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
Lin Lin 12
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
Lin Lin 13
⋅ ⋅
A Posteriori DG using Non-Polynomial Basis
Error quantificationDG approximation
Error in the broken energy norm
Goal: Find a sharp upper bound for
Lin Lin 14A Posteriori DG using Non-Polynomial Basis
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
Lin Lin A Posteriori DG using Non-Polynomial Basis 20
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.
Lin Lin A Posteriori DG using Non-Polynomial Basis 23
Estimating 𝑏𝑏𝑘𝑘
How to estimate 𝑢𝑢, 𝑣𝑣 𝜕𝜕𝜅𝜅. Importance of Lobatto grid
Here 𝑀𝑀𝑏𝑏 = �𝑊𝑊
Lin Lin A Posteriori DG using Non-Polynomial Basis 24
≈
Generalize to high dimensionsTensor product LGL grid ⇒ Tensor product Lagrange polynomials
Lin Lin A Posteriori DG using Non-Polynomial Basis 25
𝜕𝜕𝑙𝑙
Compare with asymptotic results for polynomial basis functionsFor polynomial basis functions [e.g. Houston-Schötzau-Wihler, 2007]
Lin Lin A Posteriori DG using Non-Polynomial Basis 26
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
Lin Lin A Posteriori DG using Non-Polynomial Basis 28
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
Lin Lin A Posteriori DG using Non-Polynomial Basis 34
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
Lin Lin A Posteriori DG using Non-Polynomial Basis 43
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