The CPR Method and Beyond : Prologue

In $1974$ A.R. Curtis, M.J.D. Powell, and J.K.Reid published a seminal paper on the estimation of Jacobian matrices which was later coined as the CPR method. Central to the CPR method is the effective utilization of a priori known sparsity information. It is only recently that the optimal CPR method in its general form is characterized and the theoretical underpinning for the optimality is shown. In this short note we provide an overview of the development of computational techniques and software tools for the estimation of Jacobian matrices

Time integration and steady-state continuation for 2d lubrication equations

Lubrication equations allow to describe many structurin processes of thin liquid films. We develop and apply numerical tools suitable for their analysis employing a dynamical systems approach. In particular, we present a time integration algorithm based on exponential propagation and an algorithm for steady-state continuation. In both algorithms a Cayley transform is employed to overcome numerical problems resulting from scale separation in space and time. An adaptive time-step allows to study the dynamics close to hetero- or homoclinic connections. The developed framework is employed on the one hand to analyse different phases of the dewetting of a liquid film on a horizontal homogeneous substrate. On the other hand, we consider the depinning of drops pinned by a wettability defect. Time-stepping and path-following are used in both cases to analyse steady-state solutions and their bifurcations as well as dynamic processes on short and long time-scales. Both examples are treated for two- and three-dimensional physical settings and prove that the developed algorithms are reliable and efficient for 1d and 2d lubrication equations, respectively.Comment: 33 pages, 16 figure

Non-Euclidean geometry in nature

I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random graphs. For these systems I discuss the following interlinked questions: (i) the optimal embedding of plants leaves in the three-dimensional space, (ii) the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to chaotic Hamiltonian systems is adde

SCALABLE INTEGRATED CIRCUIT SIMULATION ALGORITHMS FOR ENERGY-EFFICIENT TERAFLOP HETEROGENEOUS PARALLEL COMPUTING PLATFORMS

Integrated circuit technology has gone through several decades of aggressive scaling.It is increasingly challenging to analyze growing design complexity. Post-layout SPICE simulation can be computationally prohibitive due to the huge amount of parasitic elements, which can easily boost the computation and memory cost. As the decrease in device size, the circuits become more vulnerable to process variations. Designers need to statistically simulate the probability that a circuit does not meet the performance metric, which requires millions times of simulations to capture rare failure events. Recent, multiprocessors with heterogeneous architecture have emerged as mainstream computing platforms. The heterogeneous computing platform can achieve highthroughput energy efficient computing. However, the application of such platform is not trivial and needs to reinvent existing algorithms to fully utilize the computing resources. This dissertation presents several new algorithms to address those aforementioned two significant and challenging issues on the heterogeneous platform. Harmonic Balance (HB) analysis is essential for efficient verification of large postlayout RF and microwave integrated circuits (ICs). However, existing methods either suffer from excessively long simulation time and prohibitively large memory consumption or exhibit poor stability. This dissertation introduces a novel transient-simulation guided graph sparsification technique, as well as an efficient runtime performance modeling approach tailored for heterogeneous manycore CPU-GPU computing system to build nearly-optimal subgraph preconditioners that can lead to minimum HB simulation runtime. Additionally, we propose a novel heterogeneous parallel sparse block matrix algorithm by taking advantages of the structure of HB Jacobian matrices as well as GPU’s streaming multiprocessors to achieve optimal workload balancing during the preconditioning phase of HB analysis. We also show how the proposed preconditioned iterative algorithm can efficiently adapt to heterogeneous computing systems with different CPU and GPU computing capabilities. Extensive experimental results show that our HB solver can achieve up to 20X speedups and 5X memory reduction when compared with the state-of-the-art direct solver highly optimized for twelve-core CPUs. In nowadays variation-aware IC designs, cell characterizations and SRAM memory yield analysis require many thousands or even millions of repeated SPICE simulations for relatively small nonlinear circuits. In this dissertation, for the first time, we present a massively parallel SPICE simulator on GPU, TinySPICE, for efficiently analyzing small nonlinear circuits. TinySPICE integrates a highly-optimized shared-memory based matrix solver and fast parametric three-dimensional (3D) LUTs based device evaluation method. A novel circuit clustering method is also proposed to improve the stability and efficiency of the matrix solver. Compared with CPU-based SPICE simulator, TinySPICE achieves up to 264X speedups for parametric SRAM yield analysis without loss of accuracy

Application of symbolic computations to the constitutive modeling of structural materials

In applications involving elevated temperatures, the derivation of mathematical expressions (constitutive equations) describing the material behavior can be quite time consuming, involved and error-prone. Therefore intelligent application of symbolic systems to faciliate this tedious process can be of significant benefit. Presented here is a problem oriented, self contained symbolic expert system, named SDICE, which is capable of efficiently deriving potential based constitutive models in analytical form. This package, running under DOE MACSYMA, has the following features: (1) potential differentiation (chain rule), (2) tensor computations (utilizing index notation) including both algebraic and calculus; (3) efficient solution of sparse systems of equations; (4) automatic expression substitution and simplification; (5) back substitution of invariant and tensorial relations; (6) the ability to form the Jacobian and Hessian matrix; and (7) a relational data base. Limited aspects of invariant theory were also incorporated into SDICE due to the utilization of potentials as a starting point and the desire for these potentials to be frame invariant (objective). The uniqueness of SDICE resides in its ability to manipulate expressions in a general yet pre-defined order and simplify expressions so as to limit expression growth. Results are displayed, when applicable, utilizing index notation. SDICE was designed to aid and complement the human constitutive model developer. A number of examples are utilized to illustrate the various features contained within SDICE. It is expected that this symbolic package can and will provide a significant incentive to the development of new constitutive theories