Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa

AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20


Published: SIAM Review, 44 (2002) 631--658.

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Convergence of Adaptive Finite Element Methods

The Delaunay triangulation, in both classic and more generalized sense, is studied in this paper for minimizing the linear interpolation error (measure in L^P-norm) for a given function. The classic Delaunay triangulation can then be characterized as an optimal triangulation that minimizes the interpolation error for the isotropic function ‖x‖^2 among all the triangulations with a given set of vertices. For a more general function, a functiondependent Delaunay triangulation is then defined to be an optimal triangulation that minimizes the interpolation error for this function and its construction can be obtained by a simple lifting and projection procedure. The optimal Delaunay triangulation is the one that minimizes the interpolation error among all triangulations with the same number of vertices, i.e. the distribution of vertices are optimized in order to minimize the interpolation error. Such a function-depend entoptimal Delaunay triangulation is proved to exist for any given convex continuous function.On an optimal Delaunay triangulation associated with f, it is proved that △↓f at the interior vertices can be exactly recovered by the function values on its neighboring vertices.Since the optimal Delaunay triangulation is difficult to obtain in practice, the concept of nearly optimal triangulation is introduced and two sufficient conditions are presented for a triangulation to be nearly optimal.

Optimal delaunay triangulations

Learning to model and reconstruct humans in clothing is challenging due to articulation, non-rigid deformation, and varying clothing types and topologies. To enable learning, the choice of representation is the key. Recent work uses neural networks to parameterize local surface elements. This approach captures locally coherent geometry and non-planar details, can deal with varying topology, and does not require registered training data. However, naively using such methods to model 3D clothed humans fails to capture fine-grained local deformations and generalizes poorly. To address this, we present three key innovations: First, we deform surface elements based on a human body model such that large-scale deformations caused by articulation are explicitly separated from topological changes and local clothing deformations. Second, we address the limitations of existing neural surface elements by regressing local geometry from local features, significantly improving the expressiveness. Third, we learn a pose embedding on a 2D parameterization space that encodes posed body geometry, improving generalization to unseen poses by reducing non-local spurious correlations. We demonstrate the efficacy of our surface representation by learning models of complex clothing from point clouds. The clothing can change topology and deviate from the topology of the body. Once learned, we can animate previously unseen motions, producing high-quality point clouds, from which we generate realistic images with neural rendering. We assess the importance of each technical contribution and show that our approach outperforms the state-of-the-art methods in terms of reconstruction accuracy and inference time. The code is available for research purposes at https://qianlim.github.io/SCALE.

/pdf/scale-modeling-clothed-humans-with-a-surface-codec-of-59ncct7for.pdf

SCALE: Modeling Clothed Humans with a Surface Codec of Articulated Local Elements

We describe a discrete Laplacian suitable for any triangle mesh, including those that are nonmanifold or nonorientable (with or without boundary). Our Laplacian is a robust drop‐in replacement for the usual cotan matrix, and is guaranteed to have nonnegative edge weights on both interior and boundary edges, even for extremely poor‐quality meshes. The key idea is to build what we call a “tufted cover” over the input domain, which has nonmanifold vertices but manifold edges. Since all edges are manifold, we can flip to an intrinsic Delaunay triangulation; our Laplacian is then the cotan Laplacian of this new triangulation. This construction also provides a high‐quality point cloud Laplacian, via a nonmanifold triangulation of the point set. We validate our Laplacian on a variety of challenging examples (including all models from Thingi10k), and a variety of standard tasks including geodesic distance computation, surface deformation, parameterization, and computing minimal surfaces.

A Laplacian for Nonmanifold Triangle Meshes

This article describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Frechet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method, we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc.). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations so that they can be used in the context of tangent vector field processing.

The Vector Heat Method

Signaling pathways are a cornerstone of systems biology. Several databases store high-quality representations of these pathways that are amenable for automated analyses. Despite painstaking and manual curation, these databases remain incomplete. We present PATHLINKER, a new computational method to reconstruct the interactions in a signaling pathway of interest. PATHLINKER efficiently computes multiple short paths from the receptors to transcriptional regulators (TRs) in a pathway within a background protein interaction network. We use PATHLINKER to accurately reconstruct a comprehensive set of signaling pathways from the NetPath and KEGG databases. We show that PATHLINKER has higher precision and recall than several state-of-the-art algorithms, while also ensuring that the resulting network connects receptor proteins to TRs. PATHLINKER’s reconstruction of the Wnt pathway identified CFTR, an ABC class chloride ion channel transporter, as a novel intermediary that facilitates the signaling of Ryk to Dab2, which are known components of Wnt/β-catenin signaling. In HEK293 cells, we show that the Ryk–CFTR–Dab2 path is a novel amplifier of β-catenin signaling specifically in response to Wnt 1, 2, 3, and 3a of the 11 Wnts tested. PATHLINKER captures the structure of signaling pathways as represented in pathway databases better than existing methods. PATHLINKER’s success in reconstructing pathways from NetPath and KEGG databases point to its applicability for complementing manual curation of these databases. PATHLINKER may serve as a promising approach for prioritizing proteins and interactions for experimental study, as illustrated by its discovery of a novel pathway in Wnt/β-catenin signaling. Our supplementary website at http://bioinformatics.cs.vt.edu/~murali/supplements/2016-sys-bio-applications-pathlinker/
 provides links to the PATHLINKER software, input datasets, PATHLINKER reconstructions of NetPath pathways, and links to interactive visualizations of these reconstructions on GraphSpace. An algorithm that predicts molecular signaling pathways in humans provides a powerful way to 'join the dots' within protein networks. An important goal for systems biology is to identify the chains of reactions that carry cellular signals from receptors to the transcriptional regulators (TRs) that orchestrate gene activity. T. M. Murali, Anna Ritz, Shiv Kale and co-workers at Virginia Tech developed a method called PATHLINKER that computes the most likely paths between receptors and TRs using a large network of known human protein interactions. The researchers proved the effectiveness of PATHLINKER by correctly reconstructing 47 known signaling pathways. Most promisingly, PATHLINKER suggested a previously unknown component in the Wnt signaling pathway, which the researchers verified by experiment. PATHLINKER will be a valuable tool for choosing which proteins and interactions to study in the lab.

/pdf/pathways-on-demand-automated-reconstruction-of-human-4v9ug89j8o.pdf

Pathways on demand: automated reconstruction of human signaling networks

We present a data structure that makes it easy to run a large class of algorithms from computational geometry and scientific computing on extremely poor-quality surface meshes. Rather than changing the geometry, as in traditional remeshing, we consider intrinsic triangulations which connect vertices by straight paths along the exact geometry of the input mesh. Our key insight is that such a triangulation can be encoded implicitly by storing the direction and distance to neighboring vertices. The resulting signpost data structure then allows geometric and topological queries to be made on-demand by tracing paths across the surface. Existing algorithms can be easily translated into the intrinsic setting, since this data structure supports the same basic operations as an ordinary triangle mesh (vertex insertions, edge splits, etc.). The output of intrinsic algorithms can then be stored on an ordinary mesh for subsequent use; unlike previous data structures, we use a constant amount of memory and do not need to explicitly construct an overlay mesh unless it is specifically requested. Working in the intrinsic setting incurs little computational overhead, yet we can run algorithms on extremely degenerate inputs, including all manifold meshes from the Thingi10k data set. To evaluate our data structure we implement several fundamental geometric algorithms including intrinsic versions of Delaunay refinement and optimal Delaunay triangulation, approximation of Steiner trees, adaptive mesh refinement for PDEs, and computation of Poisson equations, geodesic distance, and flip-free tangent vector fields.

Navigating intrinsic triangulations

This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Frechet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing.

Nicholas Sharp

Papers

Pathways on demand: automated reconstruction of human signaling networks

A Laplacian for Nonmanifold Triangle Meshes

The Vector Heat Method

Navigating intrinsic triangulations

The Vector Heat Method.