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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
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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 2A Posteriori DG using Non-Polynomial Basis
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Motivation• Spatially inhomogeneous quantum systems
Ω
3Lin Lin A Posteriori DG using Non-Polynomial Basis
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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
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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
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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
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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
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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
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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
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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
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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
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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 11A Posteriori DG using Non-Polynomial Basis
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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
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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
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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
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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
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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
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Estimating constants Define
⊥ is in the sense of the inner product ⋅,⋅ ∗,𝜅𝜅Lemma. Let 𝜅𝜅 ∈ 𝒦𝒦, 𝑣𝑣 ∈ 𝐻𝐻1 𝜅𝜅 . Then
Proof:
Similar for 𝑏𝑏𝑘𝑘
Lin Lin A Posteriori DG using Non-Polynomial Basis 17
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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
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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
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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
≈
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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
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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
≈
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Generalize to high dimensionsTensor product LGL grid ⇒ Tensor product Lagrange polynomials
Lin Lin A Posteriori DG using Non-Polynomial Basis 25
𝜕𝜕𝑙𝑙
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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
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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
Lin Lin A Posteriori DG using Non-Polynomial Basis 27
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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
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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.
Lin Lin A Posteriori DG using Non-Polynomial Basis 29
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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
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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
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Computable upper bound
Lin Lin A Posteriori DG using Non-Polynomial Basis 32
Energy norm
Theorem ([LL-Stamm 2015]). Let 𝑢𝑢 ∈ 𝐻𝐻#1 Ω ⋂𝐻𝐻2(𝒦𝒦) be the true solution and 𝑢𝑢𝑁𝑁 ∈ 𝕍𝕍𝑁𝑁 the DG-approximation. Then
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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
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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
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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
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1D Poisson equation−Δ𝑢𝑢 𝑥𝑥 = sin 6𝑥𝑥
Adaptive local basis functions with 11 basis per element.
Lin Lin A Posteriori DG using Non-Polynomial Basis 36
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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
Lin Lin A Posteriori DG using Non-Polynomial Basis 39
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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
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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
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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
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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