Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems

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1 Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems Donald Estep University Interdisciplinary Research Scholar Department of Statistics Center for Interdisciplinary Mathematics and Statistics Colorado State University Research supported by Defense Threat Reduction Agency (HDTRA ), Department of Energy (DE-FG02-04ER25620, DE-FG02-05ER25699, DE-FC02-07ER54909, DE-SC , DE-SC , INL , DE SC9279), Idaho National Laboratory ( , ), Lawrence Livermore National Laboratory (B573139, B584647, B590495), National Science Foundation (DMS , DMS , DGE , MSPA-CSE , ECCS , DMS , DMS , DMS-FRG , DMS ), National Institutes of Health (#R01GM096192), Sandia Corporation (PO299784) March 26, 2014 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 1/77

2 Collaborators Todd Arbogast, UT Austin John Shadid, Sandia National Laboratory Simon Tavener, Colorado State University From my group: Varis Carey, UT Austin Jeb Collins, Colorado State University Victor Ginting, U. Wyoming Jehanzeb Hameed, Colorado State University Brendan Sheehan, Colorado State University Tim Wildey, Sandia National Laboratory Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 2/77

3 Multiphysics, Multiscale Problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 3/77

4 Multiphysics, Multiscale Systems Multiphysics, multiscale systems couple different physical processes interacting across a wide range of scales Such systems abound in science and engineering application domains Computational modeling plays a critical role in the predictive study of such systems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 4/77

5 Example 1: A MEMs Thermal Actuator V sig Contact Contact Contact Conductor 250 mm Conductor V switch (σ V ) = 0 (κ(t ) T ) = σ( V V ) ˆ (λ tr(e)i + 2µE β(t T ref )I ) = 0 E = ( ˆ d + ˆ d ) /2 electrostatic current energy displacement This is an example of parameter passing, in which the solution of one component is used to compute the parameters and/or data for another component Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 5/77

6 Example 2: Reaction-Diffusion Equations The Brusselator problem u 1 t ɛ u 1 = α βu 1 + u 2 1 u 2, u 2 t ɛ u 2 = γu 1 u 2 1 u 2, suitable initial and boundary conditions The Brusselator is a simple model of the reaction kinetics of the two main species in an autocatalytic reaction taking place in solution Tight coupling of different physical components with different scales and stability properties, e.g. reaction and diffusion, in the same equations Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 6/77

7 Example 3: Fluid-Solid Conjugate Heat Transfer Solid µ u + ρ 0 (u ) u + p + ρ 0 βt F g = ρ 0 (1 + βt 0 ) g, u = 0, k F T F + { ρ 0 c p (u T F ) = Q F, T S = T F, interface k F (n T F ) = k S (n T S ), k S T S = Q S, Physics in different regions are coupled through a common interface Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 7/77

8 Example 4: Physics in a Fusion Reactor Simplified model for core-edge coupling T c t = x χ c Tc x + S c, core region T m t = x χ m Tm x + S m ν md (T m T d ), edge midplane region T d t = x χ d T d x + S d + ν md (T m T d ) ν el T d θ(x x s ), edge divertor region The edge and core regions are coupled through a common boundary: { T c = 1 2 (T m + T d ), χ c T c n = 1 2 (χ m T m n + χ d T d n) with S c = S m + S d and χ c = χ m + χ d at the interface The physics is coupled across scales in space and time Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 8/77

9 Multiscale Operator Decomposition Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 9/77

10 Multiscale Operator Decomposition Multiscale, multiphysics systems pose severe challenges for computational solution: Scale effects Complex stability Coupling between physics Different kinds of physical descriptions Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 10/77

11 Multiscale Operator Decomposition Multiscale operator decomposition (MOD) methods are widely used for solving multiphysics, multiscale systems The system is decomposed into components representing relatively simple physics and/or single scales and a global solution is sought by iterating solutions of the components Fully Coupled Multiscale Operator Decomposition Solution Physics 1 Physics 1 1 Physics 1 2 Physics 1 3 upscale downscale upscale downscale upscale downscale upscale downscale Physics 2 Physics 2 1 Physics 2 2 Physics 2 3 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 11/77

12 MOD for a Coupled Elliptic System A simplified Thermal Actuator model: a 1 u 1 + b 1 u 1 + c 1 u 1 = f 1 (x), x Ω, a 2 u 2 + b 2 u 2 + c 2 u 2 = f 2 (x, u 1, Du 1 ), x Ω, u 1 = u 2 = 0, x Ω, Ω is a bounded domain with boundary Ω The coefficients are smooth functions and a 1, a 2 are bounded away from zero We wish to compute information depending on u 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 12/77

13 MOD for a Coupled Elliptic System Algorithm Construct discretizations T h,1, T h,2 and finite element spaces S h,1, S h,2 Compute a finite element solution U 1 S h,1 (Ω) of the first equation Project U 1 S h,1 (Ω) into the space S h,2 (Ω) Compute a finite element solution U 2 S h,2 (Ω) of the second equation The projection between the discretization spaces is a crucial step In a fully coupled system, U 2 would be projected into the space for U 1 for the next iteration Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 13/77

14 MOD for a Reaction-Diffusion Equation We consider the reaction-diffusion problem { du dt = u + F (u), 0 < t, u(0) = u 0 The diffusion component u induces stability and change over long time scales The reaction component F induces instability and change over short time scales Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 14/77

15 MOD for a Reaction-Diffusion Equation Multiscale Operator-Splitting Algorithm On (t n 1, t n ], use the fine mesh to solve { du R dt = F (u R ), t n 1 < t t n, u R (t n 1 ) = u D (t n 1 ) On (t n 1, t n ], use the coarse mesh to solve { du D dt (u D ) = 0, t n 1 < t t n, u D (t n 1 ) = u R (t n ) Advance to the next diffusion step The operator split approximation is u(t n ) u D (t n ) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 15/77

16 MOD for a Reaction-Diffusion Equation To account for the fast reaction, we use a multirate approach We solve the reaction component using many time steps inside each diffusion step Diffusion Integration: Reaction Integration: t t t t t t 0 t 1 t t 3 t 4 t 5 s1... s 1,0 s 1,M... s 2,0 s 2,M s 2 s 3... s 3,0 s 3,M... s 4,0 s 4,M Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 16/77

17 MOD for Coupling Through a Boundary A parabolic interface problem { u1 t (χ 1 u 1 ) = f 1, x Ω 1 u 2 t (χ 2 u 2 ) = f 2, x Ω 2 with coupling conditions { u 1 = u 2, χ 1 u 1 n = χ 2 u 2 n, x Γ We use different space and time discretizations for the two components We introduce projections between the discrete spaces on the two sides of the interface Γ Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 17/77

18 MOD for Coupling Through a Boundary Let T n i denote an approximation at time t n. Gauss-Seidel Algorithm Given u n 1 2, solve for u n 1 with un 1 = un 1 2 on the interface Given u n 1, solve for un 2 with χ 2 u n 2 n = χ 1 u n 1 n on the interface Move on to t n+1 There are many other ways to arrange iterations between components and timesteps Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 18/77

19 The Effects of Error from Multiscale Operator Decomposition Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 19/77

20 Issues with Multiscale Operator Decomposition Multiscale operator decomposition discretizes the instantaneous interactions between the component physics The price: Transfer of computed information between components affects the accuracy of the numerical solution New forms of instability, some of which are very subtle, arise Even if each physical component is resolved accurately, decomposition may cause loss of fidelity These consequences are often not apparent on either the component or global level. Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 20/77

21 A Simple Thermal Actuator Consider u 1 = sin(4πx) sin(πy), x Ω u 2 = b u 1 = 0, x Ω, u 1 = u 2 = 0, x Ω, b = 2 π ( ) sin(4πx) 25 sin(πy) where Ω = [0, 1] [0, 1] We consider the quantity of interest u 2 (.25,.25) We solve for u 1 first and then solve for u 2 using independent meshes Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 21/77

22 A Simple Thermal Actuator Using uniform meshes, an a posteriori error estimate yields estimate of the error in the quantity of interest.0042 true error.0048 discrepancy in estimate.0006 ( 13%) u u This arises from the operator decomposition Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 22/77

23 A Simple Thermal Actuator Adapting the mesh using only an error estimate for the second component causes the discrepancy to become alarmingly worse estimate of the error in the quantity of interest.0001 true error u Pointwise Error of u Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 23/77

24 The Brusselator Problem u 1 t u 1 = f 1 (u 1, u 2 ) = 0.6 2u 1 + u 2 1 u 2, u 2 t u 2 = f 2 (u 1, u 2 ) = 2u 1 u 2 1 u 2, suitable initial and boundary conditions Linear finite element method in space with 500 elements A standard first order splitting scheme Trapezoidal Rule with time step of.2 for the diffusion and Backward Euler with time step of.004 for the reaction Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 24/77

25 The Brusselator Problem 10 0 L 2 norm of error slope t t = 6.4 t = 16 t = 32 t = 64 t = 80 Operator Split Solution Spatial Location On moderate to long time intervals, there is a critical time step above which convergence fails Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 25/77

26 Model for Core-Edge Coupling Through a Boundary A parabolic interface problem { u1 t (χ 1 u 1 ) = f 1, x Ω 1 u 2 t (χ 2 u 2 ) = f 2, x Ω 2 with coupling conditions { u 1 = u 2, χ 1 u 1 n = χ 2 u 2 n, x Γ Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 26/77

27 Model for Core-Edge Coupling Through a Boundary 0.4 Iteration and Transfer Error Time Using t = 10 3, the total error of the approximation increases with time Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 27/77

28 Model for Core-Edge Coupling Through a Boundary Iteration Error Time Transfer Error Time The error arising from incomplete iteration on each step becomes negligible as time passes The transfer error accumulates with time and becomes the largest source of error Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 28/77

29 Model for Core-Edge Coupling Through a Boundary log( L 2 error) Fully Coupled Solution Operator Decomposition log(sqrt(degrees of freedom)) Though we use second order accurate methods for each component, the error in the operator decomposition approximation is only first order in space Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 29/77

30 A Posteriori Error Estimation using Residuals and Adjoint Problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 30/77

31 A Posteriori Error Analysis We estimate the error in a quantity of interest computed from numerical solutions Ingredients: We compute residuals to quantify the introduction of error, e.g. through discretization We solve the adjoint problem associated with the quantity of interest to quantify the effects of stability We use variational analysis to combine residuals and adjoint solutions to obtain an estimate The goal is different than convergence analysis: We seek accurate estimates of error, not general but inaccurate bounds Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 31/77

32 A Linear Algebra Example Let U be an approximate solution of Au = b We compute a quantity of interest given by a linear functional (ψ, u) The error is not computable e = u U We can compute the residual R, R = AU b Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 32/77

33 A Linear Algebra Example The adjoint problem for the generalized Green s vector is A φ = ψ Variational analysis: (ψ, e) = (A φ, e) = (φ, Ae) = (φ, R) This gives the error representation: (ψ, e) = (φ, R) We solve for φ numerically to compute the estimate Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 33/77

34 Condition Numbers and Adjoint Solutions The classic error bound is e Cκ(A) R The condition number κ(a) = A A 1 is a measure of stability The a posteriori estimate yields (e, ψ) φ R The stability factor φ is a weak condition number for the quantity of interest We can obtain κ from φ by taking the sup over all ψ = 1 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 34/77

35 Condition Numbers and Adjoint Solutions Compute (y, e 1 ) from the solution of Ay = b where A is a random matrix Condition number of A is estimate of the error in the quantity of interest a posteriori error bound for the quantity of interest traditional error bound for the error Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 35/77

36 Abstract A Posteriori Error Analysis We solve a differential equation L(u) = f for a quantity of interest given by a linear functional (u, ψ) ψ determines the quantity of interest (, ) denotes the L 2 inner product over space and time We compute a numerical approximation U u The local resolution of the approximation is measured using the residual (written formally as) R(U) = L(U) f Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 36/77

37 Abstract A Posteriori Error Analysis To build an error estimate, we solve a linear adjoint problem (DL) φ = ψ We obtain the error representation (U u, ψ) = (R(U), W(φ)) for a suitable adjoint weight W(φ) The adjoint weight reflects stability by scaling residuals to create local contributions to the error The estimate accounts for the accumulation, cancellation, and propagation of local error contributions to the global error The stability information is specific to the quantity of interest Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 37/77

38 A Posteriori Analysis for an Elliptic Problem Let U be a Galerkin finite element approximation for a u + b u + cu = f on Ω The residual in weak form is (R, v) = (a U, v)+ ( b U+cU f, v ) for all test functions v Obtained by integration by parts, the formal adjoint problem is a φ div(bφ) + cφ = ψ on Ω The estimate is (e, ψ) = ( a U, (I π h )φ) ) + ( b U + cu f, (I π h )φ ) π h is a projection into the space of U Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 38/77

39 An Estimate for an Elliptic Problem { u = 200π 2 sin(10πx) sin(10πy), (x, y) Ω = [0, 1] [0, 1], u = 0, The solution is u = sin(10πx) sin(10πy) (x, y) Ω 1 u 0-1 x y error/estimate percent error Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 39/77

40 Using Adjoints to Quantify Stability The adjoint problem quantifies the effects of stability These effects are specific to the quantity of interest ((.05 + tanh(10(x 5) (y 1) 2 )) u ) ( ) u = 1, (x, y) Ω = [0, 10] [0, 2], 0 u = 0, (x, y) Ω Convection Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 40/77

41 Using Adjoints to Quantify Stability 6 1 x y x x y x 8 10 Final meshes for an average error of 4% 24,000 elements versus 3500 elements Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 41/77

42 Analysis of Multiscale Operator Decomposition for Multiscale, Multiphysics Problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 42/77

43 Analysis of Multiscale Operator Decomposition We extend the a posteriori analysis to deal with three issues arising in MOD: The passing of error along with information between components of a multiphysics model Defining appropriate adjoint operators becomes complicated Errors are introduced in complicated ways, which complicates how error can be mitigated Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 43/77

44 Multiscale Operator Decomposition: Issue 1 Computed Information Physics 1 Physics 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 44/77

45 Multiscale Operator Decomposition: Issue 1 Computed Information Error Physics 1 Physics 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 45/77

46 Multiscale Operator Decomposition: Issue 1 Computed Information Error Physics 1 Physics 2 We define auxiliary quantities of interest corresponding to information passed between components We solve auxiliary adjoint problems to estimate the error in that information In an iterative scheme, we also estimate the history of errors passed from one iteration level to the next We can also estimate the effect of processing, e.g. up and down scaling, the information Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 46/77

47 A Linear Algebra Example Consider the triangular system ( ) ( ) A11 0 u1 Au = = b = A 21 A 22 u 2 with approximate solution U = ( U1 U 2 ) ( u1 u 2 ( b1 b 2 ) ) = u We estimate the error in a quantity of interest in u 2 given by ( ψ (1), u ) = ( ψ (1) 2, u ) 2 Estimates on u 1 are independent of u 2 We use a superscript because further estimates are required Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 47/77

48 A Linear Algebra Example The lower triangular structure of A yields A 11 u 1 = b 1, A 22 u 2 = b 2 A 21 u 1 The residuals are R 1 = b 1 A 11 U 1, R 2 = (b 2 A 21 u 1 ) A 22 U 2 The residual of U 2 depends on information from the first component Decreasing the residual of U 2 requires consideration of the accuracy of the first component Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 48/77

49 A Linear Algebra Example The adjoint problem reads ( A 11 A 12 0 A 22 The error representation is ) ( ) φ (1) 1 φ (1) = 2 ( ) 0 ψ (1) 2 ( ψ (1), e ) = ( φ (1) 2, R ) ( (1) 2 + φ 2, A ) 21e 1 This only uses φ (1) 2 and stays in the single physics paradigm The first term on the right is computable as usual The second term on the right represents the effect of errors in the first component on the second component Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 49/77

50 A Linear Algebra Example We have to compute the decomposition term ( (1) φ 2, A ) ( 21e 1 = A 21 φ (1) 2, e ) 1 The first adjoint problem does not estimate this error This is a linear functional of the error e 1 corresponding to adjoint data ( ) ( ) ψ (2) ψ (2) = 1 A = 21 φ(1) We pose a secondary adjoint problem ( A 11 A ) ( ) ( ) 12 φ (2) 0 A 1 ψ (2) 22 φ (2) = Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 50/77

51 A Linear Algebra Example The block structure of A yields φ (2) 2 = 0, A 11φ (2) 1 = ψ (2) 1 = A 21φ (1) 2 Note these solves stay in the single physics paradigm We obtain the secondary error representation ( ψ (2), e ) = ( φ (2) 1, b ) ( (2) 1 A 11 U 1 = φ 1, R ) 1 The complete error representation is ( ψ (1), e ) = ( φ (1) 2, R ) ( (2) 2 + φ 1, R ) 1 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 51/77

52 A Linear Algebra Example Compute the average of the coefficients of (u 1, u 2 ) solving: ( ) ( ) A11 0 u1 = A 21 A 22 u 2 ( b1 b 2 ), A 11, A 21, A 22, b 1, b 2 U( 1, 1) A 11 = I +.2A 11 A 21 = A 21, A 22 = I +.1A 22 We use blockwise Gauss-Seidel iteration ( ψ (1), e ) iterations for U 1 40 or more iterations for U 2 ( (1) φ 2, R ) ( (2) φ 1, R ) The error in the QoI is mainly due to transferred error If we double the iterations for U 1, the error in the QoI is 10 7 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 52/77

53 Analysis for an Elliptic System Recall the triangular problem u 1 = sin(4πx) sin(πy), x Ω u 2 = b u 1 = 0, x Ω, u 1 = u 2 = 0, x Ω, b = 2 π ( ) 25 sin(4πx) sin(πx) where Ω = [0, 1] [0, 1] We consider the quantity of interest u 2 (.25,.25) We solve for u 1 first and then solve for u 2 using independent meshes Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 53/77

54 A Simplified Model Problem u 1 = f 1 (x), x Ω, u 2 = f 2 (x, u 1, Du 1 ), x Ω u 1 = 0, u 2 = 0, x Ω We compute a quantity of interest ( ψ (1), u ) that only depends on u 2, thus ( ) 0 ψ (1) = ψ (1) 2 We require the linearization Lf 2 (w) of f 2 with respect to u 1 around the function w Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 54/77

55 The First Adjoint Problem The weak form of the primary adjoint problem is {( (1) φ 1, v ) ( 1 + Lf2 (U 1 )φ (1) 2, v 1) = 0, ( (1) φ 2, v ) ( (1) 2 = ψ 2, v 2), all test functions v 1, v 2 This yields the first error representation ( ψ (1), e ) = ( f 2 (u 1, Du 1 ), (I π h )φ (1) 2 ) ( U2, (I π h )φ (1) ) 2 The residual depends on the unknown true solution We write f 2 (u 1, Du 1 ) = f 2 (U 2, DU 2 ) + ( f 2 (u 1, Du 1 ) f 2 (U 2, DU 2 ) ) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 55/77

56 The Secondary Adjoint Problems We have a new linear functional of the error ( f2 (u 1, Du 1 ) f 2 (U 1, DU 1 ), φ (1) ) ( 2 Lf(U1 )e 1, φ (1) ) 2 = ( Lf(U 1 ) φ (1) 2, e ) ( 1 = ψ (2), e ) We pose a secondary adjoint problem {( (2) φ 1, v ) ( 1 + Lf2 (U 1 )φ (2) 2, v ) ( 1 = Lf2 (U 1 ) φ (1) 2, v ) 1 ( (2) φ 2, v 2) = 0 all test functions v 1, v 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 56/77

57 Representation Formulas We finally get the fully computable representation Theorem ( ψ (1), e ) ( f 2 (U 1, DU 1 ), (I π h )φ (1) ) ( U2, (I π h )φ (1) ) + ( f 1, (I π h )φ (2) 1 2 ) ( U1, (I π h )φ (2) 1 φ (1), φ (2) are the primary and secondary adjoint solutions The new analysis estimates the error in the information passed between components 2 ) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 57/77

58 An Elliptic System If we adapt the meshes using all the terms in the estimate, we can drive the error below We actually refine the mesh for u 1 more than the mesh for u 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 58/77

59 Further Extensions of the Representation Formula The analysis can be extended further: If we use different meshes and/or numerical methods in the different components, then there is error arising from projection between the representations We define an tertiary auxiliary quantity of interest and related adjoint problem and add another term in the estimate In a fully coupled system, there are additional terms estimating the errors passed between iterations and the error arising from incomplete iteration Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 59/77

60 Multiscale Operator Decomposition: Issue 2 The adjoint operator associated to a multiscale operator decomposition solution is generally different than the adjoint operator associated with the full problem There are difficult technical issues about defining the appropriate adjoint operators The standard approach to define an adjoint for a nonlinear problem fails Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 60/77

61 The Standard Approach for Nonlinear Problems Consider U u solving With residual R(U) = F (U) b, F (u) = b F (u) F (U) = R(U) This gives the linearized representation A e = R(U), A = F (u, U) We carry out the a posteriori analysis using the adjoint associated with A Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 61/77

62 The Standard Approach Fails for MOD A MOD approximation Ũ solves a different problem Ũ approximately solves an operator decomposed problem F (ũ) = b, R( Ũ) = F (Ũ) b Now we obtain F (u) F (Ũ) }{{} cannot linearize = R(Ũ) }{{} can be made small Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 62/77

63 The Standard Approach Fails for MOD The solution is problem-specific But in every case, this introduces an error in the adjoint operator associated with the MOD solution The estimates for MOD include terms that quantify the error in the adjoint operator This measures the impact of operator decomposition on stability Additional work is needed to obtain a computable estimate Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 63/77

64 Operator Splitting for Reaction-Diffusion Equations To treat this case, we assume that the full operator and the operator decomposed operator share a common solution, e.g. a steady-state We linearize the full and operator decomposed operator individually with respect to the common solution We carry out the analysis by comparing the resulting linear equations Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 64/77

65 Operator Splitting for Reaction-Diffusion Equations We derive a new type of hybrid a priori - a posteriori estimate Theorem Error in the Quantity of Interest Q 1 + Q 2 + Q 3 Q 1 estimates the error of the numerical solution of each component in the standard way Q 2 N n=1 (U n 1, E n 1 ), E a computable estimate for the error in the adjoint arising from operator decomposition Q 3 = O( t 2 ) is an a priori expression that is provably higher order Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 65/77

66 The Brusselator Problem Accuracy of the error estimate t = 0.01, M = 10 t = 0.01, M = 10 Error Error Component Component T = 8 T = 40 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 66/77

67 Multiscale Operator Decomposition: Issue 3 Correcting the effects of decomposition efficiently requires some thought Simply using mesh refinement may not be an efficient approach Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 67/77

68 Conjugate Heat Transfer for a Fluid-Solid System Theorem (ψ, e)= (f, φ π V φ) a 1 (u {k} h, φ π V φ 1 ) c 1 (u {k} h, u{k} h, φ π V φ) b(φ π V φ, p h ) d(t {k} F,h, φ π V φ) b(u {k} h, z π Zz) + ( Q F, θ F π WF θ F ) a2 (T {k} F,h, θ F π WF θ F ) c 2 (u {k} h, T {k} F,h, θ F π WF θ F ) + ( Q S, θ S π WS θ S ) a3 (T {k} S,h, θ S π WS θ S ) + ( T {k} S,h π ST {k} F,h, k S(n θ S ) ) + ( π Γ S T {k} I F,h T {k} F,h, k S(n θ S ) ) Γ I + ( k S (n T {k} S,h ), π ) W F θ F + ( ) ( {k} Q Γ S, π WS θ S a3 T I S,h, π ) W S θ S The estimate involves auxiliary quantities of interest and adjoint problems measuring errors passed between components and between iteration levels Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 68/77

69 Conjugate Heat Transfer One specific term in the estimate causes the observed loss of order of accuracy It arises from passing numerical fluxes at the interface We can mitigate this several ways Use higher order elements or highly refined meshes Compute a discrete boundary flux correction for the error in the derivative on the interface (Wheeler, Carey) The last post-processing approach is very cheap and can be applied easily to existing codes Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 69/77

70 Conjugate Heat Transfer Between a Fluid and a Solid Quantity of interest is the temperature at a small region in the center of the cylinder No flux correction is applied Conditions are set to simulate the flow of water past a cylinder made from stainless steel We use Taylor-Hood elements to solve the fluid and standard finite element methods for the solid with independent meshes Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 70/77

71 Conjugate Heat Transfer Between a Fluid and a Solid Using the inexpensive boundary flux correction regains second order accuracy Using FEM Flux Using Boundary Flux 4.5 log(l 2 error) log(sqrt(degrees of freedom)) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 71/77

72 Conjugate Heat Transfer Between a Fluid and a Solid Quantity of interest is the temperature at a small region in the center of the cylinder Flux correction is applied for the computation on the right Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 72/77

73 Additional Problems Additional problems that have been tackled include: Multirate solution methods for ordinary differential equations and parabolic problems IMEX (Implicit-Explicit) solution methods for parabolic problems Explicit time integration methods for ordinary differential equations Finite volume, mixed finite element methods, and mortar mixed finite element methods for elliptic and parabolic problems with coupling through a common boundary Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 73/77

74 Conclusion Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 74/77

75 Conclusions We have developed a powerful analysis framework for a posteriori analysis of multiscale operator decomposition methods It accounts for various sources of error: The solution of each component Information that is passed between components Incomplete iteration Differences between the adjoints to the original problem and a multiscale operator decomposition discretization The analysis typically involves solving auxiliary adjoint problems as a way of quantifying the effect on stability Additional work is required to obtain computable estimates Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 75/77

76 Papers A posteriori - a priori analysis of multiscale operator splitting, D. Estep, V. Ginting, D. Ropp, J. Shadid, and S. Tavener, SINUM, 46 (2008) A posteriori error estimation of approximate boundary fluxes, T. Wildey, S. Tavener, and D. Estep, Comm. Num. Meth. Engin., 24 (2008) A posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem, D. Estep, S. Tavener, T. Wildey, SINUM, 46 (2008) A posteriori error analysis of multiscale operator decomposition methods for multiphysics models, D. Estep, V. Carey, V. Ginting, S. Tavener, T. Wildey, J. Physics: Conf. Ser. 125 (2008) A posteriori analysis and adaptive error control for multiscale operator decomposition methods for coupled elliptic systems I: One way coupled systems, V. Carey, D. Estep, and S. Tavener, SINUM 47 (2009) A posteriori error analysis for a transient conjugate heat transfer problem, D. Estep, S. Tavener, T. Wildey, Fin. Elem. Analysis Design 45 (2009) Bridging the Scales in Science and Engineering, Editor J. Fish, Oxford University Press, Chapter 11: Error Estimation for Multiscale Operator Decomposition for Multiphysics Problems, D. Estep, 2010 A posteriori error estimation and adaptive mesh refinement for a multi-discretization operator decomposition approach to fluid-solid heat transfer, D. Estep, S. Tavener, T. Wildey, JCP 229 (2010) A posteriori analysis of multirate numerical methods for ordinary differential equations, SINUM, D. Estep, V. Ginting, S. Tavener, Comput. Meth. Appl. Mech. Eng (2012) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 76/77

77 Papers A posteriori analysis of an iterative multi-discretization method for reaction-diffusion systems, D. Estep. V. Ginting, J. Hameed, and S. Tavener, Comp. Meth. Appl. Mech. Engin. 267 (2013) A posteriori analysis and adaptive error control for operator decomposition solution of coupled semilinear elliptic systems, V. Carey, D. Estep, S. Tavener, Int. J. Numer. Meth. Engin. 94 (2013) A-posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with non-matching grids, T. Arbogast, D. Estep, B. Sheehan, and S. Tavener, IMA J. Numer. Analysis (2013) A posteriori error estimation for the Lax-Wendroff finite difference scheme, J. B. Collins, D. Estep, and S. Tavener, J. Comput. Appl. Math. 263C (2014) A posteriori error estimates for explicit time integration methods, J. Collins, D. Estep and S. Tavener, BIT (2012), in revision The interaction of iteration error and stability for time dependent linear partial differential equations coupled through an interface, B. Sheehan, D. Estep, S. Tavener, J. Cary, S. Kruger, A. Hakim, A. Pletzer, J. Carlsson, Numer. Alg. (2013), submitted A posteriori error analysis of IMEX time integration schemes for advection-diffusion-reaction equations, J. Chaudry, D. Estep, V. Ginting, J. Shadid, and S. Tavener, (2014), in preparation A posteriori error estimates for mixed finite element and finite volume methods for parabolic problems coupled through a boundary with non-matching discretizations, T. Arbogast, D. Estep, B. Sheehan, and S. Tavener, (2014), in preparation Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 77/77