Uncertainty Quantification: Does it need efficient linear algebra?

Transcription

1 USA trip October 2014 p. 1/62 Uncertainty Quantification: Does it need efficient linear algebra? David Silvester University of Manchester, UK Catherine Powell University of Manchester,UK

2 Yes. USA trip October 2014 p. 2/62

3 USA trip October 2014 p. 3/62 A framework for the development of implicit solvers for incompressible flow problems David Silvester University of Manchester, UK David Griffiths University of Dundee, Scotland

4 USA trip October 2014 p. 4/62 part I 1991 Incompressible flow: Navier Stokes equations fully implicit schemes and adaptive time stepping

5 USA trip October 2014 p. 5/62 part II 2011 PDEs with random data stochastic Galerkin and h-p adaptivity

6 USA trip October 2014 p. 6/62 Outline of the talk... Incompressible flow: Navier Stokes equations fully implicit schemes and adaptive time stepping PDEs with random data stochastic Galerkin and h-p adaptivity A proof-of-concept implementation: efficient linear algebra the (S)IFISS MATLAB Toolbox

7 USA trip October 2014 p. 6/62 Outline of the talk... Incompressible flow: Navier Stokes equations fully implicit schemes and adaptive time stepping PDEs with random data stochastic Galerkin and h-p adaptivity A proof-of-concept implementation: efficient linear algebra the (S)IFISS MATLAB Toolbox Howard Elman & Alison Ramage & David Silvester IFISS: A computational laboratory for investigating incompressible flow problems. SIAM Review, 56: , 2014.

8 IFISS HOME DOWNLOAD DOCUMENTATION PUBLICATIONS Incompressible Flow & Iterative Solver Software An open-source software package Summary IFISS is a graphical package for the interactive numerical study of incompressible flow problems which can be run under Matlab or Octave. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions. The package can also be used as a computational laboratory for experimenting with state-of-the-art preconditioned iterative solvers for the discrete linear equation systems that arise in incompressible flow modelling. Key Features Key features include implementation of a variety of mixed finite element approximation methods; Links Download Documentation Publications Overview Sample output Contact automatic calculation of stabilization parameters where appropriate; a posteriori error estimation for steady problems; a range of state-of-the-art preconditioned Krylov subspace solvers ; built-in geometric and algebraic multigrid solvers and preconditioners; fully implicit self-adaptive time stepping algorithms; useful visualization tools. The developers of the IFISS package are David Silvester (School of Mathematics, University of Manchester), Howard Elman (Computer Science Department, University of Maryland), and Alison Ramage (Department of Mathematics and Statistics, University of Strathclyde). The IFISS logo represents the solution of the double glazing convection-diffusion problem. It can be reproduced in IFISS via the function ifisslogo. This webpage is based on a CSS template from Free CSS Templates. USA trip October 2014 p. 7/62

9 PART I USA trip October 2014 p. 8/62

10 USA trip October 2014 p. 9/62 References I Philip Gresho & David Griffiths & David Silvester Adaptive time-stepping for incompressible flow; part I: scalar advection-diffusion SIAM J. Scientific Computing, 30: , David Kay & Philip Gresho & David Griffiths & David Silvester Adaptive time-stepping for incompressible flow; part II: Navier-Stokes equations SIAM J. Scientific Computing, 32: , Howard Elman, Milan Mihajlović and David Silvester. Fast iterative solvers for buoyancy driven flow problems J. Computational Physics, 230: , 2011.

11 USA trip October 2014 p. 10/62 Buoyancy driven flow u t + u u ν 2 u+ p = jt in W Ω (0,T) u = 0 in W T t + u T ν 2 T = 0 Boundary and initial conditions in W u = 0 on Γ [0,T]; u( x,0) = 0 in Ω. T = T g on Γ D [0,T]; T( x,0) = 0 in Ω. ν T n = 0 on Γ N [0,T];

12 USA trip October 2014 p. 10/62 Buoyancy driven flow u t + u u ν 2 u+ p = jt in W Ω (0,T) u = 0 in W T t + u T ν 2 T = 0 Boundary and initial conditions in W u = 0 on Γ [0,T]; u( x,0) = 0 in Ω. T = T g on Γ D [0,T]; T( x,0) = 0 in Ω. ν T n = 0 on Γ N [0,T]; ν = Pr/Ra, ν = 1/ Pr Ra, T g = (1 e 10t )T.

13 Rayleigh Bénard P r = 7.1, Ra = USA trip October 2014 p. 11/62

14 USA trip October 2014 p. 11/62 Rayleigh Bénard P r = 7.1, Ra = Stationary streamlines: time =

15 USA trip October 2014 p. 12/62 Smart Integrator (SI) Optimal time-stepping Black-box implementation Algorithm efficiency Solver efficiency: the linear solver convergence rate is robust with respect to the mesh size h and the flow problem parameters.

16 USA trip October 2014 p. 13/62 Rayleigh Bénard Pr = 7.1, Ra = time average vorticity time ω = u, ω Ω = 1 2A Ω ω 2

17 USA trip October 2014 p. 14/62 Rayleigh Bénard Pr = 7.1, Ra = time step time time stabilized TR ε t = 10 6 (left) and ε t = 10 5 (right).

18 USA trip October 2014 p. 15/62 Rayleigh Bénard Pr = 7.1, Ra = Isotherms: time = Isotherms: time = Isotherms: time =

19 LINEAR ALGEBRA USA trip October 2014 p. 16/62

20 USA trip October 2014 p. 17/62 Trapezoidal Rule (TR) time discretization Subdivide [0,T] into time levels {t i } N i=1. Given (un,p n,t n ) at time t n, k n+1 := t n+1 t n, compute (u n+1,p n+1,t n+1 ) via 2 k n+1 u n+1 ν 2 u n+1 +u n+1 u n+1 + p n+1 jt n+1 = u n+1 = 0 2 k n+1 u n + u n t in Ω in Ω u n+1 = 0 2 k n+1 T n+1 ν 2 T n+1 +u n+1 T n+1 = k 2 n+1 T n + T t n T n+1 = T n+1 g on Γ in Ω on Γ D ν T n+1 n = 0 on Γ N.

21 USA trip October 2014 p. 18/62 Linearization Subdivide [0,T] into time levels {t i } N i=1. Given (un,p n,t n ) at time t n, k n+1 := t n+1 t n, compute (u n+1,p n+1,t n+1 ) via 2 k n+1 u n+1 ν 2 u n+1 + w n+1 u n+1 + p n+1 jt n+1 = u n+1 = 0 2 k n+1 T n+1 ν 2 T n+1 + w n+1 T n+1 = k 2 n+1 T n + T t with w n+1 = (1+ k n+1 k n ) u n k n+1 k n u n 1. 2 k n+1 u n + u n t in Ω in Ω u n+1 = 0 on Γ. n T n+1 = T n+1 g in Ω on Γ D ν T n+1 n = 0 on Γ N,

22 USA trip October 2014 p. 19/62 Adaptive time stepping components The adaptive time step selection is based on coupled physics. Given L 2 error estimates e n+1 h and e n+1 h for the velocity and temperature respectively, the subsequent TR AB2 time step k n+2 is computed using k n+2 = k n+1 ( e n+1 h ε t 2 + e n+1 h The following parameters must be specified: 2 ) 1/3. time accuracy tolerance ε t (10 5 ) GMRES tolerance itol (10 6 ) GMRES iteration limit maxit (50)

23 USA trip October 2014 p. 20/62 Finite element matrix formulation Introducing the basis sets X h = span{ φ i } n u i=1, Velocity basis functions; M h = span{ψ j } n p j=1, Pressure basis functions. T h = span{φ k } n T k=1, Temperature basis functions; gives the method-of-lines discretized system: u M 0 0 t F B T M u p t + B 0 0 p 0 0 M 0 0 F T T t = 0 0 g with a (vertical ) mass matrix: ( M ) ij = ([0,φ i ],φ j )

24 USA trip October 2014 p. 21/62 Preconditioning strategy F B T M B 0 0 P 1 P 0 0 F α u α p α T = Given S = BF 1 B T, a perfect preconditioner is given by F B T M F 1 F 1 B T S 1 F 1 M B S F } 0 0 {{ F 1 } P 1 = f u f p f T I 0 0 BF 1 I BF 1 M F I

25 USA trip October 2014 p. 22/62 For an efficient preconditioner we need to construct a sparse approximation to the exact Schur complement S 1 = (BF 1 B T ) 1 See Chapter 11 of Howard Elman & David Silvester & Andrew Wathen Finite Elements and Fast Iterative Solvers: with applications in incompressible fluid dynamics Oxford University Press, second edition, For an efficient implementation we must also have an efficient AMG (convection-diffusion) solver...

26 USA trip October 2014 p. 23/62 HSL HSL MI20 PACKAGE SPECIFICATION HSL SUMMARY Givenann nsparsematrixaandann vectorz, HSL MI20computesthevectorx =Mz,whereMisanalgebraic multigrid(amg) v-cycle preconditioner for A. A classical AMG method is used, as described in[1](see also Section 5belowforabriefdescriptionofthealgorithm).ThematrixAmusthavepositivediagonalentriesand(mostof)the off-diagonal entries must be negative(the diagonal should be large compared to the sum of the off-diagonals). During the multigrid coarsening process, positive off-diagonal entries are ignored and, when calculating the interpolation weights, positive off-diagonal entries are added to the diagonal. Reference [1] K. Stüben. An Introduction to Algebraic Multigrid. In U. Trottenberg, C. Oosterlee, A. Schüller, eds, Multigrid, Academic Press, 2001, pp ATTRIBUTES Version: Types: Real(single, double). Uses: HSL MA48, HSL MC65, HSL ZD11, and the LAPACK routines GETRF and GETRS. Date: September Origin: J. W. Boyle, University of Manchester and J. A. Scott, Rutherford Appleton Laboratory. Language: Fortran 95, plus allocatable dummy arguments and allocatable components of derived types. Remark: The development of HSL MI20 was funded by EPSRC grants EP/C000528/1 and GR/S42170.

27 USA trip October 2014 p. 24/62 Schur complement approximation I Introducing the diagonal of the velocity mass matrix M M ij = ( φ i, φ j ), gives the least-squares commutator preconditioner : (BF 1 B T ) 1 (BM 1 B T ) 1 (BM }{{} 1 FM 1 B T )(BM 1 B T }{{} amg amg ) 1

28 USA trip October 2014 p. 25/62 Schur complement approximation II Introducing associated pressure matrices M p ( ψ i, ψ j ), mass A p ( ψ i, ψ j ), diffusion N p ( w h ψ i,ψ j ), convection F p = 2 k n+1 M p +νa p +N p, convection-diffusion gives the pressure convection-diffusion preconditioner : (BF 1 B T ) 1 Mp 1 F p A 1 p }{{} amg

29 USA trip October 2014 p. 26/62 Rayleigh Bénard Pr = 7.1, Ra = Exact PCD preconditioning t=100 t=120 t= Exact LSC preconditioning t=100 t=120 t=300 residual reduction residual reduction iteration number iteration number AMG-ILU PCD preconditioning t=100 t=120 t= AMG-ILU LSC preconditioning t=100 t=120 t=300 residual reduction residual reduction iteration number iteration number

30 USA trip October 2014 p. 27/62 What have we achieved? Black-box implementation: few parameters that have to be estimated a priori. Optimal complexity: essentially O(n) flops per iteration, where n is dimension of the discrete system. Efficient linear algebra: convergence rate is (essentially) independent of h. Given an appropriate time accuracy tolerance convergence is also robust with respect to diffusion parameters ν and ν.

31 PART II USA trip October 2014 p. 28/62

32 USA trip October 2014 p. 29/62 References II Catherine Powell & David Silvester Preconditioning steady-state Navier Stokes equations with random data. SIAM J. Scientific Computing, vol. 34, A2482 A2506, David Silvester & Alex Bespalov & Catherine Powell A framework for the development of implicit solvers for incompressible flow problems. Discrete and Continuous Dynamical Systems Series S, vol. 5, , 2012.

33 USA trip October 2014 p. 30/62 Steady-state flow with random data Problem statement u u ν 2 u+ p = 0 in Ω u = 0 in Ω u = g on Γ D ν u n p n = 0 on Γ N. We model uncertainty in the viscosity as ν(ω) = µ+σξ(ω). If ξ U( 3, 3), then ν is a uniform random variable with E[ν(ω)] = µ, Var[ν(ω)] = σ 2.

34 USA trip October 2014 p. 31/62 N S example I: flow over a step Streamlines of the mean flow field (top) and plot of the mean pressure field (bottom): µ = 1/50, σ = µ/

35 USA trip October 2014 p. 32/62 Variance of the magnitude of flow field (top) and variance of the pressure (bottom) x

36 USA trip October 2014 p. 33/62 N S example II: flow around an obstacle Streamlines of the mean flow field (top) and plot of the mean pressure field (bottom): µ = 1/100, σ = 3µ/

37 USA trip October 2014 p. 34/62 Variance of the magnitude of flow field (top) and variance of the pressure (bottom)

38 USA trip October 2014 p. 35/62 Stochastic discretisation methods Monte Carlo Methods Perturbation Methods Stochastic Galerkin Methods Stochastic Collocation Methods Stochastic Reduced Basis Methods...

39 USA trip October 2014 p. 35/62 Stochastic discretisation methods Monte Carlo Methods Perturbation Methods Stochastic Galerkin Methods Stochastic Collocation Methods Stochastic Reduced Basis Methods... Key points If the number of random variables describing the input data is small then Stochastic Galerkin and Stochastic Collocation methods can outperform Monte Carlo. If software for the deterministic problem is to be useful for Stochastic Galerkin approximation then specialised solvers need to be developed.

40 LINEAR ALGEBRA USA trip October 2014 p. 36/62

41 USA trip October 2014 p. 37/62 Stochastic Galerkin discretisation I Ingredients Picard iteration; standard finite element spaces X h E and Mh ; a suitable finite-dimensional subspace S k L 2 ρ(λ), where Λ:=ξ(Ξ), Λ y.

42 USA trip October 2014 p. 37/62 Stochastic Galerkin discretisation I Ingredients Picard iteration; standard finite element spaces X h E and Mh ; a suitable finite-dimensional subspace S k L 2 ρ(λ), where Λ:=ξ(Ξ), Λ y. Discrete formulation Find u n+1 hk X h E Sk and p n+1 hk M h S k satisfying: [ E ν(y) ( [ ( u [ u n+1 hk, v)] n (p +E hk u n+1 hk, v)] n+1 E hk, v)] = 0 for all v X h 0 Sk and q M h S k. [ E ( u n+1 hk,q)] = 0

43 USA trip October 2014 p. 38/62 Stochastic Galerkin discretisation II Discrete formulation Find u n+1 hk X h E Sk and p n+1 hk M h S k satisfying: [ E ν(y) ( [ ( u [ u n+1 hk, v)] n (p +E hk u n+1 hk, v)] n+1 E hk, v)] = 0 for all v X h 0 Sk and q M h S k. [ E ( u n+1 hk,q)] = 0

44 USA trip October 2014 p. 38/62 Stochastic Galerkin discretisation II Discrete formulation Find u n+1 hk X h E Sk and p n+1 hk M h S k satisfying: [ E ν(y) ( [ ( u [ u n+1 hk, v)] n (p +E hk u n+1 hk, v)] n+1 E hk, v)] = 0 for all v X h 0 Sk and q M h S k. [ E ( u n+1 hk,q)] = 0 Sets of basis functions X h 0 =span{ (φ i ( x),0),(0,φ i ( x)) } n u i=1 ;Mh =span{ψ j ( x)} n p j=1 ; S k =span{ϕ l (y)} k l=0.

45 USA trip October 2014 p. 39/62 Stochastic Galerkin discretisation III The linear system at the (n+1)st Picard iteration is ( )( ) ( ) Fν n B T α n f n B 0 β n = g n with F n ν = ( F n ν 0 0 F n ν ), B = ( G 0 B x1 G 0 B x2 ) and k Fν n := (µg 0 +σg 1 ) A+ H l N l, B x1, B x2 are discrete representations of the first derivatives. The system dimension is: (n u +n p )(k+1) (n u +n p )(k+1). l=0

46 USA trip October 2014 p. 40/62 (1-1) block: F n ν := (µg 0 +σg 1 ) A+ k l=0 H l N l. F n ν is a non-symmetric matrix. convection matrices N l (l = 0,...,k) are given by [N l ] ij = ( u n hl( x) φ i,φ j ) i,j = 0,...,n u. where u n hl are the spatial coefficients in the expansion of the lagged velocity field, k u n hk( x,y) = n u i=1 un il φ i ( x) ϕ }{{} l (y). l=0 u n hl( x)

47 USA trip October 2014 p. 41/62 (1-1) block: F n ν := (µg 0 +σg 1 ) A+ k l=0 H l N l. F n ν is a non-symmetric matrix. convection matrices N l (l = 0,...,k) are given by [N l ] ij = ( u n hl( x) φ i,φ j ) i,j = 0,...,n u. G 0, G 1 and H l are all (k +1) (k +1) matrices: G 0 := [G 0 ] ls = E[ϕ s (y)ϕ l (y)], G 1 := [G 1 ] ls = E[yϕ s (y)ϕ l (y)], H l := [H l ] ms = E[ϕ l (y)ϕ s (y)ϕ m (y)].

48 USA trip October 2014 p. 41/62 (1-1) block: F n ν := (µg 0 +σg 1 ) A+ k l=0 H l N l. F n ν is a non-symmetric matrix. convection matrices N l (l = 0,...,k) are given by [N l ] ij = ( u n hl( x) φ i,φ j ) i,j = 0,...,n u. G 0, G 1 and H l are all (k +1) (k +1) matrices: G 0 := [G 0 ] ls = E[ϕ s (y)ϕ l (y)], G 1 := [G 1 ] ls = E[yϕ s (y)ϕ l (y)], H l := [H l ] ms = E[ϕ l (y)ϕ s (y)ϕ m (y)]. If {ϕ l (y)} k l=0 are scaled Legendre polynomials on Λ, then G 0 = H 0 = I, G 1 = H 1 is sparse (2 non-zeros per row); H l is dense for l 2.

49 USA trip October 2014 p. 42/62 Ideal preconditioning ( F B T B 0 ) P 1 P ( ) α u α p = ( ) f u f p An ideal preconditioner is given by ( )( ) F B T F 1 F 1 B T S 1 B 0 } 0 S 1 {{ } P 1 = ( ) I 0 BF 1 I. For an efficient preconditioner we need to construct a sparse approximation to the exact Schur complement S 1 = (BF 1 B T ) 1

50 USA trip October 2014 p. 43/62 Rearrange the (1-1) block: Preconditioning I F n ν = (µg 0 +σg 1 ) A+ k l=0 H l N l = I (µa 0 +N 0 )+σg 1 A+ k l=1 H l N l and define F 0 := (µa 0 +N 0 ).

51 USA trip October 2014 p. 43/62 Rearrange the (1-1) block: Preconditioning I F n ν = (µg 0 +σg 1 ) A+ k l=0 H l N l and define = I (µa 0 +N 0 )+σg 1 A+ k l=1 H l N l F 0 := (µa 0 +N 0 ). A natural candidate for P F is the block-diagonal mean-based approximation: ( ) I F 0 0 P F = F 0 :=. 0 I F 0 This is a good approximation when σ µ is not too large.

52 USA trip October 2014 p. 44/62 Preconditioning II Replacing F n ν by F 0 in the Schur-complement gives S BF 1 0 B T = (I B x1 )(I F 1 0 )(I B T x 1 )+(I B x2 )(I F 1 0 )(I B T x 2 ) = I (B x1,b x2 )F 1 0 (B x1,b x2 ) T =: I S 0 =: S 0 = P S.

53 USA trip October 2014 p. 44/62 Preconditioning II Replacing F n ν by F 0 in the Schur-complement gives S BF 1 0 B T = (I B x1 )(I F 1 0 )(I B T x 1 )+(I B x2 )(I F 1 0 )(I B T x 2 ) = I (B x1,b x2 )F 1 0 (B x1,b x2 ) T =: I S 0 =: S 0 = P S. S 0 is the Schur-complement corresponding to the deterministic problem with viscosity µ convection coefficient u 0 hk (the mean component of velocity at the previous Picard step)

54 USA trip October 2014 p. 45/62 Preconditioning III Replacing F n ν by F 0 in the Schur-complement gives S BF 1 0 B T = (I B x1 )(I F 1 0 )(I B T x 1 )+(I B x2 )(I F 1 0 )(I B T x 2 ) = I (B x1,b x2 )F 1 0 (B x1,b x2 ) T =: I S 0 =: S 0 = P S. To apply P 1 S in each GMRES iteration requires (k +1) solves with S 0. This can be done exactly (ideal preconditioner); or inexactly with the deterministic approaches: pressure convection diffusion approximation (PCD) least squares commutator approximation (LSC).

55 USA trip October 2014 p. 46/62 Flow over a step ideal PC D L SC ideal PC D L SC residual reduction residual reduction iteration number iteration number GMRES convergence for a coarsened grid (left) and for a reference grid (right) (µ = 1/50; σ = 2µ/10).

56 USA trip October 2014 p. 47/62 Typical GMRES iteration counts Coarse grid Fine grid E[Re] k = k = σ = µ/ Ideal σ = 2µ/ σ = 3µ/ σ = µ/ PCD σ = 2µ/ σ = 3µ/ σ = µ/ LSC σ = 2µ/ σ = 3µ/

57 USA trip October 2014 p. 48/62 What have we achieved? Black-box implementation: no parameters that have to be estimated a priori. Optimal complexity: essentially O(n) flops per iteration, where n is dimension of the discrete system. Efficient linear algebra: convergence rate is independent of h. Convergence is also robust with respect to the spectral approximation parameter k as long as the variance is not too large relative to the mean.

58 Part III USA trip October 2014 p. 49/62

59 USA trip October 2014 p. 50/62 Stochastic Galerkin andh-p adaptivity What s next? Alex Bespalov, Catherine Powell & David Silvester. A posteriori error estimation for parametric operator equations with applications to PDEs with random data. SIAM J. Sci. Comput, 36:A339 A363, 2014.

60 USA trip October 2014 p. 51/62 stochastic Galerkin error estimation I B(u,v) = F(v) v V ( ) B(u XP,v) = F(v) v V XP = X P P. Thus e := u u XP V satisfies B(e,v) = F(v) B(u XP,v) }{{} = 0 v V XP v V.

61 stochastic Galerkin error estimation I B(u,v) = F(v) v V ( ) B(u XP,v) = F(v) v V XP = X P P. Thus e := u u XP V satisfies B(e,v) = F(v) B(u XP,v) }{{} = 0 v V XP v V. This motivates the estimator e V XP satisfying B(e,v) = F(v) B(u XP,v) v V XP = X P P where V XP is an enriched finite-dimensional subspace of V so that X = X Y, X Y = {0}; P P = P P P Q, P P P Q = {0}. VXP := V XP V }{{} XQ V YP V }{{}}{{} YQ X P Q Y P P Y P Q. USA trip October 2014 p. 51/62

62 USA trip October 2014 p. 52/62 stochastic Galerkin error estimation II A simple-minded estimator e VXP satisfies B(e,v) = F(v) B(u XP,v) v VXP, with VXP := V XP (X P Q ) (Y P P ) (Y P Q ).

63 USA trip October 2014 p. 52/62 stochastic Galerkin error estimation II A simple-minded estimator e VXP satisfies B(e,v) = F(v) B(u XP,v) v VXP, with VXP := V XP (X P Q ) (Y P P ) (Y P Q ). A more refined estimator e 0 series structure of ( ): find e 0 V XP is obtained by exploiting the typical such that B 0 (e 0,v) = F(v) B(u XP,v) v V XP. This leads to efficient linear algebra the associated system matrix is block diagonal with each block representing a deterministic problem associated with the mean component of the operator A(y).

64 USA trip October 2014 p. 53/62 stochastic Galerkin error estimation III The refined error estimator e 0 V XP satisfies B 0 (e 0,v) = F(v) B(u XP,v) v VXP, with VXP := V XP (X P Q ) (Y P P ) (Y P Q ).

65 USA trip October 2014 p. 53/62 stochastic Galerkin error estimation III The refined error estimator e 0 V XP satisfies B 0 (e 0,v) = F(v) B(u XP,v) v V XP, with V XP := V XP (X P Q ) (Y P P ) (Y P Q ). A simple energy estimate η is obtained by exploiting the structure of the space VXP : This is given by η := ( 2 ē XQ 2 B 0 + ē YP 2 B 0 +2 ē YQ 2 B 0 ) 1/2, where the contributing estimators ē XQ V XQ, ē YP V YP and ē YQ V YQ satisfy B 0 (ē XQ,v) = F(v) B(u XP,v) v V XQ, B 0 (ē YP,v) = F(v) B(u XP,v) v V YP, ( ) B 0 (ē YQ,v) = F(v) B(u XP,v) v V YQ.

66 USA trip October 2014 p. 54/62 error reduction indicators Recall η := ( 2 ē XQ 2 B 0 + ē YP 2 B 0 +2 ē YQ 2 B 0 ) 1/2 with ē XQ X P Q, ē YP Y P P and ē YQ Y P Q. The construction of η also gives information regarding the error reduction that would result by enriching the approximation either spatially or parametrically: λ ēyp B0 u X P u XP B Λ 1 γ 2 ē YP B0, λ ēxq B0 u XP u XP B Λ ē XQ B0. Here, λ, Λ are the constants in (B 0 ) and γ [0,1) is the constant in the strengthened Cauchy Schwarz inequality. This opens the door to h p adaptivity...

67 USA trip October 2014 p. 55/62 model problem example We solve a steady-state diffusion problem ( ) with x = (x 1,x 2 ) D := [ 1,1] 2, with data f(x) = 1 8 (2 x2 1 x2 2 ) and a(x,y) = }{{} 1 +σ 3 a 0 =E[a] m=1 λm ϕ m (x)y m. Here y Γ = m=1 [ 1,1] and {(λ m,ϕ m )} m=1 are the eigenpairs of the integral operator associated with the correlation function C[a](x,x ) = exp ( 1 2 x x l1 ), x, x [ 1,1] 2. The problem ( ) is well posed as long as σ is small enough: our reference value is σ = 0.2.

68 USA trip October 2014 p. 56/62 model problem detail spaces The detail spaces Y h and Q M,p are chosen so that X h/2 = X h Y h and P M,p+1 = P M,p Q M,p, respectively. Thus, Y h spans the set of bilinear bubble functions corresponding to the edge midpoints and element centroids of the original mesh h (defined locally, see below) whereas Q M,p spans M-variate polynomials of total degree equal to p+1. This gives a decomposition of the error estimate η with V XP := V XP (X h Q M,p ) (Y h P M,p ) (Y h Q M,p ).

69 model problem error estimation V XP := V XP (X h Q M,p ) (Y h P M,p ) (Y h Q M,p ). The smart choice of the detail space Y h gives rise to local problems defined on all elements K h. For example, we compute ē YP K Y h K P M,p satisfying B 0,K (ē YP K,v) = F K (v) + (a(x,y) u XP (x,y) ) v(x,y) dx dπ(y) Γ K [ ] 1 uxp 2 a(s, y) v(s,y) ds dπ(y), n for all v Y h K P M,p. Γ K\ D The coefficient matrix associated with this local problem is the Kronecker product of a 5 5 (stiffness) matrix and an identity matrix of dimension P = dim(p M,p ) = (p+m)! p!m!. USA trip October 2014 p. 57/62

70 USA trip October 2014 p. 58/62 example effectivity I We compute a reference solution u ref for the case M = 3, σ = 0.2, using a grid with p = 7. fixed polynomial degree: p = 2 effectivity Index : X η := η(u XP )/ u ref u XP B Grid u ref u XP B η(u XP ) X η

71 USA trip October 2014 p. 59/62 example effectivity II We compute a reference solution u ref for the case M = 3, σ = 0.2, using a grid with p = 7. fixed spatial grid: effectivity Index : X η := η(u XP )/ u ref u XP B p u ref u XP B η(u XP ) X η

72 USA trip October 2014 p. 60/62 error reduction decreasingh M = 3, σ = 0.2, fixed polynomial degree: p = YP-estimate XQ-estimate YQ-estimate η estimate estimates of the error number of d.o.f.

73 USA trip October 2014 p. 61/62 error reduction increasingp M = 3, σ = 0.2, fixed spatial grid: YP-estimate XQ-estimate YQ-estimate η estimate estimates of the error number of d.o.f.

74 USA trip October 2014 p. 62/62 What have we achieved? computable estimates of the error reduction: these exploit the tensor product structure of the approximation spaces. Efficient linear algebra: CPU time for the error estimation is commensurate with the CPU time taken to solve the original problem...