Application of FPGAs in acceleration of numerical solution of differential equations

Abstract

This thesis proposes a new application for Field Programmable Gate Array in the acceleration of the numerical solution of differential equations. By using the FPGA as a coprocessor that can be integrated with the system main processor, certain tasks that represent a computational bottleneck can be offloaded to the FPGA coprocessor and carried out more efficiently. More specific, the domain of differential equations considered in this thesis arises during the transient simulation of nonlinear circuits. The work in this thesis investigates the various computational tasks involved in the numerical solution of differential equations. A recent approach to solve differential equations numerically has been studied that requires the computation of high-order derivatives of circuit variables (e.g. node voltages, charges) with respect to a single parameter such as time. However, complex nonlinear devices, are typically characterized by complex nonlinear functions with mathematical expressions that render such computations on conventional computing platforms very time-consuming. This thesis demonstrates that using a computational platform with hardware-enabled accelerator can speed up the task of computing high-order derivatives by at least one-order-of-magnitude. The main idea of the thesis is based on using some recently derived formulas representing the high-order derivatives in terms of the lower-order ones to configure a Field Programmable Gate Arrays (FPGA) in a tree-like structure that represent the non-linear expression. The nodes of this tree will represent common non-linear terms such as exponential or logarithmic functions, which will programmed to propagate their own derivatives from the knowledge of their "children" nodes derivatives. It is shown that this scheme has the potential of relieving the central processing unit in the conventional platforms from having to fetch the tree structure from the system memory and process it in computing the derivatives and will thus lead to significant acceleration.