Understanding Krylov Subspace Methods

[@wildmanz8233]

Very useful. I'm just learning about Krylov subspace methods for problems in theoretical chemistry and this video really helped with understanding some background.

[@ssrestore]

Very good! Please continue to make videos about other iterative methods

[@wentinghsieh9961]

The video is great. Could you make the sound louder? I barely can hear you.

[@SO-dl2pv]

Thank you. Just a note: IMO, examples are mandatory for beginners, to make a mental model about concepts.

[@haji3787]

Exactly

[@nicolasfurlan8759]

As a student from first semester in Russia, you’re saving my life. Thanks 🥲❤️

[@AJ-et3vf]

Awesome video! Thank you!

[@녹차맛커피]

Professor, thank you for your good online lecture. I have some question. I understand that, Km(A,x) = span{x, Ax, AAx,,,A^mx}, so, x in Km(A,x) x = A-1b → Km(A,x) = {A-1b, b, Ab, AAb,,,A^m-1 x} AKm(A,x) = {b, Ab, AAb,,,A^mb} = Km(A,b) but, Km(A,x) = span{x, Ax, AAx,,,A^mx}, so, x in Km(A,x) (a) AKm(A,x) = {b, Ab, AAb,,,A^mb} = Km(A,b) (b) x in Km(A,b) (c) I can't link the (a),(b) and (c)

[@daniel_an]

Did you check out my recent video? That one should be easier to understand

[@benbrelje4403]

Great explanation!

[@josephm.6453]

Why does the constrained pressure residual CPR give a faster "close enough" solution than Kyrlov subspace method?

[@daniel_an]

I don't know CPR method, but a quick google search let me know that CPR uses multigrid methods, and multigrid is much faster because it solves the problem on a coarse grid to obtain an approximate solution first. Kyrlov does not assume the problem came from taking finite difference or finite element approximation of a PDE, while multigrid can be used if such a thing as "coarse grid" can be defined. There is something called algebraic multigrid method (AMG), which seems to be applicable in general situations, but I don't understand that either. Are you working on CFD?

[@josephm.6453]

@@daniel_an I am a reservoir engineer. I am trying to compare between different reservoir simulators that use different solvers. it is used to solve for transmissibility, material balance equation, temperature changes, pressure decays, fluxes... in the reservoir after certain period, in years usually, due to pressure loss because of production. the reservoir is a grid of hundred thousands to millions of cells. each cell has its own calculation for each time step. so a faster method can shorten your simulation time by hours. Kyrlov is used in one simulator, so I'm trying to find out about CPR.

[@daniel_an]

@@josephm.6453 I do CFD, and for solving pressure, we use: Fourier Transform (in case there is high symmetry) and Multigrid (if the domain is not symmetric) over Krylov solvers. So multigrid based CPR should be faster. Good luck!

[@josephm.6453]

@@daniel_an ok so if multigrids are "better", can we make an orthogonal Krylov GMRES by using Arnoldi iteration? it should be somewhat the same with CPR , right?

[@rev0cdevs38]

When I first saw this video, I couldn't understand this. After reading some books on numerical linear algebera and watching other videos, I clearly understand this video and this video nicely introduce the content. Am I an idiot not to understand this video at first try? or Is it normal for a normal person? just curious.

[@daniel_an]

Actually your experience applies to all math. It's really hard to understand something without the needed basic knowledge.

[@iagoarcas2350]

Why {v, Av, ..., A^m v} is linear Independence ? Can you help me please.

[@daniel_an]

No. {v, Av, ..., A^m v} will eventually become linearly dependent for some m. That's the beauty of this method. The smaller the m is the quicker you will solve Ax=b

[@iagoarcas2350]

Thank you @@daniel_an . One day, if it's possible for you, can you make a video about Jacobian Free Newton Krylov Method please.

[@Sporkomat]

It can be derived from the characteristic equation of A that is C(lambda)=det(lambda*E-A)=0, because every square matrix satisfies its characteristic equation C(A)=0. At some point the vector Av^m must be a linear combination of the others (that is the case for a mxm matrix)

[@iagoarcas2350]

@@Sporkomat thank you

[@ilpreterosso]

thank you!

[@zzzoldik8749]

Please the sound no peoperly heared

[@Cilanee]

God help me passing linear algebra 💀

[@jhk1702]

한국인 이세요??

[@daniel_an]

예 그렇습니다^^

[@jhk1702]

@@daniel_an 너무 멋있으세요!! 공대생인데 저도 나중에 강단에서 영어로 서술해보고 싶네요. 부럽습니다. 열심히 공부해야겠네요 ㅎㅎ

[@aerodynamico6427]

The recording is horrible; the writing on the board is totally unreadable. Completely unacceptable and no good for learning.