Porting OpenMP to GPGPU to accelerate Krylov space computations in coupled geodynamics simulator

Understanding the causality between the events leading upto and post fault slipand the earthquake recording is important for seismic design and monitoring ofunderground structures, bridges and reinforced concrete buildings as well as climatemitigation projects like carbon sequestration and energy technologies like enhancedgeothermal systems or oilfield wastewater disposal. While the events leading uptofault slip are typically governed by poroelastostatics, the events post fault slip caneasily transition into poroelastodynamics territory due to runaway fault slip velocities.An understanding of expected fault slip velocities is critical apriori, as an algorithmwhich can seamlessly transition from time marching in poroelastostatics realm toporoelastodynamics realm and vice-versa is extremely difficult to achieve. That beingsaid, every effort in the direction of accelerating the computations on the flow sideare a necessary step forward in rendering a fast coupled geodynamics simulator.In this document, we present a framework in which we study the porting of theOpenMP parallelism of the flow simulator to a GPGPU

Accuracy of the typicality approach using Chebyshev polynomials

Trace estimators allow us to approximate thermodynamic equilibrium observables with astonishing accuracy. A prominent representative is the finite-temperature Lanczos method (FTLM) which relies on a Krylov space expansion of the exponential describing the Boltzmann weights. Here we report investigations of an alternative approach which employs Chebyshev polynomials. This method turns out to be also very accurate in general, but shows systematic inaccuracies at low temperatures that can be traced back to an improper behavior of the approximated density of states with and without smoothing kernel. Applications to archetypical quantum spin systems are discussed as examples.

Krylov Methods for Large Sparse Systems: A Comprehensive Overview

In this paper are analyzed behavior and properties for different Krylov methods applied in different categories of problems. These categories often include PDEs, econometrics and network models, which are represented by large sparse systems. For our empirical analysis are taken into consideration size, the density of non-zero elements, symmetry/un-symmetry, eigenvalue distribution, also well/ill-conditioned and random systems. Convergence, approximation error and residuals are compared for the full version of methods, some restarted methods and preconditioned methods. Two preconditioners are considered respectively, ILU(0) and IC(0) by using at least five preconditioning techniques. In each case, empirical results show which technique is best to use based on properties of the system and are backed up by general theoretical information already found on Krylov space methods.

Computable upper error bounds for Krylov approximations to matrix exponentials and associated $${\varvec{\varphi }}$$-functions

An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of $$\varphi $$φ-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.

Energy Efficient Control of the Dehumidification Process in Heat Exchangers with Air Bypass

This paper deals with an energy efficient approach for the dehumidification process of supply air. The basic concept consists of an air bypass, which separates the airstream. Later the unprocessed air is mixed with the conditioned air. This mixing allows one to avoid the energy consuming reheating of the air stream. Application of this concept demands for a sophisticated controller. In this case a state space controller is designed. Therefore, the underlying model for the heat exchanger is derived and a Krylov Space based reduction method is applied. This model is broadened for the bypass. The overall linear model is derived via numerical linearization.