[petsc-users] Implementing the Sherman Morisson formula (low rank update) in petsc4py and FEniCS?
Matthew Knepley
knepley at gmail.com
Thu Feb 6 04:33:44 CST 2020
On Wed, Feb 5, 2020 at 8:53 PM Olek Niewiarowski <aan2 at princeton.edu> wrote:
> 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
>
> This is very easy to setup:
1) Create a KSP object KSPCreate(comm, &ksp)
2) Call KSPSetOperators(ksp, K, K,)
3) Solve the first system KSPSolve(ksp, -F, u_1)
4) Solve the second system KSPSolve(ksp, a, u_2)
5) Calculate beta VecDot(a, u_2, &gamma); beta = 1./(1. + k*gamma);
6) Update the guess VecDot(u_2, F, &delta); VecAXPBYPCZ(u, 1.0,
beta*delta, 1.0, u_1, u_2)
Thanks,
Matt
I don't need the Jacobian inverse, [*K*−k*aa**T*]-1 = *K*-1 - (k*K*-1 *a*
> *a**T**K*-1)/(1+k*a**T**K*-1*a*) just the solution Δ*u =* [*K*−k*aa**T*]-1
> *F *= *K*-1*F* - (k*K*-1 *a**F**K*-1*a*)/(1 + k*a**T**K*-1*a*)
> = *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/
>
>
--
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/ <http://www.cse.buffalo.edu/~knepley/>
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