this report was partially supported by Grant CCR-90-24549 from the National Science Foundation. This is a report to the National Science Foundation and other agencies; it is not a report by or of the National Science Foundation or any other agency. Participants at the Workshop on Research Directions in Integrating Numerical Analysis, Symbolic Computing, Computational Geometry, and Artificial Intelligence for Computational Science Conference Organizers

In this paper we review the current state of the problem solving environment (PSE) field and make projections for the future. First we describe the computing context, the definition of a PSE and the goals of a PSE. The state-of-the-art is summarized along with sources (books, bibliographics, web sites) of more detailed information. The principal components and paradigms for building PSEs are presented. The discussion of the future is given in three parts: future trends, scenarios for 2010/2025, and research

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xvi 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Modeling with Partial Differential Equations : : : : : : : : : : : : : : 1 1.2 Evolution of PDE Solving Software : : : : : : : : : : : : : : : : : : : 3 1.3 Problem Solving Environments : : : : : : : : : : : : : : : : : : : : : 8 1.3.1 Properties of PSEs : : : : : : : : : : : : : : : : : : : : : : : : 8 1.3.2 PSEs vs. PSE Frameworks : : : : : : : : : : : : : : : : : : : : 9 1.4 PDE Based Applications and Application PSEs : : : : : : : : : : : : 10 1.4.1 The PDELab Prototype : : : : : : : : : : : : : : : : : : : : : 11 1.5 Overview of the Thesis : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2. THE ARCHITECTURE OF A SOFTWARE FRAMEWORK FOR BUILDING PROBLEM SOLVING ENVIRONMENTS FOR PDE BASED APPLICATIONS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 2.1 Introduction : : : : : : : : : : : : : : ...

this article.

this paper we describe an interactive symbolic-numeric interface framework (editor) to the ELLPACK partial differential equation (PDE) system for building PDE solvers for a much broader range of applications. The domain of applicability of ELLPACK and its parallel version (//ELLPACK) is restricted to second order linear elliptic boundary value problems. This editor allows the specification of nonlinear initial and boundary value PDE problems. The editor applies hybrid symbolic--numeric techniques at the PDE problem level to automatically reduce them to a sequence of linear elliptic PDEs. The result of this preprocessing is recorded in the form of an ELLPACK program. Several examples are presented to demonstrate the functionality and applicability of this interface framework, and the efficiency of the underlying solution methods. 1. Introduction

. We present the design and implementation of ObjectMath, a language and environment for high-level equation-based modeling and analysis in scientific computing. The ObjectMath language integrates object-oriented modeling with mathematical language features that make it possible to express mathematics in a natural and consistent way. The implemented programming environment includes a graphical browser for visualizing and editing inheritance hierarchies, an application oriented editor for editing ObjectMath equations and formulae, a computer algebra system for doing symbolic computations, support for generation of numerical code from equations, and routines for graphical presentation. This programming environment has been successfully used in modeling and analyzing two different problems from the application domain of machine element analysis in an industrial environment. 1 Introduction The programming development process in scientific computing has not changed very much during the pas...

this paper we explore an approach which enables all of the problems listed above to be solved at a single stroke: use Lisp as the source language for the numeric and graphical code! This is not a new idea --- it was tried at MIT and UCB in the 1970's. While these experiments were modestly successful, the particular systems are obsolete. Fortunately, some of those ideas used in Maclisp [37], NIL [38] and Franz Lisp [20] were incorporated in the subsequent standardization of Common Lisp (CL) [35]. In this new setting it is appropriate to re-examine the theoretical and practical implications of writing numeric code in Lisp. The popular conceptions of Lisp's inefficiency for numerics have been based on rumor, supposition, and experience with early and (in fact) inefficient implementations. It is certainly possible to continue to write inefficient programs: As one example of the results of de-emphasizing numerics in the design, consider the situation of the basic arithmetic operators. The definitions of these functions require that they are generic, (e.g. "+" must be able to add any combination of several precisions of floats, arbitrary-precision integers, rational numbers, and complexes), The very simple way of implementing this arithmetic -- by subroutine calls -- is also very inefficient. Even with appropriate declarations to enable more specific treatment of numeric types, compilers are free to ignore declarations and such implementations naturally do not accommodate the needs of intensive number-crunching. (See the appendix for further discussion of declarations). Be this as it may, the situation with respect to Lisp has changed for the better in recent years. With the advent of ANSI standard Common Lisp, several active vendors of implementations and one active universi...

Symbolic computation is employed to automatically derive formulas in finite element analysis (FEA) and to generate parallel numeric code. Key FEA computations parallelized include element stiffness computations and solution of global system of equations. An element-by-element preconditioned conjugate gradient method is used to solve the global system of equations in parallel. Derived formulas are automatically mapped onto the shared-memory architecture. An experimental software system, P-FINGER, is being extented. P-FINGER features a specification language to describe numeric algorithms for which code is to be generated. The specifications also allow an automatic code dependence analysis mechanism to extract parallelism from the specified computational steps. A separate code translator GENCRAY is modified to render code into parallel f77. Generated parallel routines run under the control of existing FEA packages. Examples of generated code are also presented.

In this paper we describe the use of integrated symbolic--numeric computing techniques in the evolving //ELLPACK 1 [HRC + 90] Problem Solving Environment. //ELLPACK is a problem solving environment (PSE) for solving partial differential equations (PDEs) using parallel architectures. It was originally developed to use the ELLPACK[RB85] system as the numerical computing engine. The domain of applicability of ELLPACK is restricted to second order linear elliptic boundary value problems in two and three dimensions. We apply hybrid symbolic--numeric techniques to extend the domain of applicability of the //ELLPACK PSE using both ELLPACK and other numerical PDE solving systems as numerical engines. These techniques have been implemented as an interactive tool using the X Window System[SG86]. Once the PDE problem is specified, it is symbolically manipulated using the MAXIMA 2 computer algebra system to generate a //ELLPACK program to solve the problem using either ELLPACK or FIDISOL[SSM...

INTRODUCTION 2 The second version of GENTRAN was implemented at Twente University of Technology to run under REDUCE. It was designed to be interfaced with a code optimization facility created by Dr. J. A. van Hulzen. I would like to thank Dr. van Hulzen for all of his help in the implementation of GENTRAN in RLISP during a stay at his university in The Netherlands. Finally, I would like to thank Dr. Anthony Hearn of the RAND Corporation for his help in better integrating GENTRAN into the REDUCE environment. 1 INTRODUCTION Solving a problem in science or engineering is often a two-step process. First the problem is modeled mathematically and derived symbolically to provide a set of formulas which describe how to solve the problem numerically. Next numerical programs are written based on this set of formulas to e#ciently compute specific values for given sets of input. Computer algebra systems such as REDUCE provide powerful tools for use in the formula-derivation