[petsc-users] Implementing the Sherman Morisson formula (low rank update) in petsc4py and FEniCS?
Olek Niewiarowski
aan2 at princeton.edu
Wed Feb 5 19:53:51 CST 2020
Hi Barry and Matt,
Thank you for your input and for creating a new issue in the repo.
My initial question was more basic (how to configure the SNES's KSP solver as in my first message with a and k), but now I see there's more to the implementation. To reiterate, for my problem's structure, a good solution algorithm (on the algebraic level) is the following "double back-substitution":
For each nonlinear iteration:
1. define intermediate vectors u_1 and u_2
2. solve Ku_1 = -F --> u_1 = -K^{-1}F (this solve is cheap, don't actually need K^-1)
3. solve Ku_2 = -a --> u_2 = -K^{-1}a (ditto)
4. define \beta = 1/(1 + k a^Tu_2)
5. \Delta u = u_1 + \beta k u_2^T F u_2
6. u = u + \Delta u
I don't need the Jacobian inverse, [K−kaaT]-1 = K-1 - (kK-1 aaTK-1)/(1+kaTK-1a) just the solution Δu = [K−kaaT]-1F = K-1F - (kK-1 aFK-1a)/(1 + kaTK-1a)
= u_1 + beta k u_2^T F u_2 (so I never need to invert K either). (To Matt's point on gitlab, K is a symmetric sparse matrix arising from a bilinear form. ) Also, eventually, I want to have more than one low-rank updates to K, but again, Sherman Morrisson Woodbury should still work.
Being new to PETSc, I don't know if this algorithm directly translates into an efficient numerical solver. (I'm also not sure if Picard iteration would be useful here.) What would it take to set up the KSP solver in SNES like I did below? Is it possible "out of the box"? I looked at MatCreateLRC() - what would I pass this to? (A pointer to demo/tutorial would be very appreciated.) If there's a better way to go about all of this, I'm open to any and all ideas. My only limitation is that I use petsc4py exclusively since I/future users of my code will not be comfortable with C.
Thanks again for your help!
Alexander (Olek) Niewiarowski
PhD Candidate, Civil & Environmental Engineering
Princeton University, 2020
Cell: +1 (610) 393-2978
________________________________
From: Smith, Barry F. <bsmith at mcs.anl.gov>
Sent: Wednesday, February 5, 2020 15:46
To: Matthew Knepley <knepley at gmail.com>
Cc: Olek Niewiarowski <aan2 at princeton.edu>; petsc-users at mcs.anl.gov <petsc-users at mcs.anl.gov>
Subject: Re: [petsc-users] Implementing the Sherman Morisson formula (low rank update) in petsc4py and FEniCS?
https://gitlab.com/petsc/petsc/issues/557
> On Feb 5, 2020, at 7:35 AM, Matthew Knepley <knepley at gmail.com> wrote:
>
> Perhaps Barry is right that you want Picard, but suppose you really want Newton.
>
> "This problem can be solved efficiently using the Sherman-Morrison formula" Well, maybe. The main assumption here is that inverting K is cheap. I see two things you can do in a straightforward way:
>
> 1) Use MatCreateLRC() https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatCreateLRC.html to create the Jacobian
> and solve using an iterative method. If you pass just K was the preconditioning matrix, you can use common PCs.
>
> 2) We only implemented MatMult() for LRC, but you could stick your SMW code in for MatSolve_LRC if you really want to factor K. We would
> of course help you do this.
>
> Thanks,
>
> Matt
>
> On Wed, Feb 5, 2020 at 1:36 AM Smith, Barry F. via petsc-users <petsc-users at mcs.anl.gov> wrote:
>
> I am not sure of everything in your email but it sounds like you want to use a "Picard" iteration to solve [K(u)−kaaT]Δu=−F(u). That is solve
>
> A(u^{n}) (u^{n+1} - u^{n}) = F(u^{n}) - A(u^{n})u^{n} where A(u) = K(u) - kaaT
>
> PETSc provides code to this with SNESSetPicard() (see the manual pages) I don't know if Petsc4py has bindings for this.
>
> Adding missing python bindings is not terribly difficult and you may be able to do it yourself if this is the approach you want.
>
> Barry
>
>
>
> > On Feb 4, 2020, at 5:07 PM, Olek Niewiarowski <aan2 at princeton.edu> wrote:
> >
> > Hello,
> > I am a FEniCS user but new to petsc4py. I am trying to modify the KSP solver through the SNES object to implement the Sherman-Morrison formula(e.g. http://fourier.eng.hmc.edu/e176/lectures/algebra/node6.html ). I am solving a nonlinear system of the form [K(u)−kaaT]Δu=−F(u). Here the jacobian matrix K is modified by the term kaaT, where k is a scalar. Notably, K is sparse, while the term kaaT results in a full matrix. This problem can be solved efficiently using the Sherman-Morrison formula :
> >
> > [K−kaaT]-1 = K-1 - (kK-1 aaTK-1)/(1+kaTK-1a)
> > I have managed to successfully implement this at the linear solve level (by modifying the KSP solver) inside a custom Newton solver in python by following an incomplete tutorial at https://www.firedrakeproject.org/petsc-interface.html#defining-a-preconditioner :
> > • while (norm(delU) > alpha): # while not converged
> > •
> > • self.update_F() # call to method to update r.h.s form
> > • self.update_K() # call to update the jacobian form
> > • K = assemble(self.K) # assemble the jacobian matrix
> > • F = assemble(self.F) # assemble the r.h.s vector
> > • a = assemble(self.a_form) # assemble the a_form (see Sherman Morrison formula)
> > •
> > • for bc in self.mem.bc: # apply boundary conditions
> > • bc.apply(K, F)
> > • bc.apply(K, a)
> > •
> > • B = PETSc.Mat().create()
> > •
> > • # Assemble the bilinear form that defines A and get the concrete
> > • # PETSc matrix
> > • A = as_backend_type(K).mat() # get the PETSc objects for K and a
> > • u = as_backend_type(a).vec()
> > •
> > • # Build the matrix "context" # see firedrake docs
> > • Bctx = MatrixFreeB(A, u, u, self.k)
> > •
> > • # Set up B
> > • # B is the same size as A
> > • B.setSizes(*A.getSizes())
> > • B.setType(B.Type.PYTHON)
> > • B.setPythonContext(Bctx)
> > • B.setUp()
> > •
> > •
> > • ksp = PETSc.KSP().create() # create the KSP linear solver object
> > • ksp.setOperators(B)
> > • ksp.setUp()
> > • pc = ksp.pc
> > • pc.setType(pc.Type.PYTHON)
> > • pc.setPythonContext(MatrixFreePC())
> > • ksp.setFromOptions()
> > •
> > • solution = delU # the incremental displacement at this iteration
> > •
> > • b = as_backend_type(-F).vec()
> > • delu = solution.vector().vec()
> > •
> > • ksp.solve(b, delu)
> >
> > • self.mem.u.vector().axpy(0.25, self.delU.vector()) # poor man's linesearch
> > • counter += 1
> > Here is the corresponding petsc4py code adapted from the firedrake docs:
> >
> > • class MatrixFreePC(object):
> > •
> > • def setUp(self, pc):
> > • B, P = pc.getOperators()
> > • # extract the MatrixFreeB object from B
> > • ctx = B.getPythonContext()
> > • self.A = ctx.A
> > • self.u = ctx.u
> > • self.v = ctx.v
> > • self.k = ctx.k
> > • # Here we build the PC object that uses the concrete,
> > • # assembled matrix A. We will use this to apply the action
> > • # of A^{-1}
> > • self.pc = PETSc.PC().create()
> > • self.pc.setOptionsPrefix("mf_")
> > • self.pc.setOperators(self.A)
> > • self.pc.setFromOptions()
> > • # Since u and v do not change, we can build the denominator
> > • # and the action of A^{-1} on u only once, in the setup
> > • # phase.
> > • tmp = self.A.createVecLeft()
> > • self.pc.apply(self.u, tmp)
> > • self._Ainvu = tmp
> > • self._denom = 1 + self.k*self.v.dot(self._Ainvu)
> > •
> > • def apply(self, pc, x, y):
> > • # y <- A^{-1}x
> > • self.pc.apply(x, y)
> > • # alpha <- (v^T A^{-1} x) / (1 + v^T A^{-1} u)
> > • alpha = (self.k*self.v.dot(y)) / self._denom
> > • # y <- y - alpha * A^{-1}u
> > • y.axpy(-alpha, self._Ainvu)
> > •
> > •
> > • class MatrixFreeB(object):
> > •
> > • def __init__(self, A, u, v, k):
> > • self.A = A
> > • self.u = u
> > • self.v = v
> > • self.k = k
> > •
> > • def mult(self, mat, x, y):
> > • # y <- A x
> > • self.A.mult(x, y)
> > •
> > • # alpha <- v^T x
> > • alpha = self.v.dot(x)
> > •
> > • # y <- y + alpha*u
> > • y.axpy(alpha, self.u)
> > However, this approach is not efficient as it requires many iterations due to the Newton step being fixed, so I would like to implement it using SNES and use line search. Unfortunately, I have not been able to find any documentation/tutorial on how to do so. Provided I have the FEniCS forms for F, K, and a, I'd like to do something along the lines of:
> > solver = PETScSNESSolver() # the FEniCS SNES wrapper
> > snes = solver.snes() # the petsc4py SNES object
> > ## ??
> > ksp = snes.getKSP()
> > # set ksp option similar to above
> > solver.solve()
> >
> > I would be very grateful if anyone could could help or point me to a reference or demo that does something similar (or maybe a completely different way of solving the problem!).
> > Many thanks in advance!
> > Alex
>
>
>
> --
> What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.
> -- Norbert Wiener
>
> https://www.cse.buffalo.edu/~knepley/
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