A perturbation-based reduced basis method for parametric eigenvalue problems

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CERMICS, Ecole des Ponts ParisTech & Matherials

1 A perturbation-based reduced basis method for parametric eigenvalue problems Eric Cancès 1, Virginie Ehrlacher 1, David Gontier 2, Damiano Lombardi 3 and Antoine Levitt 1 1. CERMICS, Ecole des Ponts ParisTech & Matherials project-team, INRIA. 2. ETH Zürich 3. REO project-team, INRIA San Servolo, May / 26

2 Outline of the talk Motivation: electronic structure of crystals 2 / 26

3 Outline Motivation: electronic structure of crystals 3 / 26

The study of the electronic structure of perfect crystals is a central problem in solid-state physics and materials science.

4 Motivation: electronic structure of perfect crystals A perfect crystal is a material composed of an infinite number of atoms that are clamped periodically in space. The study of the electronic structure of perfect crystals is a central problem in solid-state physics and materials science. Perfect crystals constitute useful idealized systems to understand many fundamental properties of real crystalline solids (electric conductivity, dielectric permittivity of insulators and semiconductors, photo-electric effect,...). 4 / 26

5 Motivation: electronic structure of perfect crystals R: periodic lattice of the crystal, Γ: an admissible unit cell of the crystal. Within Density Functional Theory models, the electronic structure of a perfect crystal is characterized by some periodic mean-field Hamiltonian on L 2 (R 3 ), where V per L 2 per(γ) := H per = 1 2 +Vper { } u L 2 loc(r 3 ) u R-periodic. Example (cubic symmetry): R = 2πZ 3, Γ = ( π,π) 3. 5 / 26

6 Mathematical properties of H per H per defines a self-adjoint operator on L 2 (R 3 ) with domain H 2 (R 3 ); H per is bounded below; The point spectrum of H per is empty (H per has no eigenvalue); The continuous spectrum H per is composed of a union of closed intervals of R, separated by spectral gaps. Gaps Bands Spectrumof H per 6 / 26

First-Brillouin zone Since H per commutes with all R-translations, its spectral properties can be easily characterized by considering its Bloch transform [Reed,Simon,1978].

7 First-Brillouin zone Since H per commutes with all R-translations, its spectral properties can be easily characterized by considering its Bloch transform [Reed,Simon,1978]. Let R denote the dual lattice of R and Γ the Wigner-Seitz cell of R (the set of points in R 3 that are closer to 0 than to any other point in R ). Example: R = Z 3, Γ = ( 1/2, 1/2) 3. Γ is called the first Brillouin zone of the crystal and an element q Γ is called a q-point. 7 / 26

8 Bloch theory H per = 1 H Γ q dq, with H q = 1 Γ 2 i +q 2 + V per where for all q Γ, H q is a self-adjoint operator on L 2 per(γ) with domain H 2 per(γ), bounded from below and with compact resolvent. For all q-point q Γ, there exists an orthonormal basis (u n,q) n N of L 2 per(γ) and a non-decreasing sequence(λ n,q) n N such that n N, H qu n,q = λ n,qu n,q 8 / 26

9 Electronic properties of crystals Spectrum of H per: σ(h per) = σ(h q) = q Γ q Γ, n N {λ n,q}. Fermi level ǫ F : (where N el number of electrons per unit cell) N el = 1 1 Γ λn,q ǫ F dq. Γ n N Energy per unit cell: E = 1 λ n,q1 Γ λn,q ǫ F dq. Γ n N λ 2,q λ 1,q λ n,q Electronic density: ρ(x) = 1 1 Γ λn,q ǫ F u n,q(x) 2 dq. Γ n N q Γ 9 / 26

10 Back to the parameter-dependent eigenvalue problem n N, H qu n,q = λ n,qu n,q Usually solved using a Fourier discretization basis: { } X N := Span Γ 1/2 e ik x, k R, k E c, for some cut-off parameter E c > 0, so that N := dim X N 1. Let us denote by a q : Hper(Γ) H 1 per(γ) 1 C the sesquilinear form associated with H q. v, w Hper(Γ), 1 a q(v, w) := 1 ( i x + q)v ( i x + q)w + V pervw. 2 Γ Γ Galerkin problem: Find (u N n,q,λ N n,q) X N R solution to v N X N, a q(u N n,q, v N ) = λ N n,q u N n,q, v N L 2 per (Γ). What about a reduced basis method? 10 / 26

11 Fermi surface of a crystal: Fermi surfaces {q Γ, n N, λ n,q = ǫ F } 11 / 26

12 Outline Motivation: electronic structure of crystals 12 / 26

13 Litterature and principle of the approach Approximation of parameter-dependent eigenvalue problems using sparse grids: [Andreev,Schwab, 2012] POD-based method: [Quraishi, Schröder, Mehrmann, 2014] Reduced-basis method for parameter-dependent eigenvalue problem (in similar contexts): [Machiels,Maday,Oliveira,Patera,Rova,2000], [Pau,2007], [Fumagalli, Manzoni, Parolini, Verani,2015], [Horger, Wohlmuth, Dickopf,2016]... Standard reduced-basis methods work already quite well. Our approach here consists in creating bases that are q-point dependent and results in much more accurate results up to reasonable additional work in the off-line phase and low extra-costs in the on-line phase. These bases are created using a perturbation-based argument similar to the one introduced in [Cancès,Dusson,Maday,Stamm,Vohralik, 2014] for the post-treatment of the numerical solution of an eigenvalue problem using a Galerkin method. The method can be easily generalized to a more general class of elliptic parametric eigenvalue problems. 13 / 26

14 Coarse grid Λ 1 and fine grid Λ 2 of Γ The basic idea of the proposed numerical scheme is to extract local reduced bases from calculations for Q belonging to a coarse grid Λ 1 (for instance uniform of size L 1 L 1 L 1 ) of Γ. These are then used to compute the eigenmodes of H q for q on a fine grid Λ 2 (for instance uniform of size L 2 L 2 L 2, with L 2 L 1 ). Λ L1 Λ L2 for some r > 0 (for instance r = L 1 1 ). P(q) := {Q Λ 1, q Q r} 14 / 26

15 Simple reduced-basis method Let m N be an integer fixed a priori. We wish to design a reduced basis method in order to approximate the m lowest eigenvalues and associated eigenvectors of the opertaor H q for q belonging to the fine grid Λ 2. For any point Q of the coarse grid Λ 1 and N N, we will denote X N,Q := Span{u 1,Q,, u N,Q }, the subspace of X N spanned by the eigenvectors associated to the lowest N eigenvalues of the operator H Q. Assume in the following that m N N. Simple reduced-basis method (simple-rb): For all 1 n m and all Q P(q), find (u N,Q n,q,λ N,Q n,q ) X N,Q R solutions of v N,Q X N,Q, a q(u N,Q n,q, v N,Q ) = λ N,Q n,q u N,Q n,q, v N,Q L 2 per (Γ). Then, choose Q 0 P(q) so that N n=1 λn,q 0 n,q = min N Q P(q) n=1 λn,q n,q u n,q u N,Q 0 n,q, λ n,q λ N,Q 0 n,q 15 / 26

16 First-order perturbation theory (formal computations which can be made rigorous!) Let A and B be self-adjoint operators on a Hilbert space H so that (A i) 1 B L(H) = η = o(1). The operator B can be seen as a small perturbation of A. Au n = λ nu n First-order expansions: (A+B)ũ n = λ nũ n ũ n = u n + v 1 n +O(η 2 ) λ n = λ n +λ 1 n +O(η 2 ) Let us denote by X n the eigenspace of A associated with the eigenvalueλ n. Then, it holds that λ 1 n = u n, Bu n H and v 1 n = (A λ n) 1 X n Π X n Bu n where Π X n denotes the H-orthogonal projection onto X n. 16 / 26

17 Perturbation expansion Let q Λ L2 and Q P(q). H q = 1 2 i x + q 2 + V per = 1 2 iq +Vper + q 2 = A q + q 2 The rough idea is to write the exact operator A q as a perturbation of the discretized operator A N,Q q in X N,Q. More precisely, where A N,Q q A q = A Q +(q Q) ( i ) = A N,Q q + B N,Q q, = A Q +Π X N,Q(q Q) ( i )Π X N,Q = A Q (X N,Q ) + Aq X N,Q, B N,Q q = (q Q) ( i ) Π X N,Q(q Q) ( i )Π X N,Q, and Π X N,Q is the orthogonal projection on X N,Q. 17 / 26

18 It holds that Perturbation expansion ( ) B N,Q q u N,Q n,q = Π (X N,Q ) (q Q) ( i u N,Q n,q ). Then, where v N,Q n,q u n,q = u N,Q n,q + v N,Q n,q + r N,Q n,q, ( ) = (A Q µ n,q) 1 Π (X N,Q ) (X N,Q ) (q Q) ( i u N,Q n,q ) and r N,Q n,q is a remainder term whose H 1 norm is of the order of q Q 2. In practice, one has to compute an approximation of (A Q µ n,q) 1. For (X N,Q ) instance, one can compute ṽ N,Q n,q ( ) = (A Q µ n,q) 1 Π X 2N,Q (X N,Q ) (X N,Q ) (q Q) ( i u N,Q n,q ) Define ũ N,Q n,q = u N,Q n,q u N,Q n,q + ṽ N,Q n,q + ṽ N,Q n,q L 2 per (Γ). 18 / 26

19 Perturbation-based reduced basis method Two different approximations: Rayleigh: Define λ N,Q n,q := a q(ũ N,Q n,q, ũ N,Q n,q ), and u N,Q n,q := ũ N,Q n,q. and choose Q 0 := argmin Q P(q) N n=1 λn,q n,q, then set λ n,q λ N,Q 0 n,q, and u n,q u N,Q 0 n,q. Corrected: Solve a Galerkin eigenvalue problem similar to the simple reduced basis method, but with the new discretization space { } := Span,, ũn,q. X N,Q q ũ N,Q 1,q instead of X N,Q. For all 1 n m and all Q P(q), find (û N,Q n,q, λ N,Q n,q ) X N,Q R solutions of v N,Q X N,Q, a q(û N,Q n,q, v N,Q N,Q ) = λ n,q û N,Q n,q, v N,Q L 2 per (Γ). N,q and choose Q 0 := argmin Q P(q) N n=1 λ N,Q n,q, then set λ n,q λ N,Q 0 n,q, and u n,q û N,Q 0 n,q. 19 / 26

20 Numerical results Silicium (Here, N 5000). n = 9, N = 60 Error on λ n,q for q [(0, 0, 0),(1/2, 0, 0)]. 20 / 26

21 Silicium Numerical results Error on λ N,Q n,q λ n,q L (Λ 2 ) for different values of N and different sizes of the coarse grid Λ 1. Λ 2 = / 26

22 Numerical results Silicium [Gontier, 2015] Λ 1 = 6 6 6, Λ 2 = / 26

23 Numerical results Aluminium [Gontier, 2015] Λ 1 = 6 6 6, Λ 2 = / 26

24 Conclusion Perturbation-based reduced basis method for parametric eigenvalue problem: the idea is to write the exact operator as a perturbation of the discretized operator in a standard reduced basis. Can be extended to families of operators that are self-adjoint holomorphic of type A with compact resolvent. [Kato,1976], [Rellich, 1953] The method gives more accurate results up to some moderate extra-cost in the off-line and in the on-line phase of the reduced basis method. Application to q-point integration for the computation of the electronic structure of perfect crystals. Open questions: Error analysis of the method (a priori and a posteriori) : adaptive refinement Real test cases Kolmogorov n-widths of parametric eigenvalue problems (non necessarily self-adjoint)? In general, branches of analytic functions with at most algebraic singularities / 26

25 References Methods of Modern Mathematical Physics. Analysis of Operators, vol. IV, M. Reed and B. Simon, Academic Press, Perturbation theory for linear operators, T. Kato, Grundlehren 132, 1976, Springer-Verlag, Berlin, New York. Perturbation Theory of Eigenvalue Problems, F. Rellich, 1953 New York University Lecture Notes reprinted by Gordon and Breach,1968. Sparse Tensor Approximation of Parametric Eigenvalue Problems, R. Andreev and C. Schwab, Lecture notes in Computational Science and Engineering, Output Bounds for Reduced-Basis Approximations of Symmetric Positive Definite Eigenvalue Problems, L. Machiels, Y. Maday, I.B. Oliveira, A.T. Patera and D.V. Rovas, CR Acad Sci Paris, Solution of large scale parametric eigenvalue problems arising from brake squeal modeling, S. Quraishi, C. Schröder and V. Mehrmann, Proceedings in Applied Mathematics and Mechanics, Reduced-basis method for band structure calculations, G.S.H. Pau, Phys. Rev. E 76 (2007), Feasability and competitiveness of a reduced basis approach for rapid electronic structure calculations in quantum chemistry, E. Cancès, C. Le Bris, Y. Maday, A.T. Patera and G.S.H. Pau, CRM Proceedings and lecture notes, A perturbation-method-baseda posteriori estimator for the planewave discretization of nonlinear Schrödinger equations, E. Cancès, G. Dusson, M. Maday, B. Stamm, and M. Vohralik, C. R. Math. 352 (2014), no. 11, 941?946. Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems, T. Horger, B. Wohlmuth and T. Dickopf, ESAIM: M2AN (2016). Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems, I. Fumagalli, A. Manzoni, N. Parolini and M. Verani, MOX-Report No. 16 (2015). Contributions mathématiques aux calculs de structures électronique, D. Gontier, PhD thesis (2015). 25 / 26

26 Thank you for your attention! 26 / 26

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