Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

In this article the concept of saturation of an arbitrary regularization method is formalized based upon the original idea of saturation for spectral regularization methods introduced by A. Neubauer in 1994. Necessary and sufficient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the total error must be optimal in two senses, namely as optimal order of convergence over a certain set which at the same time, must be optimal (in a very precise sense) with respect to the error. Finally, two converse results are proved and the theory is applied to find sufficient conditions which ensure the existence of global saturation for spectral methods with classical qualification of finite positive order and for methods with maximal qualification. Finally, several examples of regularization methods possessing global saturation are shown.Comment: 29 page

Mathematical modelling of curtain coating

We present a simple mathematical model for the fluid flow in the curtain coating process, exploiting the small aspect ratio, and examine the model in the large-Reynolds-number limit of industrial interest. We show that the fluid is in free fall except for a region close to the substrate, but find that the model can not describe the turning of the curtain onto the substrate. We find that the inclusion of a viscous bending moment close to the substrate allows the curtain to “turn the corner”

Optimisation of patch distribution strategies for AMR applications

As core counts increase in the world's most powerful supercomputers, applications are becoming limited not only by computational power, but also by data availability. In the race to exascale, efficient and effective communication policies are key to achieving optimal application performance. Applications using adaptive mesh refinement (AMR) trade off communication for computational load balancing, to enable the focused computation of specific areas of interest. This class of application is particularly susceptible to the communication performance of the underlying architectures, and are inherently difficult to scale efficiently. In this paper we present a study of the effect of patch distribution strategies on the scalability of an AMR code. We demonstrate the significance of patch placement on communication overheads, and by balancing the computation and communication costs of patches, we develop a scheme to optimise performance of a specific, industry-strength, benchmark application

Resident block-structured adaptive mesh refinement on thousands of graphics processing units

Block-structured adaptive mesh refinement (AMR) is a technique that can be used when solving partial differential equations to reduce the number of cells necessary to achieve the required accuracy in areas of interest. These areas (shock fronts, material interfaces, etc.) are recursively covered with finer mesh patches that are grouped into a hierarchy of refinement levels. Despite the potential for large savings in computational requirements and memory usage without a corresponding reduction in accuracy, AMR adds overhead in managing the mesh hierarchy, adding complex communication and data movement requirements to a simulation. In this paper, we describe the design and implementation of a resident GPU-based AMR library, including: the classes used to manage data on a mesh patch, the routines used for transferring data between GPUs on different nodes, and the data-parallel operators developed to coarsen and refine mesh data. We validate the performance and accuracy of our implementation using three test problems and two architectures: an 8 node cluster, and 4,196 nodes of Oak Ridge National Laboratory’s Titan supercomputer. Our GPU-based AMR hydrodynamics code performs up to 4.87x faster than the CPU-based implementation, and is scalable on 4,196 K20x GPUs using a combination of MPI and CUDA

Generalized Qualification and Qualification Levels for Spectral Regularization Methods

The concept of qualification for spectral regularization methods for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error. In this article, the definition of qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualification extends the definition introduced by Mathe and Pereverzev in 2003, mainly in the sense that the functions associated to orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a SRM to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found. Examples of all three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced by Mathe and Perevezev are shown. In particular, spectral regularization methods having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented. Finally several implications of this theory in the context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown.Comment: 20 pages, 1 figur

Performance optimisation of inertial confinement fusion codes using mini-applications

Despite the recent successes of nuclear energy researchers, the scientific community still remains some distance from being able to create controlled, self-sustaining fusion reactions. Inertial Confinement Fusion (ICF) techniques represent one possible option to surpass this barrier, with scientific simulation playing a leading role in guiding and supporting their development. The simulation of such techniques allows for safe and efficient investigation of laser design and pulse shaping, as well as providing insight into the reaction as a whole. The research presented here focuses on the simulation code EPOCH, a fully relativistic particle-in-cell plasma physics code concerned with faithfully recreating laser-plasma interactions at scale. A significant challenge in developing large codes like EPOCH is maintaining effective scientific delivery on successive generations of high-performance computing architecture. To support this process, we adopt the use of mini-applications -- small code proxies that encapsulate important computational properties of their larger parent counterparts. Through the development of a mini-application for EPOCH (called miniEPOCH), we investigate a variety of the performance features exhibited in EPOCH, expose opportunities for optimisation and increased scientific capability, and offer our conclusions to guide future changes to similar ICF codes