PLATFORM FOR BUILDING PDEBASED PROBLEMSOLVING ENVIRONMENTS Mo Mu

PLATFORM FOR BUILDING PDE-BASED PROBLEM-SOLVING ENVIRONMENTS Mo Mu Department of Mathematics Center for Scientific Computation Hong Kong University of Science & Technology Collaborators: E. Houstis, J. Rice (Computer Sci. Dept, Purdue Univ. ) E. Vavalis (Dept. of Math. , Crete Univ. , Greece)

WHAT ARE PSEs "A PSE is a computer system that provides all the computational facilities needed to solve a target class of problems. These features include advanced solution methods, automatic and semiautomatic selection of solution methods, and ways to easily incorporate novel solution methods. Moreover, PSEs use the language of the target class of problems, so users can run them without specialized knowledge of the underlying computer hardware or software. By exploiting modern technologies such as interactive color graphics, powerful processors, and networks of specialized services, PSEs can track extended problem solving tasks and allow users to review them easily. Overall, they create a framework that is all things to all people: they solve simple or complex problems, support rapid prototyping or detailed analysis, and can be used in introductory education or at the frontiers of science. " -"Computer as Thinker/Doer: Problem-Solving Environments for Computational Science" by E. Gallopoulos, E. Houstis and J. Rice (IEEE Computational Science and Engineering, Summer 1994).

WHY PSEs • Researchers in scientific computing and simulation have witnessed the evolution of computer systems from subroutines to libraries, then to software packages/solvers, and now to PSEs. • This evolution is possible and necessary because computational techniques have become more sophisticated, the computing environment has become more powerful with many varieties of architectures, and above all, the driving force, computational problems have become more complex.

EXAMPLES OF PSEs • Workbench frameworks: users enter commands or create the application – – – – Matlab (high-level interface to toolboxes) Maple, Mathematica PELLPACK PETSc (PDE-based scientific applications) Net. Solve Air. Shed Modeler Soliton Explorer

EXAMPLES OF PSEs • Component composition systems: users can wire together components to create a complete application – SCIRun – Component Architecture Toolkit – Workbench for Interactive Simulation of Ecosystems

EXAMPLES OF PSEs • Code composition/generation frameworks: these are distinguished by having as an end product code in a programming language – – POOMA/PAWS ATHAPASCAN Sci. Napse Falcon

EXAMPLES OF PSEs • Collaboration frameworks: they provide frameworks that allow multiple users at widely separated sites to work together on a single problem – Tech. Talk (supporting shared Matlab, Maple) – Shastra – Intelligent Archive

LESSONS • • LARGE COMPLEX HETEROGENEOUS ENORMOUS EFFORTS

CHALLENGES AND OPPORTUNITIES • How to build these very large and complex computer systems effectively and efficiently? We face many technological difficulties that have not yet been faced in the development of conventional scientific software packages. • Meeting this challenge will lead to many research issues in software methodology, computational frameworks, and scientific modeling and will create great opportunities for exploring novel concepts and methodologies in these areas.

PDE-BASED PSEs • Necessary to have a platform for building PSEs in a plugand-play fashion • Realistic for PDE applications

RESEARCH ASPECTS • Scientific -- model integration (application users) • Mathematical -- solution integration • Technical -- software/shareware integration (real & virtual)

COLLABORATING PDE SOLVERS • Solve submodels independently with supplied boundary conditions • Adjust boundary conditions to better satisfy interface constraints • Iterate until convergence

INTERFACE RELAXATION • Originating from the spirit of the classical idea by Southwell in the 1930 s for solving linear algebraic systems • Appearing in various formulations such as valueaveraging, least-squares minimization, Newton iteration, energy minimization, Dirichlet-Neumann alternating, etc. • Effective for a wide range of applications • Analyzed for a class of model problems • Reduced to domain decomposition in special cases • Accurate and reliable • Flexible

AGENT-BASED ARCHITECTURE • Agents--single submodel solvers/PDE packages • Mediators--interface relaxers / central convergence handler

ALL PURPOSE DOMAIN PROCESSOR • Internal domain processing: It supports one, two, three, and even higher-dimensional spatial domains. It is able to handle reasonably complex geometric shapes on both symbolic and numerical levels. – Horizontally, it supports various mesh structures including tensorproduct grids, polar meshes, finite element meshes, unstructured meshes. – Vertically, it supports mesh coarsening/refinement, multi-level meshes, adaptive meshes, mesh restructuring, mesh partitioning, mesh overlapping, time-dependent moving meshes, moving boundaries.

DOMAIN PROCESSOR (continued) • Inter-domain processing: It provides facilities to support communication and plugging with other domains. – Domain identification – Domain joining (selecting, moving, gluing, interface-domain mapping, cross-point handling, etc. ) – Global/local coordinate systems mapping – Meshing consistency

DOMAIN PROCESSOR (continued) • Data structures and operations: – Hierarchical/paging techniques are used in our data structures for the purposes of flexibility and efficiency. – For ease of implementation of native PDE solvers, we provide utility procedures including interpolation, restriction, extrapolation, differentiation, neighboring values, blending, smoothing, functions of boundary value, normal derivatives, interface values and/or jumps, values at previous time levels, mean-values at overlapping points, such as cross-points and those in an overlapping region.

APPLICATIONS PROGAMMER INTERFACES • Standards--necessary for software/shareware integration • Efficiency--optimized implementation by experts • Publicity--participation of computer vendors and software firms • Successful example: BLAS/BLACS • Upcoming example: DFTAPI • On low level of subroutines • We see the need and importance to have a widely accepted standard Applications Programmer Interface (API) for PDE-based PSE components

STANDARD PDE APIs • Several levels of the PDE-API are defined. • The lowest level starts with a set of basic building blocks for PDE solvers. • The next level contains basic and commonly used PDE solvers defined on simple shaped domains. • The more advanced level has general linear PDE solvers. • The highest level covers general nonlinear PDEs, systems of PDEs, complex-valued PDEs, and so on. , in order to model a reasonably complicated, yet still well understood and solvable single physical or chemical phenomenon.

STANDARD INTERFACE APIs • This covers commonly used interface conditions and relaxation/matching schemes

Wrapper. Lib • Optimized implementation of core PDE solvers • Browser foreign solvers • Wrapper. Lib--a collection of wrappers that map the user interfaces of commonly used PDE solvers and shareware to the standard PDE-API.

PDE BROWSER • Browse available PDE solvers according to user’s specification • Solver selection by invoking the expert system • Plug-in (real or virtual) through PDE-API/Wrapper. Lib

APPLICATION EXAMPLE • Long Range Transport of Air Pollution (LRtap) • Multiple models: convection, diffusion, deposition, emission, chemical reaction • Sci. Agents implementation • Simulation for the island of Crete

PDE-Mart: Features • PDE based PSE • Networked (the user’s point of view: GUI) • Distributed (the computing point of view: LIB, network/parallel computing)

PDE-Mart: System Structure • PDE-Mart GUI • PDE-Mart library

PDE-Mart GUI • The interface is a platform for specifying applications, constructing PDE solvers, evaluating mathematical models and numerical methods, as well as scientific simulation • The engine (a preprocessor or interpreter) translates the user interface to the internal system interface

PDE-Mart GUI: practical and technical issues • Invoke mode: web-based versus host-based – web-based • convenient for anonymous users through browser access to the software system • could be inefficient due to network traffic • network security: especially for performing I/O (access to the file systems at both ends) – untrusted users to the PDE-Mart server – untrusted shareware to the user’s computer

PDE-Mart GUI: practical and technical issues • web-based versus host-based (continued) – host-based • suitable for advanced users who need an efficient GUI and intensive I/O, are experienced with the software system, and are mutually trusted with the PDE-Mart server • remote host: use the PDE-Mart server as the host – directly connect to the server through an explicit network protocol such as telnet or other gateways – no regular access to the local machine • local host: user’s computer as the host – download the GUI and run the Java application on the local machine – with full access to the local host – communicate with the PDE-Mart server through a suitable networking technology such as CGI or the Java client-server mode

PDE-Mart GUI: practical and technical issues • I/O mode: batch versus interactive – batch mode • program text based, such as the Ellpack approach • necessary file access and possible implementation – on the remote server • browse and load sample programs (e. g. prog. e) - simple for HTML - Java applet (coming from the remote server): network connection or interaction with CGI) • review and edit/save intermediate results (e. g. prog. f, out. dat) - HTML/CGI or Java/CGI - limited access for untrusted users

PDE-Mart GUI: practical and technical issues (batch mode) • necessary file access and possible implementation (continued) – on user’s machine • file access – browse and load sample or previous programs (e. g. prog. e) – edit/save intermediate and final results (e. g. prog. e, prog. f, out. dat) • implementation – HTML or Java application running on the local host – Java applet is not allowed to do so. Must go through HTML or using Java client-server to walk around • a form protocol is sufficient for job submission and retrieving data, such as HTML/CGI or Java/CGI • efficient interactive post-processing on the local machine is possible after the solution is computed and fetched back to the user’s end

PDE-Mart GUI: practical and technical issues • I/O mode: batch versus interactive – conclusions of the web-based, batch version: the HTML/CGI approach is chosen in our implementation • browser enabled • simple • enough graphics for basic UI purposes

PDE-Mart GUI: practical and technical issues • I/O mode: batch versus interactive – web-based interactive mode: Java applet • web browser enabled • object-oriented – – – domain class PDE class domain processor class solver class post-processing class • reusability • graphics support (Java AWT, Java Swing) • Do users often need to save the current application (domain, PDE, ect. ) on client machines, reload and edit it? And how (via a template)?

PDE-Mart GUI: practical and technical issues • Security policies – on the user’s side • HTML--save by users • Java applet? Current approach--through HTML – on the server’s side • with limited access to the server for intermediate I/O in order to control anonymous users, except for authorized ones

PDE-Mart: GUI (html) WELCOME TO PDE-MART PDE-PSE ELLPACK OTHER PDE SYSTEMS

PDE-Mart: GUI A PROBLEM-SOLVING ORIENTED GUI FOR PDE-PSE (Java applet) • Problem specification: – DOMAIN – PDE • Solver specification – – • DOMAIN DISCRETIZATION PDE DISCRETIZATION SOLUTION ANALYSIS Function buttons – – submit compile run load

PDE-Mart: GUI DOMAIN SPECIFICATION-2 D DOMAIN OF SPECIAL SHAPES POLYGONS DOMAIN WITH PIECEWISELY DEFINED BOUNDARIES BY PARAMETRIC FUNCTIONS

PDE-Mart: GUI DOMAIN OF SPECIAL SHAPES RECTANGLE DISC ELLIPSE

PDE-Mart: GUI RECTANGLE Please input the coordinates for the four edges: (xa, xb), (ya, yb)

PDE-Mart: GUI DISC Please input the coordinates of the center (a, b) and the radius r

PDE-Mart: GUI ELLIPSE Please input the …….

PDE-Mart: GUI POLYGON Please input the vertices list

PDE-Mart: Technical issues of Library • Interact with other software parts – data structure aspect: PDE-API – language: JNI (Java Native Interface) • Interact with other servers – Java Networking in JDK with TCP and UDP – Java RMI (Remote Method Invocation)

PDE-Mart Library • • Domain class PDE class Discrete domain class Discretization class Solution class Evaluation class Display class

PDE-Mart Library: domain class • • • disc square rectangle triangle polygon general domain with piecewise boundary defined by parametric functions (supplied by users through GUI)

PDE-Mart Library: PDE class • equation class – elliptic • • Poisson equation second order linear equation with constant coefficients second order linear equation with non-constant coefficients forth order equation – parabolic • heat equation – hyperbolic • wave equation • Euler equation • N-S equation

PDE-Mart Library: PDE class • boundary condition class • interface condition class