Nonlinear Overlapping Domain Decomposition Methods

Transcription

1 Nonlnear Overlappng Doman Decomposton Methods Xao-Chuan Ca 1 Department of Computer Scence, Unversty of Colorado at Boulder, Boulder, CO 80309, ca@cs.colorado.edu Summary. We dscuss some overlappng doman decomposton algorthms for solvng sparse nonlnear system of equatons arsng from the dscretzaton of partal dfferental equatons. All algorthms are derved usng the three basc algorthms: Newton for local or global nonlnear systems, Krylov for the lnear Jacoban system nsde Newton, and Schwarz for lnear and/or nonlnear precondtonng. The two key ssues wth nonlnear solvers are robustness and parallel scalablty. Both ssues can be addressed f a good combnaton of Newton, Krylov and Schwarz s selected, and the rght selecton s often dependent on the partcular type of nonlnearty and the computng platform. 1 Introducton For solvng partal dfferental equatons on large scale parallel computers, doman decomposton s a natural choce. Overlappng Schwarz methods and non-overlappng teratve substructurng methods are the two major classes of doman decomposton methods [13, 14, 15]. In ths paper we only consder overlappng methods for solvng large sparse nonlnear system of equatons arsng from the dscretzaton of nonlnear partal dfferental equatons,.e., for a gven nonlnear functon F : R n R n, we compute a vector u R n, such that F (u) = 0, (1) startng from an ntal guess u (0) R n. Here F = (F 1,..., F n ) T, F = F (u 1,..., u n ), and u = (u 1,..., u n ) T. One of the popularly used technques for solvng (1) s the so-called nexact Newton algorthms (IN) whch are descrbed brefly here. Suppose u (k) s the current approxmate soluton and J = F (u (k) ), a new approxmate soluton u (k+1) can be computed through The research was supported n part by DOE under DE-FC02-01ER25479 and DE- FC02-04ER25595, and n part by NSF under grants ACI , CNS , CCF , and CNS

2 2 Xao-Chuan Ca the followng steps: frst fnd an nexact Newton drecton p (k) by solvng the Jacoban system Jp (k) = F (u (k) ) (2) such that F (u (k) ) Jp (k) η k F (u (k) ), then compute the new approxmate soluton u (k+1) = u (k) λ (k) p (k). (3) Here η k [0, 1) s a scalar that determnes how accurately the Jacoban system needs to be solved usng, for example, Krylov subspace methods. λ (k) s another scalar that determnes the step length n the selected nexact Newton drecton. Sometmes when J s not explctly avalable, one can use the matrxfree verson [11]. IN has several well-known features. (a) Fast convergence. If the ntal guess s close enough to the desred soluton then the convergence s very fast (quadratc) provded that the η k s are suffcently small. (b) Non-robustness. The convergence, or fast convergence, happens only f a good ntal guess s avalable. Generally t s dffcult to obtan such an ntal guess especally for nonlnear equatons that have unbalanced nonlneartes [12]. The step length λ (k) s often determned by the components wth the strongest nonlneartes, and ths may lead to an extended perod of stagnaton n the nonlnear resdual curve. We say that the nonlneartes are unbalanced when λ (k), n effect, s determned by a subset of the overall degrees of freedom. (c) Scalablty. The parallel scalablty of the method s mostly determned by how the Jacoban system (2) s solved. There are a number of strateges [7, 8, 10], such as lnesearch, trust regon, contnuaton or better ways to choose the forcng term, to make the algorthm more robust or converge faster, however, these strateges are all based on certan global knowledge of F or J. In other words, all equatons n the system are treated equally as f they were some of the worst equatons n the system. Other ways to look at the global nature of IN are (d) To advance from u (k) to u (k+1), all n varables and equatons need to be updated even though n many stuatons n can be very large, but only a small number of components of u (k) receve sgnfcant updates. (e) If a small number of components of the ntal guess u (0) are not acceptable, the entre u (0) s declared bad. (f) There are two global control varables η k and λ (k). Any slght change of F may result n the change of η k or λ (k), and any slght change of η k or λ (k) may result n some global functon evaluatons and/or the solvng of global Jacoban systems. For example, f the search drecton p (k) has one unacceptable component, then the entre steplength s reduced. Note that these global operatons can be expensve when n s large and when the number of processors s large. Usng doman decomposton methods, more

3 Nonlnear Doman Decomposton Methods 3 localzed treatments can be appled based on the locaton or the physcal nature of the nonlneartes, and the number of global operatons can be made small n some stuatons. We should pont out that the words local and global have dfferent meanngs n the context of doman decomposton methods [15] than n the context of nonlnear equaton solvers [7], among others. In nonlnear solvers, local means a small neghborhood of the exact soluton of the nonlnear system, and global means a relatvely large neghborhood of the exact soluton of the nonlnear system. In doman decomposton, local means some subregons n the computatonal doman and global means the whole computatonal doman. All the algorthms to be dscussed n the paper are constructed wth a combnaton of the three basc technques: Newton, Krylov and Schwarz. Newton s the basc nonlnear solver that s used for ether the system defned on the whole space or some subspaces (subdoman subspace or coarse subspace). Krylov s the basc lnear solver that s used nsde a Newton solver. Schwarz s a precondtoner for ether the lnear or the nonlnear solver. Many algorthms can be derved wth dfferent combnatons of the three basc algorthms. For a gven class of problems and computng platform, a specal combnaton mght be necessary n order to obtan the best performance. The three basc algorthms are all well understood ndvdually, however, the constructon of the best combnaton remans a challenge. The same can be sad for the software. All software components are readly avalable n PETSc [1], but some of the advanced combnatons have to be programmed by the user. We next defne (nformally) some notatons for descrbng doman decomposton methods. u s understood as a dscrete (or coeffcents of a fnte element) functon defned on the computatonal doman Ω whch s already parttoned nto a set of subdomans {Ω1, δ, ΩN δ }. Here Ωδ s an δ-extenson of Ω, and the collecton of {Ω } s a non-overlappng partton of Ω. We defne R δ as a restrcton operator assocated wth Ωδ and R0 as the restrcton operator assocated wth Ω. We denote u Ω δ as the restrcton of u on Ω δ, and u Ω δ as the restrcton of u on the boundary of Ω δ. Here we use the word doman to denote the mesh ponts n the nteror of the doman and boundary to denote the mesh ponts on the boundary of the doman. Smlarly, we may restrct the nonlnear functon to a subdoman, such as. For boundary value problems consdered n ths paper, we assume (u) = (u Ω δ, u Ω δ ). That s to say that there are no global equatons n the system that may couple the equatons defned at a mesh pont to equatons defned outsde a small neghborhood. The rest of the paper s organzed as follows. In Secton 2, we dscuss the most popular overlappng nonlnear doman decomposton method, Newton- Krylov-Schwarz algorthm, and n Sectons 3-6, we dscuss some more advanced nonlnear methods. Some fnal remarks are gven n Secton 7.

4 4 Xao-Chuan Ca 2 Newton-Krylov-Schwarz algorthms Newton-Krylov-Schwarz (NKS) s smply the applcaton of a lnear Schwarz precondtoner for solvng the Jacoban equaton (2) n the nexact Newton algorthm [2, 3]. Dependng on what type of Schwarz precondtoner s used (addtve, multplcatve, restrcted, one-level, two-level, etc), there are several NKS algorthms. Let us defne the subdoman precondtoners as J = R δ J(R δ ) T, = 1,..., N, then the addtve Schwarz precondtoner can be wrtten as M 1 N AS = (R δ ) T J 1 R δ. =1 Because of ts smplcty, NKS has become one of the most popular doman decomposton methods for solvng nonlnear PDEs and s the default nonlnear solver n PETSc [1]. The nonlnear propertes of NKS are exactly the same as that of nexact Newton. For example, the ntal guess has to be suffcently close to the soluton n order to obtan convergence, and fast convergence can be acheved when the nonlnearty s well balanced. NKS addresses the scalablty ssue (c) of IN well, but not the other ssues (a, b, d-f). 3 The classcal Schwarz alternatng algorthms Let (u (0),..., u (0) ) be the ntal guess for all subdomans. The classcal Ω1 δ ΩN δ Schwarz alternatng algorthm (SA) can be descrbed as follows: k = 1,..., tll convergence condton s satsfed = 1,..., N defne u (k) usng {u (k 1), 1 j N} or {u (k) Ω δ Ωj δ Ωj δ compute u (k) Ω δ by solvng (u (k), u (k) ) = 0. Ω δ Ω δ, 1 j < } The algorthm doesn t belong to the class of IN algorthms and, n general, not share propertes (a-f). The method s usually not used by tself as a nonlnear solver because of ts slow convergence, but n some cases when the nonlneartes are solated wthn some of the subdomans, the method can be a good alternatve to IN. Note that SA doesn t nvolve any global operatons. 4 Nonlnear addtve Schwarz precondtoned nexact Newton algorthms The basc dea of nonlnearly precondtoned nexact Newton algorthms ([4, 9]) s to fnd the soluton u R n of (1) by solvng an equvalent system

5 Nonlnear Doman Decomposton Methods 5 F(u) = 0 (4) usng IN. Systems (1) and (4) are sad to be equvalent f they have the same soluton. For any gven v R n, we defne a subdoman projecton T (v), whch s a functon wth support n Ω δ, as the soluton of the followng subspace nonlnear system (v T (v)) = 0, for = 1,..., N. Then a nonlnearly precondtoned functon s defned as F(u) = N T (u). =1 It can be shown that, under certan condtons, for ths partcular F, (1) and (4) offer the same soluton subject to the error due to dfferent stoppng condtons and precondtoners. Ths algorthm s often referred to as the addtve Schwarz precondtoned nexact Newton algorthm (ASPIN). Sometmes we call t a left precondtoned IN because n the lnear case (.e., F (u) = Ju b) F(u) = ( N =1 (Rδ )T J 1 R δ )(Ju b). When usng IN to solve (4), the Jacoban of F, or ts approxmaton, s needed. Because of the specal defnton of the functon F, ts Jacoban can only be gven as the sum of matrx-vector products and the explct elements of F are not avalable. It s known that for left precondtoned lnear teratve methods, the stoppng condton s often nfluenced by the precondtoner. The mpact of the precondtoner on the stoppng condton can be removed f the precondtoner s appled to the rght. Unlke lnear precondtonng, the swtch from left to rght s not trval n the nonlnear case. A rght nonlnear precondtoner wll be dscuss n a later secton of the paper. 5 Nonlnear elmnaton algorthms The nonlnear elmnaton algorthm (NE) was ntroduced n [12] for nonlnear algebrac systems wth local hgh nonlneartes. It was not ntroduced as a doman decomposton method, but we nclude t n the paper because t s the man motvaton for the algorthm to be dscussed n the next secton. Suppose that the functon F s more nonlnear n the subdoman Ω δ, then we can elmnate all unknowns n ths partcular subdoman and let Newton work on the rest of the varables and equatons. Let y = u Ω δ and x = u Ω\Ω δ, then usng the mplct functon theorem, under some assumptons, we can solve for y n terms of x;.e., solve (x, y) = 0

6 6 Xao-Chuan Ca for y, whch symbolcally equals to y = F 1 (x). After the elmnaton, we Ω δ can use the regular Newton method for the rest of the system whch s more balanced, at least n theory, ( ) F Ω\Ω δ x, F 1 (x) = 0. Ω δ The algorthm has some obvous advantages. We menton some of ts dsadvantages as a motvaton for the algorthm to be dscussed n the next secton. In practce, t s often dffcult to tell whch components are more nonlnear than the others, and the stuaton may change from teraton to teraton. The algorthm may ntroduce sharp jumps n the resdual functon near the nterface of x and y. Such jumps may lead to slow convergence or dvergence. Some mproved versons are gven n [6]. In the next secton, we combne the deas of ASPIN and NE nto a rght precondtoned Newton method. 6 Nonlnear restrcted addtve Schwarz algorthms In [5], a rght precondtoned nexact Newton algorthm was ntroduced as follows: Fnd the soluton u R n of (1) by frst solvng a precondtoned nonlnear system F (G(v)) = 0 for v, and then obtan u = G(v). For any gven v R n, we defne a subdoman projecton T (v), whch s a functon wth support n Ω δ, as the soluton of the followng subspace nonlnear system (v + T (v)) = 0, for = 1,..., N. Then the nonlnear precondtonng functon s defned as G(v) = v + N R 0 T (v). =1 Here the non-overlappng restrcton operator R 0 effectvely removes the sharp jumps on the nterfaces of the overlappng subdomans. In the lnear case ( N ) G(v) = v (R 0 ) T J 1 R δ (Jv b), =1 whch can be regarded as a restrcted addtve Schwarz precondtoned Rchardson method. Ths precondtoner doesn t have to be appled at every outer Newton teraton. It s used only when some local hgh nonlneartes are sensed, somehow. Below we descrbe the overall algorthm (NKS-RAS). The goal s to solve equaton (1) wth a gven ntal guess u (0). Suppose u (k) s the current soluton.

7 Nonlnear Doman Decomposton Methods 7 Step 1 (The Nonlnearty Checkng Step): Check local and global stoppng condtons. If the global condton s satsfed, stop. If local condtons ndcate that nonlneartes are not balanced, go to Step 2. If local condtons ndcate that nonlneartes are balanced, set ũ (k) = u (k), go to Step 3. Step 2 (The RAS Step): Solve local nonlnear problems on the overlappng subdomans to obtan the subdoman correctons T (u (k) ) ) (u (k) + T (u (k) ) = 0, for = 1,, N. Drop the soluton n the overlappng part of the subdoman and compute the global functon G(u (k) ) and set ( ũ (k) = G u (k)). Go to Step 3. Step 3 (The NKS Step): Compute the next approxmate soluton u (k+1) by solvng the followng system F (u) = 0 wth one step of NKS usng ũ (k) as the ntal guess. Go to Step 1. The nonlnearty checkng step s mportant. However, we only have a few ad hoc technques such as computng the resdual norm subdoman by subdoman (or feld by feld n the case of mult-physcs applcatons). If some of the subdoman (or sub-feld) norms are much larger than for other subdomans, we label these subdomans as hghly nonlnear subdomans and proceed wth the RAS elmnaton step. Otherwse, when the nonlnearty s more or less balanced we bypass the RAS step and go drectly to the global NKS step. The subdoman nonlnear systems n Step 2 do not need to be solved very accurately snce the solutons are used only to construct an ntal guess for Step 3. In NKS-RAS, a nonlnear system s set up on each subdoman, but n practce, not all subdoman nonlnear problem needs to be solved. In the nottoo-nonlnear regons, the solver may declare to have converged n 0 teraton. 7 Concludng remarks In ths paper, we have gven a quck overvew of overlappng doman decomposton methods for solvng nonlnear partal dfferental equatons. The two key ssues of nonlnear methods are robustness and scalablty. Both ssues

8 8 Xao-Chuan Ca can be addressed by usng some combnatons of the three basc algorthms: Newton, Krylov and Schwarz. Several algorthms are presented n the paper together wth some of ther advantages and dsadvantages. Dependng on the partcular types of nonlneartes and the computng platform, dfferent combnatons of the three basc algorthms may be needed n order to obtan the best performance and robustness. Due to page lmt, applcatons have not been dscussed n the paper. Some of them can be found n the references. References 1. S. Balay, K. Buschelman, W. D. Gropp, D. Kaushk, M. Knepley, L. C. McInnes, B. F. Smth, and H. Zhang, PETSc Users Manual, ANL, X.-C. Ca, W. Gropp, D. Keyes, R. Melvn, and D. P. Young, Parallel Newton- Krylov-Schwarz algorthms for the transonc full potental equaton, SIAM J. Sc. Comput., 19 (1998), pp X.-C. Ca, W. D. Gropp, D. E. Keyes, and M. D. Tdrr, Newton-Krylov- Schwarz methods n CFD, Proceedngs of the Internatonal Workshop on the Naver-Stokes Equatons, Notes n Numercal Flud Mechancs, R. Rannacher, eds. Veweg Verlag, Braunschweg, X.-C. Ca and D. E. Keyes, Nonlnearly precondtoned nexact Newton algorthms, SIAM J. Sc. Comput., 24 (2002), pp X.-C. Ca and X. L, Inexact Newton methods wth nonlnear restrcted addtve Schwarz precondtonng for problems wth hgh local nonlneartes, n preparaton 6. X.-C. Ca and X. L, A doman decomposton based parallel nexact Newtons method wth subspace correcton for ncompressble Naver-Stokes equatons, Lecture Notes n Computer Scence, Sprnger, J. E. Denns and R. B. Schnabel, Numercal Methods for Unconstraned Optmzaton and Nonlnear Equatons, SIAM, S. C. Esenstat and H. F. Walker, Choosng the forcng terms n an nexact Newton method, SIAM J. Sc. Comput., 17 (1996), pp F.-N. Hwang and X.-C. Ca, A parallel nonlnear addtve Schwarz precondtoned nexact Newton algorthm for ncompressble Naver-Stokes equatons, J. Comput. Phys., 204 (2005), pp C. T. Kelley and D. E. Keyes, Convergence analyss of pseudo-transent contnuaton, SIAM J. Numer. Anal., 35 (1998), pp D. Knoll and D. E. Keyes, Jacoban-free Newton-Krylov methods: a survey of approaches and applcatons, J. Comput. Phys., 193 (2004), pp P. J. Lanzkron, D. J. Rose, and J. T. Wlkes, An analyss of approxmate nonlnear elmnaton, SIAM J. Sc. Comput., 17 (1996), pp B. Smth, P. Bjørstad, and W. Gropp, Doman Decomposton. Parallel Multlevel Methods for Elptc Partal Dfferental Equatons, Cambrdge Unversty Press, New York, A. Quarteron and A. Vall, Doman Decomposton Methods for Partal Dfferental Equatons, Oxford Unversty Press, Oxford, A. Tosell and O. Wdlund, Doman Decomposton Methods Algorthms and Theory, Sprnger, Berln, 2005