Browsing by Subject "Dense linear algebra"
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Item A calculus of loop invariants for dense linear algebra optimization(2013-12) Low, Tze Meng; Van de Geijn, Robert A.Loop invariants have traditionally been used in proofs of correctness (e.g. program verification) and program derivation. Given that a loop invariant is all that is required to derive a provably correct program, the loop invariant can be thought of as being the essence of a loop. Being the essence of a loop, we ask the question “What other information is embedded within a loop invariant?” This dissertation provides evidence that in the domain of dense linear algebra, loop invariants can be used to determine the behavior of the loops. This dissertation demonstrates that by understanding how the loop invariant describes the behavior of the loop, a goal-oriented approach can be used to derive loops that are not only provably correct, but also have the desired performance behavior.Item Design by transformation : from domain knowledge to optimized program generation(2014-05) Marker, Bryan Andrew; Van de Geijn, Robert A.; Batory, Don S., 1953-Expert design knowledge is essential to develop a library of high-performance software. This includes how to implement and parallelize domain operations, how to optimize implementations, and estimates of which implementation choices are best. An expert repeatedly applies his knowledge, often in a rote and tedious way, to develop all of the related functionality expected from a domain-specific library. Expert knowledge is hard to gain and is easily lost over time when an expert forgets or when a new engineer starts developing code. The domain of dense linear algebra (DLA) is a prime example with software that is so well designed that much of experts' important work has become tediously rote in many ways. In this dissertation, we demonstrate how one can encode design knowledge for DLA so it can be automatically applied to generate code as an expert would or to generate better code. Further, the knowledge is encoded for perpetuity, so it can be reused to make implementing functionality on new hardware easier or it can be used to teach how software is designed to a non-expert. We call this approach to software engineering (encoding expert knowledge and automatically applying it) Design by Transformation (DxT). We present our vision, the methodology, a prototype code generation system, and possibilities when applying DxT to the domain of dense linear algebra.Item Finite element modeling of electromagnetic radiation and induced heat transfer in the human body(2013-08) Kim, Kyungjoo; Demkowicz, Leszek; Eijkhout, Victor; Van de Geijn, Robert A.This dissertation develops adaptive hp-Finite Element (FE) technology and a parallel sparse direct solver enabling the accurate modeling of the absorption of Electro-Magnetic (EM) energy in the human head. With a large and growing number of cell phone users, the adverse health effects of EM fields have raised public concerns. Most research that attempts to explain the relationship between exposure to EM fields and its harmful effects on the human body identifies temperature changes due to the EM energy as the dominant source of possible harm. The research presented here focuses on determining the temperature distribution within the human body exposed to EM fields with an emphasis on the human head. Major challenges in accurately determining the temperature changes lie in the dependence of EM material properties on the temperature. This leads to a formulation that couples the BioHeat Transfer (BHT) and Maxwell equations. The mathematical model is formed by the time-harmonic Maxwell equations weakly coupled with the transient BHT equation. This choice of equations reflects the relevant time scales. With a mobile device operating at a single frequency, EM fields arrive at a steady-state in the micro-second range. The heat sources induced by EM fields produce a transient temperature field converging to a steady-state distribution on a time scale ranging from seconds to minutes; this necessitates the transient formulation. Since the EM material properties depend upon the temperature, the equations are fully coupled; however, the coupling is realized weakly due to the different time scales for Maxwell and BHT equations. The BHT equation is discretized in time with a time step reflecting the thermal scales. After multiple time steps, the temperature field is used to determine the EM material properties and the time-harmonic Maxwell equations are solved. The resulting heat sources are recalculated and the process continued. Due to the weak coupling of the problems, the corresponding numerical models are established separately. The BHT equation is discretized with H¹ conforming elements, and Maxwell equations are discretized with H(curl) conforming elements. The complexity of the human head geometry naturally leads to the use of tetrahedral elements, which are commonly employed by unstructured mesh generators. The EM domain, including the head and a radiating source, is terminated by a Perfectly Matched Layer (PML), which is discretized with prismatic elements. The use of high order elements of different shapes and discretization types has motivated the development of a general 3D hp-FE code. In this work, we present new generic data structures and algorithms to perform adaptive local refinements on a hybrid mesh composed of different shaped elements. A variety of isotropic and anisotropic refinements that preserve conformity of discretization are designed. The refinement algorithms support one- irregular meshes with the constrained approximation technique. The algorithms are experimentally proven to be deadlock free. A second contribution of this dissertation lies with a new parallel sparse direct solver that targets linear systems arising from hp-FE methods. The new solver interfaces to the hierarchy of a locally refined mesh to build an elimination ordering for the factorization that reflects the h-refinements. By following mesh refinements, not only the computation of element matrices but also their factorization is restricted to new elements and their ancestors. The solver is parallelized by exploiting two-level task parallelism: tasks are first generated from a parallel post-order tree traversal on the assembly tree; next, those tasks are further refined by using algorithms-by-blocks to gain fine-grained parallelism. The resulting fine-grained tasks are asynchronously executed after their dependencies are analyzed. This approach effectively reduces scheduling overhead and increases flexibility to handle irregular tasks. The solver outperforms the conventional general sparse direct solver for a class of problems formulated by high order FEs. Finally, numerical results for a 3D coupled BHT with Maxwell equations are presented. The solutions of this Maxwell code have been verified using the analytic Mie series solutions. Starting with simple spherical geometry, parametric studies are conducted on realistic head models for a typical frequency band (900 MHz) of mobile phones.