Open course ware

INTRODUCTION: The following content is provided by MIT OpenCourseWare under a Creative Commons license. Additional information about our license and MIT OpenCourseWare in general is available at ocw.mit.edu. PROFESSOR: Specific problem, and it's a pure linear least squares problem, but it's got two terms. So we're used to minimizing A*u minus b square. That gives us the least squares solution u hat to a linear system. And usually the reason we have to go to the least square thing is that there's no exact solution. Probably A has more equations than unknowns. A is long and thin, and there's no exact solution, so we look for the best solution, and we call it u hat. OK. But there are a lot of problems in which a second square appears. There's also a B*u equal d hiding in the background. And so we really have like two sets of equations. And we multiply that second square by some factor alpha and that wise choice of alpha is usually a big part of the pr...

INTRODUCTION: The following content is provided by MIT OpenCourseWare under a Creative Commons license. Additional information about our license and MIT OpenCourseWare in general is available at ocw.mit.edu. PROFESSOR: OK. Should we go? Great. Well I hope you enjoyed Patriots' Day, and now we'll go for the final weeks of the course. And I wanted start with just a thought about scientific computing, and the people who work in it, and the problems. So just sort of an orientation. And maybe my point is that in part of the subject, the situation started like, just give me the matrix, tell me the equation. I don't want to know the physics behind it, just the form of the equation, and I'll discuss how it could be solved. And to contrast that with what you may often be meeting in your theses, even in projects, where the details, the physical consequences, the actual detail behavior of this solution is what you're looking for. So somehow there are those two pa...

INTRODUCTION: The following content is provided by MIT OpenCourseWare under a Creative Commons license. Additional information about our license and MIT OpenCourseWare in general is available at OCW.MIT.edu. PROFESSOR: OK. Ready for today's discussion, which I'm quite sort of happy about. I hadn't really seen this coming. First, we haven't said anything about how do you estimate the error in a problem. Because we're often taking a continuous problem coming from a differential equation, typically, and discretizing it, and solving it -- solving the discrete problem. And we want some idea. It's not just some math question of course, because, you know, in engineering design or analysis, an idea of what the error is critical. And secondly, it gives us an idea of where we can improve it. You know, if we see what's controlling the error then we know what's going on. So I'm speaking now about steady-state problems. We discussed error for initia...