We will discuss the solution of eigenvalue problems associated with partial differential equations (PDE)s that can be written in the generalised form Ax = λMx, where the matrices A and/or M may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilised formulations are used for the numerical approximation of PDEs. With the help of classical numerical examples we will show that the presence of one (or both) parameters can produce unexpected results.

Hessian-dependent functionals play a pivotal role in a wide latitude of problems in mathematics. Arising in the context of differential geometry and probability theory, this class of problems find applications in the mechanics of deformable media (mostly in elasticity theory) and the modelling of slow viscous fluids. We study such functionals from three distinct perspectives.

We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently proposed in [1, 2], as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems. In a very general abstract framework.

Analogies between financial markets and critical phenomena have long been observed empirically. So far, no convincing theory has emerged that can explain these empirical observations. Here, we take a step towards such a theory by modeling financial markets as a lattice gas.

Wave propagation and scattering problems are of huge importance in many applications in science and engineering - e.g., in seismic and medical imaging and more generally in acoustics and electromagnetics.

In this short course, we will introduce some elements in deriving the hp a posteriori error estimate for a high-order unfitted finite element method for elliptic interface problems. The key ingredient is an hp domain inverse estimate, which allows us to prove a sharp lower bound of the hp a posteriori error estimator.

HPC accelerates! This year's motto of the Supercomputing conference...

Scientific applications from diverse sources rely on dense matrix operations. These operations arise in: Schur complements, integral equations, covariances in spatial statistics, ridge regression, radial basis functions from unstructured meshes, and kernel matrices from machine learning, among others. This thesis demonstrates how to extend the problem sizes that may be treated and reduce their execution time. Sometimes, even forming the dense matrix can be a bottleneck – in computation or storage.

 

Heterogeneity in Hardware: Opportunities and Challenges for Software and Applications

Abstract

With the hardware technology scaling and the trend on heterogeneous chip design, the exis

We present Exascale GeoStatistics (ExaGeoStat) software, a high-performance library implemented on a wide variety of contemporary hybrid distributed-shared supercomputers whose primary target is climate and environmental prediction applications.

Out-of-Core simulation systems often produce a massive amount of data that cannot fit on the aggregate fast memory of the compute nodes, and they also require to read back these data for computation. As a result, I/O data movement can be a bottleneck in large-scale simulations. Advances in memory architecture have made it feasible to integrate hierarchical storage media on large-scale systems, starting from the traditional Parallel File Systems to intermediate fast disk technologies (e.g., node-local and remote-shared NVMe and SSD-based Burst Buffers) and up to CPU’s main memory and GPU’s High Bandwidth Memory. However, while adding additional and faster storage media increases I/O bandwidth, it pressures the CPU, as it becomes responsible for managing and moving data between these layers of storage. Simulation systems are thus vulnerable to being blocked by I/O operations. The Multilayer Buffer System (MLBS) proposed in this research demonstrates a general method for overlapping I/O with computation that helps to ameliorate the strain on the processors through asynchronous access. The main idea consists in decoupling I/O operations from computational phases using dedicated hardware resources to perform expensive context switches. By continually prefetching up and down across all hardware layers of the memory/storage subsystems, MLBS transforms the original I/O-bound behavior of evaluated applications and shifts it closer to a memory-bound or compute-bound regime.

The SLATE (Software for Linear Algebra Targeting Exascale) library is being developed to provide fundamental dense linear algebra capabilities for current and upcoming distributed high-performance systems, both accelerated CPU–GPU based and CPU based.