A Posteriori Error Estimates For Discontinuous Galerkin Methods Using

Non-polynomial Basis Functions

Lin LinDepartment of Mathematics, UC Berkeley; Computational Research Division, LBNL

Joint work with Benjamin Stamm

Dimension Reduction: Mathematical Methods and Applications, Penn State University, March, 2015

Supported by DOE SciDAC Program and CAMERA Program

1Lin Lin A Posteriori DG using Non-Polynomial Basis

Outline• Introduction: Adaptive local basis functions

• Computable upper bound for Poisson’s equation

• Computable upper / lower bound for indefinite equations

• Numerical examples

• Conclusion and future work

Lin Lin 2A Posteriori DG using Non-Polynomial Basis

Motivation• Spatially inhomogeneous quantum systems

Ω

3Lin Lin A Posteriori DG using Non-Polynomial Basis

Kohn-Sham density functional theory

• Efficient: Always solve an equation in 𝑅𝑅3, regardless of the number of electrons 𝑁𝑁.

• Accurate: Exact ground state energy for exact 𝑉𝑉𝑥𝑥𝑥𝑥[𝜌𝜌], [Hohenberg-Kohn,1964], [Kohn-Sham, 1965]

• Best compromise between efficiency and accuracy. Most widely used electronic structure theory for condensed matter systems and molecules

• Nobel Prize in Chemistry, 1998

4

𝐻𝐻 𝜌𝜌 𝜓𝜓𝑖𝑖 𝑥𝑥 = −12Δ + 𝑣𝑣𝑒𝑒𝑥𝑥𝑒𝑒 𝑥𝑥 + ∫ 𝑑𝑑𝑥𝑥′

𝜌𝜌 𝑥𝑥′

𝑥𝑥 − 𝑥𝑥′+ 𝑉𝑉𝑥𝑥𝑥𝑥 𝜌𝜌 𝜓𝜓𝑖𝑖 𝑥𝑥 = 𝜀𝜀𝑖𝑖𝜓𝜓𝑖𝑖 𝑥𝑥

𝜌𝜌 𝑥𝑥 = 2�𝑖𝑖=1

𝑁𝑁/2

𝜓𝜓𝑖𝑖 𝑥𝑥 2 , ∫ 𝑑𝑑𝑥𝑥 𝜓𝜓𝑖𝑖∗ 𝑥𝑥 𝜓𝜓𝑗𝑗 𝑥𝑥 = 𝛿𝛿𝑖𝑖𝑗𝑗 , 𝜀𝜀1 ≤ 𝜀𝜀2 ≤ ⋯

Lin Lin A Posteriori DG using Non-Polynomial Basis

Discretization costBasis Example DOF / atom Construction

Uniform basis PlanewaveFinite differenceFinite element

500~10000 or more

Simple and systematic

Quantum chemistry basis

Gaussian orbitals Atomic orbitals

4~100 Fine tuning

Non-systematic convergence

Q: Combine the advantage of both?

5Lin Lin A Posteriori DG using Non-Polynomial Basis

Adaptive local basis functions• Idea: Use local eigenfunctions as basis functions

• How to patch the basis functions together?

6Lin Lin A Posteriori DG using Non-Polynomial Basis

Discontinuous Galerkin method

Kohn-Sham

New terms

• [LL-Lu-Ying-E, J. Comput. Phys. 231, 2140 (2012)] • Interior penalty method [Arnold, 1982]

7Lin Lin A Posteriori DG using Non-Polynomial Basis

Why a posteriori error estimator• Measuring the accuracy of eigenvalues and densities

without performing an expensive converged calculation, or benchmarking with another code.

• Optimal allocation of basis functions for inhomogeneous systems.

8Lin Lin A Posteriori DG using Non-Polynomial Basis

Residual based a posteriori error estimatorVast literature for second order PDE and eigenvalue problems

• Polynomial basis functions, finite element:

[Verfürth,1996] [Larson, 2000] [Durán-Padra-Rodríguez, 2003] [Chen-He-Zhou, 2011]...

• Polynomial basis functions, discontinuous Galerkin:

[Karakashian-Pascal, 2003], [Houston-Schötzau-Wihler, 2007], [Schötzau-Zhu, 2009], [Giani-Hall, 2012] ...

9Lin Lin A Posteriori DG using Non-Polynomial Basis

Difficulty• A posteriori error analysis relies on the detailed

knowledge of asymptotic approximation properties of the basis set

• Difficult for “equation-aware” basis functions Adaptive local basis functions Heterogeneous multiscale method (HMM) [E-Engquist

2003] Multiscale finite element [Hou-Wu 1997] Multiscale discontinuous Galerkin [Wang-Guzmán-

Shu, 2011] etc

Lin Lin 10A Posteriori DG using Non-Polynomial Basis

Outline• Introduction: Adaptive local basis functions

• Computable upper bound for Poisson’s equation

• Computable upper / lower bound for indefinite equations

• Numerical examples

• Conclusion and future work

Lin Lin 11A Posteriori DG using Non-Polynomial Basis

Model problem

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Discontinuous space (broken Sobolev space)

𝜅𝜅Ω

𝕍𝕍𝑁𝑁 ⊂ 𝐻𝐻2(𝒦𝒦)

𝐹𝐹

A Posteriori DG using Non-Polynomial Basis

Piecewise constant function belongs to 𝕍𝕍𝑁𝑁

DG discretizationBilinear form (𝜃𝜃 = 1 corresponds to the symmetric form)

Define the inner products

Average and jump operators

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⋅ ⋅

A Posteriori DG using Non-Polynomial Basis

Error quantificationDG approximation

Error in the broken energy norm

Goal: Find a sharp upper bound for

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Upper bound of errorTheorem ([LL-Stamm 2015]). Let 𝑢𝑢 ∈ 𝐻𝐻#1 Ω ⋂𝐻𝐻2(𝒦𝒦) be the true solution and 𝑢𝑢𝑁𝑁 ∈ 𝕍𝕍𝑁𝑁 the DG-approximation. Then

where

The key is to find the dependence of 𝑎𝑎𝜅𝜅 , 𝑏𝑏𝜅𝜅 , 𝑐𝑐𝜅𝜅 w.r.t. 𝕍𝕍𝑁𝑁.

Lin Lin 15

ResidualJump of gradientJump of function

A Posteriori DG using Non-Polynomial Basis

Projection operator𝐿𝐿2 𝜅𝜅 -projection operatorInner product

Projection operator onto basis space

Therefore

Lin Lin 16A Posteriori DG using Non-Polynomial Basis

Similar to𝐻𝐻1(𝜅𝜅) norm

Estimating constants Define

⊥ is in the sense of the inner product ⋅,⋅ ∗,𝜅𝜅Lemma. Let 𝜅𝜅 ∈ 𝒦𝒦, 𝑣𝑣 ∈ 𝐻𝐻1 𝜅𝜅 . Then

Proof:

Similar for 𝑏𝑏𝑘𝑘

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Numerical procedure for computing the constants• Basic idea: estimate the constants by iteratively solving

generalized eigenvalue problems on an infinite dimensional space

• 1D demonstration, generalizable to any d-dimension. • Consider 𝜅𝜅 = 0,ℎ , spectral discretization with Legendre-

Gauss-Lobatto (LGL) quadrature:

Integration points 𝑦𝑦𝑗𝑗 𝑗𝑗=1𝑁𝑁𝑔𝑔 , integration weights 𝜔𝜔𝑗𝑗 𝑗𝑗=1

𝑁𝑁𝑔𝑔

Lin Lin 18A Posteriori DG using Non-Polynomial Basis

0 ℎ𝑦𝑦𝑗𝑗

Numerical representation of inner productLGL grid points defines associated Lagrange polynomials of degree 𝑁𝑁𝑔𝑔 − 1

Approximate any 𝑣𝑣 ∈ 𝐻𝐻1 𝜅𝜅

Define

Inner product

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Numerical representation of inner product‖ ⋅ ‖∗,𝜅𝜅 requires differentiation matrix

Differentiation becomes matrix-vector multiplication

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Numerical representation of inner productProjection onto constant

In sum

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Estimating 𝑎𝑎𝑘𝑘

Lin Lin A Posteriori DG using Non-Polynomial Basis 22

Here

Handling the orthogonal constraint by projection𝑄𝑄 = 𝐼𝐼 − Π𝑁𝑁𝜅𝜅

≈

Estimating 𝑎𝑎𝑘𝑘

This is a generalized eigenvalue problem

Solve with iterative method, such as the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method [Knyazev 2001]

Only require matrix-vector multiplication.

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Estimating 𝑏𝑏𝑘𝑘

How to estimate 𝑢𝑢, 𝑣𝑣 𝜕𝜕𝜅𝜅. Importance of Lobatto grid

Here 𝑀𝑀𝑏𝑏 = �𝑊𝑊

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≈

Generalize to high dimensionsTensor product LGL grid ⇒ Tensor product Lagrange polynomials

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𝜕𝜕𝑙𝑙

Compare with asymptotic results for polynomial basis functionsFor polynomial basis functions [e.g. Houston-Schötzau-Wihler, 2007]

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ℎ = 1

Penalty parameterParameter {𝛾𝛾𝜅𝜅}• Large enough for coercivity of the bilinear form• “magic parameter” in interior penalty method [Arnold

1982]

Define

Lemma. If 𝛾𝛾𝜅𝜅 ≥12

1 + 𝜃𝜃 2𝑑𝑑𝜅𝜅2, then the bilinear form is coercive

Automatic guarantee of stability

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Penalty parameterComputation of 𝑑𝑑𝜅𝜅 through eigenvalue problem

By setting 𝑣𝑣𝑁𝑁 = Φ𝑐𝑐, span Φ = 𝕍𝕍𝑁𝑁 𝜅𝜅 . Can be solved with direct method

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Upper bound estimatorThe last constant

𝑑𝑑𝜅𝜅𝑢𝑢(𝑢𝑢𝑁𝑁) involves the true solution 𝑢𝑢 and therefore is the only constant that cannot be explicitly computed.

However, numerical result shows that 𝑑𝑑𝜅𝜅𝑢𝑢 𝑢𝑢𝑁𝑁 ≈ 𝑑𝑑𝜅𝜅

is a good approximation.

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Outline• Introduction: Adaptive local basis functions

• Computable upper bound for Poisson’s equation

• Computable upper / lower bound for indefinite equations

• Numerical examples

• Conclusion and future work

Lin Lin 30A Posteriori DG using Non-Polynomial Basis

Model problemIndefinite equation

𝑉𝑉 ∈ 𝐿𝐿∞ Ω and −Δ + 𝑉𝑉 has no zero eigenvalue.

Bilinear form

DG approximation

Lin Lin 31A Posteriori DG using Non-Polynomial Basis

Computable upper bound

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Energy norm

Theorem ([LL-Stamm 2015]). Let 𝑢𝑢 ∈ 𝐻𝐻#1 Ω ⋂𝐻𝐻2(𝒦𝒦) be the true solution and 𝑢𝑢𝑁𝑁 ∈ 𝕍𝕍𝑁𝑁 the DG-approximation. Then

Computable lower boundTheorem ([LL-Stamm 2015]). Let 𝑢𝑢 ∈ 𝐻𝐻#1 Ω ⋂𝐻𝐻2(𝒦𝒦) be the true solution and 𝑢𝑢𝑁𝑁 ∈ 𝕍𝕍𝑁𝑁 the DG-approximation. Then

where

Lin Lin 33A Posteriori DG using Non-Polynomial Basis

All constants other than 𝑑𝑑𝜅𝜅𝑢𝑢are computable

Computable lower boundBubble function 𝑏𝑏𝜅𝜅

For instance, 𝑏𝑏𝜅𝜅 𝑥𝑥 = 4 𝑥𝑥 1 − 𝑥𝑥 , 𝜅𝜅 = 1

Lemma.

where

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Outline• Introduction: Adaptive local basis functions

• Computable upper bound for Poisson’s equation

• Computable upper / lower bound for indefinite equations

• Numerical examples

• Conclusion and future work

Lin Lin 35A Posteriori DG using Non-Polynomial Basis

1D Poisson equation−Δ𝑢𝑢 𝑥𝑥 = sin 6𝑥𝑥

Adaptive local basis functions with 11 basis per element.

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Effectiveness of upper/lower estimtaorMeasure local effectiveness (𝐶𝐶𝜂𝜂 ≥ 1, 𝐶𝐶𝜉𝜉 ≤ 1)

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1D indefinite−Δ𝑢𝑢 𝑥𝑥 + 𝑉𝑉 𝑥𝑥 𝑢𝑢(𝑥𝑥) = sin 6𝑥𝑥

Adaptive local basis functions with 11 basis per element.

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Effectiveness of upper/lower estimtaor

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2D Helmholtz−Δ𝑢𝑢 + 𝑉𝑉𝑢𝑢 = 𝑓𝑓,

𝑉𝑉 = −16.5, 𝑓𝑓 𝑥𝑥,𝑦𝑦 = 𝑒𝑒−2 𝑥𝑥−𝜋𝜋 2−2 𝑦𝑦−𝜋𝜋 2

Adaptive local basis functions with 31 basis per element.

40Lin Lin A Posteriori DG using Non-Polynomial Basis

Effectiveness for upper/lower bound

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Validate the approximation for 𝑑𝑑𝜅𝜅𝑢𝑢

Note that

Although 𝑑𝑑𝜅𝜅𝑢𝑢 is not known, it is only sufficient to have

𝑑𝑑𝜅𝜅𝑢𝑢 ≈ 𝑑𝑑𝜅𝜅 or 𝑑𝑑𝜅𝜅𝑢𝑢 ≪ 𝑏𝑏𝜅𝜅𝛾𝛾𝜅𝜅

1D:

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Validate the approximation for 𝑑𝑑𝜅𝜅𝑢𝑢

2D indefinite

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Conclusion• Systematic derivation of a posteriori error estimation for

general non-polynomial basis function

• Explicitly computable constants for upper/lower estimator.

• The only one non-computable constant can be reasonably estimated by known ones.

Lin Lin 44A Posteriori DG using Non-Polynomial Basis

Future work• Eigenvalue problem

• Nonlinearity, atomic force, linear response properties

• Implementation in DGDFT

• Other basis functions, including MsFEM, HMM, MsDG etc.

Ref:LL and B. Stamm, A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part I: Second order linear PDE, arXiv:1502.01738

Thank you for your attention!

Lin Lin 45A Posteriori DG using Non-Polynomial Basis

A Posteriori Error Estimates For �Discontinuous Galerkin Methods Using �Non-polynomial Basis FunctionsOutlineMotivationKohn-Sham density functional theoryDiscretization costAdaptive local basis functionsDiscontinuous Galerkin methodWhy a posteriori error estimatorResidual based a posteriori error estimatorDifficultyOutlineModel problemDG discretizationError quantificationUpper bound of errorProjection operatorEstimating constants Numerical procedure for computing the constantsNumerical representation of inner productNumerical representation of inner productNumerical representation of inner productEstimating 𝑎 𝑘 Estimating 𝑎 𝑘 Estimating 𝑏 𝑘 Generalize to high dimensionsCompare with asymptotic results for polynomial basis functionsPenalty parameterPenalty parameterUpper bound estimatorOutlineModel problemComputable upper boundComputable lower boundComputable lower boundOutline1D Poisson equationEffectiveness of upper/lower estimtaor1D indefiniteEffectiveness of upper/lower estimtaor2D HelmholtzEffectiveness for upper/lower boundValidate the approximation for 𝑑 𝜅 𝑢 Validate the approximation for 𝑑 𝜅 𝑢 ConclusionFuture work