3014:
42:
1814:
2797:
118:
along the time axis. Both ideas, multigrid in time as well as adopting multiple shooting for time integration, go back to the 1980s and 1990s. Parareal is a widely studied method and has been used and modified for a range of different applications. Ideas to parallelize the solution of initial value
3888:
A method with improved parallel efficiency based on a combination of
Parareal with spectral deferred corrections (SDC) has been proposed by M. Minion. It limits the choice for coarse and fine integrator to SDC, sacrificing flexibility for improved parallel efficiency. Instead of the limit of
4061:
The multigrid reduction in time method (MGRIT) generalises the interpretation of
Parareal as a multigrid-in-time algorithms to multiple levels using different smoothers. It is a more general approach but for a specific choice of parameters it is equivalent to Parareal. The
271:
4662:
Steiner, Johannes; Ruprecht, Daniel; Speck, Robert; Krause, Rolf (2015-01-01). "Convergence of
Parareal for the Navier-Stokes Equations Depending on the Reynolds Number". In Abdulle, Assyr; Deparis, Simone; Kressner, Daniel; Nobile, Fabio; Picasso, Marco (eds.).
3813:
3018:
These two bounds illustrate the trade off that has to be made in choosing the coarse method: on the one hand, it has to be cheap and/or use a much larger time step to make the first bound as large as possible, on the other hand the number of iterations
2322:
3879:
will become more accurate. This will lead to faster convergence. This version of
Parareal can also stably integrate linear hyperbolic partial differential equations. An extension to nonlinear problems based on the reduced basis method exists as well.
1549:
958:
1939:. If this critertion is not satisfied, subsequent iterations can then be run by applying the fine solver in parallel and then the predictor-corrector. Once the criterion is satisfied, however, the algorithm is said to have converged in
3009:{\displaystyle S_{p}={\frac {c_{\text{fine}}}{c_{\text{parareal}}}}={\frac {1}{(k+1){\frac {N_{c}}{N_{f}}}{\frac {\tau _{c}}{\tau _{f}}}+{\frac {k}{P}}}}\leq \min \left\{{\frac {N_{f}\tau _{f}}{N_{c}\tau _{c}}},{\frac {P}{k}}\right\}.}
1417:
3591:
2505:
steps of the coarse integrator. This includes in particular the assumption that all time slices are of identical length and that both coarse and fine integrator use a constant step size over the full simulation. Second, denote by
1540:
1909:
4628:
Staff, Gunnar
Andreas; Rønquist, Einar M. (2005-01-01). Barth, Timothy J.; Griebel, Michael; Keyes, David E.; Nieminen, Risto M.; Roose, Dirk; Schlick, Tamar; Kornhuber, Ralf; Hoppe, Ronald; Périaux, Jacques (eds.).
2788:
2395:
of
Parareal can be derived. Although in applications these assumptions can be too restrictive, the model still is useful to illustrate the trade offs that are involved in obtaining speedup with Parareal.
3106:
137:
3309:. Originally, the idea was formulated for the parallel implicit time-integrator PITA, a method closely related to Parareal but with small differences in how the correction is done. In every iteration
2103:
4770:
Farhat, Charbel; Cortial, Julien; Dastillung, Climène; Bavestrello, Henri (2006-07-30). "Time-parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses".
2238:
1156:(and therefore at much lower computational cost). Having a coarse solver that is much less computationally expensive than the fine solver is the key to achieving parallel speed-up with Parareal.
3382:
3971:
2647:
3213:
equations. Even though the formal analysis by Gander and
Vandewalle covers only linear problems with constant coefficients, the problem also arises when Parareal is applied to the nonlinear
2328:
2327:
2324:
2323:
3429:
2329:
3685:
388:
3877:
3307:
3273:
4867:
Farhat, Charbel; Chandesris, Marion (2003-11-07). "Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid–structure applications".
443:
499:
2360:
1937:
1818:
At this stage, one can use a stopping criterion to determine whether the solution values are no longer changing each iteration. For example, one may test this by checking if
3653:
3473:
1809:{\displaystyle U_{j+1}^{k}={\mathcal {G}}(t_{j},t_{j+1},U_{j}^{k})+{\mathcal {F}}(t_{j},t_{j+1},U_{j}^{k-1})-{\mathcal {G}}(t_{j},t_{j+1},U_{j}^{k-1}),\quad j=0,\ldots ,N-1.}
595:
2313:
2289:
2265:
1154:
1107:
829:
685:
4293:
Ketcheson, David; Waheed, Umair bin (2014-06-13). "A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and parallel".
663:
using a serial time-stepping method (e.g. Runge-Kutta) that has high numerical accuracy (and therefore high computational cost). We refer to this method as the fine solver
962:
The problem with this (and the reason for attempting to solve in parallel in the first place) solution is that it is computationally infeasible to calculate in real-time.
2560:
the computing time required for a single step of the fine and coarse methods, respectively, and assume that both are constant. This is typically not exactly true when an
2326:
2161:
2558:
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836:
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1963:
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2017:
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739:
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323:
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1991:
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293:
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551:
4197:
Jack
Dongarra; Jeffrey Hittinger; John Bell; Luis Chacon; Robert Falgout; Michael Heroux; Paul Hovland; Esmond Ng; Clayton Webster; Stefan Wild (March 2014).
1993:
iterations. For
Parareal to provide speedup, however, it has to converge in a number of iterations significantly smaller than the number of time slices, i.e.
5291:
1431:
295:
is assumed to be a smooth (possibly nonlinear) function. It can also correspond to the spatial discretization of a partial differential equation in a
3483:
1823:
3210:
71:
5121:
Speck, R.; Ruprecht, D.; Krause, R.; Emmett, M.; Minion, M.; Winkel, M.; Gibbon, P. (2012-11-01). "A massively space-time parallel N-body solver".
4824:
Chen, Feng; Hesthaven, Jan S.; Zhu, Xueyu (2014-01-01). "On the Use of
Reduced Basis Methods to Accelerate and Stabilize the Parareal Method". In
970:
Instead of using a single processor to solve the initial value problem (as is done with classical time-stepping methods), Parareal makes use of
4067:
2696:
5138:
4488:
4029:
the typically greater number of iterations of the parallel hybrid method. The Parareal-SDC hybrid has been further improved by addition of a
4996:
Dutt, Alok; Greengard, Leslie; Rokhlin, Vladimir (2000-06-01). "Spectral Deferred Correction Methods for Ordinary Differential Equations".
2325:
87:
1973:
Parareal should reproduce the solution that is obtained by the serial application of the fine solver and will converge in a maximum of
1085:
Parareal makes use of a second time-stepping method to solve this initial value problem in parallel, referred to as the coarse solver
4851:
4690:
4646:
4399:
2315:
can also use a coarser spatial discretization, but this can negatively impact convergence unless high order interpolation is used.
266:{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=f(t,u)\quad {\text{over}}\quad t\in \quad {\text{with}}\quad u(t_{0})=u^{0}.}
3048:
4046:
2025:
115:
5162:
Falgout, R.; Friedhoff, S.; Kolev, T.; MacLachlan, S.; Schroder, J. (2014-01-01). "Parallel Time Integration with Multigrid".
4667:. Lecture Notes in Computational Science and Engineering. Vol. 103. Springer International Publishing. pp. 195–202.
5301:
2166:
4053:/P system JUGENE showed that PFASST could produce additional speedup beyond saturation of the spatial tree parallelisation.
3332:
75:
4473:. Contributions in Mathematical and Computational Sciences. Vol. 9 (1 ed.). Springer International Publishing.
3209:
is large enough to make Parareal stable, no speedup is possible. This also means that Parareal is typically unstable for
3119:
3214:
1109:. The coarse solver works the same way as the fine solver, propagating an initial value over a time interval of length
5296:
5286:
4951:
Ruprecht, D.; Krause, R. (2012-04-30). "Explicit parallel-in-time integration of a linear acoustic-advection system".
3922:
2596:
3189:
is always going to be smaller than one. So either the number of iterations is small and Parareal is unstable or, if
38:, Maday and Turinici. Since then, it has become one of the most widely studied parallel-in-time integration methods.
3808:{\displaystyle {\mathcal {K}}_{\Delta t}(u)={\mathcal {F}}_{\delta t}(P_{k}u)+{\mathcal {G}}_{\Delta t}((I-P_{k})u)}
3387:
4706:
Dai, X.; Maday, Y. (2013-01-01). "Stable Parareal in Time Method for First- and Second-Order Hyperbolic Systems".
4042:
67:
1031:
328:
119:
problems go back even further: the first paper proposing a parallel-in-time integration method appeared in 1964.
4351:
Gander, Martin J.; Vandewalle, Stefan (2007). "Analysis of the Parareal Time-Parallel Time-Integration Method".
3848:
3278:
3244:
3233:
There are multiple algorithms that are directly based or at least inspired by the original Parareal algorithm.
90:. Because Parareal computes the numerical solution for multiple time steps in parallel, it is categorized as a
55:
4198:
1426:
Next, run the fine solver on each of the time slices, in parallel, from the most up-to-date solution values:
2561:
393:
98:
like parallel Runge-Kutta or extrapolation methods, where independent stages can be computed in parallel or
83:
59:
4063:
5232:
5179:
4668:
4360:
4079:
3225:
too large. Different approaches exist to stabilise Parareal, one being Krylov-subspace enhanced Parareal.
448:
5215:
Gander, M.; GĂĽttel, S. (2013-01-01). "PARAEXP: A Parallel Integrator for Linear Initial-Value Problems".
4838:. MS&A - Modeling, Simulation and Applications. Springer International Publishing. pp. 187–214.
4633:. Lecture Notes in Computational Science and Engineering. Springer Berlin Heidelberg. pp. 449–456.
4128:
3623:
31:
2336:
1965:
iterations. Note that other stopping criterion do exist and have been successfully tested in Parareal.
1916:
3629:
5224:
5171:
4876:
4779:
4725:
4108:
3434:
556:
5237:
5184:
4673:
4365:
2294:
2270:
2246:
1135:
1088:
810:
666:
4082:
within Parareal. While limited to linear problems, it can produce almost optimal parallel speedup.
501:. Carrying out this discretisation we obtain a partitioned time interval consisting of time slices
35:
953:{\displaystyle U_{j+1}={\mathcal {F}}(t_{j},t_{j+1},U_{j}),\quad {\text{where}}\quad U_{0}=u^{0}.}
5144:
5021:
4978:
4960:
4900:
4803:
4749:
4715:
4577:
4527:
4416:
4328:
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method is used, because then runtimes vary depending on the number of iterations required by the
2128:
1079:
79:
63:
27:
23:
2243:
Typically, some form of Runge-Kutta method is chosen for both coarse and fine integrator, where
4442:
Kiehl, Martin (1994). "Parallel multiple shooting for the solution of initial value problems".
2536:
2509:
1209:
5250:
5197:
5134:
5123:
2012 International Conference for High Performance Computing, Networking, Storage and Analysis
5103:
5062:
5013:
4892:
4847:
4795:
4741:
4686:
4642:
4569:
4484:
4395:
4320:
4275:
2365:
1112:
630:
3241:
Early on it was recognised that for linear problems information generated by the fine method
1942:
777:
744:
5242:
5189:
5126:
5093:
5052:
5005:
4970:
4931:
4884:
4839:
4825:
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4474:
4451:
4370:
4312:
4267:
4232:
4171:
4116:
4034:
2565:
1996:
111:
4258:
Iserles, A.; NøRSETT, S. P. (1990-10-01). "On the Theory of Parallel Runge—Kutta Methods".
4005:
3978:
3821:
3658:
3598:
3165:
2481:
2454:
1162:
717:
690:
603:
4829:
4428:
4141:
3222:
1412:{\displaystyle U_{j+1}^{0}={\mathcal {G}}(t_{j},t_{j+1},U_{j}^{0}),\quad j=0,\ldots ,N-1.}
296:
3892:
2392:
302:
5228:
5175:
4880:
4783:
4729:
4112:
4155:
Pentland, Kamran; Tamborrino, Massimiliano; Samaddar, Debasmita; Appel, Lynton (2022).
3312:
3192:
3145:
3125:
3022:
2674:
2654:
2574:
2108:
1976:
1189:
1061:
1037:
1030:
smaller initial value problems (one on each time slice) in parallel. For example, in a
1013:
993:
973:
278:
4455:
4157:"Stochastic parareal: an application of probabilistic methods to time-parallelization"
4120:
2402:
1249:
504:
5280:
4807:
4183:
5025:
4982:
4974:
4904:
4753:
4581:
4332:
4244:
1544:
Now update the parareal solution values sequentially using the predictor-corrector:
45:
Illustration of the first iteration in Parareal (adapted from the original version).
5148:
4546:
4531:
3476:
4682:
5082:"Toward an efficient parallel in time method for partial differential equations"
4843:
2333:
Visualization of the Parareal algorithm. The coarse propagator here is labelled
1535:{\displaystyle {\mathcal {F}}(t_{j},t_{j+1},U_{j}^{k-1}),\quad j=0,\ldots ,N-1.}
4920:"Analysis of a Krylov subspace enhanced parareal algorithm for linear problems"
3586:{\displaystyle S_{k}:=\left\{U_{j}^{k'}:0\leq k'\leq k,j=0,\ldots ,N-1\right\}}
1904:{\displaystyle |U_{j}^{k}-U_{j}^{k-1}|<\varepsilon \quad \forall \ j\leq N,}
5098:
5081:
5057:
5040:
5009:
4613:
4596:
4479:
4389:
4316:
4271:
4236:
2571:
Under these two assumptions, the runtime for the fine method integrating over
5254:
5201:
5107:
5066:
5017:
4896:
4799:
4745:
4638:
4573:
4324:
4279:
807:. The goal is to calculate the solution (with high numerical accuracy) using
4050:
3218:
4833:
4565:
4936:
4919:
4522:
4505:
3122:. It typically only converges toward the very last iterations, that is as
2783:{\displaystyle c_{\text{parareal}}=(k+1)PN_{c}\tau _{c}+kN_{f}\tau _{f}.}
4175:
3118:
The vanilla version of Parareal has issues for problems with imaginary
3040:
2291:. If the initial value problem stems from the discretization of a PDE,
2022:
In the Parareal iteration, the computationally expensive evaluation of
78:
focussed on the spatial discretization, in view of the challenges from
5246:
5193:
5130:
4737:
4374:
1246:
Firstly, run the coarse solver serially over the entire time interval
4791:
2240:
means that the coarse correction has to be computed in serial order.
1055:
5271:
4888:
41:
4965:
4720:
4506:"Parallel methods for integrating ordinary differential equations"
4307:
3039:
has to be kept low to keep the second bound large. In particular,
86:
have been identified as a possible way to increase concurrency in
4002:
being the number of iterations of the serial SDC base method and
3595:
is defined and updated after every Parareal iteration. Denote as
3917:, the bound on parallel efficiency in the hybrid method becomes
5086:
Communications in Applied Mathematics and Computational Science
5045:
Communications in Applied Mathematics and Computational Science
4601:
Communications in Applied Mathematics and Computational Science
4295:
Communications in Applied Mathematics and Computational Science
4099:
Lions, Jacques-Louis; Maday, Yvon; Turinici, Gabriel (2015). .
3682:. Then, replace the coarse method with the improved integrator
62:
methods, some of the computations in Parareal can be performed
4835:
Reduced Order Methods for Modeling and Computational Reduction
4665:
Numerical Mathematics and Advanced Applications - ENUMATH 2013
299:
approach. We wish to solve this problem on a temporal mesh of
4041:(PFASST). Performance of PFASST has been studied for PEPC, a
3110:
that is by the inverse of the number of required iterations.
3855:
3760:
3721:
3692:
3339:
3285:
3251:
3101:{\displaystyle E_{p}={\frac {S_{p}}{P}}\leq {\frac {1}{k}},}
2300:
2276:
2252:
2172:
2031:
1711:
1642:
1579:
1437:
1320:
1141:
1094:
861:
816:
672:
2391:
Under some assumptions, a simple theoretical model for the
1285:
to calculate an approximate initial guess to the solution:
1132:, however it does so at much lower numerical accuracy than
70:
method. While historically most efforts to parallelize the
4869:
International Journal for Numerical Methods in Engineering
4772:
International Journal for Numerical Methods in Engineering
132:
The goal is to solve an initial value problem of the form
3275:
can be used to improve the accuracy of the coarse method
2098:{\displaystyle {\mathcal {F}}(t_{j},t_{j+1},U_{j}^{k-1})}
1159:
Henceforth, we will denote the Parareal solution at time
5041:"A hybrid parareal spectral deferred corrections method"
4597:"A Hybrid Parareal Spectral Deferred Corrections Method"
2267:
might be of lower order and use a larger time step than
2233:{\displaystyle {\mathcal {G}}(t_{j},t_{j+1},U_{j}^{k})}
600:
The objective is to calculate numerical approximations
4008:
3981:
3925:
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4223:
Burrage, Kevin (1997). "Parallel methods for ODEs".
4039:
parallel full approximation scheme in space and time
3377:{\displaystyle {\mathcal {F}}_{\delta t}(U_{j}^{k})}
4199:
Applied Mathematics Research for Exascale Computing
4101:Comptes Rendus de l'Académie des Sciences, Série I
4021:
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4547:"Convergence of Parareal with spatial coarsening"
4066:library implementing MGRIT is being developed by
4049:. Simulations using all 262,144 cores on the IBM
3818:As the number of iterations increases, the space
2125:processing units. By contrast, the dependency of
4554:Proceedings in Applied Mathematics and Mechanics
3966:{\displaystyle E_{p}\leq {\frac {k_{s}}{k_{p}}}}
2929:
2642:{\displaystyle c_{\text{fine}}=PN_{f}\tau _{f}.}
94:method. This is in contrast to approaches using
5080:Emmett, Matthew; Minion, Michael (2012-03-28).
1054:would be the number of processes, while in an
4045:tree code based particle solver developed at
3884:Hybrid Parareal spectral deferred corrections
3424:{\displaystyle u_{j}^{k}\in \mathbb {R} ^{d}}
8:
4346:
4344:
4342:
66:and Parareal is therefore one example of a
383:{\displaystyle (t_{0},t_{1},\ldots ,t_{N})}
4471:50 years of Time Parallel Time Integration
16:Parallel algorithm from numerical analysis
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3899:
3894:
3872:{\displaystyle {\mathcal {K}}_{\Delta t}}
3860:
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3302:{\displaystyle {\mathcal {G}}_{\Delta t}}
3290:
3284:
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3280:
3268:{\displaystyle {\mathcal {F}}_{\delta t}}
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1139:
1137:
1114:
1093:
1092:
1090:
1063:
1039:
1015:
995:
975:
941:
928:
918:
905:
886:
873:
860:
859:
844:
838:
815:
814:
812:
785:
779:
752:
746:
725:
719:
698:
692:
671:
670:
668:
644:
632:
611:
605:
558:
528:
515:
506:
483:
474:
450:
420:
401:
395:
371:
352:
339:
330:
304:
280:
254:
238:
222:
206:
187:
154:
144:
141:
139:
2362:whereas the fine propagator is labelled
2320:
40:
4091:
4424:
4414:
4137:
4126:
4068:Lawrence Livermore National Laboratory
3845:will grow and the modified propagator
3221:coefficient becomes too small and the
438:{\displaystyle t_{j+1}=t_{j}+\Delta T}
4819:
4817:
4765:
4763:
4225:Advances in Computational Mathematics
4037:. This led to the development of the
3114:Instability for imaginary eigenvalues
7:
5217:SIAM Journal on Scientific Computing
5164:SIAM Journal on Scientific Computing
4708:SIAM Journal on Scientific Computing
4353:SIAM Journal on Scientific Computing
4164:SIAM Journal on Scientific Computing
2478:steps of the fine integrator and of
2399:First, assume that every time slice
687:, which propagates an initial value
494:{\displaystyle \Delta T=(T-t_{0})/N}
50:Parallel-in-time integration methods
4631:Stability of the Parareal Algorithm
4057:Multigrid reduction in time (MGRIT)
3861:
3766:
3698:
3291:
1883:
1116:
452:
429:
155:
145:
110:Parareal can be derived as both a
102:methods like waveform relaxation.
14:
4260:IMA Journal of Numerical Analysis
4204:(Report). US Department of Energy
3475:. Based on this information, the
3237:Krylov-subspace enhanced Parareal
2355:{\displaystyle {\bar {\varphi }}}
1932:{\displaystyle \varepsilon >0}
990:processors. The aim to is to use
5292:Numerical differential equations
3648:{\displaystyle \mathbb {R} ^{d}}
2671:processing units and performing
2105:can be performed in parallel on
1078:would be equal to the number of
4975:10.1016/j.compfluid.2012.02.015
4545:Ruprecht, Daniel (2014-12-01).
3468:{\displaystyle j=0,\ldots ,N-1}
2591:time slices can be modelled as
1882:
1778:
1504:
1381:
923:
917:
590:{\displaystyle j=0,\ldots ,N-1}
227:
221:
192:
186:
34:. It was introduced in 2001 by
5039:Minion, Michael (2011-01-05).
4918:Gander, M.; Petcu, M. (2008).
3802:
3796:
3777:
3774:
3751:
3735:
3712:
3706:
3371:
3353:
3041:Parareal's parallel efficiency
2859:
2847:
2725:
2713:
2651:The runtime of Parareal using
2438:
2406:
2346:
2308:{\displaystyle {\mathcal {G}}}
2284:{\displaystyle {\mathcal {F}}}
2260:{\displaystyle {\mathcal {G}}}
2227:
2177:
2092:
2036:
1872:
1828:
1772:
1716:
1703:
1647:
1634:
1584:
1498:
1442:
1375:
1325:
1272:
1253:
1149:{\displaystyle {\mathcal {F}}}
1102:{\displaystyle {\mathcal {G}}}
911:
866:
824:{\displaystyle {\mathcal {F}}}
680:{\displaystyle {\mathcal {F}}}
650:
637:
540:
508:
480:
461:
377:
332:
244:
231:
218:
199:
183:
171:
76:partial differential equations
1:
4456:10.1016/S0167-8191(06)80013-X
4121:10.1016/S0764-4442(00)01793-6
4047:Juelich Supercomputing Centre
96:parallelism across the method
30:and used for the solution of
4683:10.1007/978-3-319-10705-9_19
4391:Parabolic multi-grid methods
4388:Hackbusch, Wolfgang (1985).
2792:Speedup of Parareal then is
68:parallel-in-time integration
4844:10.1007/978-3-319-02090-7_7
4595:Minion, Michael L. (2010).
2156:{\displaystyle U_{j+1}^{k}}
5318:
4469:Gander, Martin J. (2015).
100:parallel across the system
5099:10.2140/camcos.2012.7.105
5058:10.2140/camcos.2010.5.265
4998:BIT Numerical Mathematics
4614:10.2140/camcos.2010.5.265
4510:Communications of the ACM
4504:Nievergelt, JĂĽrg (1964).
4480:10.1007/978-3-319-23321-5
4317:10.2140/camcos.2014.9.175
4031:full approximation scheme
2553:{\displaystyle \tau _{c}}
2526:{\displaystyle \tau _{f}}
1231:{\displaystyle U_{j}^{k}}
92:parallel across the steps
4639:10.1007/3-540-26825-1_46
2375:{\displaystyle \varphi }
1125:{\displaystyle \Delta T}
656:{\displaystyle u(t_{j})}
5010:10.1023/A:1022338906936
4272:10.1093/imanum/10.4.463
4237:10.1023/A:1018997130884
4080:exponential integrators
3384:is computed for values
3215:Navier–Stokes equations
1958:{\displaystyle k\leq N}
800:{\displaystyle t_{j+1}}
767:{\displaystyle U_{j+1}}
84:temporal discretization
82:, parallel methods for
4953:Computers & Fluids
4566:10.1002/pamm.201410490
4023:
3996:
3967:
3911:
3873:
3839:
3809:
3676:
3649:
3616:
3587:
3469:
3425:
3378:
3323:
3303:
3269:
3203:
3183:
3156:
3136:
3102:
3033:
3010:
2784:
2685:
2665:
2643:
2585:
2554:
2527:
2499:
2472:
2445:
2383:
2376:
2356:
2309:
2285:
2261:
2234:
2157:
2119:
2099:
2013:
2012:{\displaystyle k\ll N}
1987:
1959:
1933:
1905:
1810:
1536:
1413:
1279:
1232:
1200:
1180:
1150:
1126:
1103:
1072:
1048:
1024:
1004:
984:
954:
825:
801:
768:
735:
708:
681:
657:
627:to the exact solution
621:
591:
547:
495:
439:
384:
325:equally spaced points
319:
289:
267:
46:
32:initial value problems
5302:Computational science
4523:10.1145/355588.365137
4033:as used in nonlinear
4024:
4022:{\displaystyle k_{p}}
3997:
3995:{\displaystyle k_{s}}
3968:
3912:
3874:
3840:
3838:{\displaystyle S_{k}}
3810:
3677:
3675:{\displaystyle S_{k}}
3650:
3624:orthogonal projection
3617:
3615:{\displaystyle P_{k}}
3588:
3470:
3426:
3379:
3324:
3304:
3270:
3204:
3184:
3182:{\displaystyle S_{p}}
3157:
3137:
3103:
3034:
3011:
2785:
2686:
2666:
2644:
2586:
2555:
2528:
2500:
2498:{\displaystyle N_{c}}
2473:
2471:{\displaystyle N_{f}}
2446:
2377:
2357:
2332:
2310:
2286:
2262:
2235:
2158:
2120:
2100:
2014:
1988:
1960:
1934:
1906:
1811:
1537:
1422:Subsequent Iterations
1414:
1280:
1233:
1201:
1181:
1179:{\displaystyle t_{j}}
1151:
1127:
1104:
1073:
1049:
1025:
1005:
985:
955:
831:such that we obtain
826:
802:
769:
736:
734:{\displaystyle t_{j}}
709:
707:{\displaystyle U_{j}}
682:
658:
622:
620:{\displaystyle U_{j}}
592:
548:
496:
440:
385:
320:
290:
268:
114:in time method or as
44:
5272:parallel-in-time.org
4394:. pp. 189–197.
4006:
3979:
3923:
3893:
3849:
3822:
3686:
3659:
3630:
3599:
3484:
3435:
3388:
3333:
3313:
3279:
3245:
3193:
3166:
3146:
3126:
3049:
3023:
2798:
2697:
2675:
2655:
2597:
2575:
2537:
2510:
2482:
2455:
2451:consists of exactly
2403:
2366:
2337:
2295:
2271:
2247:
2167:
2129:
2109:
2026:
1997:
1977:
1943:
1917:
1824:
1550:
1432:
1291:
1250:
1210:
1190:
1163:
1136:
1113:
1089:
1062:
1038:
1014:
1010:processors to solve
994:
974:
837:
811:
778:
745:
741:to a terminal value
718:
691:
667:
631:
604:
557:
505:
449:
394:
329:
303:
279:
275:The right hand side
138:
54:In contrast to e.g.
5229:2013SJSC...35C.123G
5176:2014SJSC...36C.635F
4937:10.1051/proc:082508
4881:2003IJNME..58.1397F
4784:2006IJNME..67..697F
4730:2013SJSC...35A..52D
4113:2001CRASM.332..661L
3910:{\displaystyle 1/k}
3524:
3405:
3370:
2226:
2152:
2091:
1913:and some tolerance
1870:
1846:
1771:
1702:
1633:
1573:
1497:
1374:
1314:
1227:
318:{\displaystyle N+1}
5297:Parallel computing
5287:Numerical analysis
4924:ESAIM: Proceedings
4444:Parallel Computing
4176:10.1137/21M1414231
4019:
3992:
3963:
3907:
3869:
3835:
3805:
3672:
3645:
3612:
3583:
3505:
3465:
3421:
3391:
3374:
3356:
3319:
3299:
3265:
3199:
3179:
3162:, and the speedup
3152:
3132:
3098:
3029:
3006:
2780:
2681:
2661:
2639:
2581:
2550:
2523:
2495:
2468:
2441:
2384:
2372:
2352:
2305:
2281:
2257:
2230:
2212:
2153:
2132:
2115:
2095:
2071:
2009:
1983:
1955:
1929:
1901:
1850:
1832:
1806:
1751:
1682:
1619:
1553:
1532:
1477:
1409:
1360:
1294:
1275:
1228:
1213:
1196:
1176:
1146:
1122:
1099:
1068:
1044:
1020:
1000:
980:
950:
821:
797:
764:
731:
704:
677:
653:
617:
587:
543:
491:
435:
380:
315:
285:
263:
88:numerical software
80:exascale computing
72:numerical solution
47:
28:numerical analysis
24:parallel algorithm
5247:10.1137/110856137
5194:10.1137/130944230
5140:978-1-4673-0805-2
5131:10.1109/SC.2012.6
5125:. pp. 1–11.
4826:Quarteroni, Alfio
4738:10.1137/110861002
4490:978-3-319-23321-5
4375:10.1137/05064607X
3961:
3322:{\displaystyle k}
3202:{\displaystyle k}
3155:{\displaystyle N}
3135:{\displaystyle k}
3093:
3080:
3032:{\displaystyle k}
2996:
2983:
2924:
2921:
2908:
2884:
2836:
2833:
2823:
2707:
2684:{\displaystyle k}
2664:{\displaystyle P}
2607:
2584:{\displaystyle P}
2349:
2330:
2118:{\displaystyle N}
1986:{\displaystyle N}
1888:
1199:{\displaystyle k}
1071:{\displaystyle N}
1047:{\displaystyle N}
1023:{\displaystyle N}
1003:{\displaystyle N}
983:{\displaystyle N}
921:
288:{\displaystyle f}
225:
190:
163:
116:multiple shooting
5309:
5259:
5258:
5240:
5223:(2): C123–C142.
5212:
5206:
5205:
5187:
5170:(6): C635–C661.
5159:
5153:
5152:
5118:
5112:
5111:
5101:
5077:
5071:
5070:
5060:
5036:
5030:
5029:
4993:
4987:
4986:
4968:
4948:
4942:
4941:
4939:
4915:
4909:
4908:
4875:(9): 1397–1434.
4864:
4858:
4857:
4830:Rozza, Gianluigi
4821:
4812:
4811:
4792:10.1002/nme.1653
4767:
4758:
4757:
4723:
4703:
4697:
4696:
4676:
4659:
4653:
4652:
4625:
4619:
4618:
4616:
4592:
4586:
4585:
4560:(1): 1031–1034.
4551:
4542:
4536:
4535:
4525:
4501:
4495:
4494:
4482:
4466:
4460:
4459:
4439:
4433:
4432:
4426:
4422:
4420:
4412:
4410:
4408:
4385:
4379:
4378:
4368:
4348:
4337:
4336:
4310:
4290:
4284:
4283:
4255:
4249:
4248:
4220:
4214:
4213:
4211:
4209:
4203:
4194:
4188:
4187:
4161:
4152:
4146:
4145:
4139:
4134:
4132:
4124:
4096:
4028:
4026:
4025:
4020:
4018:
4017:
4001:
3999:
3998:
3993:
3991:
3990:
3972:
3970:
3969:
3964:
3962:
3960:
3959:
3950:
3949:
3940:
3935:
3934:
3916:
3914:
3913:
3908:
3903:
3878:
3876:
3875:
3870:
3868:
3867:
3859:
3858:
3844:
3842:
3841:
3836:
3834:
3833:
3814:
3812:
3811:
3806:
3795:
3794:
3773:
3772:
3764:
3763:
3747:
3746:
3734:
3733:
3725:
3724:
3705:
3704:
3696:
3695:
3681:
3679:
3678:
3673:
3671:
3670:
3654:
3652:
3651:
3646:
3644:
3643:
3638:
3621:
3619:
3618:
3613:
3611:
3610:
3592:
3590:
3589:
3584:
3582:
3578:
3541:
3523:
3522:
3513:
3496:
3495:
3474:
3472:
3471:
3466:
3430:
3428:
3427:
3422:
3420:
3419:
3414:
3404:
3399:
3383:
3381:
3380:
3375:
3369:
3364:
3352:
3351:
3343:
3342:
3328:
3326:
3325:
3320:
3308:
3306:
3305:
3300:
3298:
3297:
3289:
3288:
3274:
3272:
3271:
3266:
3264:
3263:
3255:
3254:
3208:
3206:
3205:
3200:
3188:
3186:
3185:
3180:
3178:
3177:
3161:
3159:
3158:
3153:
3141:
3139:
3138:
3133:
3107:
3105:
3104:
3099:
3094:
3086:
3081:
3076:
3075:
3066:
3061:
3060:
3038:
3036:
3035:
3030:
3015:
3013:
3012:
3007:
3002:
2998:
2997:
2989:
2984:
2982:
2981:
2980:
2971:
2970:
2960:
2959:
2958:
2949:
2948:
2938:
2925:
2923:
2922:
2914:
2909:
2907:
2906:
2897:
2896:
2887:
2885:
2883:
2882:
2873:
2872:
2863:
2842:
2837:
2835:
2834:
2831:
2825:
2824:
2821:
2815:
2810:
2809:
2789:
2787:
2786:
2781:
2776:
2775:
2766:
2765:
2750:
2749:
2740:
2739:
2709:
2708:
2705:
2690:
2688:
2687:
2682:
2670:
2668:
2667:
2662:
2648:
2646:
2645:
2640:
2635:
2634:
2625:
2624:
2609:
2608:
2605:
2590:
2588:
2587:
2582:
2566:iterative solver
2559:
2557:
2556:
2551:
2549:
2548:
2532:
2530:
2529:
2524:
2522:
2521:
2504:
2502:
2501:
2496:
2494:
2493:
2477:
2475:
2474:
2469:
2467:
2466:
2450:
2448:
2447:
2444:{\displaystyle }
2442:
2437:
2436:
2418:
2417:
2381:
2379:
2378:
2373:
2361:
2359:
2358:
2353:
2351:
2350:
2342:
2331:
2314:
2312:
2311:
2306:
2304:
2303:
2290:
2288:
2287:
2282:
2280:
2279:
2266:
2264:
2263:
2258:
2256:
2255:
2239:
2237:
2236:
2231:
2225:
2220:
2208:
2207:
2189:
2188:
2176:
2175:
2162:
2160:
2159:
2154:
2151:
2146:
2124:
2122:
2121:
2116:
2104:
2102:
2101:
2096:
2090:
2079:
2067:
2066:
2048:
2047:
2035:
2034:
2018:
2016:
2015:
2010:
1992:
1990:
1989:
1984:
1964:
1962:
1961:
1956:
1938:
1936:
1935:
1930:
1910:
1908:
1907:
1902:
1886:
1875:
1869:
1858:
1845:
1840:
1831:
1815:
1813:
1812:
1807:
1770:
1759:
1747:
1746:
1728:
1727:
1715:
1714:
1701:
1690:
1678:
1677:
1659:
1658:
1646:
1645:
1632:
1627:
1615:
1614:
1596:
1595:
1583:
1582:
1572:
1567:
1541:
1539:
1538:
1533:
1496:
1485:
1473:
1472:
1454:
1453:
1441:
1440:
1418:
1416:
1415:
1410:
1373:
1368:
1356:
1355:
1337:
1336:
1324:
1323:
1313:
1308:
1284:
1282:
1281:
1278:{\displaystyle }
1276:
1265:
1264:
1242:Zeroth Iteration
1237:
1235:
1234:
1229:
1226:
1221:
1205:
1203:
1202:
1197:
1185:
1183:
1182:
1177:
1175:
1174:
1155:
1153:
1152:
1147:
1145:
1144:
1131:
1129:
1128:
1123:
1108:
1106:
1105:
1100:
1098:
1097:
1077:
1075:
1074:
1069:
1053:
1051:
1050:
1045:
1029:
1027:
1026:
1021:
1009:
1007:
1006:
1001:
989:
987:
986:
981:
959:
957:
956:
951:
946:
945:
933:
932:
922:
919:
910:
909:
897:
896:
878:
877:
865:
864:
855:
854:
830:
828:
827:
822:
820:
819:
806:
804:
803:
798:
796:
795:
773:
771:
770:
765:
763:
762:
740:
738:
737:
732:
730:
729:
713:
711:
710:
705:
703:
702:
686:
684:
683:
678:
676:
675:
662:
660:
659:
654:
649:
648:
626:
624:
623:
618:
616:
615:
596:
594:
593:
588:
552:
550:
549:
546:{\displaystyle }
544:
539:
538:
520:
519:
500:
498:
497:
492:
487:
479:
478:
444:
442:
441:
436:
425:
424:
412:
411:
389:
387:
386:
381:
376:
375:
357:
356:
344:
343:
324:
322:
321:
316:
294:
292:
291:
286:
272:
270:
269:
264:
259:
258:
243:
242:
226:
223:
211:
210:
191:
188:
164:
162:
158:
152:
148:
142:
112:multigrid method
5317:
5316:
5312:
5311:
5310:
5308:
5307:
5306:
5277:
5276:
5268:
5263:
5262:
5238:10.1.1.800.5938
5214:
5213:
5209:
5185:10.1.1.701.2603
5161:
5160:
5156:
5141:
5120:
5119:
5115:
5079:
5078:
5074:
5038:
5037:
5033:
4995:
4994:
4990:
4950:
4949:
4945:
4917:
4916:
4912:
4889:10.1002/nme.860
4866:
4865:
4861:
4854:
4823:
4822:
4815:
4769:
4768:
4761:
4705:
4704:
4700:
4693:
4674:10.1.1.764.6242
4661:
4660:
4656:
4649:
4627:
4626:
4622:
4594:
4593:
4589:
4549:
4544:
4543:
4539:
4516:(12): 731–733.
4503:
4502:
4498:
4491:
4468:
4467:
4463:
4441:
4440:
4436:
4423:
4413:
4406:
4404:
4402:
4387:
4386:
4382:
4366:10.1.1.154.6042
4350:
4349:
4340:
4292:
4291:
4287:
4257:
4256:
4252:
4222:
4221:
4217:
4207:
4205:
4201:
4196:
4195:
4191:
4170:(3): S82–S102.
4159:
4154:
4153:
4149:
4135:
4125:
4098:
4097:
4093:
4088:
4076:
4059:
4009:
4004:
4003:
3982:
3977:
3976:
3951:
3941:
3926:
3921:
3920:
3891:
3890:
3886:
3852:
3847:
3846:
3825:
3820:
3819:
3786:
3757:
3738:
3718:
3689:
3684:
3683:
3662:
3657:
3656:
3633:
3628:
3627:
3602:
3597:
3596:
3534:
3515:
3504:
3500:
3487:
3482:
3481:
3433:
3432:
3409:
3386:
3385:
3336:
3331:
3330:
3311:
3310:
3282:
3277:
3276:
3248:
3243:
3242:
3239:
3231:
3223:Reynolds number
3191:
3190:
3169:
3164:
3163:
3144:
3143:
3124:
3123:
3116:
3067:
3052:
3047:
3046:
3021:
3020:
2972:
2962:
2961:
2950:
2940:
2939:
2936:
2932:
2898:
2888:
2874:
2864:
2846:
2826:
2816:
2801:
2796:
2795:
2767:
2757:
2741:
2731:
2700:
2695:
2694:
2673:
2672:
2653:
2652:
2626:
2616:
2600:
2595:
2594:
2573:
2572:
2540:
2535:
2534:
2513:
2508:
2507:
2485:
2480:
2479:
2458:
2453:
2452:
2422:
2409:
2401:
2400:
2389:
2364:
2363:
2335:
2334:
2321:
2318:
2293:
2292:
2269:
2268:
2245:
2244:
2193:
2180:
2165:
2164:
2127:
2126:
2107:
2106:
2052:
2039:
2024:
2023:
1995:
1994:
1975:
1974:
1971:
1941:
1940:
1915:
1914:
1822:
1821:
1732:
1719:
1663:
1650:
1600:
1587:
1548:
1547:
1458:
1445:
1430:
1429:
1424:
1341:
1328:
1289:
1288:
1256:
1248:
1247:
1244:
1208:
1207:
1188:
1187:
1166:
1161:
1160:
1134:
1133:
1111:
1110:
1087:
1086:
1060:
1059:
1036:
1035:
1012:
1011:
992:
991:
972:
971:
968:
937:
924:
901:
882:
869:
840:
835:
834:
809:
808:
781:
776:
775:
748:
743:
742:
721:
716:
715:
694:
689:
688:
665:
664:
640:
629:
628:
607:
602:
601:
555:
554:
524:
511:
503:
502:
470:
447:
446:
416:
397:
392:
391:
367:
348:
335:
327:
326:
301:
300:
297:method of lines
277:
276:
250:
234:
202:
153:
143:
136:
135:
130:
125:
108:
52:
17:
12:
11:
5:
5315:
5313:
5305:
5304:
5299:
5294:
5289:
5279:
5278:
5275:
5274:
5267:
5266:External links
5264:
5261:
5260:
5207:
5154:
5139:
5113:
5092:(1): 105–132.
5072:
5051:(2): 265–301.
5031:
5004:(2): 241–266.
4988:
4943:
4910:
4859:
4852:
4813:
4778:(5): 697–724.
4759:
4714:(1): A52–A78.
4698:
4691:
4654:
4647:
4620:
4607:(2): 265–301.
4587:
4537:
4496:
4489:
4461:
4450:(3): 275–295.
4434:
4425:|journal=
4400:
4380:
4359:(2): 556–578.
4338:
4301:(2): 175–200.
4285:
4266:(4): 463–488.
4250:
4215:
4189:
4147:
4107:(7): 661–668.
4090:
4089:
4087:
4084:
4075:
4072:
4058:
4055:
4016:
4012:
3989:
3985:
3958:
3954:
3948:
3944:
3938:
3933:
3929:
3906:
3902:
3898:
3885:
3882:
3866:
3863:
3857:
3832:
3828:
3804:
3801:
3798:
3793:
3789:
3785:
3782:
3779:
3776:
3771:
3768:
3762:
3756:
3753:
3750:
3745:
3741:
3737:
3732:
3729:
3723:
3717:
3714:
3711:
3708:
3703:
3700:
3694:
3669:
3665:
3642:
3637:
3609:
3605:
3581:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3540:
3537:
3533:
3530:
3527:
3521:
3518:
3512:
3508:
3503:
3499:
3494:
3490:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3418:
3413:
3408:
3403:
3398:
3394:
3373:
3368:
3363:
3359:
3355:
3350:
3347:
3341:
3318:
3296:
3293:
3287:
3262:
3259:
3253:
3238:
3235:
3230:
3227:
3198:
3176:
3172:
3151:
3131:
3115:
3112:
3097:
3092:
3089:
3084:
3079:
3074:
3070:
3064:
3059:
3055:
3043:is bounded by
3028:
3005:
3001:
2995:
2992:
2987:
2979:
2975:
2969:
2965:
2957:
2953:
2947:
2943:
2935:
2931:
2928:
2920:
2917:
2912:
2905:
2901:
2895:
2891:
2881:
2877:
2871:
2867:
2861:
2858:
2855:
2852:
2849:
2845:
2840:
2829:
2819:
2813:
2808:
2804:
2779:
2774:
2770:
2764:
2760:
2756:
2753:
2748:
2744:
2738:
2734:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2703:
2691:iterations is
2680:
2660:
2638:
2633:
2629:
2623:
2619:
2615:
2612:
2603:
2580:
2547:
2543:
2520:
2516:
2492:
2488:
2465:
2461:
2440:
2435:
2432:
2429:
2425:
2421:
2416:
2412:
2408:
2388:
2385:
2371:
2348:
2345:
2302:
2278:
2254:
2229:
2224:
2219:
2215:
2211:
2206:
2203:
2200:
2196:
2192:
2187:
2183:
2179:
2174:
2150:
2145:
2142:
2139:
2135:
2114:
2094:
2089:
2086:
2083:
2078:
2074:
2070:
2065:
2062:
2059:
2055:
2051:
2046:
2042:
2038:
2033:
2008:
2005:
2002:
1982:
1970:
1967:
1954:
1951:
1948:
1928:
1925:
1922:
1900:
1897:
1894:
1891:
1885:
1881:
1878:
1874:
1868:
1865:
1862:
1857:
1853:
1849:
1844:
1839:
1835:
1830:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1777:
1774:
1769:
1766:
1763:
1758:
1754:
1750:
1745:
1742:
1739:
1735:
1731:
1726:
1722:
1718:
1713:
1708:
1705:
1700:
1697:
1694:
1689:
1685:
1681:
1676:
1673:
1670:
1666:
1662:
1657:
1653:
1649:
1644:
1639:
1636:
1631:
1626:
1622:
1618:
1613:
1610:
1607:
1603:
1599:
1594:
1590:
1586:
1581:
1576:
1571:
1566:
1563:
1560:
1556:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1503:
1500:
1495:
1492:
1489:
1484:
1480:
1476:
1471:
1468:
1465:
1461:
1457:
1452:
1448:
1444:
1439:
1423:
1420:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1380:
1377:
1372:
1367:
1363:
1359:
1354:
1351:
1348:
1344:
1340:
1335:
1331:
1327:
1322:
1317:
1312:
1307:
1304:
1301:
1297:
1274:
1271:
1268:
1263:
1259:
1255:
1243:
1240:
1225:
1220:
1216:
1195:
1186:and iteration
1173:
1169:
1143:
1121:
1118:
1096:
1067:
1043:
1019:
999:
979:
967:
964:
949:
944:
940:
936:
931:
927:
916:
913:
908:
904:
900:
895:
892:
889:
885:
881:
876:
872:
868:
863:
858:
853:
850:
847:
843:
818:
794:
791:
788:
784:
761:
758:
755:
751:
728:
724:
701:
697:
674:
652:
647:
643:
639:
636:
614:
610:
586:
583:
580:
577:
574:
571:
568:
565:
562:
542:
537:
534:
531:
527:
523:
518:
514:
510:
490:
486:
482:
477:
473:
469:
466:
463:
460:
457:
454:
434:
431:
428:
423:
419:
415:
410:
407:
404:
400:
379:
374:
370:
366:
363:
360:
355:
351:
347:
342:
338:
334:
314:
311:
308:
284:
262:
257:
253:
249:
246:
241:
237:
233:
230:
220:
217:
214:
209:
205:
201:
198:
195:
185:
182:
179:
176:
173:
170:
167:
161:
157:
151:
147:
129:
126:
124:
121:
107:
104:
51:
48:
15:
13:
10:
9:
6:
4:
3:
2:
5314:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5284:
5282:
5273:
5270:
5269:
5265:
5256:
5252:
5248:
5244:
5239:
5234:
5230:
5226:
5222:
5218:
5211:
5208:
5203:
5199:
5195:
5191:
5186:
5181:
5177:
5173:
5169:
5165:
5158:
5155:
5150:
5146:
5142:
5136:
5132:
5128:
5124:
5117:
5114:
5109:
5105:
5100:
5095:
5091:
5087:
5083:
5076:
5073:
5068:
5064:
5059:
5054:
5050:
5046:
5042:
5035:
5032:
5027:
5023:
5019:
5015:
5011:
5007:
5003:
4999:
4992:
4989:
4984:
4980:
4976:
4972:
4967:
4962:
4958:
4954:
4947:
4944:
4938:
4933:
4929:
4925:
4921:
4914:
4911:
4906:
4902:
4898:
4894:
4890:
4886:
4882:
4878:
4874:
4870:
4863:
4860:
4855:
4853:9783319020891
4849:
4845:
4841:
4837:
4836:
4831:
4827:
4820:
4818:
4814:
4809:
4805:
4801:
4797:
4793:
4789:
4785:
4781:
4777:
4773:
4766:
4764:
4760:
4755:
4751:
4747:
4743:
4739:
4735:
4731:
4727:
4722:
4717:
4713:
4709:
4702:
4699:
4694:
4692:9783319107042
4688:
4684:
4680:
4675:
4670:
4666:
4658:
4655:
4650:
4648:9783540225232
4644:
4640:
4636:
4632:
4624:
4621:
4615:
4610:
4606:
4602:
4598:
4591:
4588:
4583:
4579:
4575:
4571:
4567:
4563:
4559:
4555:
4548:
4541:
4538:
4533:
4529:
4524:
4519:
4515:
4511:
4507:
4500:
4497:
4492:
4486:
4481:
4476:
4472:
4465:
4462:
4457:
4453:
4449:
4445:
4438:
4435:
4430:
4418:
4403:
4401:9780444875976
4397:
4393:
4392:
4384:
4381:
4376:
4372:
4367:
4362:
4358:
4354:
4347:
4345:
4343:
4339:
4334:
4330:
4326:
4322:
4318:
4314:
4309:
4304:
4300:
4296:
4289:
4286:
4281:
4277:
4273:
4269:
4265:
4261:
4254:
4251:
4246:
4242:
4238:
4234:
4231:(1–2): 1–31.
4230:
4226:
4219:
4216:
4200:
4193:
4190:
4185:
4181:
4177:
4173:
4169:
4165:
4158:
4151:
4148:
4143:
4130:
4122:
4118:
4114:
4110:
4106:
4102:
4095:
4092:
4085:
4083:
4081:
4078:ParaExp uses
4073:
4071:
4069:
4065:
4056:
4054:
4052:
4048:
4044:
4040:
4036:
4032:
4014:
4010:
3987:
3983:
3973:
3956:
3952:
3946:
3942:
3936:
3931:
3927:
3918:
3904:
3900:
3896:
3883:
3881:
3864:
3830:
3826:
3816:
3799:
3791:
3787:
3783:
3780:
3769:
3754:
3748:
3743:
3739:
3730:
3727:
3715:
3709:
3701:
3667:
3663:
3640:
3625:
3607:
3603:
3593:
3579:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3538:
3535:
3531:
3528:
3525:
3519:
3516:
3510:
3506:
3501:
3497:
3492:
3488:
3479:
3478:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3416:
3406:
3401:
3396:
3392:
3366:
3361:
3357:
3348:
3345:
3316:
3294:
3260:
3257:
3236:
3234:
3228:
3226:
3224:
3220:
3216:
3212:
3196:
3174:
3170:
3149:
3129:
3121:
3113:
3111:
3108:
3095:
3090:
3087:
3082:
3077:
3072:
3068:
3062:
3057:
3053:
3044:
3042:
3026:
3016:
3003:
2999:
2993:
2990:
2985:
2977:
2973:
2967:
2963:
2955:
2951:
2945:
2941:
2933:
2926:
2918:
2915:
2910:
2903:
2899:
2893:
2889:
2879:
2875:
2869:
2865:
2856:
2853:
2850:
2843:
2838:
2827:
2817:
2811:
2806:
2802:
2793:
2790:
2777:
2772:
2768:
2762:
2758:
2754:
2751:
2746:
2742:
2736:
2732:
2728:
2722:
2719:
2716:
2710:
2701:
2692:
2678:
2658:
2649:
2636:
2631:
2627:
2621:
2617:
2613:
2610:
2601:
2592:
2578:
2569:
2567:
2563:
2545:
2541:
2518:
2514:
2490:
2486:
2463:
2459:
2433:
2430:
2427:
2423:
2419:
2414:
2410:
2397:
2394:
2386:
2369:
2343:
2319:
2316:
2241:
2222:
2217:
2213:
2209:
2204:
2201:
2198:
2194:
2190:
2185:
2181:
2148:
2143:
2140:
2137:
2133:
2112:
2087:
2084:
2081:
2076:
2072:
2068:
2063:
2060:
2057:
2053:
2049:
2044:
2040:
2020:
2006:
2003:
2000:
1980:
1968:
1966:
1952:
1949:
1946:
1926:
1923:
1920:
1911:
1898:
1895:
1892:
1889:
1879:
1876:
1866:
1863:
1860:
1855:
1851:
1847:
1842:
1837:
1833:
1819:
1816:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1775:
1767:
1764:
1761:
1756:
1752:
1748:
1743:
1740:
1737:
1733:
1729:
1724:
1720:
1706:
1698:
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4407:August 29,
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