Thomas Gerstner

Numerical integration using sparse grids

We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction.

Dimension-adaptive tensor-product quadrature

We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high–dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower–dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself. The dimension–adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate–dimensional problems. The dimension–adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm. The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.

Topology preserving and controlled topology simplifying multiresolution isosurface extraction

Multiresolution methods are becoming increasingly important tools for the interactive visualization of very large data sets. Multiresolution isosurface visualization allows the user to explore volume data using simplified and coarse representations of the isosurface for overview images, and finer resolution in areas of high interest or when zooming into the data. Ideally, a coarse isosurface should have the same topological structure as the original. The topological genus of the isosurface is one important property which is often neglected in multiresolution algorithms. This results in uncontrolled topological changes which can occur whenever the level-of-detail is changed. The scope of this paper is to propose an efficient technique which allows preservation of topology as well as controlled topology simplification in multiresolution isosurface extraction.

Multiresolution Compression and Visualization of Global Topographic Data

We present a multiresolution model for terrain surfaces which is able to handle large-scale global topographic data. It is based on a hierarchical decomposition of the sphere by a recursive bisection triangulation in geographic coordinates. Error indicators allow the representation of the data at various levels of detail and enable data compression by local omission of data values. The resulting adaptive hierarchical triangulation is stored using a bit code of the underlying binary tree and additionally, relative pointers which allow a selective tree traversal. This way, it is possible to work directly on the compressed data. We show that significant compression rates can be obtained already for small threshold values. In a visualization application, adaptive triangulations which consist of hundreds of thousands of shaded triangles are extracted and drawn at interactive rates.

Multi-Resolutional Parallel Isosurface Extraction based on Tetrahedral Bisection

A variety of multi-resolution visualisation methods have been designed to serve as tools for interactive visualisation of large datasets. The local resolution of the generated visual objects, such as isosurfaces, is thereby steered by error indicators which measure the error due to a locally coarser approximation of the data. On one hand, post-processing methods can be applied to already extracted surfaces and can turn them into multi-resolutional objects, which can then be interactively inspected [1–4]. On the other hand, we can also adaptively extract the considered isosurfaces from the 3D dataset. Thereby, starting at a coarse approximation of the data, we recursively add details in areas where some error indicator points out a local error with respect to the exact data values. If the error is below a user prescribed threshold, the algorithm locally stops the successive refinement and extracts the surface on the current level. Different approaches have been presented to solve the outstanding continuity problem, i.e., to avoid cracks in the adaptive isosurfaces. In the Delaunay approach by Cignoni et al. [5] and the nested mesh method by Grosso et al. [6], the successive remeshing during the refinement guarantees the continuity. On the other hand, Shekhar et al. [7] rule out hanging nodes by inserting additional points on faces with a transition from finer to coarser elements due to an adaptive stopping criterion.

Error indicators for multilevel visualization and computing on nested grids

Abstract Nowadays computing and post processing of simulation data is often based on efficient hierarchical methods. While multigrid methods are already established standards for fast simulation codes, multiresolution visualization methods have only recently become an important ingredient of real–time interactive post processing. Both methodologies use local error indicators which serve as criteria where to refine the data representation on the physical domain. In this article we give an overview on different types of error measurement on nested grids and compare them for selected applications in 2D as well as in 3D. Furthermore, it is pointed out that a certain saturation of the considered error indicator plays an important role in multilevel visualization and computing on implicitly defined adaptive grids.

Feasible and Successful: Genome-Wide Interaction Analysis Involving All 1.9 × 1011 Pair-Wise Interaction Tests

The Genome-Wide Association Study (GWAS) is the study design of choice for detecting common genetic risk factors for multifactorial diseases. The performance of full Genome-Wide Interaction Analyses (GWIA) has always been considered computationally challenging. Two-stage strategies to reduce the amount of numerical analysis require the detection of single marker effects or prior pathophysiological hypotheses before the analysis of interaction. This prevents the detection of pure epistatic effects. Our case-control study in idiopathic generalized epilepsy demonstrates that a full GWIA is feasible through use of data compression, specific data representation, interleaved data organization, and parallelization of the analysis on a multi-processor system. Following extensive quality control of the genotypes, our final list of top interaction hits contains only pairs of interacting SNPs with negligible marginal effects. The TOP HIT interaction was between a SNP-pair intragenic to gene DNER (chr 2) and gene CTNNA3 (chr 10). Both of these genes are functionally involved in neuronal migration, synaptogenesis, and the formation of neuronal circuits. Our results therefore indicate a possible interaction between these two genes in epileptogenesis. Results from GWAS are beginning to reveal a ‘missing heritability’ in complex traits and diseases. Systematic, hypothesis-free analysis of epistatic interaction (GWIA) may help to close this increasingly recognized gap in heritability.

Fast multiresolution extraction of multiple transparent isosurfaces

In this paper, we present a multiresolution algorithm which is capable to render multiple transparent isosurfaces under real–time constraints. To this end, the underlying 3D data set is covered with a hierarchical tetrahedral grid. The multiresolution extraction algorithm is then based on an adaptive traversal of the tetrahedral grid with the help of error indicators. The display of transparent isosurfaces using alpha blending requires a back–to–front rendering of the isosurface triangles. This is achieved by a hierarchical sorting procedure of the tetrahedra and the hierarchical computation of data gradients. We will also comment on the automated selection of suitable isovalues for visualization applications.

Multiresolution extraction and rendering of transparent isosurfaces

Abstract Multiresolution methods are often needed for the interactive visualization of large volumetric data sets. Here, we present a multiresolution algorithm which is designed to extract and render multiple transparent isosurfaces very quickly. To this end, the underlying data set is covered with a hierarchical tetrahedral grid. The multiresolution extraction is based on an adaptive traversal of the tetrahedral grid with the help of error indicators. The display of transparent isosurfaces using alpha blending requires the back-to-front rendering of the isosurface triangles. This is achieved by a hierarchical sorting procedure of the tetrahedra and the isosurface components inside each tetrahedron. The sorting of the isosurface components requires the data gradients, though. These gradients can in principle be precomputed but are quite expensive to store. We therefore show how these gradients can be very efficiently computed on-the-fly during the grid traversal. We will also comment on the automated selection of suitable isovalues for visualization applications.

A case study on multiresolution visualization of local rainfall from weather radar measurements

Weather radars can measure the backscatter from rain drops in the atmosphere. A complete radar scan provides three-dimensional precipitation information. For the understanding of the underlying atmospheric processes interactive visualization of these data sets is necessary. This is a challenging task due to the size, structure and required context of the data. In this case study, a multiresolution approach for real-time simultaneous visualization of radar measurements together with the corresponding terrain data is illustrated.