Block Structured Adaptive Mesh and Time Refinement for Hybrid, Hyperbolic + N-body Systems

We present a new numerical algorithm for the solution of coupled collisional and collisionless systems, based on the block structured adaptive mesh and time refinement strategy (AMR). We describe the issues associated with the discretization of the system equations and the synchronization of the numerical solution on the hierarchy of grid levels. We implement a code based on a higher order, conservative and directionally unsplit Godunov's method for hydrodynamics; a symmetric, time centered modified symplectic scheme for collisionless component; and a multilevel, multigrid relaxation algorithm for the elliptic equation coupling the two components. Numerical results that illustrate the accuracy of the code and the relative merit of various implemented schemes are also presented.Comment: 40 pages, 10 figures, JPC in press. Extended the code test section, new convergence tests, several typos corrected. Full resolution version available at http://www.exp-astro.phys.ethz.ch/miniati/charm.pd

Proceedings for the ICASE Workshop on Heterogeneous Boundary Conditions

Domain Decomposition is a complex problem with many interesting aspects. The choice of decomposition can be made based on many different criteria, and the choice of interface of internal boundary conditions are numerous. The various regions under study may have different dynamical balances, indicating that different physical processes are dominating the flow in these regions. This conference was called in recognition of the need to more clearly define the nature of these complex problems. This proceedings is a collection of the presentations and the discussion groups

A multi-resolution, non-parametric, Bayesian framework for identification of spatially-varying model parameters

This paper proposes a hierarchical, multi-resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output. Such parameters are frequently encountered in PDE-based models and correspond to quantities such as density or pressure fields, elasto-plastic moduli and internal variables in solid mechanics, conductivity fields in heat diffusion problems, permeability fields in fluid flow through porous media etc. The proposed model has all the advantages of traditional Bayesian formulations such as the ability to produce measures of confidence for the inferences made and providing not only predictive estimates but also quantitative measures of the predictive uncertainty. In contrast to existing approaches it utilizes a parsimonious, non-parametric formulation that favors sparse representations and whose complexity can be determined from the data. The proposed framework in non-intrusive and makes use of a sequence of forward solvers operating at various resolutions. As a result, inexpensive, coarse solvers are used to identify the most salient features of the unknown field(s) which are subsequently enriched by invoking solvers operating at finer resolutions. This leads to significant computational savings particularly in problems involving computationally demanding forward models but also improvements in accuracy. It is based on a novel, adaptive scheme based on Sequential Monte Carlo sampling which is embarrassingly parallelizable and circumvents issues with slow mixing encountered in Markov Chain Monte Carlo schemes

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

Assessment of a photogrammetric approach for urban DSM extraction from tri-stereoscopic satellite imagery

Built-up environments are extremely complex for 3D surface modelling purposes. The main distortions that hamper 3D reconstruction from 2D imagery are image dissimilarities, concealed areas, shadows, height discontinuities and discrepancies between smooth terrain and man-made features. A methodology is proposed to improve automatic photogrammetric extraction of an urban surface model from high resolution satellite imagery with the emphasis on strategies to reduce the effects of the cited distortions and to make image matching more robust. Instead of a standard stereoscopic approach, a digital surface model is derived from tri-stereoscopic satellite imagery. This is based on an extensive multi-image matching strategy that fully benefits from the geometric and radiometric information contained in the three images. The bundled triplet consists of an IKONOS along-track pair and an additional near-nadir IKONOS image. For the tri-stereoscopic study a densely built-up area, extending from the centre of Istanbul to the urban fringe, is selected. The accuracy of the model extracted from the IKONOS triplet, as well as the model extracted from only the along-track stereopair, are assessed by comparison with 3D check points and 3D building vector data

Enabling technology for non-rigid registration during image-guided neurosurgery

In the context of image processing, non-rigid registration is an operation that attempts to align two or more images using spatially varying transformations. Non-rigid registration finds application in medical image processing to account for the deformations in the soft tissues of the imaged organs. During image-guided neurosurgery, non-rigid registration has the potential to assist in locating critical brain structures and improve identification of the tumor boundary. Robust non-rigid registration methods combine estimation of tissue displacement based on image intensities with the spatial regularization using biomechanical models of brain deformation. In practice, the use of such registration methods during neurosurgery is complicated by a number of issues: construction of the biomechanical model used in the registration from the image data, high computational demands of the application, and difficulties in assessing the registration results. In this dissertation we develop methods and tools that address some of these challenges, and provide components essential for the intra-operative application of a previously validated physics-based non-rigid registration method.;First, we study the problem of image-to-mesh conversion, which is required for constructing biomechanical model of the brain used during registration. We develop and analyze a number of methods suitable for solving this problem, and evaluate them using application-specific quantitative metrics. Second, we develop a high-performance implementation of the non-rigid registration algorithm and study the use of geographically distributed Grid resources for speculative registration computations. Using the high-performance implementation running on the remote computing resources we are able to deliver the results of registration within the time constraints of the neurosurgery. Finally, we present a method that estimates local alignment error between the two images of the same subject. We assess the utility of this method using multiple sources of ground truth to evaluate its potential to support speculative computations of non-rigid registration

The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body