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Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. 

Generalized Moving Least Squares: Approximation Theory and Applications

M.Perego, P. Bochev, P. Kuberry, N. Trask

Humboldt University, Berlin, Germany, June 12, 2019

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Outline:

● Intro to (Classic) Moving least Square● Intro to Generalized Moving least Square● GMLS approximation theory / tools● Meshless discretization

● A collocation method● A Galerkin method

Motivation:

● GMLS allows to reconstruct functions/functionals given samples on scattered data points ● It can be use for data transfer between different meshes / point clouds● It can be use for meshless discretizations of differential problems

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Moving Least Squares

point cloud:

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Moving Least Squares

point cloud:

filling distance:

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Moving Least Squares

point cloud:

filling distance:

separation distance:

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Moving Least Squares

point cloud:

filling distance:

separation distance:

the point cloud is quasi-uniform if

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Least Square formulation:

Moving Least Squares

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Least Square formulation:

Moving Least Squares

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Least Square formulation:

Moving Least Squares

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Least Square formulation:

Moving Least Squares

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Least Square formulation:

Moving Least Squares

diffuse derivative

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Least Square formulation:

Moving Least Squares

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Least Square formulation:

Moving Least Squares

Constrained Optimization formulation:

Reproducibility:

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Recipes (2 equivalent formulations):

2. Constrained Optimization formulation:

1. Least Square formulation:

Generalized Moving Least Squares

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Generalized Moving Least Squares(approximation theory)

Goal: collect and consolidate approximation results* for GMLS

In particular we are interested in these generalizations:

* Mirzaei, Narcowich, Rieger, Schaback, Ward, Wendland

Approximation space: we want to consider vector polynomials and subspaces of complete polynomial spaces (e.g. div-free, curl-free spaces) Sampling/target functionals: we want to consider differential forms and in particular volume integrals, fluxes over (virtual) faces and line integral over (virtual) edges

In the following we present generalization of definition of “local polynomial reproduction” existence and approximation results for different sampling/target functionals

Applications: PDE discretizations and data transfer.

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Generalized Moving Least Squares

- a function space (e.g. continuous functions) and its dual

- a finite dimensional approximation space, e.g. polynomials

- a finite set of sampling functionals (e.g. point evaluations) FE analog: degrees of freedom

- a target functional (or a family of target functionals)

Ingredients:

Example, MLS case:

- a window function correlating functionals (e.g. a radial kernel) determines the smoothness of reconstruction

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Properties of Generalized Moving Least Squares

● Approximation property:

● Unisolvency:

Main questions: When is the reconstruction possible? How good is the reconstruction?

* H. Wendland, Scattered data approximation, Vol. 17. Cambridge university press, 2004 ** Christian Rieger, PhD thesis, personal communication.*** D. Mirzaei et Al., IMA J. Num. Analysis, 2011, D. Mirzaei, Comp. And App. Math. 2016

➢ Results has been generalized by Mirzaei***, for point derivatives as target functionals:

and for weak derivatives in Sobolev spaces***.

➢Sufficient results for unisolvancy and approx. error have been provided by Wendland* for (non generalized) MLS.

➢Non local smapling operators**

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Result (local polynomial reproduction) [MLS]

Result (generalized local reproduction) [GMLS]

Towards a general theory for GMLS: local reproduction

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Reconstructing vector functions(from projections along given directions)

Sampling functional:

Other sampling functionals we considered:

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GMLS approximation results:

Basic technique:

reconstruction property

GMLS definition

Holds for any target functional and approximation space:

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GMLS approximation results: derivatives as target functionals

** D. Mirzaei et Al., IMA J. Num. Analysis, 2011.

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GMLS approximation results: div free spaces

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GMLS approximation results: integral target functionals (e.g. used for nonlocal problems)

** D. Mirzaei et Al., IMA J. Num. Analysis, 2011.

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Application to PDE: staggered scheme for Darcy

Solution of PDE w/ jumps in permeability

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Sum over mesh elements

Finite dimensional space

Finite dimensional formulation

Galerkin Discretization

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Finite dimensional formulation

Recall that:

Galerkin Discretization

Linearity of bilinear form

Linearity of bilinear form

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Galerkin Discretization (DG)

Basis functions

Example (1D)

1D2D

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Conclusion

● GMLS are fun (?)

● Lot of flexibility

● Lot to do

● Prove well posedness of GMLS DG formulation

● 2D/3D implementation● ...

Thank you!