Reduced Order and Surrogate Models for Gravitational Waves

We present an introduction to some of the state of the art in reduced order and surrogate modeling in gravitational wave (GW) science. Approaches that we cover include Principal Component Analysis, Proper Orthogonal Decomposition, the Reduced Basis approach, the Empirical Interpolation Method, Reduced Order Quadratures, and Compressed Likelihood evaluations. We divide the review into three parts: representation/compression of known data, predictive models, and data analysis. The targeted audience is that one of practitioners in GW science, a field in which building predictive models and data analysis tools that are both accurate and fast to evaluate, especially when dealing with large amounts of data and intensive computations, are necessary yet can be challenging. As such, practical presentations and, sometimes, heuristic approaches are here preferred over rigor when the latter is not available. This review aims to be self-contained, within reasonable page limits, with little previous knowledge (at the undergraduate level) requirements in mathematics, scientific computing, and other disciplines. Emphasis is placed on optimality, as well as the curse of dimensionality and approaches that might have the promise of beating it. We also review most of the state of the art of GW surrogates. Some numerical algorithms, conditioning details, scalability, parallelization and other practical points are discussed. The approaches presented are to large extent non-intrusive and data-driven and can therefore be applicable to other disciplines. We close with open challenges in high dimension surrogates, which are not unique to GW science.Comment: Invited article for Living Reviews in Relativity. 93 page

Systematic Design Methods for Efficient Off-Chip DRAM Access

Typical design flows for digital hardware take, as their input, an abstract description of computation and data transfer between logical memories. No existing commercial high-level synthesis tool demonstrates the ability to map logical memory inferred from a high level language to external memory resources. This thesis develops techniques for doing this, specifically targeting off-chip dynamic memory (DRAM) devices. These are a commodity technology in widespread use with standardised interfaces. In use, the bandwidth of an external memory interface and the latency of memory requests asserted on it may become the bottleneck limiting the performance of a hardware design. Careful consideration of this is especially important when designing with DRAMs, whose latency and bandwidth characteristics depend upon the sequence of memory requests issued by a controller. Throughout the work presented here, we pursue exact compile-time methods for designing application-specific memory systems with a focus on guaranteeing predictable performance through static analysis. This contrasts with much of the surveyed existing work, which considers general purpose memory controllers and optimized policies which improve performance in experiments run using simulation of suites of benchmark codes. The work targets loop-nests within imperative source code, extracting a mathematical representation of the loop-nest statements and their associated memory accesses, referred to as the ‘Polytope Model’. We extend this mathematical representation to represent the physical DRAM ‘row’ and ‘column’ structures accessed when performing memory transfers. From this augmented representation, we can automatically derive DRAM controllers which buffer data in on-chip memory and transfer data in an efficient order. Buffering data and exploiting ‘reuse’ of data is shown to enable up to 50× reduction in the quantity of data transferred to external memory. The reordering of memory transactions exploiting knowledge of the physical layout of the DRAM device allowing to 4× improvement in the efficiency of those data transfers

An Active-Library Based Investigation into the Performance Optimisation of Linear Algebra and the Finite Element Method

In this thesis, I explore an approach called "active libraries". These are libraries that take part in their own optimisation, enabling both high-performance code and the presentation of intuitive abstractions. I investigate the use of active libraries in two domains. Firstly, dense and sparse linear algebra, particularly, the solution of linear systems of equations. Secondly, the specification and solution of finite element problems. Extending my earlier (MEng) thesis work, I describe the modifications to my linear algebra library "Desola" required to perform sparse-matrix code generation. I show that optimisations easily applied in the dense case using code-transformation must be applied at a higher level of abstraction in the sparse case. I present performance results for sparse linear system solvers generated using Desola and compare against an implementation using the Intel Math Kernel Library. I also present improved dense linear-algebra performance results. Next, I explore the active-library approach by developing a finite element library that captures runtime representations of basis functions, variational forms and sequences of operations between discretised operators and fields. Using captured representations of variational forms and basis functions, I demonstrate optimisations to cell-local integral assembly that this approach enables, and compare against the state of the art. As part of my work on optimising local assembly, I extend the work of Hosangadi et al. on common sub-expression elimination and factorisation of polynomials. I improve the weight function presented by Hosangadi et al., increasing the number of factorisations found. I present an implementation of an optimised branch-and-bound algorithm inspired by reformulating the original matrix-covering problem as a maximal graph biclique search problem. I evaluate the algorithm's effectiveness on the expressions generated by our finite element solver

A unified approach to evaluation algorithms for multivariate polynomials

Abstract. We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the Bernstein-Bézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of L-bases, a class of bases that include the Bernstein-Bézier, multinomial, and a rich subclass of Lagrange and Newton bases. All of the known evaluation algorithms can be generated either by considering up recursive evaluation algorithms for L-bases or by examining change of basis algorithms for L-bases. For polynomials of degree n in s variables, the class of up recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity O(n s+1), a nested multiplication algorithm with computational complexity O(n s log n) and a ladder recurrence algorithm with computational complexity O(n s). These algorithms also generate a new generalization of the Aitken-Neville algorithm for evaluation of multivariate polynomials expressed in terms of Lagrange L-bases. The second class of algorithms, based on certain change of basis algorithms between L-bases, include a nested multiplication algorithm with computational complexity O(n s), a divided difference algorithm, a forward difference algorithm, and a Lagrange evaluation algorithm with computational complexities O(n s), O(n s)andO(n) per point respectively for the evaluation of multivariate polynomials at several points. 1