We propose and analyze a numerical method for time-dependent linear Schrödinger equations with uncertain parameters in both the potential and the initial data. The random parameters are discretized by stochastic collocation on a sparse grid, and the sample solutions in the nodes are approximated with the Strang splitting method. The computational work is reduced by a multi-level strategy, i.e. by combining information obtained from sample solutions computed on different refinement levels of the discretization. We prove new error bounds for the time discretization which take the finite regularity in the stochastic variable into account, and which are crucial to obtain convergence of the multi-level approach. The predicted cost savings of the multi-level stochastic collocation method are verified by numerical examples.
CHARLIE Right, haha. 10. 11. DAVID Well, mom was just wondering, you want to invite a little friend tomorrow? Keep you company? CHARLIE I don't need a "little friend," I'm not like 5! I'll be fine! DAVID Ah, okay then. Well, see you bright and early in the morning! CHARLIE Goodnight. David goes to leave, but, thinking better of it turns back. DAVID Look. I know you're not exactly excited for tomorrow, but I'm glad we'll be sharing it with you. Anyways, that's all. Goodnight! David exits, leaving Charlie looking slightly confused. INT. CAR -DAY 5 5 Charlie, in a suit, sits in back. He pulls at his tie, uncomfortable. His parents sit up front, just as formal. REBECCA You look great, kiddo. Excited? CHARLIE Excited? I don't even know whose bar mitzvah this is! DAVID Bat mitzvah. Bar is for boys. REBECCA You don't remember Emma? CHARLIE Not a bit. DAVID Janie and Ivan's kid! REBECCA You think he's gonna remember Janie and Ivan if he can't remember Emma? 11. 12. David looks back at Charlie, who shakes his head no. DAVID Well, Ivan's quite the character! Big Jewfro, suspenders, likes to wear a watch on each arm. CHARLIE I can't tell if you're kidding. DAVID Well, you'll see when we get there! CHARLIE Right. And what am I doing again? REBECCA Follow our lead and you'll be ok. DAVID Unless the Rabbi calls you up for an aliyah. Hey, maybe he'll even have you do the hagbah! CHARLIE What!? REBECCA Would you stop messing with him and focus on the road? CHARLIE So you're kidding, right? REBECCA And look, you just missed the turn. DAVID No this way is faster! Trust me! CHARLIE Right? You're kidding? REBECCA Yes Charlie, he's kidding! Just give us a second here. (To David) Look, if you just get on I-90 here...and you missed it. CHARLIE Can someone just tell me what I'm gonna have to do? 12. 13. DAVID Bud, in a minute, the grown-ups are talking. (To Rebecca) And look, I didn't miss anything. The backroads'll be faster, nobody's gonna be on them! REBECCA Yeah, because everyone else actually knows what they're doing. DAVID Have you met people? You think they know what they're doing? CHARLIE Hello? REBECCA I guess we'll see, won't we. Charlie crosses his arms. REBECCA (CONT'D) It's your family, your funeral. DAVID Actually, its a bat mitzvah.Charlie rolls his eyes. He pulls out an mp3 player and headphones, putting them on and hitting play. The sound fades up, drowning out the conversation between Rebecca and David.Gas stations, houses, trees fly by. Charlie stares out the window, face blank. After a stretch of fencing, the temple appears. Charlie sits up, interested by the new sight.He glances across the temple, taking in the massive building as well as its name, Temple Beth Israel. Staring out, Charlie presses his hands up against the window, face so close his breath fogs it up. REBECCACharlie! Charlie snaps back to reality, pulling down his headphones. REBECCA (CONT'D) I said let's go, we're going to be late! No thanks to dad's driving. 13. 14. INT. TEMPLE BETH ISRAEL FOYER -DAY 6 6 Charlie and his parents walk up to the sanctua...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.