on tetrahedral meshes. The first combines face and interior weighted moments (face-volume strategy), the second uses only volumetric quadratic moments (purely volumetric strategy), and the third enriches the quadratic space through edge-supported probabilistic moments (edge-face strategy). All constructions are based on integral functionals defined by suitable probability densities and orthogonal polynomials within quadratic trial spaces. We provide a comprehensive analysis that establishes unisolvence and derives necessary and sufficient conditions on the densities to guarantee well-posedness. Representative density families, including two-parameter symmetric Dirichlet laws and convexly blended volumetric families, are examined in detail, and a general procedure for constructing the associated quadratic basis functions is outlined. For all admissible densities, an adaptive algorithm automatically selects optimal parameters. Extensive numerical experiments confirm that the proposed strategies yield substantial accuracy improvements over the classical linear histopolation scheme.