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 Department of Mathematics 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 ), Lawrence Livermore National Laboratory (B573139, B584647, B590495), National Aeronautics and Space Administration (NNG04GH63G), National Institutes of Health (1R01GM ), National Science Foundation (DMS , DMS , DGE , MSPA-CSE , ECCS ), Idaho National Laboratory ( ), Sandia Corporation (PO299784) September 7, 2010 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 1/80

2 Collaborators Varis Carey Department of Mathematics Colorado State University Victor Ginting Department of Mathematics University Wyoming Simon Tavener Department of Mathematics Colorado State University Tim Wildey Institute for Computational Engineering and Sciences University of Texas, Austin Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 2/80

3 Multiphysics, Multiscale Problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 3/80

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 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 4/80

5 A Multiphysics Model of a Thermal Actuator A thermal actuator is a MEMS scale electric switch V sig Contact Contact Contact Conductor Conductor 250 mm Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 5/80

6 A Multiphysics Model of a Thermal Actuator A thermal actuator is a MEMS scale electric switch V sig V switch Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 5/80

7 A Multiphysics Model of a Thermal Actuator A thermal actuator is a MEMS scale electric switch V sig V switch Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 5/80

8 A Multiphysics Model of a Thermal Actuator A thermal actuator is a MEMS scale electric switch V sig V switch Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 5/80

9 Example 1: A Thermal Actuator Electrostatic current equation (J = σ V ) Steady-state energy equation (σ V ) = 0 (κ(t ) T ) = σ( V V ) Steady-state displacement (linear elasticity) ˆ (λ tr(e)i + 2µE β(t T ref )I ) = 0 E = ( ˆ d + ˆ d ) /2 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 6/80

10 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 7/80

11 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 8/80

12 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 9/80

13 Challenges Posed by Solving Multiphysics Systems Computational modeling plays a critical role in the predictive study of such systems These systems pose challenges for computational solution: Accurately and efficiently computing information that depends on behaviors at very different scales Dealing with a complex stability picture resulting from a fusion of the stability properties of different physics Understanding the significance of linkages between physical components and how those affect model output Combining different modeling paradigms, e.g. stochastic and deterministic Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 10/80

14 Multiscale Operator Decomposition Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 11/80

15 Multiscale Operator Decomposition Multiscale operator decomposition 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 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 12/80

16 Multiscale Operator Decomposition Multiscale operator decomposition methods are widely used for solving multiphysics, multiscale systems Fully Coupled Physics 1 upscale downscale Physics 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 13/80

17 Multiscale Operator Decomposition Multiscale operator decomposition methods are widely used for solving multiphysics, multiscale systems Multiscale Operator Decomposition Solution Physics 1 1 Physics 1 2 Physics 1 3 upscale downscale upscale downscale upscale downscale Physics 2 1 Physics 2 2 Physics 2 3 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 14/80

18 Multiscale Operator Decomposition Multiscale operator decomposition methods are widely used for solving multiphysics, multiscale systems Reasons: Many multiphysics systems are naturally built in a componentwise fashion MOD accommodates multiple scales, physical descriptions, and discretization methods in one computation There is a good understanding of how to solve a broad spectrum of single physics problems accurately and efficiently MOD provides a way to utilize the investment in code developed for single physics problems and to achieve efficiency on HPC platforms Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 15/80

19 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 16/80

20 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 17/80

21 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 18/80

22 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 19/80

23 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 20/80

24 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 21/80

25 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 22/80

26 What Can Go Wrong with Multiscale Operator Decomposition? Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 23/80

27 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 24/80

28 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 25/80

29 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 26/80

30 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 27/80

31 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 28/80

32 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 29/80

33 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 30/80

34 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 31/80

35 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 32/80

36 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 33/80

37 The Key is Stability But What is Stability?... and Stability of What? Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 34/80

38 Instability of the Lorenz Problem The chaotic Lorenz problem u 1 = 10u u 2, u 2 = 28u 1 u 2 u 1 u 3, 0 < t, u 3 = 8 3 u 3 + u 1 u 2, Chaotic behavior affects numerical solutions as well Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 35/80

39 Accurate and Inaccurate Solutions of Lorenz 2% error on [0,30] 100% error at t= U 3 U U U U U 2 30 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 36/80

40 Stability of the Lorenz Problem Distance Between Solutions E-3 1E-5 Separatix E Time Time 10 Errors Decrease Errors Increase The distance between the two numerical solutions The errors follow an increasing trend, but decrease as well as increase Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 37/80

41 Stability of the Lorenz Problem The rate that errors grow depends strongly on the information being computed We consider the average distance from a solution to the origin over a long time interval u 3 distance Distance from Origin Coarse Fine Time Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 38/80

42 Stability of the Lorenz Problem The rate that errors grow depends strongly on the information being computed We compute the average distance over time We compare to an ensemble average of 100 accurate solutions computed using time step.0001 for 15 time units Coarse Solutions Fine Solutions Ensemble Ave End Time Ave Var Ave Var Ave Var Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 39/80

43 What About Classic A Priori Stability Analysis? Classic general stability analysis uses a Gronwall argument to obtain effect of perturbation at time t C e Lt size of perturbation C and L are constants, L is typically large Such bounds are non-descriptive past a very short initial transient Consider the chaotic Lorenz problem An analogous difficulty holds for a priori stability analysis of stationary problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 40/80

44 The Basic Tools for Quantifying Stability Properties: Functionals and Adjoints Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 41/80

45 Goal-Oriented Solution In application, it is often (always?) the case that computing particular information is the goal But, it is rarely the case (in my experience) that global pointwise space-time solution behavior is the application goal This is somewhat at odds with much of theoretical numerical analysis, which focuses on convergence in norm We focus on computing particular information from a solution Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 42/80

46 Functionals and Computing Information We focus on computing particular information from a solution A bounded linear functional l is a continuous linear map from a vector space X to the reals R Generally, it is easier to compute an accurate functional value than a solution that is accurate everywhere Often, we require only a small set of functional values The Fourier coefficients of a function are linear functionals Applications typically use a truncated Fourier series Nonlinear functionals require additional effort Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 43/80

47 Goal-Oriented Solution Let L : X Y be a continuous linear map between vector spaces The goal is to compute a linear functional value of the output l ( L(x) ) Some important questions: Can we find a way to compute the functional value efficiently? What is the error in the functional value if approximations are involved? Given functional values, what can we say about x? Given a collection of functional values, what can we say about L? Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 44/80

48 Definition of the Adjoint Operator Let L be a continuous linear map between normed vector spaces X and Y Each functional y on Y implicitly defines a functional x on X x L(x) y*(l(x)) x*(x) The adjoint L is the map that takes each functional y on Y to the corresponding functional x on X x = L (y ) If L has matrix L then L has matrix L Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 45/80

49 The Fundamental Role of the Adjoint There is a close connection between the stability properties of an operator and its adjoint The singular values of a matrix L are the eigenvalues of the square, symmetric transformations L L or LL The solvability of L(y) = b is closely related to the solvability of L (φ) = ψ In a Hilbert space, the existence of sufficiently many solutions of L (φ) = 0 implies that L(y) = b can have at most one solution Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 46/80

50 Adjoint Operators for Differential Equations For differential equations in Sobolev spaces, the adjoint is obtained by integrating by parts in the weak formulation to move all derivatives onto the test function Boundary terms involving functions and derivatives must be resolved The formal adjoint neglects the boundary terms Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 47/80

51 Examples of Formal Adjoint Operators L(u) = a u + b u + cu L (v) = a v div(b v) + cv For the operator for an initial value problem L(u) = u t a u + b u + cu, 0 < t T we have the operator for the final value problem L (v) = v a v div(b v) + cv = 0, T > t 0 t Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 48/80

52 Using Duality and Adjoints to Estimate the Error in Numerical Solutions Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 49/80

53 A Posteriori Error Analysis Problem: Estimate the error in a quantity of interest computed from a numerical solution of a differential equation We assume that the quantity of interest can be represented as a linear functional of the solution We use the adjoint problem associated with the linear functional Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 50/80

54 A Linear Algebra Example Problem Ay = b Quantity of interest (y, ψ) Approximate solution Y y Error (e, ψ) = (y Y, ψ) Residual R = AY b Adjoint problem A φ = ψ Analysis (e, ψ) = (e, A φ) = (Ae, φ) = (R, φ) We solve for φ numerically to compute the estimate Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 51/80

55 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 52/80

56 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 53/80

57 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 R(U) = L(U) f Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 54/80

58 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 55/80

59 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 56/80

60 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 57/80

61 Effect of Stability on a Stationary Problem ((.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 58/80

62 Goal-Oriented Adaptive Error Control 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 59/80

63 Analysis of Multiscale Operator Decomposition for Multiscale, Multiphysics Problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 60/80

64 Multiscale Operator Decomposition: Issue 1 Computed Information Physics 1 Physics 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 61/80

65 Multiscale Operator Decomposition: Issue 1 Computed Information Error Physics 1 Physics 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 62/80

66 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 63/80

67 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 64/80

68 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 65/80

69 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 66/80

70 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 67/80

71 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 There is a tertiary adjoint problem and another term in the estimate if the different discretizations are used for the two components 2 ) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 68/80

72 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 69/80

73 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 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 70/80

74 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 We assume that the full operator and the decomposed operator share a common solution, e.g. a steady-state, and linearize near this solution to form the adjoint Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 71/80

75 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 72/80

76 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 73/80

77 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 74/80

78 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 75/80

79 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 76/80

80 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 77/80

81 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 78/80

82 Conclusion Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 79/80

83 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 80/80