Christophe Prud homme 8 avril 2014

Transcription

1 Reduced Basis methods: an introduction Christophe Prud homme CeMosis - IRMA Université de Strasbourg 8 avril 2014

2 1 Motivations and Framework 2 3 Non Compliant Output and/or Non-Symmetric Elliptic Problems 4 5 6

3 Collaborators on the framework Cecile Daversin (IRMA - UdS, Strasbourg and CNRS/LNCMI, Grenoble) Christophe Prud homme (IRMA - UdS, Strasbourg) Alexandre Ancel (IRMA - UdS, Strasbourg) Christophe Trophime (CNRS/LNCMI, Grenoble) Stephane Veys (LJK - UJF, Grenoble) Karen Veroy (RWTH-Aachen) Martin Grepl (RWTH-Aachen) and former collaborators : S. Vallaghe, E. Schenone. Non-intrusive RB Rachida Chakir (LJLL - UPMC, Paris) Yvon Maday (LJLL - UPMC, Paris) Current Funding ANR/CHORUS, ANR/HAMM, FRAE/RB4FASTSIM, ANR/Vivabrain, Labex IRMIA, IDEX

4 Motivations Introduction Feel++ Features User Point of View Wrappers Context and Motivations Context : parametrized PDE input-parameter examples : geometric configuration, physical properties, boundary conditions, sources. output examples : mean temperature over a subdomain, flux on a boundary, etc. Motivation : rapid and reliable evaluation of input-output relationships. Real-time context : parameter-estimation, control. Many-query context : sensitivity analysis, multi-model simulation.

5 Motivations Introduction Feel++ Features User Point of View Wrappers OPUS Heat Transfer Benchmark

6 Motivations Introduction Feel++ Features User Point of View Wrappers Thermal Testcase Description y Hot air outflow Overview Ω43 Γ32 = Γ23 = Ω2 Ω3 hpcb Ω42 Γ31 = Γ13 = Ω1 Ω3 Ω41 Γ1 Γ1 epcb Γ3 eic Ω2(IC2) hic x2 Γ2 eic Ω1(IC1) hic x1 Γ4 ea Cooling air inflow (fan) Ω4(Air) = 4 i=1 Ω4i Ω44 Ω3(PCB) x Heat-Transfer with conduction and convection possibly coupled with Navier-Stokes Simple but complex enough to contain all difficulties to test the certified reduced basis non symmetric, non compliant steady/unsteady physical and geometrical parameters coupled models Testcase can be easily complexified

7 Thermal Testcase Description Ω43 Γ32 = Γ23 = Ω2 Ω3 hpcb y Γ1 x2 Hot air outflow Γ3 eic Ω2(IC2) hic Γ2 Ω4(Air) = 4 i=1 Ω4i Motivations Introduction Feel++ Features User Point of View Wrappers Heat transfer equation ρc i ( T t +v T ) (k i T ) = Q i, Inputs i = 1, 2, 3, 4 Ω42 eic Ω44 µ = {e a ; k Ic ; D; Q; r}. Γ1 Γ31 = Γ13 = Ω1 Ω3 Ω41 epcb Ω1(IC1) hic x1 Γ4 ea x Ω3(PCB) Outputs s 1 (µ) = 1 T e IC h IC Ω 2 Cooling air inflow (fan) s 2 (µ) = 1 e a Ω 4 Γ 3 T

8 Motivations Introduction Feel++ Features User Point of View Wrappers Thermal Testcase Description y Hot air outflow Γ3 Ω43 Fluid model Poiseuille flow or Navier-Stokes flow eic Ω4(Air) = 4 i=1 Ω4i Boundary conditions Γ32 = Γ23 = Ω2 Ω3 hpcb Ω42 Γ31 = Γ13 = Ω1 Ω3 Ω41 Γ1 Γ1 epcb Ω2(IC2) hic x2 Γ2 eic Ω1(IC1) hic x1 Γ4 ea Cooling air inflow (fan) x Ω44 Ω3(PCB) on Γ 3 Ω 3, a zero flux on Γ 3 Ω 4, outflow on Γ 4, (0 x e Pcb + e a, y = 0) temperature is set Γ 1 and Γ 2 periodic at interfaces between the ICs and PCB, thermal discontinuity (conductance) on other internal boundaries, the continuity of the heat flux and C. Prud homme Reduced temperature Basis methods: an introduction

9 Thermal Testcase Description Ω43 Γ32 = Γ23 = Ω2 Ω3 hpcb Ω42 Γ31 = Γ13 = Ω1 Ω3 Ω41 y Γ1 Γ1 epcb x2 x1 Hot air outflow Γ3 eic Ω2(IC2) eic Ω1(IC1) Γ4 ea hic hic Γ2 Cooling air inflow (fan) Ω4(Air) = 4 i=1 Ω4i x Ω44 Ω3(PCB) Motivations Introduction Feel++ Features User Point of View Wrappers Finite element method P k, k = 1,..., 4 Lagrange elements Weak treatment of Dirichlet conditions CIP Stabilisation Locally Discontinuous FEM functions Validation Comparison between Comsol(EADS) and Feel++ Extensive testing and comparisons Implementation validated, ref. config. max rel error < 1% Diff. : mesh, stabilisation, Dirichlet

10 Motivations Introduction Feel++ Features User Point of View Wrappers Thermal Testcase Description y Hot air outflow Ω43 Γ3 eic Ω4(Air) = 4 i=1 Ω4i Γ32 = Γ23 = Ω2 Ω3 Ω2(IC2) hic x2 hpcb Γ1 Γ2 Ω42 Ω44 eic Γ1 Γ31 = Γ13 = Ω1 Ω3 Ω1(IC1) hic x1 Ω41 epcb Γ4 ea x Ω3(PCB) Figure : Temperature plot for e a {2.5e 3, 8e 3, 5e 2} Cooling air inflow (fan)

11 Motivations Introduction Feel++ Features User Point of View Wrappers HiFiMagnet project High Field Magnet Modeling

12 Motivations Introduction Feel++ Features User Point of View Wrappers Laboratoire National des Champs Magnétiques Intenses Large scale user facility in France High magnetic field : from 24 T Grenoble : continuous magnetic field (36 T) Toulouse : pulsed magnetic field (90 T) Application domains Magnetoscience Solide state physic Chemistry Biochemistry Magnetic Field Earth : T Supraconductors : 24T Continuous field : 36T Pulsed field : 90T Access Call for Magnet Time : 2 per year 140 projects per year

13 Motivations Introduction Feel++ Features User Point of View Wrappers High Field Magnet Modeling

14 Motivations Introduction Feel++ Features User Point of View Wrappers Why use Reduced Basis Methods? Challenges Modeling : multi-physics non-linear models, complex geometries, genericity Account for uncertainties : uncertainty quantification, sensitivity analysis Optimization : shape of magnets, robustness of design Objective 1 : Fast Complex geometries Large number of dofs Uncertainty quantification Large number of runs Objective 2 : Reliable Field quality Design optimization Certified bounds Reach material limits

15 Motivations Introduction Feel++ Features User Point of View Wrappers An open reduced basis framework Objectives Provide an open framework for certified reduced basis methods Provide a rapid prototyping framework using the Feel++ language for the standard finite/spectral element methods Provide interfaces to various open "mathematical" programming environments such as Python/OpenTURNS(UQ) or Octave Where to get it? sources are available at available as Debian/Ubuntu packages

16 Motivations Introduction Feel++ Features User Point of View Wrappers Feel++ Features Galerkin methods (fem,sem, cg, dg) in 1D, 2D and 3D on simplices and hypercubes Interfaces to PETSc/SLEPc Language embedded in C++ close to variational formulation language that shortens tremendously the time to results // T h = {elements} auto mesh = loadmesh ( new Mesh < Simplex <3 > > ); // X h = {v C 0 (Ω) v K P 2(K), K T h } auto Xh = Pch <2 >( mesh ); auto u = Xh -> element (), v = Xh -> element (); auto a = form2 ( _test =Xh, _trial =Xh, ); // a(u, v) = u v Ω a= integrate ( elements ( mesh ), gradt (u)* trans ( grad (v ))); auto b= form2 ( _test =Xh, _trial = Xh ); // a(u, v) = u v Ω b = integrate ( elements ( mesh ), idt (u)* id(v) );

17 Motivations Introduction Feel++ Features User Point of View Wrappers Feel++ Reduced Basis Features Lower bound for coercivity/inf-sup OK 90% CRB Linear Elliptic case OK 100% (coercive) 90% (non-coercive) CRB Linear Parabolic case OK 100% Automatic differentiation OK 90% EIM (space/time) OK 100% CRB Nonlinear case Ok 80% CRB automated affine decomposition Not OK 50% CRB for multiphysics OK 80%

18 Motivations Introduction Feel++ Features User Point of View Wrappers Feel++ Framework : the User point of view Parametric FEM Cmd line Python Specifications Geometry, PDE,... µ0, µ1,..., µp, t Work in progress Affine decomposition to be automated Code generator CRB SCM EIM Offline/Online Database handling Auto. diff.... Octave Gmsh/OneLab

19 Motivations Introduction Feel++ Features User Point of View Wrappers Wrappers : Python/OpenTURNS Wrappers are automatically generated by the framework Python Code (OpenTURNS wrapper for UQ) # fem code modelfem = NumericalMathFunction (" modelfem ") # crb code modelrb = NumericalMathFunction (" modelrb ") mu [0] = 10 mu [1] = 7e -3 outputfem = modelfem ( mu) outputrb = modelrb ( mu)

20 Motivations Introduction Feel++ Features User Point of View Wrappers Wrapper : Octave Wrappers are automatically generated by the framework Octave code # kic : thermal conductivity ( default : 2) inp (1) = 1.0 e +1; # D : fluid flow rate ( default : 5e -3) inp (2) = 7.0e -3; # Q : heat flux ( default : 1 e6) inp (3) = 1.0 e +6; # r : conductance ( default : 100) inp (4) = 1.0 e +2; # ea : length air flow channel ( default : 4e -3) inp (5) = 4.0e -3; for i =1: N inp (1)= 0.2+( i -1)*( )/ N; D =[ D inp (1)]; s= [ s opuseadscrb ( inp )]; end

21 Motivations Introduction Feel++ Features User Point of View Wrappers Reduced Basis Methodology

22 RB Approx. A Posteriori Error Sampling Linear Compliant Elliptic Problems

23 RB Approx. A Posteriori Error Sampling Notations and Definitions Notations ( ) N finite element approximation ( ) N reduced basis approximation µ input parameter (physical, geometrical,...) s(t; µ) s N (t; µ) s N (t; µ) output approximations µ s(t; µ) input-output relationship Definitions Ω R d spatial domain µ P-uplet D µ R P parameter space s output, l, f functionals u field variable X function space H 1 0 (Ω)ν X H 1 (Ω) ν (ν = 1 for simplicity) (, ) X scalar product and X norm associated to X

24 RB Approx. A Posteriori Error Sampling Problem Statement The formal problem statement reads : Given µ D µ, evaluate where u(x; µ) X satisfies s(µ) = l(u(µ); µ) a(u(µ), v; µ) = f (v; µ), v X Remark We consider first the case of linear affine compliant elliptic problem and then complexify

25 RB Approx. A Posteriori Error Sampling Hypothesis : Reference Geometry In these notes Ω is considered parameter independent To apply the reduced basis methodology exposed later, we need to setup a reference spatial domain Ω ref We introduce an affine mapping T ( ; µ) : Ω( Ω ref = Ω o ( µ)) Ω o (µ) such that a(u, v; µ) = a o (u o T µ, v o T µ ; µ)

26 RB Approx. A Posteriori Error Sampling Hypothesis : Continuity, stability, compliance We consider the following µ PDE a(, ; µ) f ( ; µ), l( ; µ) bilinear symmetric continuous coercive ( µ D µ ) linear bounded ( µ D µ ) and in particular, to start, the compliant case a symmetric f ( ; µ) = l( ; µ) µ D µ

27 RB Approx. A Posteriori Error Sampling Hypothesis : Affine dependence in the parameter We require for the RB methodology where for q = 1,..., Q a Q a a(u, v; µ) = Θ q a(µ) a q (u, v), q=1 Θ q a : D µ R µ dependent functions a q : X X R µ independent bilinear forms Remark similar decomposition is required for l(v; µ) and f (v; µ), and denote Q l and Q f the corresponding number of terms applicable to a large class of problems including geometric variations can be relaxed (see non affine/non linear problems)

28 RB Approx. A Posteriori Error Sampling Inner Products and Norms We next define the energy inner product and associated norm (parameter dependent) (((w, v))) µ = a(w, v; µ) v µ = a(v, v; µ) u, v X v X X -inner product and associated norm (parameter independent) (w, v) X = (((w, v))) µ ( a(w, v; µ)) v X = v µ ( a(v, v; µ)) u, v X v X

29 RB Approx. A Posteriori Error Sampling Coercivity and Continuity Constants Recall that coercivity constant Continuity constant a(v, v; µ) (0 <)α(µ) inf v X v 2 X a(w, v; µ) γ(µ) sup sup (< ) w X v X w X v X

30 RB Approx. A Posteriori Error Sampling Example Thermal Block : Heat Transfer Γ top (Zero Dirichlet) µ 7 µ 8 µ 9 µ 4 µ 5 µ 6 Γ sides (Insulated) µ 1 µ 2 µ 3 Γ 0 (Heat Flux)

31 RB Approx. A Posteriori Error Sampling Example Thermal Block : Problem statement Given µ (µ 1,...µ P ) D µ [µ min, µ max ] P, evaluate (recall that l = f ) s(µ) = f (u(µ)) where u(µ) X {v H 1 (Ω), v Γtop = 0} satisfies a(u(µ), v; µ) = f (v; µ) v X we have P = 8 and given 1 < µ r < we set such that µ max /µ min = µ r. µ min = 1/ µ r, µ max = µ r

32 RB Approx. A Posteriori Error Sampling Example Thermal Block Recall we are in the compliant case l = f, we have f (v) = v X Γ 0 v and a(u, v; µ) = P i=1 µ i u v + 1 u v Ω i Ω P+1 u, v X where Ω = P+1 i=1 Ω i.

33 RB Approx. A Posteriori Error Sampling Example Thermal Block The inner product is defined as follows (u, v) X = P i=1 µ i u v + 1 u v Ω i Ω P+1 where µ i is a reference parameter. We have readily that a is symmetric parametrically coercive 0 < 1 µr min(µ 1 / µ 1,..., µ P / µ P, 1) α(µ) and continuous γ(µ) max(µ 1 / µ 1,..., µ P / µ P, 1) µ r < and the linear form f is bounded.

34 RB Approx. A Posteriori Error Sampling Example Thermal Block : Affine decomposition We obtain the affine decomposition P+1 a(u, v; µ) = Θ q (µ)a q (u, v) q=1 with Θ 1 (µ) = µ 1 a 1 (u, v) = u v Ω 1. Θ P (µ) = µ P a P (u, v) = u v Ω P Θ P+1 (µ) = 1 a P+1 (u, v) = u v Ω P+1

35 Example Thermal Block Homogeneous parameters Maximum parameters values. RB Approx. A Posteriori Error Sampling Minimum parameters values. Heterogeneous parameters

36 RB Approx. A Posteriori Error Sampling Truth FEM Approximation Let µ D µ, evaluate where u N (µ) X N satisfies s N (µ) = l(u N (µ)), a(u N (µ), v; µ) = f (v), v X N. Here X N X is a Truth finite element approximation of dimension N 1 equiped with an inner product (, ) X and induced norm X. Denote also X and associated norm. l X, l X sup v X l(v) v X

37 RB Approx. A Posteriori Error Sampling Purpose Equate u(µ) and u N (µ) in the sense that u(µ) u N (µ) X tol µ D µ Build the reduced basis approximation using the FEM approximation Measure the error associated with the reduced basis approximation relative to the FEM approximation

38 RB Approx. A Posteriori Error Sampling Reduced Basis Objectives For any given accuracy ɛ, evaluate Accuracy µ D µ s N (µ)( s N (µ)) and s N(µ) that provably achieves the desired accuracy Reliability s N (µ) s N (µ) s N(µ) ɛ for a very low cost t comp Efficiency Independent of N as N where t comp is the time to perform the input-output relationship µ (s N (µ), s N(µ))

39 RB Approx. A Posteriori Error Sampling Reduced Basis Objective : Rapid Convergence Build a rapidly convergent approximation of s N (µ) R and u N (µ) X N X N X such that for all µ, we have s N (µ) s N (µ) and u N (µ) u N (µ) rapidly as N = dimx N (= ) (and independently of N )

40 RB Approx. A Posteriori Error Sampling Reduced Basis Objective : Reliability and Sharpness Provide a posteriori error bound N (µ) and s N (µ) : N (µ) 1 u N (µ) u N (µ) X E and 1 for all N = 1... N max and µ D µ. s N (µ) s N (µ) s N (µ) E

41 RB Approx. A Posteriori Error Sampling Reduced Basis Objective : Efficiency Develop a two stage strategy : Offline/Online Offline : very expensive pre-processing, we have typically that for a given µ D µ tcomp offline >> t µ sn (µ) comp Online : very rapid convergent certified reduced basis input-output relationship independent of N t online comp Remark N may/should be chosen conservatively

42 RB Approx. A Posteriori Error Sampling Approximation opportunities : Low-Dimension Manifold Y H 1 (Ω R d ) To approximate u(µ), and thus s(µ), we need not represent all functions in Y

43 RB Approx. A Posteriori Error Sampling Approximation opportunities : Low-Dimension Manifold W = {u(µ) Y ; µ D µ } Y H 1 (Ω R d ) To approximate u(µ), and thus s(µ), we need only approximate functions in low-dimensional manifold W = {u(µ) Y ; µ D µ }

44 RB Approx. A Posteriori Error Sampling Approximation opportunities : Low-Dimension Manifold W N = {u(µ i ) Y ; µ i D µ } u(µ 2 ) u(µ 3 ) u(µ 4 ) x u(µ 1 x ) x x Y H 1 (Ω R d ) To approximate u(µ), and thus s(µ), we construct the approximation space W N = {u(µ i ) Y ; (µ i ) i=1..n D µ }

45 RB Approx. A Posteriori Error Sampling Spaces Parameter Samples : with Sample : S N = {µ 1 D µ,..., µ N D µ } 1 N N max, Lagrangian Hierarchical Space with Sampling strategies? S 1 S 2... S Nmax 1 S Nmax D µ W N = span {ζ n u(µ n ), n = 1,..., N}. }{{} u N (µ n ) W 1 W 2... W Nmax X N X Equidistributed points in D µ (curse of dimensionality) Log-random distributed points in D µ See later for more efficient, adaptive strategies

46 RB Approx. A Posteriori Error Sampling Spaces : Remarks Remark We could include µ-derivatives (evaluation of the derivatives at given parameters) into W N to obtain Hermite-like spaces (not covered here). Remark The basis functions ζ n = u(µ n ), n = 1...N max of W N are orthonormalized via a Gram-Schmidt process. We then denote X N = Gram-Schmidt(W N )

47 RB Approx. A Posteriori Error Sampling Formulation (Linear Compliant Case) : a Galerkin method Galerkin Projection Given µ D µ evaluate where u N (µ) X N satisfies s N (µ) = f (u N (µ); µ) (1) a(u N (µ), v; µ) = f (v; µ), v X N.

48 RB Approx. A Posteriori Error Sampling Formulation (Linear Compliant Case) : Optimality For any µ D µ, we have the following optimality results (thanks to Galerkin) u(µ) u N (µ) µ = inf u(µ) v N (µ) µ, v N X N γ(µ) u(µ) u N (µ) X inf u(µ) v N (µ) X, α(µ) v N X N and and finally s(µ) s N (µ) = u(µ) u N (µ) 2 µ, = inf v N X N u(µ) v N (µ) 2 µ, 0 s(µ) s N (µ) γ(µ) inf v N X N u(µ) v N (µ) 2 X

49 RB Approx. A Posteriori Error Sampling Formulation (Linear Compliant Case) : offline-online decomposition Expand our RB approximations : N u N (µ) = u N j (µ) ζ j (2) Express s N (µ) j=1 j=1 N Q f s N (µ) = u N j (µ) Θ q f (µ) f q (ζ j ) q=1 where u N i (µ), 1 i N satisfies { N Qa } Θ q a(µ) a q (ζ i, ζ j ) u N j (µ) = j=1 q=1 Q f q=1 Θq f (µ) f q (ζ i ), (3) 1 i N (4) (5)

50 RB Approx. A Posteriori Error Sampling Formulation (Linear Compliant Case) : matrix form Solve where A N (µ) u N (µ) = F N (A N ) i j (µ) = Q a q=1 Θ q a(µ) a q (ζ i, ζ j ), F N i = Q f q=1 Θ q f (µ)f q (ζ i ). 1 i, j N, 1 i N

51 RB Approx. A Posteriori Error Sampling Formulation (Linear Compliant Case) : complexity analysis Offline : independent of µ Solve : N FEM system depending on N Form and store : f q (ζ i ) Form and store : a q (ζ i, ζ j ) Online : independent of N Given a new µ D µ Form and solve A N (µ) : O(QN 2 ) and O(N 3 ) Compute s N (µ) Online : N << N Online we realize often orders of magnitude computational economies relative to FEM in the context of many µ-queries

52 RB Approx. A Posteriori Error Sampling Formulation (Linear Compliant Case) : Condition number Proposition Thanks to the orthonormalization of the basis function, we have that the condition number of A N (µ) is bounded by the ratio γ(µ)/α(µ). Démonstration. Write the Rayleigh Quotient Express vn T A N(µ)v N vn T v, v N R N N v N = N v Nn ζ n n=1 Use coercivity, continuity and orthonormality.

53 RB Approx. A Posteriori Error Sampling Questions What is the accuracy of u N (µ) and s N (µ)? Online u(µ) u N (µ) X ɛ tol, µ D µ s(µ) s N (µ) X ɛ s tol, µ D µ What is the best value for N? Offline/Online N too large computational inefficiency N too small unacceptable error How should we build S N? is there an optimal construction? Offline Good approximation of the manifold M through the RB space, but need for well conditioned RB matrices

54 RB Approx. A Posteriori Error Sampling A Posteriori Error Estimation : Requirements We shall develop the following error bounds N (µ) and s N (µ) with the following properties rigorous s(µ) s N (µ) s N(µ), µ D µ 1 N N max u(µ) u N (µ) X N (µ), µ D µ reasonably sharp 1 N N max N (µ) u(µ) u N (µ) X C, s N (µ) s(µ) s N (µ) C, C 1 efficient Online cost depend only on Q and N and not N

55 RB Approx. A Posteriori Error Sampling u N (µ) : Error equation and residual dual norm Given our RB approximation u N (µ), we have that satisfies e(µ) u(µ) u N (µ) a(e(µ), v; µ) = r(u N (µ), v; µ), v X where r(u N (µ), v; µ) = f (v) a(u N (µ), v; µ) is the residual. We have then from coercivity and the definitions above that e(µ) X r(u N(µ), v; µ) X α(µ) = ε N(µ) α(µ)

56 RB Approx. A Posteriori Error Sampling A Posteriori error estimation : Dual norm of the residual Proposition Given µ D µ, the dual norm of r(u N (µ), ; µ) is defined as follows where ê(µ) X satisfies r(u N (µ), ; µ) X r(u N (µ), v; µ) sup v X v X = ê(µ) X (ê(µ), v) X = r(u N (µ), v; µ) The error residual equation can then be rewritten a(e(µ), v; µ) = (ê(µ), v) X, v X

57 RB Approx. A Posteriori Error Sampling u N (µ) : Definitions of energy error bounds and effectivity Given α LB (µ) a nonnegative lower bound of α(µ) : α(µ) α LB (µ) ɛ α α(µ), ɛ α ]0, 1[, µ D µ (6) Denote ε N (µ) = ê(µ) X = r(u N (µ), v; µ X Definition : Energy error bound N (µ) ε N(µ) αlb (µ) (7) Definition : Effectivity η N (µ) N(µ) e(µ) µ (8)

58 RB Approx. A Posteriori Error Sampling u N (µ) : Rigorous sharp error bounds One can prove that Remarks 1 η N (µ) γ UB (µ) α LB (µ), 1 N N max, µ D µ (9) Rigorous : Left inequality ensures rigorous upper bound measured in X, i.e. e(µ) X N (µ), µ D µ Sharp : Right inequality states that N (µ) overestimates the true error by at most γ(µ)/α LB (µ)

59 RB Approx. A Posteriori Error Sampling s N (µ) : error bounds It follows from (100) and (102) s(µ) s N (µ) s N(µ), µ D µ (10) where s N(µ) = N(µ) 2 α LB (µ) (11) Rapid convergence of the error in the output Note that the error in the output vanishes quadratically

60 RB Approx. A Posteriori Error Sampling Offline-Online decomposition Denote ê(µ) X ê(µ) X = ε N (µ) = r(u N (µ), ; µ) X such that And recall that (ê(µ), v) X = r(u N (µ), v; µ), v X Q N r(u N (µ), v; µ) = f (v) q=1 n=1 Θ q (µ) u N n (µ) a q (ζ n, v), v X

61 RB Approx. A Posteriori Error Sampling Offline-Online decomposition It follows next that ê(µ) X satisfies (ê(µ), v) X = f (v) Q N q=1 n=1 Θ q (µ) u N n (µ) a q (ζ n, v), v X Observe then that the rhs is the sum of products of parameter dependent functions and parameter independent linear functionals, thus invoking linear superposition Q N ê(µ) = C Θ q (µ) u N n (µ) L q n where C X satisfies L X satisfies q=1 n=1 (C, v) = f (v), v X (L q n, v) X = a q (ζ n, v), v X, 1 n N, 1 q Q which are parameter C. Prud homme independent Reduced problems Basis methods: an introduction

62 RB Approx. A Posteriori Error Sampling Offline-Online decomposition : Error bounds From (20) we get ê(µ) 2 X = (C, C) X + Q q=1 N n=1 Θq (µ) u N n (µ) { 2(C, L q n) X + Q N q =1 n =1 (µ) u Θq N n (µ) (L q n, L q n ) X } (12) Remark In (12), ê(µ) 2 X is the sum of products of parameter dependent (simple/known) functions and parameter independent inner-product, the offline-online for the error bounds is now clear.

63 RB Approx. A Posteriori Error Sampling Offline-Online decomposition : steps and complexity Offline : Solve for C and L q n, 1 n N, 1 q Q Form and save (C, C) X, (C, L q n) X and (L q n, L q n ) X, 1 n, n N, 1 q, q Q Online Given a new µ D µ Evaluate the sum ê(µ) 2 X (12) in terms of Θq (µ) and u N n (µ) Complexity in O(Q 2 N 2 ) independent of N

64 RB Approx. A Posteriori Error Sampling Offline-Online Scenarii Offline Given a tolerance τ, build S N and W N s.t. µ P D µ, N (µ) < τ Online Given µ and a tolerance τ, find N and thus s N (µ) s.t. N = arg max N ( N (µ) < τ) or given µ and a max execution time T, find N and thus s N (µ) s.t. N = arg min N ( N (µ) and execution time < T )

65 RB Approx. A Posteriori Error Sampling S N and W N Generation Strategies Offline Generation Given a tolerance ɛ, set N = 0 and S 0 = While max N N = N + 1 > ɛ If N == 1 ; then Pick ((log-)randomly) µ 1 D µ Build S N := {µ N } S N 1 Build W N := {ξ = u(µ N )} W N 1 Compute max N := max µ D µ N (µ) µ N+1 := arg max µ D µ N (µ) End While Condition number recall that the ζ n are orthonormalized, this ensures that the condition number will stay bounded by γ(µ)/α(µ)

66 RB Approx. A Posteriori Error Sampling Online Algorithm I µ adaptive online Given µ D µ, compute (s N (µ), N (µ)) such that N (µ) < τ. N = 2 While N (µ) > τ Compute (s N (µ), N (µ)) using (S N, W N ) N = N 2 use the (very) fast convergence properties of RB End While

67 RB Approx. A Posteriori Error Sampling Online Algorithm II Offline While i <= Imax >> 1 Pick log-randomly µ D µ Store in table T, N (µ) if worst case for N = 1,..., N max i = i + 1 ; End While Online Algorithm II µ adaptive online worst case Given µ D µ, compute (s N (µ), N (µ)) such that N (µ) < τ. N := argmax T N (µ) < τ Use W N to compute (s N (µ), N (µ))

68 RB Approx. A Posteriori Error Sampling Online Algorithm II Offline While i <= Imax >> 1 Pick log-randomly µ D µ Store in table T, N (µ) if worst case for N = 1,..., N max i = i + 1 ; End While Online Algorithm II µ adaptive online worst case Given µ D µ, compute (s N (µ), N (µ)) such that N (µ) < τ. N := argmax T N (µ) < τ Use W N to compute (s N (µ), N (µ))

69 RB Approx. A Posteriori Error Sampling Online Algorithm II Offline While i <= Imax >> 1 Pick log-randomly µ D µ Store in table T, N (µ) if worst case for N = 1,..., N max i = i + 1 ; End While Online Algorithm II µ adaptive online worst case Given µ D µ, compute (s N (µ), N (µ)) such that N (µ) < τ. N := argmax T N (µ) < τ Use W N to compute (s N (µ), N (µ))

70 RB Approx. A Posteriori Error Sampling Lower bound for coercivity constant We require a lower bound α LB (µ) for α(µ) = α c (µ), µ D µ Two strategies are available : Min Theta approach if a is parametrically coercive (i.e. the coercivity constant depends solely on µ) and more generally the Successive Constraint Method(SCM) which can also be applied in case of Inf-Sup stable problems (Stokes, Helmholtz,...)

71 RB Approx. A Posteriori Error Sampling Min Theta approach : Lower bound for α(µ) Θ q (µ) > 0, µ D µ and a q (v, v) 0, v X, 1 q Q We define Θ min, µ Θ q (µ) a (µ) = min q=1...q Θ q ( µ) α(µ) for µ D µ which was used to define the X -inner product and induced norm (u, v) X = a(u, v; µ), v X = (u, v) X, u, v X v X

72 RB Approx. A Posteriori Error Sampling Min Theta approach : Upper bound for γ(µ) Similarly we develop an upper bound γ UB (µ) for γ(µ). We define > Θ max, µ Θ q (µ) a (µ) = max q=1...q Θ q ( µ) γ(µ) Remark γ UB (µ) is actually not required in practice but relevant in the theory.

73 RB Approx. A Posteriori Error Sampling Min Theta approach : Summary if a is parametrically coercive we then choose the coercivity constant lower bound to be α LB (µ) Θ min, µ a (µ) and the continuity constant upper bound to be (a symmetric) Remark γ UB (µ) Θ max, µ a (µ) Online cost to evaluate α LB (µ) : O(Q a ) Choice of inner product important (u, v) X = a(u, v; µ) (see multiple inner products approach) Extends to non-symmetric problems by considering the symmetric part a s (u, v; µ) = 1 (a(u, v; µ) + a(v, u; µ)) 2

74 RB Approx. A Posteriori Error Sampling Stability estimates We wish to compute α LB : D R such that 0 < α LB (µ) α N (µ), µ D (13) and it computation is rapid O(1) where α N (µ) = inf w X N a(w, w; µ) w 2 X (14) Computation of α N (µ) α N (µ) is the minimum eigenvalue of the following generalized eigenvalue problem a(w, v; µ) = λ(µ) m(w, v; µ), (Aw = λbw) (15) where m(, ) is the bi-linear form associated with X and B is the associated matrix.

75 RB Approx. A Posteriori Error Sampling Reformulation The problem as a minimization one First recall Q a a(w, v; µ) = θ q (µ) a q (w, v) Hence we have q=1 α N Q (µ) = inf w X N a θ q (µ) a q(w, w) w 2 X q=1 and we note J obj (w; µ) = Q a q=1 θ q (µ) a q(w, w) w 2 X

76 RB Approx. A Posteriori Error Sampling Reformulation We have the following optimisation problem where and Y = α N (µ) = inf y Y J obj (µ; y) J obj (µ; y) Q a q=1 θ q (µ)y q { y R Qa w X N s.t. y q = a q(w, w) w 2 X N, 1 q Q a } We now need to characterize Y, to do this we construct two sets Y LB and Y UB such that Y UB Y Y LB over which finding α N (µ) is feasible.

77 RB Approx. A Posteriori Error Sampling First we set the design space for the minimisation problem (69). We introduce Q a B = q=1 [ inf w X N Ξ = a q (w, w) w 2 ; sup w X N X { } µ i D; i = 1,..., J a q (w, w) ] w 2 X and C K = { } µ i Ξ; i = 1,..., K Ξ Ξ is constructed using a 1 2 p division of D : in 1D, 0, 1; 1 2 ; 1 4, 3 4 ;... C K will be constructed using a greedy algorithm. Finally we shall denote P M (µ; E) the set of M points closest to µ in the set E. We shall need this type of set to construct the lower bounds.

78 Computing α LB (µ; C K ) is in fact a linear program with Q a design variables, y q, and 2Q a + M α + M + constraints online. It requires the construction of C K offline. Motivations and Framework RB Approx. A Posteriori Error Sampling Lower bounds : Y LB Given M α, M + N we are now ready to define Y LB { Y LB (µ; C K ) = y R Qa y B, Q a q=1 Q a q=1 θ q (µ )y q α N (µ ), µ P Mα (µ; C K ) } θ q (µ )y q α LB (µ ; C K 1 ), µ P M+ (µ; Ξ\C K ) We now set α LB (µ; C K ) = inf y YLB(µ;CK ) J obj (µ; y)

79 RB Approx. A Posteriori Error Sampling Let with We set Y UB (C K ) = { } y (µ k ), 1 k K y (µ) = argmin y Y J obj (µ; y) α UB (µ; C K ) = inf y YUB (C K ) J obj (µ; y) Y UB requires K eigensolves to compute the eigenmode η k associated with w k, k = 1,..., K and KQN inner products to compute the yq (w k ) = aq(η k,η k ;µ), k = 1,..., K offline. Then computing α η k 2 UB (µ; C K ) is X a simple enumeration N online.

80 RB Approx. A Posteriori Error Sampling [C Kmax ] = Greedy(Ξ, ɛ) Given Ξ and ɛ [0; 1] While max µ Ξ α UB (µ;c K ) α LB (µ;c K ) α UB (µ;c K ) > ɛ µ K+1 = argmax µ Ξ α UB (µ;c K ) α LB (µ;c K ) α UB (µ;c K ) C K+1 = C K {µ K+1 } K K + 1 set K max = K

81 RB Approx. A Posteriori Error Sampling Offline-Online Offline 2Q a + M α + M + eigensolves α N (µ), y (µ k ) n Ξ K max LP(Q, M α, M + ) to build C Kmax K max Q inner products over X N Y UB [α LB(µ)] = Online(µ, C Kmax, M α, M +) Given µ D sort over C Kmax P Mα (µ; C Kmax ) and P M+ (µ; Ξ\C Kmax ) (M α + M + + 2)Q a evaluation of θ q (µ ) M α lookups to get µ α N (µ ) LP(Q a, M α, M+) to get α LB (µ)

82 RB Approx. A Posteriori Error Sampling Example Thermal Block Configuration : dofs ; Preconditionner : LU Solver : MUMPS Ξ : parameter sampling of dimension Plot max µ Ξ s N (µ) s N (µ) s N (µ) 10 1 Relative error s N (µ) sn(µ) s N (µ) s N (µ) N

83 RB Approx. A Posteriori Error Sampling Example Thermal Block More parameters there are, more rich the problem is ; Notations : e i (µ) is the relative error on the output when i parameters vary. Relative error e 1 (µ) e 2 (µ) e 3 (µ) e 4 (µ) e 5 (µ) e 6 (µ) e 7 (µ) e 8 (µ) N

84 RB Primal Only RB Primal-Dual Non Compliant Output and/or Non-Symmetric Elliptic Problems

85 RB Primal Only RB Primal-Dual Truth FEM Approximation Let µ D µ, evaluate s N (µ) = l(u N (µ)), where u N (µ) X N X satisfies a(u N (µ), v; µ) = f (v), v X N. and we suppose that a(, ; µ) is bilinear, f ( ; µ and l( ; µ) are linear f and l are bounded l f (non-compliance) and/or a is non-symmetric

86 RB Primal Only RB Primal-Dual Truth FEM Approximation : Hypothesis We assume that a : X N X N R is coercive Continuous a(v, v; µ) (0 <)α(µ) inf v X v 2 X a(w, v; µ) γ(µ) sup sup (< ) w X v X w X v X and enjoys affine parametric dependence Q a a(u, v; µ) = Θ q a(µ) a q (u, v), q=1 u, v X

87 RB Primal Only RB Primal-Dual Truth FEM Approximation : Inner products and Norms We next define the energy inner product and associated norm (parameter dependent) (((w, v))) µ = a s (w, v; µ) v µ = a S (v, v; µ) u, v X v X X -inner product and associated norm (parameter independent) (w, v) X = (((w, v))) µ ( a S (w, v; µ)) v X = v µ ( a S (v, v; µ)) u, v X v X where a s denotes the symmetric part of a.

88 RB Primal Only RB Primal-Dual Spaces Parameter Samples : with Sample : S N = {µ 1 D µ,..., µ N D µ } 1 N N max, Lagrangian Hierarchical Space S 1 S 2... S Nmax D µ W N = span {ζ n u(µ n ), n = 1,..., N}. }{{} u N (µ n ) with W 1 W 2... W Nmax X N X

89 RB Primal Only RB Primal-Dual Formulation : Galerkin method Galerkin Projection Given µ D µ evaluate where u N (µ) X N satisfies s N (µ) = l(u N (µ); µ) a(u N (µ), v; µ) = f (v; µ), v X N. RB Space X N = GramSchmidt(W N ) Well posed problem (there exists a unique solution : coercivity, continuity, linear independence)

90 RB Primal Only RB Primal-Dual Formulation : a priori convergence For any µ D µ, we have the following optimality(thanks to Galerkin) results ( u(µ) u N (µ) X 1 + γ(µ) ) inf u(µ) v N (µ) X, α(µ) v N X N and Remember that ( ) symmetry : 1 + γ(µ) α(µ) s(µ) s N (µ) C u(µ) v N (µ) X γ(µ) α(µ) ; compliance : quadratic convergence in the output (not the case anymore)

91 RB Primal Only RB Primal-Dual Formulation : a posteriori error estimation We wish to develop rigorous, sharp and efficient online a posteriori error estimation N (µ), s N (µ) such that µ Dµ u(µ) u N (µ) X N (µ) s(µ) s N (µ) s N(µ) Coercivity Lower Bound OK Error Bounds OK (using a s ) However two issues remain.

92 RB Primal Only RB Primal-Dual Formulation : coercivity lower bound For a non-symmetric we introduce Q as a s (u, v; µ) = Θ q a s (µ) as q (u, v), q=1 u, v X where a s (u, v; µ) = 1 (a(u, v; µ) + a(v, u; µ)) 2 We then apply either the min Θ approach if a s is parametrically coercive or the SCM (a s ) α LB (µ) Θ min, µ a s = min q {1...Q as Θ q a s (µ) Θ q a s ( µ)

93 RB Primal Only RB Primal-Dual Formulation : a posteriori error bounds Given our RB approximation u N (µ), we have that satisfies e(µ) u(µ) u N (µ) a(e(µ), v; µ) = r(u N (µ), v; µ), v X where r(u N (µ), v; µ) = f (v) a(u N (µ), v; µ) is the residual. We have then from coercivity and the definitions above that e(µ) X r(u N(µ), v; µ) X α(µ) = ε N(µ) α(µ)

94 RB Primal Only RB Primal-Dual A Posteriori error estimation : Dual norm of the residual Proposition Given µ D µ, the dual norm of r(u N (µ), ; µ) is defined as follows where ê(µ) X satisfies r(u N (µ), ; µ) X r(u N (µ), v; µ) sup v X v X = ê(µ) X (ê(µ), v) X = r(u N (µ), v; µ) The error residual equation can then be rewritten a(e(µ), v; µ) = (ê(µ), v) X, v X

95 RB Primal Only RB Primal-Dual A Posteriori error estimation : Dual norm of the residual Then we can define Definition : Energy error bound N (µ) ê(µ) X α LB (µ) Definition : Effectivity η N (µ) N(µ) e(µ) X Proposition for N = 1...N max, the effectivity η N (µ) verifies 1 η N (µ) γ UB(µ) α LB (µ), µ Dµ.

96 RB Primal Only RB Primal-Dual A Posteriori error estimation : Output error bound Then we can define Definition : Output error bound s N(µ) l(, µ) X N (µ) Definition : Output Effectivity η s N(µ) s N (µ) s(µ) s N (µ) Proposition for N = 1...N max, the error s(µ) s N (µ) verifies s(µ) s N (µ) s N(µ), µ D µ

97 RB Primal Only RB Primal-Dual A Posteriori error estimation : Remarks and Motivations for a Primal/Dual formulation Very similar to the compliant case : need only l(, µ) X find ê l X (Riesz representation) such that (ê l, v) X = l(v, µ), v X and apply offline-online decomposition similarly to other terms Rigorous error bounds Best approach if many outputs (little overhead), however in case of few outputs a primal-dual formulation is preferable 1 Loss of quadratic convergence 2 Effectivities possibly unbounded, ηn(µ) s l(, µ) X γ(µ) u(µ) u N (µ) X from output error bound (taking l = f ) and energy error bound.

98 RB Primal Only RB Primal-Dual Formulation (Linear Case) Sample : S N = {µ 1 D µ,..., µ N D µ }. Sample : S du N = {µ du D µ,..., µ du du N D µ }. du Space : W N = span {ζ n u(µ n ), n = 1,..., N}. }{{} u N (µ n ) Space : = span {ζn du Ψ(µ du n ) Sampling strategies? W du N du }{{} Ψ N (µ du n ), n = 1,..., Ndu }. Equidistributed points in D µ (curse of dimensionality) Log-random distributed points in D µ See later for more efficient, adaptive strategies

99 RB Primal Only RB Primal-Dual Formulation (Linear Case) : a Galerkin method Galerkin Projection Given µ D µ evaluate s N (µ) = l(u N (µ)) r(u N (µ), Ψ N du(µ); µ) ; where u N (µ) X N and Ψ N du(µ) X du N satisfy du and a(u N (µ), v; µ) = f (v), v X N. a(v, Ψ N du(µ) ; µ) = l(v; µ), v X du N du. Note that RB Space X N = GramSchmidt(W N ), XN du du = GramSchmidt(WN ) In general N N du (primal and dual are different problems)

100 RB Primal Only RB Primal-Dual Formulation (Linear Case) : back to the compliant case Recall that in compliance a is symmetric l = f such that Ψ(µ) = u(µ). We may take N du = N, SN du = S N and XN du = X N and get Ψ N (µ) = u N (µ) Compliant case The dual problem is never formed/solved We simply identify Ψ N (µ) = u N (µ) We get a 50% cost reduction

101 RB Primal Only RB Primal-Dual Formulation : A priory convergence Proposition For any µ D µ, we have ( ) s(µ) s N (µ) C inf u(µ) v N (µ) X v N X N ( ) inf ψ(µ) ψn du (µ) X vn du X du N Recovery of quadratic convergence for the output! Alternative : build RB space comprising both primal and dual basis functions ( output dual correction not needed however more costly and conditioning issues)

102 RB Primal Only RB Primal-Dual Formulation (Linear Case) : offline-online decomposition Expand our RB approximations : Express s N (µ) u N (µ) = Ψ N du(µ) = N j=1 N du j=1 u N j (µ) ζ j Ψ N j (µ) ζ du j s N (µ) = N u N j (µ) l(ζ j ) j=1 + N N du Q j=1 j =1 q=1 N du j=1 Ψ N j (µ) f (ζ du j ) u N j (µ) Ψ N j (µ) Θ q (µ) a q (ζ j, ζ du j )

103 RB Primal Only RB Primal-Dual Formulation (Linear Case) : offline-online decomposition u N i (µ), 1 i N and Ψ N i (µ), 1 i N du satisfy { N Q } Θ q (µ) a q (ζ i, ζ j ) u N j (µ) = f (ζ i ), j=1 { N du Q j=1 q=1 q=1 Θ q (µ) a q (ζi du, ζj du ) } Ψ N du j(µ) = 1 i N l(ζ du i ), 1 i N du

104 RB Primal Only RB Primal-Dual Formulation (Linear Case) : matrix form Solve A N (µ) u N (µ) = F N and where A du N du(µ) Ψ N du(µ) = L N (A N ) i j (µ) = Q q=1 Θq (µ) a q (ζ i, ζ j ), F N i = f (ζ i ). 1 i, j N 1 i N and (A du N ) du i j (µ) = Q q=1 Θq (µ) a q (ζi du, ζj du ), L N i = l(ζi du ). 1 i, j N du 1 i N du

105 Online : N, N du << N Online we realize often orders of magnitude computational economies relative to FEM in the context of many µ-queries Motivations and Framework RB Primal Only RB Primal-Dual Formulation (Linear Case) : complexity analysis Offline : independent of µ Solve : N + N du FEM system depending on N Form and store : f (ζ i ), l(ζ i ), f (ζi du ), l(ζi du ) Form and store : a q (ζ i, ζ j ), a q (ζi du, ζj du ), a q (ζ i, ζ du Online : independent of N Given a new µ D µ Form and solve A N (µ) : O(QN 2 ) and O(N 3 ) j ) Form and solve A N du du (µ) : O(QN du2 ) and O(N du3 ) Compute s N (µ)

106 RB Primal Only RB Primal-Dual u N (µ) : Error equation and residual dual norm Given our RB approximation u N (µ), we have that satisfies e(µ) u(µ) u N (µ) a(e(µ), v; µ) = r(u N (µ), v; µ), v X where r(u N (µ), v; µ) = f (v) a(u N (µ), v; µ) in the linear case is the residual. We have then e(µ) X r(u N(µ), v; µ) X α(µ) = ε N(µ) α(µ)

107 RB Primal Only RB Primal-Dual u N (µ) : Defintions of energy error bounds and effectivity Given α LB (µ) a nonnegative lower bound of α(µ) : α(µ) α LB (µ) ɛ α α(µ), ɛ α ]0, 1[, µ D µ Definition : Energy error bound N (µ) ε N(µ) α LB (µ) Definition : Effectivity η N (µ) N(µ) e(µ) X

108 RB Primal Only RB Primal-Dual u N (µ) : Rigorous sharp error bounds One can prove that Remarks 1 η N (µ) γ(µ) α LB (µ), 1 N N max, µ D µ Rigorous : Left inequality ensures rigorous upper bound measured in X, i.e. e(µ) X N (µ), µ D µ Sharp : Right inequality states that N (µ) overestimates the true error by at most γ(µ)/α LB (µ)

109 RB Primal Only RB Primal-Dual Ψ N (µ) : error bounds We have a similar result for the dual problem Ψ(µ) Ψ N du X du N (µ), 1 N du N du max, µ D µ where du N (µ) ε du N (µ) is the dual norm of the residual. εdu N (µ) α LB (µ) l( ) a(, Ψ N du(µ); µ) X α LB (µ)

110 RB Primal Only RB Primal-Dual s N (µ) : error bounds From primal and dual energy error bounds we have where s(µ) s N (µ) s N(µ), µ D µ s N(µ) ε N (µ) du N (µ)(µ) = α LB (µ) N (µ) du N (µ) Rapid convergence of the error in the output Note that the error in the output vanishes as the product of the error in the primal and dual error Back to compliance : a symmetric and l = f We obtain s N(µ) ε2 N (µ), µ Dµ α LB (µ)

111 RB Primal Only RB Primal-Dual Offline-Online decomposition (Primal problem) «««< HEADDual problem : similar treatmentdenote ê(µ) Y ê(µ) Y = ε N (µ) = g(u N (µ), ; µ) Y (16) such that (ê(µ), v) Y = g(u N (µ), v; µ), v Y (17) Recall that Q N g(u N (µ), v; µ) = f (v) Θ q (µ) u N n (µ) a q (ζ n, v), q=1 n=1 v X (18)

112 RB Primal Only RB Primal-Dual Offline-Online decomposition (Primal problem) It follows next that ê(µ) Y satisfies (ê(µ), v) Y = f (v) Q N q=1 n=1 Θ q (µ) u N n (µ) a q (ζ n, v), v X (19) Observe then that the rhs is the sum of products of parameter dependent functions and parameter independent linear functionals, thus invoking linear superposition Q N ê(µ) = C Θ q (µ) u N n (µ) L q n (20) C Y satisfies L Y satisfies q=1 n=1 (C, v) = f (v), v Y (21) (L q n, v) Y = a q (ζ n, v), v Y, 1 n N, 1 q Q (22) (22) are parameter C. Prud homme independentreduced Poisson Basis problems methods: an introduction

113 RB Primal Only RB Primal-Dual Offline-Online decomposition : Error bounds From (20) we get ê(µ) 2 Y = (C, C) Y + Q q=1 N n=1 Θq (µ) u N n (µ) { 2(C, L q n) Y + Q N q =1 n =1 (µ) u Θq N n (µ) (L q n, L q n ) Y } (23) Remark In (23), ê(µ) 2 Y is the sum of products of parameter dependent (simple/known) functions and parameter independent inner-product, the offline-online for the error bounds is now clear.

114 RB Primal Only RB Primal-Dual Offline-Online decomposition : steps and complexity Offline : Solve for C and L q n, 1 n N, 1 q Q Form and save (C, C) Y, (C, L q n) Y and (L q n, L q n ) Y, 1 n, n N, 1 q, q Q Online Given a new µ D µ Evaluate the sum (23) in terms of Θ q (µ) and u N n (µ) Complexity in O(Q 2 N 2 ) independent of N

115 RB Primal Only RB Primal-Dual The linear symmetric coercive case We require a lower bound β LB (µ) for β(µ) = α c (µ), µ D µ If Primal-Dual problem : similar treatment as in Primal-only formulation New ingredient introduced is the correction term for the output r(u N (µ), Ψ N du(µ); µ) = f (ψn du(µ); µ) a(u N(µ), ψn du (µ); µ) which requires cross terms between primal and dual problems f q (ζ du n ), 1 n N du, 1 q Q f a q (ζ n, ζ du n ), 1 n N, 1 n N du, 1 q Q a

116 Motivations Empirical Interpolation Method Non-Affine decomposition

117 Motivations Empirical Interpolation Method Non-Affine decomposition Notations NonLinear µ-parametrized PDE Set of parameters : µ D µ R p, Solution of the nonlinear µ-pde : u(µ) X H 1 (Ω R d ), PDE weak formulation : We look for u(µ) X such that g(u(µ), v; µ) = 0, v X.

118 Motivations Empirical Interpolation Method Non-Affine decomposition Truth approximation "Truth" FEM approximation X N X : finite element approximation of dimension N >> 1. u N (µ) X N is solution of g(u N (µ), v; µ) = 0, v X N. Solution strategies such as Newton or Picard iterations, e.g. given 0 u N, build the nonlinear iterates 1 u N,..., k u N,... where j(δ k u N (µ), v; k u N (µ), µ) = g( k u N (µ), v; µ) We recover the linear case. Equate u(µ) and u N (µ), i.e. δ k u N (µ) = k+1 u N (µ) k u N (µ). u(µ) u N (µ) X tol, µ D µ.

119 Motivations Empirical Interpolation Method Non-Affine decomposition Recovering Affine decomposition (Nonlinear-case) We wish to build a reduced basis approximation u N (µ), to apply the methodology recall that we require affine dependence. Generalized affine decomposition Suppose that we can build g AD (u(µ), v; µ) such that g (u(µ), v; µ) g AD (u(µ), v; µ) = and similarly for Q g M g q q=1 m=1 β qm g (µ; u(µ)) g qm (v), j (u(µ), v; µ; w(µ)) j AD (u(µ), v; µ; w(µ)) = Q j M j q q=1 m=1 β qm j (µ; w(µ)) j qm (u(µ), v), (24)

120 Motivations Empirical Interpolation Method Non-Affine decomposition Generalized affine decomposition Finite element or reduced basis approximations The Newton algorithm now reads offline(n ) and online (N) of the reduced basis methodology ) ( ) j AD (δ k u N N (µ), v; µ; k u N N (µ) = g k AD u N N (µ), v; µ, (25) for the increment δ k u N N (µ) defined by ( ) δ k u N N (µ) = k+1 u N N (µ) k u N N (µ). (26)

121 Motivations Empirical Interpolation Method Non-Affine decomposition Empirical Interpolation Method : Objective We are given a parametrized function g that depend on space x Ω and a parameter µ R P, σ(x; µ) and possibly a field u(µ) solution of a µ-pde σ(u(µ); x; µ) EIM : build σ M σ We wish to build the expansion σ M (u(µ); x; µ) = M β m (µ) q m (x) σ(u(µ); x; µ) m=1

122 Motivations Empirical Interpolation Method Non-Affine decomposition Empirical Interpolation Method : Objective One we have σ M we can for example compute integrals with varying µ Ω σ(x; µ) Ω σ M (x; µ) = M m=1 β m (µ) q m (x) Ω The last term Ω β(µ)) q m(x) (independent of µ) can be precomputed and hence it provides a very efficient method recover affine decomposition for bilinear/linear forms Ω σ(x; µ) u v Ω σ M (x; µ) u v = M m=1 β m (µ) q m (x) u v Ω

123 Motivations Empirical Interpolation Method Non-Affine decomposition Empirical Interpolation Method : Examples Consider the following function with one parameter µ and 2D space dimension σ(x; µ) = exp( ((x 0.5) 2 + (y 0.5) 2 )/(2µ 2 )) Consider the following function with the triplet parameter µ = (µ 1, µ 2, µ 3 ) σ(u(µ); x; µ) = where u(µ) X is solution of the µ-pde µ µ 2 (u(µ) µ 3 ) a(u(µ), v; µ) = f (v; µ) v X

124 Motivations Empirical Interpolation Method Non-Affine decomposition Empirical Interpolation Method [Barrault et al., 2004] Ingredients Training set Ξ µ train Dµ Offline step Sample S M = {µ 1 Ξ µ train,..., µ M Ξ µ train }, Interpolation points t 1,..., t M Ω Approximation space W M = span{q 1(x),..., q M (x)} Residual r m(x) = σ(u N (µ m), x, µ m) σ m(u N (µ m), x, µ m) q m+1 (x) = rm(x) (matrix (B r m(t m) i,j ) = q j (t i ) lower triangular) Online step : Compute approximation coefficients β m (u N N (µ); µ) σ M (u N N (µ); t i ; µ) = M β m (u N N (µ); µ) q m (t i ) = σ(u N N (µ); t i ; µ) m=1 i = 1,..., M

125 Motivations Empirical Interpolation Method Non-Affine decomposition A language embedded in C++ : EIM expansion Let u be the solution of g(u, v; µ) = 0 v X N and σ(u) the non linear expression involving a field. parameterspace_ptrtype D; parameter_type mu; //µ D µ auto Pset = Dmu - > sampling (); int eim_ sampling_ size = 1000; Pset -> randomize ( eim_sampling_size ); // expression we want EIM expansion auto sigma = ref (mu (0))/(1+ ref (mu (2))*( idv (u)-u0 )); // call Feel ++ function eim auto eim_sigma = eim ( _model = solve ( g(u, v; µ; x) = 0 ), _element =u, // u N (µ) _parameter =mu, // µ _expr = sigma, // σ(u) _space =X N, _name =" eim_sigma ", _sampling = Pset ); // then we can have access to β coefficients // of EIM expansion M m=1 β(µ, un (µ)) qm(x) std :: vector < double > C. Prud homme beta_sigma Reduced = Basis eim_sigma methods: an -> introduction beta ( mu );

126 Motivations Empirical Interpolation Method Non-Affine decomposition Non-affine Non-Linear decomposition : Wrap-up Build EIM approximations of non-linear terms using the initial finite element approximation Generate databases of N independent terms and N dependent terms Optimisation opportunities in EIM right hand side online step evaluation σ(u N N (µ); t i ; µ) by storing the (FEM or RB) basis functions associated to u N N (µ) at the t i. Build the generalized affine decomposition of the non-linear problem (Newton or Picard iterations) Compute the RB approximations (and associated reduced quantities) using the FEM approximation of the generalized affine problem Store all the N -independent terms in database (we get rid of the finite element space dimension) and depend solely N, Q and the complexity of the generalized affine expansion.

127 Motivations Empirical Interpolation Method Non-Affine decomposition Affine decomposition in Feel++ given Q g, Q j and Q l the EIM determines (Mq g ) q=1,..,qg and (Mq) l q=1,..,ql such that we can write :, (M j q) q=1,..,qj and Q g M g q ( g k AD u(µ), v; µ ) = q=1 m=1 Q j M j q ( j AD u(µ), v; µ; k u(µ) ) = q=1 m=1 M l q Q l l AD (v; µ) = q=1 m=1 β qm g (µ; k u(µ)) g qm (v), β qm j (µ; k u(µ)) j qm (u(µ), v), β qm l (µ) l qm (v). The standard affine decomposition is in fact a special case of the generalized one.

128

129 Truth FEM Approximation Let I = (0, T f ), with T f is the final time. Let µ D µ and t I, evaluate where u N (t; µ) X N X satisfies s N (t; µ) = l(t; u N (µ)), ( u N ) (µ) m, v; µ + a(u N (µ), v; µ) = f (t; v), v X N t I. t a(, ; µ) and m(, ; µ) are bilinear, f ( ; µ) and l( ; µ) are linear f and l are bounded

130 Truth FEM Approximation : Inner products and Norms Let t the time step defined by t = T f K. w, v X and 1 k K we next define the energy inner product and associated norm (parameter dependent) ((( ))) w(t k ), v(t k ) µ v(t k ) = µ = m(w(t k ), v(t k ); µ)+ k a S (w(t k ), v(t k ); µ) t k =1 m(v(t k ), v(t k ); µ) + a S (v(t k ), v(t k ); µ) t X -inner product and associated norm (parameter independent) ( ) ((( ))) w(t k ), v(t k ) = w(t k ), v(t k ) w, v X, 1 k K X µ v(t k ) = v(t k ) v X, 1 k K X where a s denotes the symmetric part of a. µ

131 Spaces Parameter Samples : Sample : S N = {µ 1 D µ,..., µ N D µ } 1 N N max, with S 1 S 2... S Nmax D µ { } Let Snap(µ) = u N (t k, µ), 1 k K a snapshot set with µ S N { ( 1 W N = span ζ n arg inf v Snap(µ) K K u N (t k, µ) v ) } 2, n = 1,..., N. X k=1 with W 1 W 2... W Nmax X N X

132 A posteriori error estimation Given µ D µ we define ε N (t k ; µ) = r(u N (t k ; µ), v; µ) X, 1 k K. Let e(t k, µ) = u N (t k, µ) u N (t k, µ) X, we can write e(t k, µ) N (t k, µ) t α LB (µ) k ε N (t k ; µ) 2, 1 k K. k =1 µ D µ and 1 k K the output error bound is given by s N (t k, µ) s N (t k, µ) s N (t k, µ) N (t k, µ) du N (t k, µ).

133 Example Heat Shield : Problem statement { u + u t = 0 in Ω, boundaries conditions on Ω. Ωiso (10,4) (3,3) Ωext (1,1) (4,1) Ωint (0,0) Ω Boundaries conditions Ω ext : heat transfert with T air = 1 u n = Biot ext (u T air ); Ω int : heat transfert, T air = 0 u n = Biot int (u T air ); Ω iso : Insulated. 2 parameters Biot ext and Biot int.

134 Example Heat Shield final time : 20 s ; time step : 0.2 s ; Minimum parameters values. Maximum parameters values.

135 Example Heat Shield Configuration : dofs ; Preconditionner : LU Solver : MUMPS Ξ : parameter sampling of dimension 430. Let e s (T f ; µ) = sn (T f ;µ) s N (T f ;µ) s N (T f ;µ), plot max µ Ξ e s (T f ; µ) e s (T f ; µ) s N (Tf ; µ)/sn (T f ; µ) Relative error N

136 Example Heat Shield : Scalability Study the time to build the first reduced basis Configuration : dofs for FEM approximation. 10 RB construction Theoretical Speed up Number of processors

137 HiFiMagnet HiFiMagnet project High Field Magnet Modeling

138 HiFiMagnet Laboratoire National des Champs Magnétiques Intenses Large scale user facility in France High magnetic field : from 24 T Grenoble : continuous magnetic field (36 T) Toulouse : pulsed magnetic field (90 T) Application domains Magnetoscience Solide state physic Chemistry Biochemistry Magnetic Field Earth : T Supraconductors : 24T Continuous field : 36T Pulsed field : 90T Access Call for Magnet Time : 2 per year 140 projects per year

139 HiFiMagnet High Field Magnet Modeling

140 HiFiMagnet Why use Reduced Basis Methods? Challenges Modeling : multi-physics non-linear models, complex geometries, genericity Account for uncertainties : uncertainty quantification, sensitivity analysis Optimization : shape of magnets, robustness of design Objective 1 : Fast Complex geometries Large number of dofs Uncertainty quantification Large number of runs Objective 2 : Reliable Field quality Design optimization Certified bounds Reach material limits

141 HiFiMagnet Electro-thermal model V : electrical potential T : temperature { (σ(t ) V ) = 0 on Ω (k(t ) T ) = σ(t ) V V on Ω Boundary conditions Applied potential V = 0 on Bottom V = V Top on Top Water / Glue electrically isolant σ(t ) V n = 0 No thermic exchange with air k(t ) T n = 0 Thermic exchange (h) with cooling water (T w ) k(t ) T n = h(t T w ) Non-linearity Electrical conductivity σ 0 σ(t ) = 1 + α(t T 0 ) σ 0 = σ(ref = 20 C) T 0 = 20 C α = temperature coeff. Thermal conductivity k(t ) = LT σ(t ) L = Lorentz number

142 HiFiMagnet Electro-thermal model : Inputs/Outputs Parameters Material properties Electrical conductivity (σ 0 ) Temperature coeff (α) Lorentz number (L) µ = (σ 0, α, L, V Top, h, T w ) Operating conditions Applied potential (V D ) Heat transfert coeff. (h) Water temperature (T w ) Outputs s(µ) = l(v (µ), T (µ)) Possibilities for l : Mean temperature in the domain Magnetic field on specific point (Biot & Savart s law) Power of the magnet

143 HiFiMagnet Variationnal formulation { (σ(t ) V ) = 0 on ΩV (k(t ) T ) = σ(t ) V V on Ω T { V = VD on D V σ(t ) V n = V N on N V { σ(t ) T n = 0 on NT σ(t ) T n = T R1 T + T R2 on R T Electrical Potential Find V X H 1(Ω) such that φ V X : σ(t ) V φ V Ω Temperature Find T X H 1(Ω) such that φ T X : Ω σ(t )( V n)φ V + σ(t )( γ V φ V ( φ V n)v ) D V D V h s σ(t )( γ V D φ V ( φ V n)v D ) = 0 D V h s k(t ) T φ T + T R1 T φ T = σ(t ) V V φ T T R2 φ T R T Ω R T

144 HiFiMagnet Non-affine parameter dependance Considering first term of electrical potential formulation : σ 0 a v = σ(t ) V φ V = 1 + α(t T 0 ) V φ V Ω As σ 0 and α are input parameters : Ω a v q, θ v q a v (V, T ; µ) = q Θ v q(µ)a v q(v, T, φ V, φ T ) EIM : Empirical Interpolation Method Build an affine approximation a aff v of a v such that : a aff v = q Θ v q(µ)a v q(v, T, φ V, φ T ) exact on a set of interpolation points {t i } : av aff (t i ) = a v (t i ) i.

145 Electrical potential σ(t ) V φ V Ω σ(t ) = D V σ(t ) HiFiMagnet ( ( V n)φ V + γ ) V φ V ( φ V n)v h s D V σ(t )V D ( γ h s φ V ( φ V n)) = 0 σ α(t T 0 ) σaff (T ) = M σ m σ=0 β mσ (µ)q mσ (T ) Temperature k(t ) T φ T + T R1 T φ T = Ω R T k(t ) = σ(t )LT k aff (T ) = Ω σ(t ) V V φ T T R2 φ T R T M k m k =0 J(V, T ) = σ(t ) V V J aff (V, T ) = β mk (µ)q mk (T ) M J m J =0 β mj (µ)q mj (V, T )

146 HiFiMagnet Coupled formulation Find (V, T ) X X [H 1(Ω)] 2 such that (φ V, φ T ) X X : a((v, T ), (φ V, φ T ); µ) = f ((φ V, φ T ); µ) (φ V, φ T ) X X Affine decomposition Q a Mσ Θ q a(µ)a q ((V, T ), (φ V, φ T )) = q a=1 [ β mσ (µ) mσ =1 q mσ (T ) D V + = Mσ mσ =1 M k m k =1 M J m J =1 Q f q f =1 Θ q f (µ)f q ((φ V, φ T )) q mσ (T ) V φ V Ω ( ( V n)φ V + γ ) ] V φ V ( φ V n)v h s β mσ (µ)v D D V q mσ (T ) ( ) γ φ V ( φ V n) h s β mk (µ) q mk (T ) T φ T + T R1 T φ T Ω R T β mj (µ) q mj (V, T )φ T T R2 φ T Ω R T

147 HiFiMagnet Reduced Electro-thermal model From small towards large simulations

148 HiFiMagnet Parametric study on Bitter magnet 100 Maximum current density without thermal 80 damages? 60 Parametric study with j [30; 90].10 6 A.m 2 40 (other µ i fixed) Mean temperature ( C) Current density (A.m 2 ) 10 7 FEM CRB Figure : Bitter magnet 40 C to 60 C : + 1 Tesla

149 HiFiMagnet Performances Simulation characteristics Number of dofs : processors Number of dofs / proc : Number of inputs : 6 σ 0, α, L, U, h, T w Non-Linear model (20 Picard iter.) Number od Reduced Basis : 10 Error FEM / RB : FEM time 20min RB online time 16sec Gain factor : 72 Figure : Helix mesh

150 HiFiMagnet Error FEM/CRB Error estimation is not yet available for non-linear problems No error estimations : parameters are chosen randomly Parameter chosen : σ 0 = (S.m 1 ) α = (K 1 ) L = U = (V ) h = 82152(W.m 2.K 1 ) T w = (K) Error FEM/RB Number of basis (N) Figure : Error FEM/RB depending on reduced basis space size (N)

151 HiFiMagnet Sensitivity Analysis : CRB on helix sector Sobol indices : S i = V (E[Y X i]) V (Y ) σ 0 : α : L : U : 0.12 h : 0.24 T w : 0.62 Temperature range Mean of outputs : T = 367.8K 95C Standard deviation : 6, 2 Range for T : [361.6; 374](K) = [88.6; 101](C) Quantiles Determine a threshold q(γ) such that P(Y < q(γ)) > γ 99.0 % : 380 K = 107 C 80.0 % : K = C