502 research outputs found
Point spread function approximation of high rank Hessians with locally supported non-negative integral kernels
We present an efficient matrix-free point spread function (PSF) method for
approximating operators that have locally supported non-negative integral
kernels. The method computes impulse responses of the operator at scattered
points, and interpolates these impulse responses to approximate integral kernel
entries. Impulse responses are computed by applying the operator to Dirac comb
batches of point sources, which are chosen by solving an ellipsoid packing
problem. Evaluation of kernel entries allows us to construct a hierarchical
matrix (H-matrix) approximation of the operator. Further matrix computations
are performed with H-matrix methods. We use the method to build preconditioners
for the Hessian operator in two inverse problems governed by partial
differential equations (PDEs): inversion for the basal friction coefficient in
an ice sheet flow problem and for the initial condition in an
advective-diffusive transport problem. While for many ill-posed inverse
problems the Hessian of the data misfit term exhibits a low rank structure, and
hence a low rank approximation is suitable, for many problems of practical
interest the numerical rank of the Hessian is still large. But Hessian impulse
responses typically become more local as the numerical rank increases, which
benefits the PSF method. Numerical results reveal that the PSF preconditioner
clusters the spectrum of the preconditioned Hessian near one, yielding roughly
5x-10x reductions in the required number of PDE solves, as compared to
regularization preconditioning and no preconditioning. We also present a
numerical study for the influence of various parameters (that control the shape
of the impulse responses) on the effectiveness of the advection-diffusion
Hessian approximation. The results show that the PSF-based preconditioners are
able to form good approximations of high-rank Hessians using a small number of
operator applications
Wavelet and Multiscale Methods
[no abstract available
esys-Escript User’s Guide: Solving Partial Differential Equations with Escript and Finley Release - 3.2.1 (r3613)
esys.escript is a python-based environment for implementing mathematical models, in particular those based on coupled, non-linear, time-dependent partial differential equations. It consists of four major components • esys.escript core library • finite element solver esys.finley (which uses fast vendor-supplied solvers or our paso linear solver library) • the meshing interface esys.pycad • a model library. The current version supports parallelization through both MPI for distributed memory and OpenMP for distributed shared memory. Please see Chapter 2 for changes to the way to launch esys.escript scripts. For more info on this and other changes from previous releases see Appendix B. If you use this software in your research, then we would appreciate (but do not require) a citation. Some relevant references can be found in Appendix D
esys-Escript User’s Guide: Solving Partial Differential Equations with Escript and Finley Release - 3.3.1 (r4302)
esys.escript is a python-based environment for implementing mathematical models, in particular those based on coupled, non-linear, time-dependent partial differential equations. It consists of five major components • esys.escript core library • finite element solver esys.finley (which uses fast vendor-supplied solvers or our paso linear solver library) • the meshing interface esys.pycad • a model library. • an inversion library. The current version supports parallelization through both MPI for distributed memory and OpenMP for distributed shared memory. In this release there are a number of small changes which are not backwards compatible. Please see Appendix B to see if your scripts will be affected. If you use this software in your research, then we would appreciate (but do not require) a citation. Some relevant references can be found in Appendix D. For Python3 support, see Appendix E
esys-Escript User’s Guide: Solving Partial Differential Equations with Escript and Finley Release - 3.4.2 (r4925)
esys.escript is a python-based environment for implementing mathematical models, in particular those based on coupled, non-linear, time-dependent partial differential equations. It consists of five major components • esys.escript core library • finite element solver esys.finley (which uses fast vendor-supplied solvers or our paso linear solver library) • the meshing interface esys.pycad • a model library. • an inversion library. The current version supports parallelization through both MPI for distributed memory and OpenMP for shared memory. In this release there are a number of small changes which are not backwards compatible. Please see Appendix B to see if your scripts will be affected. If you use this software in your research, then we would appreciate (but do not require) a citation. Some relevant references can be found in Appendix D. For Python 3 support, see Appendix E
esys-Escript User’s Guide: Solving Partial Differential Equations with Escript and Finley Release - 3.4 (r4488)
esys.escript is a python-based environment for implementing mathematical models, in particular those based on coupled, non-linear, time-dependent partial differential equations. It consists of five major components • esys.escript core library • finite element solver esys.finley (which uses fast vendor-supplied solvers or our paso linear solver library) • the meshing interface esys.pycad • a model library. • an inversion library. The current version supports parallelization through both MPI for distributed memory and OpenMP for distributed shared memory. In this release there are a number of small changes which are not backwards compatible. Please see Appendix B to see if your scripts will be affected. If you use this software in your research, then we would appreciate (but do not require) a citation. Some relevant references can be found in Appendix D. For Python3 support, see Appendix E
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