An adaptive procedure, based on Automatic Differentiation, for estimating &quot;good&quot; values for the relaxation paramaters for general multi-dimensional problems is proposed

Rice, John R.

Tsompanopoulou, P.

Vavalis, E.

English

Purdue E-Pubs

Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1999 Automated Estimation of Relaxation Parameters for Interface Relaxation John R. Rice Purdue University, jrr@cs.purdue.edu P. Tsompanopoulou E. Vavalis Report Number: 99-015 Rice, John R.; Tsompanopoulou, P.; and Vavalis, E., "Automated Estimation of Relaxation Parameters for Interface Relaxation" (1999). Department of Computer Science Technical Reports. Paper 1446. https://docs.lib.purdue.edu/cstech/1446 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. AUTOMATED ESTIMATION OF RELAXATIONPARAMETERS FOR INTERFACE RELAXATIONJohn R. RiceP. TsompanopoulouE. VavalisDepartment of Computer SciencesPurdue UniversityWest LafayeUe, IN 47907CSD-TR #99-015April 1999Automated estimation of relaxation parametersfor interface relaxation 1J.R. Rice P. Tsompanopoulou 2 and E. Vavalis 2Purdue University, Computer Science Department, West Lafayette, IN 47907.AbstractAn adaptive procedure, based on Automatic Differentiation, for e..'itimating" good"values for the relaxation paramaters for general multi-dimensional problems is pro-posed.1 IntroductionInterface Relaxation (IR) methods [ll] have been recently attracted a lot ofattention. This is due to their increased capability in splitting Multi-PhysicsMulti~Domain (MPMD) PDE problems into a collaborative pool of simpleproblems [8]. This splitting is done in a natural way and leads to effectivenumerical methods which fully utilize existing PDE solvers through softwarereuse practices. The resulting PDE sub-problems are coupled together thoughtrelaxation mechanisms that try to mimic the physical world. There alreadyexist several such interface relaxation techniques in the literature [11]. Most ofthem involve relaxation parameters that can significantly improve the conver-gence properties of the various IR methods if their values are properly chosen.These parameters depend on many components (PDE operator, PDE domain,splitting, ... ) of the original problem. This, unfortunately makes the selectionof II optimum" values for the relaxation parameters a hard and challengingproblem. On the other hand the local subdomain discretization schemae donot affect the convergence properties of the IR schemae giving them the greatversatility to let one select the most appropriate discretization parameters ormethod for the differential problem define on each subdomain.1 Work supported in part by PENED grants 95-602 and 95-107, NSF grants CCR-9202536 and CDA-9123502, and AFOSR FM 49620-92-J-0069.2 Author's permanent address: University of Crete, Mathematics Department, 71409 Heraklion, Greece and IACM, FORTH, 711 10 Heraklion, Greece.Preprint submitted to Elsevier Science 3 December 1998Several papers have been devoted in theoretically obtaining optimum valuesfor the interface relaxation parameters [14,16]. Nevertheless the analysis isdone for model problems and subdomain splittings and although they provideimportant information fail to assist a non-expert user to select values for thoseparameters for problems with moderate complexion. When this complicationis increased even experts can not effectively select parameters.The main objective of our study is to provide an adaptive heuristic mecha-nism for automatically selecting" good" relaxation parameters for general dif-ferential equations and domain decompositions. These proposed mechanismuses Automatic Differentiation (AD) and utilizes the existing analytically es-timated valuesThe rest of this paper is organized as following. In the next section we brieflypresent the general methodology of IR. An adaptive procedure for estimatingn good" values for the relaxation parameters is proposed in section 3.2 Domain decomposition with iterated interface relaxationCurrently the domain decomposition world consists of two parts -overlappingand non-overlapping- both living in prosperity. Overlapping, known also asSchwartz, methods were the first considered and have already proved them-selves as very efficient numerical procedures enjoying certain very desirableconvergence properties. Nevertheless it has been also observed that they mighthave several serious drawbacks which will prohibit their use for certain appli-cations. For example, almost all of the many proposed domain decompositionmethods for solving wave propagation models (that consist of the Helmholtzequation coupled with various absorbing or reflecting boundary conditions)are non-overlapping and of interface relaxation type [1,3,7,10].Non-overlapping methods exhibit certain advantages compared to overlappingones. Specifically:They are not sensitive to jumps on the operator coefficients. Their convcr-gence behavior and theoretical crror estimates remain the same even if thedifferential operator includcs discontinuous coefficients provided that thejumps occur along the interface lines [15].- They have smaller communication overhead in a parallel implementation ondistributed memory multiprocessor systems. Their communication overheadis proportional to the length of the interface lines while it is proportional tothe overlapping area in the case of overlapping methods [4J.The bookkeeping is rather easy for the decomposition and manipulationsof the associated data structures compared to the more complicated and2r----'u.new II Il. __ ....__ J,Relaxation:Q.Ir----'b ..new I..... r - - - - ,lJ I ................ r U.new Il. J 1 J I1. .J(au .new au .new)new new I ]gijUi ,uj 'an 'an aFig. 1. The interface relaxation mechanismcostly bookkeeping of the overlapping methods [4].There are two principal viewpoints of non-overlapping methods, precondition-ing and interface relaxation. For an in depth and up-to-date survey of non-overlapping domain decomposition methods considered and analyzed from thepreconditioning viewpoint the reader is referred to [13] and for a general for-mulation and analysis of interface rela.xation methods to [8]. We give a briefpresentation of the interface relaxation method philosophy and practice, inorder to identify its main characteristics.Interface relaxation is a step beyond non-overlapping domain decomposition;it follows Southwell's relaxation of the 1930's - but at the PDE instead of thelinear algebra level - to formulate relaxation as iterated interface smoothingprocedures. A complex physical phenomenon consists of a collection of simpleparts with each one of them obeying a single physical law locally and ad-justing its interface conditions with neighbors. Interface relaxation partitionsthe domain on a set of non-overlapping subdomains, imposes some boundaryconditions on the interface among subdomains lines. Given an initial guess,it imitates the physics of the real world by solving the local problems exactlyon each subdomain and relaxing boundary values to get better estimates ofcorrect interface conditions. This is illustrated in Figure 1 where the genericrelaxation formula gi,j (based on the current solutions u['ew and Ujew ofthe two local to the neighboring subdomains 0 1 and OJ) calculates successiveapproximations b~ew to the solution on the interface ri,j between them.To formally describe this method we consider the differential problemDu :=:: f in fl, Bu :=:: c on ao3(1)where D is an elliptic, non-linear in general, differential operator and B a con-dition operator defined on the boundary an of a domain nERd, d = 1,2" , ..This domain is partitioned into p subdomains ni,i = 1,., "p such thaton = ~=lni and ri=l ni= 0. For reasons related either to the physical char-acteristics of this problem or to the computing resources available, one wouldlike to replace (1) with the following system of loosely coupled differentialproblems(2)where i = 1, ... , p. These differential problems are coupled through the inter-face conditions Giu = 0 and involve the restrictions Di and B i of the globaldifferential and boundary operators, D and B, respectively, on each subdomainwith some of them possibly linear and some others nonlinear. The functions Iiand Cj are similar restrictions of functions 1 and c. The local interface operatorGi is associated with the interface relaxation method and different selectionsfor the G/s lead to different relaxation schemes. In this study we considerseveral interface relaxation methods that have the following characteristics:- They first decompose the problem (1) at differential level and then discretizethe resulting differential subproblems (2).They have the versatility to use the most appropriate discretization schemefor each subproblem.- They do not overlap the subdomains Oi.- Using good relaxation parameters in Gi , they are fast enough so no precon-ditioning is needed.- They simplify the geometry and physics of the computation by consideringthe subproblems (2) instead of the global differential problem (1).They can utilize software parts technology by reusing existing "legacy" soft-ware parts for solving the individual subproblems (2).- They are general and robust.There are several challenging questions concerning practical applications ofsuch methods (e.g. find the most suitable relaxer for a particular problemof application, determine what is the domain of applicability of each one ofthem, explain the interaction between the mathematical iteration and the nu~merical solving method, select" good" or n optimal" values for the relaxationparameters involved, ...), It is worth to point out that since all the methodsdecompose and relax interface values at continuum level the convergence anal-ysis of these methods need to be carried out at PDE (continuum) level andtherefore is a mathematical and not a numerical analysis problem (sec [8] fora discussion).43 Estimating relaxation paramaters in the general caseAs it has been allready observed [11 ,5] the value of the relaxation paramatersplays a crousial role in the convergence of the various interface relaxationmethods. Fine tuning of these parameters can greatly improove the rate ofconvergence while a bad choice might lead to unacceptable slow convergenceor even divergence. They depend on both the eigenvalues of the differentialoperator and the geometric parameters of the domain. Unfortunately analyticexpressions of this dependence are not available and docs not seem easy to de-rive them for general cases. Therefore general and robust heuristic mechanismsthat automaticaly compute "good" values for the relaxation paramaters areneeded. We next present a framework to adaptively adjust the values of therelaxation paramaters O:i and fh of the averaging interface relaxation methodtowards its optimal value. The methodology of the proposed scheme is generalenough to treat complicated (possibly non-linear) differential operator and ar-bitrary domain decompositions in 1,2 or 3 dimensions, has allready be appliedwith success to the successive over-relaxation (SOR) iterative method [6] andcan be readily applied to virtualy any interface relaxation methods.Consider the particular interface ri,j E Rd , d = 0, 1, 2 between subdomains r.!iand njo The interface relaxation equations on ri,j are given by(3)where :v represents the outward normal derivative to ri,j and(4)We arc interested in investigating how different values of the relaxation pawrameters affect the quality of the iterants u(x) and u/(x). Vie can achieve thatby computing the derivative of a certain quantity, that reflects the qualityof iterants, with respect to these relaxation parameters and use it to updatetheir values. We can consider two such quantities: the 2-norm of the residualr =11 Du(x) - J(x) [[ on the interface ri,j or the size of the Gerr disc associ-ated with that particular interface as considered in the previous section. Weshould mark that the first choice will not work for interfaces where there arediscontinuities and the second one assume that that the iteration matrix 111[121 has been obtained.We first conccdrate on the first option Assuming that we are able to computethe sensitivity of the residual with respect to O:i,j as its partial derivativer0 = a~~ _ after performing the Neumann (3) rela.xation then we can use a..,5secant method to update the value of the relaxation parameter in the nextiteration. More specifficaly, if we assume that with the superscripts below wedenote the iteration step then we obtain the following simple adaptive formula(k) (k-l)(k+l) _ (k) (k) cxi,j - CXi,jai,j - cxi,j - To: (k) (k 1)'ro: - To:while the equivalent formula associated with the Dirichlet (??(k) (k-l)PlkH) _ plk} _ (k)f3;j - Pi,;ij - i,j Tp (k) .(k 1)'Tp -1{J(5)(6)can be derived similarly. Obviously there is no way to analyticaly calculate r 0:since there is no closed form of r, as a function of the cx's and {3's, avaialble.Nevertheless such function is given in a form of an algorithm while relaxingeach interface. Therefore a possible solution is to use automatic differentiation[9] to obtain" good" values for the relaxation parameters. In the automaticdifferentiation framework one ap1lies a process that generates code which useslookup tables and mechanicaly apply the chain rule to compute the derivativeof a function given in a form of a computer program. Among the varioussoftware infrastructures that exploit the automatic differentiation idea we haveselect the ADIFOR package [2] for our implementation.To move to our second option we consider the Gerschorin disk associated withthe ith interface and try to simultaniously minimize its area and move itscenter as close to the origin as possible. This can be done by minimizing thesum of the absolute value of the elements in the ith row of the iteration matrix!vI with respect to cx and {3.Thus we can use automatic differention to calculate the Jacobian matrix andsolve the associated equations for the optimum values of the relaxation pa-rameters.References[1] A. Bamberger, R. Glowinski, and Q.H. TraIl. A domain decomposition methodfor the acoustic wave equation with discontinuous coefficients and grid change.SIAM J. Numer. Anal., 34:777-77, 1997.[2] C. Bischof, A. Carle, P. Khademi, and Mauer A. ADIFOR 2.0: Automaticdifferentiation of Fortran 77 programs. IEEE Computational Science £3Engineering, 3:18-32, 1996.6[3J B. Despres. Domain decomposition method and the Helmholtz problem. InL. Halpern G. Cohen and P. Joly, editors, Mathematical and Numerical Aspectsof Wave Propagation Phenomena, pages 44-52. SIAM, 1991.[4] T.T. Drashansky. An agent-based approach to building multidisciplinaryproblem solving enviroments. PhD thesis, Purdue University, Computer ScienceDepartment, December 1996.[5J D. Funaro, A. Quarteroni, and P. Zanolli. An iterative procedure withinterface relaxation for domain decomposition methods. SIAM J. Numer.Anal., 25(6P213-1236, 1988.[6] P. Hovland and M. Heath. Adaptive SOR: A case study in automaticdefferentiation of algorithm parameters. Technical Report ANL/MCS-P672-0697, Argonne National Laboratory, 1997.[7] S. Kim. A parallelizable iterative procecedure for the Helmholtz problem. Appl.Numer. Math., 17:411-'!7, 1995.[8] M. Mu. Solving composite problems with interface relaxation. SIAM J. Sci.Comput., 1998 (to appear).[9] L.B. RaIl. Automatic differentiation: Techniques and applications, volume 120of Lecture Notes in Computer Science. Springer-Verlag, New York, 1981.[10] J.R. Rice, P. Tsombanopoulou, E. Vavalis, and D. Yang. Domain decompositionmethods for underwater acoustics problems. Technical report, ComputerScience Department, Purdue University, W. Lafayette, IN, in preparation.[11] J.R. Rice, P. Tsompanopoulou, and E.A. Vavalis. Review and performanceinterface relaxation methods for elliptic pdes. Technical Report CSD-TR-97-004, Purdue University, W. Lafayette. IN, 1997.[12] J.R. Rice, P. Tsompanopoulou, and E.A. Vavalis. Fine tuning interfacerelaxation methods for elliptic differential equations. Technical Report CSD-TR-98-017, Purdue University, W. Lafayette. IN, 1998.[13] J. Xu and J. Zon. Non-overlapping domain decomposition methods. Technicalreport, Mathematics Department, Pennsylvania State University, UniversityPark, PA, 1996.[14] D. Yang. A parallel iterative nonovelapping domain decomposition procedurefor elliptic problems. IMA J. Numer. Anal., 16:75-91, 1996.[15] D. Yang. A non-overlapping domain decomposition method for elliptic interfaceprroblems. Technical Report IMA # 1472, University of Minnesota, 1997.[16] D. Yang. A parallel domain decomposition algorithm for elliptic problems. J.Compo Math., 16:141-151, 1998.7

Automated Estimation of Relaxation Parameters for Interface Relaxation

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Automated Estimation of Relaxation Parameters for Interface Relaxation

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