Efficient fault tolerance for selected scientific computing algorithms on heterogeneous and approximate computer architectures

Scientific computing and simulation technology play an essential role to solve central challenges in science and engineering. The high computational power of heterogeneous computer architectures allows to accelerate applications in these domains, which are often dominated by compute-intensive mathematical tasks. Scientific, economic and political decision processes increasingly rely on such applications and therefore induce a strong demand to compute correct and trustworthy results. However, the continued semiconductor technology scaling increasingly imposes serious threats to the reliability and efficiency of upcoming devices. Different reliability threats can cause crashes or erroneous results without indication. Software-based fault tolerance techniques can protect algorithmic tasks by adding appropriate operations to detect and correct errors at runtime. Major challenges are induced by the runtime overhead of such operations and by rounding errors in floating-point arithmetic that can cause false positives. The end of Dennard scaling induces central challenges to further increase the compute efficiency between semiconductor technology generations. Approximate computing exploits the inherent error resilience of different applications to achieve efficiency gains with respect to, for instance, power, energy, and execution times. However, scientific applications often induce strict accuracy requirements which require careful utilization of approximation techniques. This thesis provides fault tolerance and approximate computing methods that enable the reliable and efficient execution of linear algebra operations and Conjugate Gradient solvers using heterogeneous and approximate computer architectures. The presented fault tolerance techniques detect and correct errors at runtime with low runtime overhead and high error coverage. At the same time, these fault tolerance techniques are exploited to enable the execution of the Conjugate Gradient solvers on approximate hardware by monitoring the underlying error resilience while adjusting the approximation error accordingly. Besides, parameter evaluation and estimation methods are presented that determine the computational efficiency of application executions on approximate hardware. An extensive experimental evaluation shows the efficiency and efficacy of the presented methods with respect to the runtime overhead to detect and correct errors, the error coverage as well as the achieved energy reduction in executing the Conjugate Gradient solvers on approximate hardware

CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations

CP2K is an open source electronic structure and molecular dynamics software package to perform atomistic simulations of solid-state, liquid, molecular, and biological systems. It is especially aimed at massively parallel and linear-scaling electronic structure methods and state-of-the-art ab initio molecular dynamics simulations. Excellent performance for electronic structure calculations is achieved using novel algorithms implemented for modern high-performance computing systems. This review revisits the main capabilities of CP2K to perform efficient and accurate electronic structure simulations. The emphasis is put on density functional theory and multiple post–Hartree–Fock methods using the Gaussian and plane wave approach and its augmented all-electron extension

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

In-memory computing with emerging memory devices: Status and outlook

Supporting data for "In-memory computing with emerging memory devices: status and outlook", submitted to APL Machine Learning

Large-Scale Atomistic Simulations of Complex and Functional Properties of Ferroic Materials

Ferroelectric (FE) nanostructures have attracted considerable attention as our abilities improve to synthesize them and to predict their properties by theoretical means. Depolarizing field effects at interfaces of FE heterostructures are particularly notable for causing topological defects such as FE vortices and negative dielectric responses in superlattices. In this thesis, I employ two large-scale atomistic techniques, the first-principles-based effective Hamiltonian (HEff) method and the linear-scaling three-dimensional fragment (LS3DF) method. I use these methods to explore optical rotation in FE vortices, electro-optic effects in FE vortices and skyrmions, and voltage amplification via negative capacitance in ferroelectric-paraelectric superlattices. We employ HEff in Monte Carlo and molecular dynamics schemes to maximize spontaneous optical rotation in a BaTiO$_3$/SrTiO$_3$ nanocomposite. For a small bias field, maximal optical rotation was realized at room temperature. The result has acquired greater relevance since Ramesh and coworkers observed ``emergent chirality in FE vortex arrays in PbTiO$_3$/SrTiO$_3$ superlattices. In a similar nanocomposite as above, we use the combined HEff and LS3DF method to study how band gap and band alignment evolves along the path from a polar-toroidal to an electrical skyrmion state. Temperature control of the vortex provides substantially larger range of control of bandgap and band alignment than field control of the skyrmion. Using temperature and electric fields to manipulate polarization and bond angle distortion in both constituent materials provides an additional handle for bandgap engineering in such nanostructures. We then use HEff to study BaTiO$_3$/SrTiO$_3$ superlattices as a platform for negative differential capacitance. We implement an atomistic framework amenable to simulation of negative capacitance in strained superlattices. In these systems, we predict misfit epitaxial strain control allows for broadly extending the operable temperature range for negative. By manipulating this strain, we observed switching of negative capacitance behavior between both constituent materials of the superlattice at low temperature

Computational modeling and simulation of nonlinear electromagnetic and multiphysics problems

In this dissertation, nonlinear electromagnetic and multiphysics problems are modeled and simulated using various three-dimensional full-wave methods in the time domain. The problems under consideration fall into two categories. One is nonlinear electromagnetic problems with the nonlinearity embedded in either the permeability or the conductivity of the material's constitutive properties. The other is multiphysics problems that involve interactions between electromagnetic and other physical phenomena. A numerical solution of nonlinear magnetic problems is formulated using the three-dimensional time-domain finite element method (TDFEM) combined with the inverse Jiles-Atherton vector hysteresis model. A second-order nonlinear partial differential equation (PDE) that governs the nonlinear magnetic problem is constructed through the magnetic vector potential in the time domain, which is solved by applying the Newton-Raphson method. To solve the ordinary differential equation (ODE) representing the magnetic hysteresis accurately and efficiently, several ODE solvers are specifically designed and investigated. To improve the computational efficiency of the Newton-Raphson method, the multi-dimensional secant methods are incorporated in the nonlinear TDFEM solver. A nonuniform time-stepping scheme is also developed using the weighted residual approach to remove the requirement of a uniform time-step size during the simulation. Breakdown phenomena during high-power microwave (HPM) operation are investigated using different physical and mathematical models. During the breakdown process, the bound charges in solid dielectrics and air molecules break free and are pushed to move by the Lorentz force produced by the electromagnetic fields. The motion of free electrons produces plasma currents, which generate secondary electromagnetic fields that couple back to the externally applied fields and interact with the free electrons. When the incident field intensity is high enough, this will lead to an exponential increase of the charged particles known as breakdown. Such a process is first described by a nonlinear conductivity of the solid dielectric as a function of the electric field to model the dielectric breakdown phenomenon. The air breakdown problem encountered with HPM operation is then simulated with the plasma current modeled by a simplified plasma fluid equation. Both the dielectric and air breakdown problems are solved with the TDFEM together with a Newton's method, where the dielectric breakdown is treated as a pure nonlinear electromagnetic problem, while the air breakdown is treated as a multiphysics problem. To describe the plasma behavior more accurately, the plasma density and velocity are modeled by the equations of diffusion and motion, respectively. This results in a multiphysics and multiscale system depicted by the nonlinearly coupled full-wave Maxwell and plasma fluid equations, which are solved by a nodal discontinuous Galerkin time-domain (DGTD) method in three dimensions. The air breakdown during the HPM operation and the resulting plasma formation and shielding are modeled and simulated. Several important numerical issues in the simulation of nonlinear electromagnetic and multiphysics problems have been investigated and discussed. A continuity-preserving and divergence-cleaning scheme for electromagnetic problems involving inhomogeneous materials has been proposed based on the purely and damped hyperbolic Maxwell equations. A divergence-cleaning method is presented to enforce Gauss's laws and normal flux continuity by introducing auxiliary variables and damping terms into the original Maxwell's equations, which result in artificial propagation and dissipation of the numerical errors. Based on the DGTD method, dynamic h- and p-adaptation algorithms are developed for a full-wave analysis of electromagnetic and multiphysics problems. The dynamic h-adaptation algorithm can dynamically refine the mesh to resolve the local variation of the fields during the wave propagation, while the dynamic p-adaptation algorithm can determine and adjust the basis order in real time during the simulation. Both algorithms developed and investigated in this dissertation are highly flexible and efficient, and are powerful simulation tools in the solution of nonlinear electromagnetic and multiphysics problems