Equivalence Theorems and Their Applications

Transcription

1 Equivalence Theorems and Their Applications Tan Bui-Thanh, Center for Computational Geosciences and Optimization Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin, USA September 13, 2010

2 Time Domain Electromagnetic Waves Spherical Cavity electromagnetic scattering

3 Outline Consistency, Stability and Convergence An Equivalence Theorem for y = Tx: T linear Lax Equivalence Theorem for u (t) = Au(t) Nonlinear maps y = Tx Equivalence Theorem for Ay = x A Discontinuous Spectral Element Method for Hyperbolic Equations?? (Next presentation) Stability of a Discontinuous Spectral Element Method for Wave Propagation Problems (Next presentation)

4 Consistency, Stability and Convergence An Equivalence Theorem for y = Tx: T linear Lax Equivalence Theorem for u (t) = Au(t) Nonlinear maps y = Tx Equivalence Theorem for Ay = x A Discontinuous Spectral Element Method for Hyperbolic Equations?? (Next presentation) Stability of a Discontinuous Spectral Element Method for Wave Propagation Problems (Next presentation)

5 Motivation Question What is the first thing you need to do when you derive/invent a new numerical method?

6 Motivation Question What is the first thing you need to do when you derive/invent a new numerical method? consistency stability convergence

7 Consistency, Stability and Convergence What is consistency?

8 Consistency, Stability and Convergence What is consistency? Consistency is a measure of how close a discretization is to the continuous problem = how good you approximate operators and functions

9 Consistency, Stability and Convergence What is consistency? Consistency is a measure of how close a discretization is to the continuous problem = how good you approximate operators and functions What is stability?

10 Consistency, Stability and Convergence What is consistency? Consistency is a measure of how close a discretization is to the continuous problem = how good you approximate operators and functions What is stability? Stability means that the propagated error is controlled by the error in the data = continuity of solution w.r.t the data, uniform boundedness of the discrete operator

11 Consistency, Stability and Convergence What is consistency? Consistency is a measure of how close a discretization is to the continuous problem = how good you approximate operators and functions What is stability? Stability means that the propagated error is controlled by the error in the data = continuity of solution w.r.t the data, uniform boundedness of the discrete operator What is convergence?

12 Consistency, Stability and Convergence What is consistency? Consistency is a measure of how close a discretization is to the continuous problem = how good you approximate operators and functions What is stability? Stability means that the propagated error is controlled by the error in the data = continuity of solution w.r.t the data, uniform boundedness of the discrete operator What is convergence? Convergence means that the discrete solution converges to the exact solution = error between the exact and discrete solutions converges to zero

13 The importance of equivalence theorems Which of the three (consistency, stability, convergence) is the most difficult? Why?

14 The importance of equivalence theorems Which of the three (consistency, stability, convergence) is the most difficult? Why? Convergence: needs knowledge about the exact solution

15 The importance of equivalence theorems Which of the three (consistency, stability, convergence) is the most difficult? Why? Convergence: needs knowledge about the exact solution Alternate route for convergence: Equivalence theorems consistency + stability convergence

16 The importance of equivalence theorems Which of the three (consistency, stability, convergence) is the most difficult? Why? Convergence: needs knowledge about the exact solution Alternate route for convergence: Equivalence theorems consistency + stability convergence Peter Lax 1953 Well-posedness of the original differential equation problem and consistency imply the equivalence between stability and convergence of difference methods stability convergence

17 More on Stability The easiest among the three?

18 More on Stability The easiest among the three? purely the property of the discrete problem: Knowledge about the exact solution/operators is not needed

19 More on Stability The easiest among the three? purely the property of the discrete problem: Knowledge about the exact solution/operators is not needed An analogy: uniqueness implies existence (matrix theory, Fredolm theory). Think about the proof of Banach fixed point theorem. (next talk about inverse problem theory)

20 More on Stability The easiest among the three? purely the property of the discrete problem: Knowledge about the exact solution/operators is not needed An analogy: uniqueness implies existence (matrix theory, Fredolm theory). Think about the proof of Banach fixed point theorem. (next talk about inverse problem theory) Extremely important for computer implementation?

21 More on Stability The easiest among the three? purely the property of the discrete problem: Knowledge about the exact solution/operators is not needed An analogy: uniqueness implies existence (matrix theory, Fredolm theory). Think about the proof of Banach fixed point theorem. (next talk about inverse problem theory) Extremely important for computer implementation? round-off errors

22 General settings Definitions Let V, W be Banach spaces, and T, T h : V W i) Wellposedness: continuity of T, T h ii) Consistency: T h is said to be consistent with T if lim h 0 (T h T ) v 0 = 0, v 0 D V, D dense in V, iii) Stability: T h is called stable if sup h T h <. (Uniform boundedness) iv) Convergence: T h is said to converge to T, if lim h 0 (T h T ) v = 0, v V Replace h by n if n is more natural

23 Equivalence Theorem for Linear Operators Theorem A consistent family of T h is convergent if and only if it is stable.

24 Equivalence Theorem for Linear Operators Theorem A consistent family of T h is convergent if and only if it is stable. Proof. ) Since lim h 0 (T h T ) v = 0, v V, the family T h is pointwise uniformly bounded continuous linear operators. The uniform boundedness principle yields sup h T h <, which is exactly stability.

25 Equivalence Theorem for Linear Operators Theorem A consistent family of T h is convergent if and only if it is stable. Proof. ) Since lim h 0 (T h T ) v = 0, v V, the family T h is pointwise uniformly bounded continuous linear operators. The uniform boundedness principle yields sup h T h <, which is exactly stability. ) By triangle inequality (three-ɛ argument), v 0 D, Tv T h v T v v 0 + (T T h ) v 0 + T h v 0 v. The proof is complete by the following two facts. First, the consistency implies h 0 : h h 0 such that (T T h ) v 0 ɛ/3. Second, the density of D allows us to pick v 0 such that v v 0 ɛ 3 max{ T,sup h T h }.

26 A suitable setting V = (C [a, b], ) Numerical Integrations Tf = b a f dx, f V, then T is linear and bounded, T f (b a) f T n f = n i=1 w if i, f V, then T n is linear and bounded, T n f ( n i=1 w i) f The family T n is consistent if f P, P space of polynomials (P V, dense?), lim n T n f Tf = 0. The family T n is stable if sup n T n <. The family T n is convergent if f V lim n T n f Tf = 0 Examples Stable : Trapezoidal, Simpson, Gauss quadrature, and etc Unstable : Newton-Cotes

27 Numerical Derivatives A suitable setting V = ( C k [a, b], C k), W = (C [a, b], ) D (k) f, f V is linear and bounded, D (k) f f C k Denote D (k) h f, f V the numerical derivative The family ( D (k) h is consistent ) if lim h 0 D (k) h D(k) f = 0 f D V, D is dense The family D (k) h D (k) is stable if sup h <. The family D (k) h is convergent if f V D (k) lim h 0 h f D(k) f = 0 Examples: Stability of forward differentiation D (1) f(x+h) f(x) = sup f C 1 =1 sup x = h sup f C 1 =1 sup x f (x + θh) 1 h h

28 Numerical Derivatives Convergence is independent of Stability lim D (1) h 0 h f D(1) f = lim sup h 0 x lim sup h 0 x f(x + h) f(x) f (x) h = f (x + θh) f (x) = 0

29 Linear time dependent problems One step method Consider the problem u (t) = Au(t), one step method u(0) = u 0, and an abstract v(h) = B h u 0, v(nh) = B n h u 0. Well-posedness of the continuous problem Let S : V V, u(t) = S(t)u 0, we require u(t) is continuous w.r.t t and sup t S(t) <. Well-posedness of the discrete problem For each 0 h h 0, we require sup h B h <

30 Linear time dependent problems Consistency v 0 D V, D is dense, lim B hu(t) S(t + h)v 0 = 0 h 0 Stability B n h <, h, n : nh T Convergence lim k B n k h k v 0 S(t)v 0 = 0, where limk n k h k = t

31 Stability Convergence Stability Convergence Using triangle inequality and three-ɛ trick we have v(nh) u(t) = B n h u 0 S(t)u 0 Bh n } v {{ u 0 + B n h } v S(t)v + S(t) v u }{{} 0 0, }{{} stability + density consistency wellposedness + density

32 Stability Convergence Convergence Stability If n is finite then by the discrete wellposedness we have sup h B n h sup B h n <. h Now k : lim k n k h k = t, by convergence we have for each u 0 V, the family B n k h k is uniformly bounded. Then by the uniform boundedness principle, we have sup B n k h k <. k In both cases, the stability condition is proved.

33 A nonlinear setting for y = Tx T is nonlinear T : V W, and V, W are Banach Wellposedness: T is continuous Consistency: convergence on a dense subspace D v D V : lim n T nv Tv = 0 Stability ɛ, v v 0 δ, such that T n v T n v 0 ɛ (replace uniform boundedness by equi-continuity) Convergence v V, n 0 : n n 0 lim n T nv Tv = 0

34 Equivalence Theorem for nonlinear Operators Theorem A consistent family of T h is convergent if and only if it is stable.

35 Equivalence Theorem for nonlinear Operators Theorem A consistent family of T h is convergent if and only if it is stable. Proof. ) Again the three-ɛ trick, v v 0 < δ T n v T n v 0 T n v Tv + Tv Tv }{{} 0 + Tv }{{} 0 T n v 0 ɛ }{{} convergence Wellposedness convergence

36 Equivalence Theorem for nonlinear Operators Theorem A consistent family of T h is convergent if and only if it is stable. Proof. ) Again the three-ɛ trick, v v 0 < δ T n v T n v 0 T n v Tv + Tv Tv }{{} 0 + Tv }{{} 0 T n v 0 ɛ }{{} convergence Wellposedness convergence ) By the three-ɛ trick, v V, by density v 0 D, v v 0 < δ Tv T n v Tv Tv 0 + Tv }{{} 0 T n v 0 + T }{{} n v 0 T n v ɛ }{{} wellposedness Consistency Stability

37 Linear equation Ay = x Wellposedness Let A : W V, and V, W are Banach. The problem is wellposed if A is continuous and bijective. We know that the inverse exists, i.e. T = A 1, and T is continuous (Why??). The problem can be converted to the form y = Tx which we have already discussed.

38 Consistency, Stability and Convergence An Equivalence Theorem for y = Tx: T linear Lax Equivalence Theorem for u (t) = Au(t) Nonlinear maps y = Tx Equivalence Theorem for Ay = x A Discontinuous Spectral Element Method for Hyperbolic Equations?? (Next presentation) Stability of a Discontinuous Spectral Element Method for Wave Propagation Problems (Next presentation)

39 Consistency, Stability and Convergence An Equivalence Theorem for y = Tx: T linear Lax Equivalence Theorem for u (t) = Au(t) Nonlinear maps y = Tx Equivalence Theorem for Ay = x A Discontinuous Spectral Element Method for Hyperbolic Equations?? (Next presentation) Stability of a Discontinuous Spectral Element Method for Wave Propagation Problems (Next presentation)