A Posteriori Existence in Numerical Computations

Size: px

Start display at page:

Download "A Posteriori Existence in Numerical Computations"

Tyler Tate
5 years ago
Views:

1 Oxford University Computing Laboratory OXMOS: New Frontiers in the Mathematics of Solids OXPDE: Oxford Centre for Nonlinear PDE January, 2007

2 Introduction Non-linear problems are handled via the implicit function theorem.

3 Example: (u 3 x u x ) x + u = 0, u(0) = u(1) =

4 Example: u xx = u , u(0) = u(1) =

5 Example: u xx = u , u(0) = u(1) =

6 Introduction Abstract Setting: For, X, Y Banach spaces, F : X Y, solve F(u) = 0

7 Introduction Abstract Setting: For, X, Y Banach spaces, F : X Y, solve F(u) = 0 In general, nonlinear problems exhibit non-uniqueness of solutions (global or local) non-existence of solutions spurious solutions in numerical approximations

8 Introduction Abstract Setting: For, X, Y Banach spaces, F : X Y, solve F(u) = 0 In general, nonlinear problems exhibit non-uniqueness of solutions (global or local) non-existence of solutions spurious solutions in numerical approximations Given a computed approximate solution U, does an exact solution u exist which is near U?

9 Introduction Basic Idea for an A Posteriori Existence Proof: Let U X be a computed approximation. U solves v F(v) F(U) = 0 u X satisfies F(u) = 0 if it solves (this perturbs the above problem) v F(v) F(U) = F(U) Estimate F(U) Y and F (U) 1 L(Y,X ) and apply the Inverse Function Theorem.

10 History Monotonicity Methods: inclusion in an function interval u(x) [u 0 (x), u 1 (x)]; [Collatz, ArchMath, 1952] Methods based on interval arithmetic [Nakao et al., from about 1988] Fixed point methods: applications mostly to nonlinear Poisson problems, rigorous computational proofs; [Plum et al.,from about 1990] Shadowing: for dynamical systems; [Hammel et al., Complexity, 1987]

11 History Monotonicity Methods: inclusion in an function interval u(x) [u 0 (x), u 1 (x)]; [Collatz, ArchMath, 1952] Methods based on interval arithmetic [Nakao et al., from about 1988] Fixed point methods: applications mostly to nonlinear Poisson problems, rigorous computational proofs; [Plum et al.,from about 1990] Shadowing: for dynamical systems; [Hammel et al., Complexity, 1987] Bibliography on Enclosure Methods by G. Bohlender

12 A Posteriori Existence : Abstract Result Suppose that F is Fréchet differentiable and that F (v) F (w) L(X,Y) g( w X ; v w X ), where g(s; ) is continuous and increasing. Proposition Suppose that U X and R > 0 satisfy (i) F(U) Y η (ii) F (U) 1 L(Y,X ) 1/σ (iii) η + (iv) R 0 g ( U X ; r ) dr σr g ( U X ; R ) < σ Then there exists a unique u B(U, R) such that F(u) = 0. Proof: Track constants in proof of Inverse Function Theorem.

13 A Posteriori Existence : Proof 1. Define Fixed-Point Map: N : X X [ ] F (U)(N (v) U) = F(U) F(v) F(U) F (U)(v U) F(u) = 0 if, and only if, N (u) = u.

14 A Posteriori Existence : Proof 1. Define Fixed-Point Map: N : X X [ ] F (U)(N (v) U) = F(U) F(v) F(U) F (U)(v U) F(u) = 0 if, and only if, N (u) = u. 2. N (B(U, R)) B(U, R): N (v) U X F (U) 1 L(Y,X ) reduces to σ 1 (η + R 0 g(r)dr) R ( F(U) Y + + F(v) F(U) F (U)(v U) Y )! R

15 A Posteriori Existence : Proof 1. Define Fixed-Point Map: N : X X [ ] F (U)(N (v) U) = F(U) F(v) F(U) F (U)(v U) F(u) = 0 if, and only if, N (u) = u. 2. N (B(U, R)) B(U, R): N (v) U X F (U) 1 L(Y,X ) reduces to σ 1 (η + R 0 g(r)dr) R ( F(U) Y + + F(v) F(U) F (U)(v U) Y )! R 3. N is a contraction: reduces to the condition to σ 1 g(r) < 1

16 Example: Semi-linear Poisson Problem Strong form: Ω convex domain in R 2 u = f (u), in Ω; u = 0, on Ω.

17 Example: Semi-linear Poisson Problem Strong form: Ω convex domain in R 2 u = f (u), in Ω; u = 0, on Ω. Weak form: Find u H 1 0 (Ω) such that [ ] u w f (u)w dx = 0 Ω w H 1 0 (Ω) X = H 1 0 (Ω), Y = H 1 (Ω) F(u), w = [ ] Ω u w f (u)w dx F (u)v, w = [ Ω v w f (u)vw ] dx

18 Discretization T : regular subdivision of Ω. S 0 (T ) H 1 0 (Ω): conforming finite element space. Galerkin Projection: F(U), W = Ω [ U W + f (U)W (solved using Newton s method) ] dx = 0 W S 0 (T ).

19 Discretization T : regular subdivision of Ω. S 0 (T ) H 1 0 (Ω): conforming finite element space. Galerkin Projection: F(U), W = Ω [ U W + f (U)W (solved using Newton s method) ] dx = 0 W S 0 (T ). We need: I. Residual Estimate: F(U) Y η II. Stability Estimate: F (U) 1 L(Y,X ) 1/σ III. Bound the Modulus of Continuity of F

20 III. Modulus of Continuity of F Example: u = u 2 + λ on Ω = (0, 1) 2 F (u)v = v 2uv ( v 2u 1 v) ( v 2u 2 v) H 1 2(u 1 u 2 )v H 1 = g(r) = 2C 3 s R where C s 0.55 is the constant for u L 3 C s u L 2.

21 III. Modulus of Continuity of F Example: u = u 2 + λ on Ω = (0, 1) 2 F (u)v = v 2uv ( v 2u 1 v) ( v 2u 2 v) H 1 2(u 1 u 2 )v H 1 = g(r) = 2C 3 s R where C s 0.55 is the constant for u L 3 C s u L 2. More general: u = f (u) If f has p-growth (in 2D) then g(s; R) = C(1 + s + R) p R where C depends on embedding constants which can be computed explicitly and on f

22 I. Residual Estimate version 1 Standard Residual Estimate: If U is an exact solution of the finite element discretization then [ F(U) H 1 C(T ) ht 2 U + f (U) 2 L 2 (T ) T T + e E h e [ U] 2 L 2 (e) ] 1/2 Advantages: efficient to compute analytically well-understood (lower bounds, optimality, etc.) Problems: Difficult to obtain a sharp constant C(T ) U is in general not an exact discrete solution, so also need to estimate the discrete residual

23 I. Residual Estimate version 2 Alternative Idea: Let G H(div) [ [ U w fw] dx = ( U G) w + G w fw] dx = ( U G) w dx (divg + f )w dx U G L 2 w L 2 + C p divg + f L 2 w L 2

24 I. Residual Estimate version 2 Alternative Idea: Let G H(div) [ [ U w fw] dx = ( U G) w + G w fw] dx = ( U G) w dx (divg + f )w dx U G L 2 w L 2 + C p divg + f L 2 w L 2 Residual Estimate: [Repin], [Plum, LinearAlgebraApp, 2001] u + f (u) H 1 U G L 2 + C p divg + f L 2

25 I. Residual Estimate version 2 Alternative Idea: Let G H(div) [ [ U w fw] dx = ( U G) w + G w fw] dx = ( U G) w dx (divg + f )w dx U G L 2 w L 2 + C p divg + f L 2 w L 2 Residual Estimate: [Repin], [Plum, LinearAlgebraApp, 2001] u + f (u) H 1 U G L 2 + C p divg + f L 2 Compute G by solving a quadratic optimization problem: U G 2 + C L p divg 2 + f 2 min 2 L 2 In my examples: G minimizes this quadratic in S 0 (T ) n

26 II. Stability Estimate version 2 Residual Estimate: u + f (u) H 1 U G L 2 + C p divg + f L 2 =: η Advantages of this Estimate: Requires only the computation of the Poincaré constant Seems quite sharp for many problems Requires no information on U or G, so can be made rigorous in exact arithmetic (Mathematica or Maple) or interval arithmetic Disadvantage: G may be more expensive to compute than the solution

27 II. Stability Estimate Optimal Constant: σ opt = inf sup v =1 w =1 F (U)v, w

28 II. Stability Estimate Optimal Constant: σ opt = Approximate σ opt by Σ = inf sup v =1 w =1 inf sup V =1 W =1 F (U)v, w F (U)V, W

29 II. Stability Estimate Optimal Constant: σ opt = Approximate σ opt by Σ = Challenges: inf sup v =1 w =1 inf sup V =1 W =1 F (U)v, w F (U)V, W In general, Σ > σ opt, so we require an error estimate H 1 -eigenvalue problem is ill-posed since eigenvalues cluster at 1.? Can Σ be computed reliably?

30 II. Stability Estimate Some Possibilities: Weinstein bounds [Chatelin, Academic Press, 1983] Kato bounds [Kato, JPhysSocJapan, 1949] A posteriori error estimate for eigenvalue problem e.g. [Larsen, SINUM, 2000],...

31 II. Stability Estimate Some Possibilities: Weinstein bounds [Chatelin, Academic Press, 1983] Kato bounds [Kato, JPhysSocJapan, 1949] A posteriori error estimate for eigenvalue problem e.g. [Larsen, SINUM, 2000],...!!! all require a priori information: a lower bound on λ N+1.!!! Homotopy Algorithm to derive this bound: [Goerisch, 1987], [Plum, ZAMP, 1990]

32 II. Stability Estimate Some Possibilities: Weinstein bounds [Chatelin, Academic Press, 1983] Kato bounds [Kato, JPhysSocJapan, 1949] A posteriori error estimate for eigenvalue problem e.g. [Larsen, SINUM, 2000],...!!! all require a priori information: a lower bound on λ N+1.!!! Homotopy Algorithm to derive this bound: [Goerisch, 1987], [Plum, ZAMP, 1990] Here: direct a priori error estimate for Σ!

33 II. Stability Estimate step 1 There exist v H 1 0 (Ω) and λ {±σ opt} such that v + f (U)v = λ( v)

34 II. Stability Estimate step 1 There exist v H 1 0 (Ω) and λ {±σ opt} such that v + f (U)v = λ( v) v = f (U)v λ 1 Eigenvalues are clustered at 1, so best possible case is σ opt = 1.

35 II. Stability Estimate step 1 There exist v H 1 0 (Ω) and λ {±σ opt} such that v + f (U)v = λ( v) v = f (U)v λ 1 Eigenvalues are clustered at 1, so best possible case is σ opt = 1. Fix ρ < 1! If σ opt ρ take σ = ρ. If σ opt < ρ use regularity to estimate σ opt in terms of Σ.

36 II. Stability Estimate step 1 There exist v H 1 0 (Ω) and λ {±σ opt} such that v + f (U)v = λ( v) v = f (U)v λ 1 Eigenvalues are clustered at 1, so best possible case is σ opt = 1. Fix ρ < 1! If σ opt ρ take σ = ρ. If σ opt < ρ use regularity to estimate σ opt in terms of Σ. Suppose that Ω is convex, then 2 v L 2 f L v L 2 1 λ C p f L 1 ρ

37 II. Stability Estimate step 2 Let V S 0 (T ) be the Ritz projection of v, then [ ] V Σ sup V W f VW dx W =1

38 II. Stability Estimate step 2 Let V S 0 (T ) be the Ritz projection of v, then [ ] V Σ sup V W f VW dx W =1 sup W =1 [ v W f vw ] dx + f L v V L 2 W L 2 σ opt + C p f L v V L 2

39 II. Stability Estimate step 2 Let V S 0 (T ) be the Ritz projection of v, then [ ] V Σ sup V W f VW dx W =1 sup W =1 [ v W f vw ] dx + f L v V L 2 W L 2 σ opt + C p f L v V L 2 Standard H 1 and L 2 error estimates: (v V ) L 2 C i h T 2 v L 2 C ic p f L 1 ρ h T v V L 2 C 2 i h 2 T 2 v L 2 C2 i C p f L 1 ρ h 2 T (Note: C i = 1/(4 2) in my computations)

40 II. Stability Estimate Choose ρ < 1 and find 0 < σ σ opt : If σ opt ρ take σ = ρ. If σ opt < ρ take σ = Σ error estimate.

41 II. Stability Estimate Choose ρ < 1 and find 0 < σ σ opt : If σ opt ρ take σ = ρ. If σ opt < ρ take σ = Σ error estimate. Proposition where σ opt min [ ρ, Σ (1 ɛ/(1 ρ)) ɛ 2 /(1 ρ) ] =: σ, and the optimal ρ is given by ɛ = f L C i C p h T, ρ = 1 + Σ (1 Σ) 2 + ɛ + ɛ 2 2

42 Example: u = u 2, Ω = (0, 1) 2

Example: u = u 2, Ω = (0, 1) 2 η σ R a.p.e.c. T 6.3 0.3 21 23 512 4.4 0.3 15 16 1189 2.8 0.33 8.6 8.4 2574 1.9 0.34 5.5 5.2 5665 1.3 0.35 3.

43 Example: u = u 2, Ω = (0, 1) 2 η σ R a.p.e.c. T

44 Example: u = u , Ω = (0, 1) 2

45 Example: u = u , Ω = (0, 1) y 0 0 x η σ R a.p.e.c. T

46 Example: u = u , Ω = (0, 1) 2 η σ R a.p.e.c. T

47 1D Example: u xx = u 2 + λ

48 Computation in 1D: u xx = u 2 + λ λ = 22.6 it. T η Σ σ a.p.e.c. err.est

49 Computation in 1D: u xx = u 2 + λ λ = 22.6 it. T η Σ σ a.p.e.c. err.est λ = it. T η Σ σ a.p.e.c. err.est

50 Quasilinear BVPs in 1D Model Problem: ( a(u x ) ) x = f (u), u(0) = u(1) = 0. If solutions are expected to be Lipschitz then take X = W 1, 0 (0, 1) and Y = W 1, (0, 1)

51 Quasilinear BVPs in 1D Model Problem: ( a(u x ) ) x = f (u), u(0) = u(1) = 0. If solutions are expected to be Lipschitz then take X = W 1, 0 (0, 1) and Y = W 1, (0, 1) 1. Residual Estimate: ( a(u x ) ) x + f (u) W 1, η := 1 2 max ( h k a(ux ) ) k x + f (u) L (x k 1,x k )

52 Quasilinear BVPs in 1D Model Problem: ( a(u x ) ) x = f (u), u(0) = u(1) = 0. If solutions are expected to be Lipschitz then take X = W 1, 0 (0, 1) and Y = W 1, (0, 1) 1. Residual Estimate: ( a(u x ) ) x + f (u) W 1, η := 1 2 max ( h k a(ux ) ) k x + f (u) L (x k 1,x k ) 2. Stability Estimate: F (u) 1 1 L(W 1,,W 1, 0 ) σ := (2 + 1 π f L 2/σ 2 )/a 0, where a 0 = ess.inf.a (u x ) and σ 2 is the smallest H 1 -singular value.

53 Quasilinear BVPs in 1D: The Stability Constant 1. inf-sup condition a 0 2 inf sup v x L =1 w x L 1=1 1 0 a (u x )v x w x dx

54 Quasilinear BVPs in 1D: The Stability Constant 1. inf-sup condition 2. This implies a 0 2 inf 1 2 a 0 v x L sup w x L 1=1 sup v x L =1 w x L 1=1 [ a (u x )v x w x dx ( ) a (u x )v x w x f (u)vw dx + f (u) L 2 v L 2 w L (a (u x )v x ) x + f (u)v W 1, + 1 2π f (u) L 2 v x L 2 ]

55 Quasilinear BVPs in 1D: The Stability Constant 1. inf-sup condition 2. This implies a 0 2 inf 1 2 a 0 v x L sup w x L 1=1 sup v x L =1 w x L 1=1 3. Use singular value estimate [ a (u x )v x w x dx ( ) a (u x )v x w x f (u)vw dx + f (u) L 2 v L 2 w L (a (u x )v x ) x + f (u)v W 1, + 1 2π f (u) L 2 v x L 2 ] σ 2 v x L 2 (a (u x )v x ) x +f (u)v H 1 ( (a (u x )v x ) x +f (u)v W 1, )

56 Example: (u 3 x u x ) x + u =

57 Conclusion Examples of A Posteriori Existence: Nonlinear Poisson Problem: u = f (x, u) Quasilinear BVPs in 1D Idea: Residual Estimate + Local Stability Estimate (regularity theory for the PDE) + Inverse Function Theorem implies existence of exact solution. Crucial additional step on top of existing methodology: estimating the local stability constant Some Possible Extensions: Rigorous computation of solution branches: [Plum, JComputApplMath, 1995] Time-dependent problems ( shadowing ): e.g. [Coomes et al., NumerMath, 1995]? Singular solutions (corner singularities, vortices, dislocations,... )? Quasilinear BVPs in 2D and 3D

58 Bibliography M. Plum, Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra and its Applications 324 (2001) M. Plum, Guaranteed numerical bounds for eigenvalues, in D. Hinton and P.W. Schaefer (Eds.), Spectral Theory and Computational Methods of Sturm Liouville Problems, Marcel Dekker, New York, 1997 C. Ortner, Preprint (in progress)

A Posteriori Existence in Adaptive Computations

Report no. 06/11 A Posteriori Existence in Adaptive Computations Christoph Ortner This short note demonstrates that it is not necessary to assume the existence of exact solutions in an a posteriori error