Equivalence Theorems and Their Applicationstanbui/Equivalence...(Uniform boundedness)...

Equivalence Theorems and Their Applications

Tan Bui-Thanh,

Center for Computational Geosciences and OptimizationInstitute for Computational Engineering and Sciences (ICES)

The University of Texas at Austin, USA

September 13, 2010

Time Domain Electromagnetic WavesSpherical Cavity

electromagnetic scattering

Outline

Consistency, Stability and ConvergenceAn Equivalence Theorem for y = Tx: T linearLax Equivalence Theorem for u′(t) = Au(t)Nonlinear maps y = TxEquivalence Theorem for Ay = x

A Discontinuous Spectral Element Method for HyperbolicEquations?? (Next presentation)

Stability of a Discontinuous Spectral Element Method for WavePropagation Problems (Next presentation)

Motivation

QuestionWhat is the first thing you need to do when you derive/invent anew numerical method?

I consistency

I stability

I convergence

Motivation

QuestionWhat is the first thing you need to do when you derive/invent anew numerical method?

I consistency

I stability

I convergence

Consistency, Stability and Convergence

What is consistency?

I Consistency is a measure of how close a discretization is tothe “continuous” problem = how good you approximateoperators and functions

What is stability?

I Stability means that the propagated error is controlled by theerror in the data = continuity of solution w.r.t the data,uniform boundedness of the discrete operator

What is convergence?

I Convergence means that the discrete solution converges tothe exact solution = error between the exact and discretesolutions converges to zero

What is stability?

The importance of equivalence theorems

Which of the three (consistency, stability, convergence) is themost difficult? Why?

I 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 equationproblem and consistency imply the equivalence betweenstability and convergence of difference methods”

stability ⇔ convergence

More on Stability

The “easiest” among the three?

I purely the property of the discrete problem: Knowledge aboutthe exact solution/operators is not needed

I An analogy: uniqueness implies existence (matrix theory,Fredolm theory). Think about the proof of Banach fixed pointtheorem. (next talk about inverse problem theory)

Extremely important for computer implementation?

I round-off errors

More on Stability

I round-off errors

More on Stability

I round-off errors

More on Stability

I round-off errors

More on Stability

I round-off errors

General settings

DefinitionsLet V ,W be Banach spaces, and T,Th : V →W

i) Wellposedness: continuity of T,Th

ii) Consistency: Th is said to be consistent with T iflimh→0 ‖(T h − T ) v0‖ = 0, ∀v0 ∈D ⊂ V , D dense in V ,

iii) Stability: Th is called stable if suph ‖Th‖ <∞. (Uniformboundedness)

iv) Convergence: Th is said to converge to T, iflimh→0 ‖(T h − T ) v‖ = 0, ∀v ∈ V

Replace h by n if n is more natural

Equivalence Theorem for Linear Operators

TheoremA consistent family of Th is convergent if and only if it is stable.

Proof.

⇒) Since limh→0 ‖(T h − T ) v‖ = 0, ∀v ∈ V , the family Th ispointwise uniformly bounded continuous linear operators. Theuniform boundedness principle yields suph ‖Th‖ <∞, which isexactly stability.

⇐) By triangle inequality (three-ε argument), v0 ∈D,

‖Tv −Thv‖ ≤ ‖T‖ ‖v − v0‖+ ‖(T−Th) v0‖+ ‖Th‖ ‖v0 − v‖ .

The proof is complete by the following two facts. First, theconsistency implies ∃h0 : ∀h ≤ h0 such that‖(T−Th) v0‖ ≤ ε/3. Second, the density of D allows us topick v0 such that ‖v − v0‖ ≤ ε

3max{‖T‖,suph‖Th‖} .

Proof.

Numerical Integrations

A suitable settingI V = (C [a, b] , ‖·‖∞)

I Tf =∫ ba f dx, f ∈ V , then T is linear and bounded,

|Tf | ≤ (b− a) ‖f‖∞I Tnf =

∑ni=1wifi, f ∈ V , then Tn is linear and bounded,

|Tnf | ≤ (∑n

i=1wi) ‖f‖∞I The family Tn is consistent if ∀f ∈ P, P space of

polynomials (P ⊂ V , dense ?), limn→∞ |Tnf −Tf | = 0.

I The family Tn is stable if supn ‖Tn‖ <∞.

I The family Tn is convergent if ∀f ∈ Vlimn→∞ |Tnf −Tf | = 0

Examples

Stable : Trapezoidal, Simpson, Gauss quadrature, and etc

Unstable : Newton-Cotes

Numerical Derivatives

A suitable settingI V =

(Ck [a, b] , ‖·‖Ck

), W = (C [a, b] , ‖·‖∞)

I D(k)f, f ∈ V is linear and bounded,∥∥D(k)f

∥∥∞ ≤ ‖f‖Ck

I Denote D(k)h f, f ∈ V the numerical derivative

I The family D(k)h is consistent if

limh→0

∥∥∥(D(k)h −D

(k))f∥∥∥∞

= 0 ∀f ∈D ⊂ V , D is dense

I The family D(k)h is stable if suph

∥∥∥D(k)h

∥∥∥∞<∞.

I The family D(k)h is convergent if ∀f ∈ V

limh→0

∥∥∥D(k)h f −D(k)f

∥∥∥∞

Examples: Stability of forward differentiation∥∥∥D(1)h

∥∥∥ = sup‖f‖C1=1 supx∣∣∣f(x+h)−f(x)

∣∣∣ =sup‖f‖C1=1 supx |f ′(x+ θh)| ≤ 1

Numerical Derivatives

Convergence is independent of Stability

limh→0

∥∥∥D(1)h f −D(1)f

∥∥∥∞

= limh→0

∣∣∣∣f(x+ h)− f(x)h

− f ′(x)∣∣∣∣ =

limh→0

∣∣f ′(x+ θh)− f ′(x)∣∣ = 0

Linear time dependent problems

One step method

Consider the problem u′(t) = Au(t), u(0) = u0, and an abstractone step method

v(h) = Bhu0,

v(nh) = Bnhu0.

Well-posedness of the continuous problem

Let S : V → V , u(t) = S(t)u0, we require u(t) is continuous w.r.tt and supt ‖S(t)‖ <∞.

Well-posedness of the discrete problem

For each 0 ≤ h ≤ h0, we require suph ‖Bh‖ <∞

Linear time dependent problems

Consistency

∀v0 ∈D ⊂ V , D is dense,

limh→0‖Bhu(t)− S(t+ h)v0‖ = 0

Stability

‖Bnh‖ <∞, ∀h, n : nh ≤ T

Convergence

limk→∞

∥∥∥Bnkhkv0 − S(t)v0

∥∥∥ = 0, where limk→∞ nkhk = t

Stability ⇔ Convergence

Stability ⇒ Convergence

Using triangle inequality and three-ε trick we have

‖v(nh)− u(t)‖ = ‖Bnhu0 − S(t)u0‖ ≤

‖Bnh‖ ‖v − u0‖︸︷︷︸

stability + density

+ ‖Bnhv − S(t)v‖︸︷︷︸consistency

+ ‖S(t)‖ ‖v − u0‖︸︷︷︸wellposedness + density

→ 0,

Stability ⇔ Convergence

Convergence ⇒ Stability

If n is finite then by the discrete wellposedness we have

suph‖Bn

h‖ ≤ suph‖Bh‖n <∞.

Now k →∞ : limk nkhk = t, by convergence we have for eachu0 ∈ V , the family Bnk

hkis uniformly bounded. Then by the

uniform boundedness principle, we have

∥∥∥Bnkhk

∥∥∥ <∞.In both cases, the stability condition is proved.

A nonlinear setting for y = Tx

T is nonlinearT : V →W , and V ,W are Banach

I Wellposedness: T is continuous

I Consistency: convergence on a dense subspace D

∀v ∈D ⊂ V : limn→∞

‖Tnv −Tv‖ = 0

I Stability ∀ε, ‖v − v0‖ ≤ δ, such that ‖Tnv −Tnv0‖ ≤ ε(replace uniform boundedness by equi-continuity)

I Convergence

∀v ∈ V ,∃n0 : n ≥ n0 limn→∞

‖Tnv −Tv‖ = 0

Equivalence Theorem for nonlinear Operators

Proof.

⇒) Again the three-ε trick, ‖v − v0‖ < δ

‖Tnv −Tnv0‖ ≤‖Tnv −Tv‖︸︷︷︸

convergence

+ ‖Tv −Tv0‖︸︷︷︸Wellposedness

+ ‖Tv0 −Tnv0‖︸︷︷︸convergence

≤ ε

⇐) By the three-ε trick, ∀v ∈ V , by density ∃v0 ∈D,‖v − v0‖ < δ

‖Tv −Tnv‖ ≤‖Tv −Tv0‖︸︷︷︸wellposedness

+ ‖Tv0 −Tnv0‖︸︷︷︸Consistency

+ ‖Tnv0 −Tnv‖︸︷︷︸Stability

≤ ε

Proof.

convergence

≤ ε

Proof.

convergence

≤ ε

Linear equation Ay = x

Wellposedness

Let A : W → V , and V ,W are Banach. The problem iswellposed if A is continuous and bijective. We know that theinverse exists, i.e. T = A−1, and T is continuous (Why??).

The problem can be converted to the form y = Tx which we havealready discussed.

Equivalence Theorems and Their Applicationstanbui/Equivalence...(Uniform boundedness)...

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