Numerical verification methodand it’s applicationsand it s applications

K t K b hiKenta Kobayashi

Kanazawa University,

Ishikawa, Japan.

Table of contentsTable of contents

1. Reliability of the numerical computations.

2 Introduction to the numerical verification method2. Introduction to the numerical verification method.

3. Application to the uniqueness proof of Stokes’ wave3. Application to the uniqueness proof of Stokes wave of extreme form.

4. Possibility for the application to the equation in mathematical finance.

Accuracy of the numerical computation

Phenomenon（motion of water, phase transition, finance, etc．）

Suitability in modeling and simplification

Navier-Stokes eq., Allen-Cahn eq., Black-Scholes eq. etc．

T ti di ti tiTruncation error, discretization errorDifference between infinite dimension and finite dimension

Discretization（Difference method, finite element method, etc．）

finite dimension

Rounding error

Numerical result

Rounding error

Usually, double precision computation (15～16 digits) is considered to be enough But

bAx =

is considered to be enough. But...

01

,1025589615.4186952015901872164919121

bA ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=

21122211

212

21122211

221 ,

aaaaax

aaaaax

−−

=−

=

⎠⎝⎠⎝

2112221121122211

Exact solution： Numerical result：

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛541869520

102558961,

83739041205117922

⎟⎠

⎜⎝

⎟⎠

⎜⎝ 5.4186952083739041

Discretization error

Sometimes, ghost numerical solutions appear in discretizations

Example：Emden equation

discretizations.

⎩⎨⎧

Ω∂=×=Ω=Δ−

on0),/1,0(),0(in2

uaauu

Exact solution Ghost solution

Why we need numerical verifications？

１ We want to guarantee the quality of１．We want to guarantee the quality ofthe numerical result.

2．We want to prove the existence and/orthe uniqueness of the equation.

We can realize them bynumerical verification method

Introduction to the numerical verification method

We cannot avoid rounding error in floating point calculation.

Let’s imagine 4-digits computer: 1.235×2.597＝3.207

Exact solution is 3 207295Exact solution is 3.207295.

It is impossible to obtain exact resultIt is impossible to obtain exact result.But it is possible to evaluate the error explicitly.

We introduce interval arithmetic.

Interval arithmeticInterval arithmetic

We define interval of real number [a,b]We define interval of real number [a,b]as following closed set.

}|{],[ bxaxba ≤≤=

We define operation ＊ betweentwo interval X and Y as follows:

},|{ YyXxyxYX ∈∈∗=∗

where ＊ denotes one of ＋，－， ・，or ／.

]3334.0,3333.0[31∈Example１

3

→ We can treat any real number.

Example2

]6667.0,1428.0[]666666.0,142857.0[]7,3[]2,1[ ⊂= LL／

→ We cannot realize exact results of on a computere ca ot ea e e act esu ts o o a co putebut we can obtain the intervals which enclose the results.

We calculate upper bounds of intervals by using round-up modeWe calculate upper bounds of intervals by using round up modeand lower bounds by round-down mode.

The IEEE 754 standard which is followed by many computersThe IEEE-754 standard which is followed by many computers (and CPU’s such as Intel Pentium etc.) has four different rounding modes

d t t d d d d d t d 0round-to-nearest, round-up, round-down, and round-toward 0.Default mode is round-to-nearest.

Fixed point theorem

If the compact operator satisfies ,

)()(

uFuUUFF

=⊂

the solution of exists)(uFu =the solution of exists.

（Schauder’s fix point theorem)

If the operator satisfiesF

)()10(,)()(

uFuUyxyxyFxF

=<≤

∈∀−≤−

λλλ

for some , then has a )()(unique solution. （Banach’s fix point theorem)

A fixed point theorem plays an essential role in the numerical verification method.We can prove the fixed point theorem on a computer by usingWe can prove the fixed point theorem on a computer by using interval arithmetic.

Candidate setWe can enclose a subset of the infinite dimensional space with finite number of variables.

Ex.１ ( )⎩⎨⎧

∈∈∈+= ∑∞

=2211

0,,,,cossin | nn

kkk AaAaAakbkaU Lθθ

⎭⎬⎫

<+∈∈∈ ∑∞

+= 1

221100 )(,,,,

nkkknn rbaBbBbBb L

rBBBAAA nn ,,,,,,, 2121 LL ：Intervals ：positive number

Ex.2 ruruuHuU hHh⎭⎬⎫

⎩⎨⎧

≤−∈= 10

|10 ：FEM sol. ：positive number

Ex.3 uuxuxuxuxuU ,)()()()( |⎭⎬⎫

⎩⎨⎧

≤≤= ：piecewise liner functions|⎭⎩

Newton like operator

In order to apply a fixed point theorem, must be a contraction mapping.

Fpp g

It’s not always true, but if we define Newton like operator

))(())~('()( 1 uFuuFIuuN −−−= −

y , p

approximate solution

Frechet derivative:)~(':~

uFu

then holds and is )()()( uNuuNuuF =⇔=expected to be a contraction mapping.

By using this transformation, we can treat unstable solutions.

)()()(

Actual process of numerical verification

Enclose candidate set (subset of some infinite dimensional space) by finite number of variables.

U

Convert the target equation to the fixed point formulation )(uFu =Convert the target equation to the fixed point formulation .)(uFu =

If necessary, c, convert the operator to a Newton like operator.F

Verify on a computer by using interval arithmetic.UUF ⊂)(y p y g)(

Existence of the solution has been proved by the computer.

Applications

・ Existence and (local) uniqueness proof for non-liner PDEsuch as Navier-Stokes equation.

・ Existence proof for the bifurcation point.

・ Existence proof for the strange chaos attractor.

・ etc…

General numerical verification method for linear and non-linearGeneral numerical verification method for linear and non linear

elliptic PDE’s is well constructed by prof. M. T. Nakao’s group.

However for now at least we need one-by-one approach forHowever, for now at least, we need one-by-one approach for

the other equations.

Water wave equation１．We consider two-dimensional motion.

２．The fluid is incompressive, inviscidand irrotationaland irrotational.

３．The pressure of the atmosphere isstable.stable.

４．The flow is infinitely deep.

５ The wave moves at constant speed y５．The wave moves at constant speed,and wave profile is stationary andperiodic.

y

４．The water wave has only one peakand one trough per period.

LL

x

５．The shape of the wave is symmetric. 2x +=

2x −=

In a coordinate system moving with the waveIn a coordinate system moving with the wave,the wave profile is assumed to be stationary

Outline of the derivation of the water wave equationy

⎪⎧ ∂∂

+∂ 1 puvuu

)(

x

xhEuler equation for incompressive irrotational fluid

⎪⎪⎪⎪

−∂

−=∂

+∂

∂−=

∂+

∂1 gpvvvu

xmyv

xu

22LxLx

x

=−=

⎧

⎪⎪

⎪⎪⎨

=∂∂

+∂∂

∂=

∂+

∂

0vu

gymy

vx

u

⎪⎪⎪

⎨

⎧

±==

−∞→→

20

)()0,(),(Lxonv

ycvu

⎪⎪⎪⎪

⎩=

∂∂

−∂∂

∂∂

0yu

xv

yx

⎪⎪⎪

⎩

⎨

−−=−=

),(),(),(),(

2

yxvyxvyxuyxu

⎪⎩ ∂∂ yx

(u,v)：velocity m：density g：gravitational acceleration

⎩ ),(),( yy

xV

yUv

yV

xUu

∂∂

−=∂∂

=∂∂

=∂∂

=There exist single valued function and

U is called velocity potential and V is called stream function.

iyxzzfiVU +==+ ),( f is turn out to be an iyxzzfiVU ++ ),(analytical function of z

By integrating through the stream line, we have

constmpgy

dzdf

=++2

21

（Bernoulli’s theorem）

)( 0 TKppxhy +==On , from Laplace’s relation, holds.f th t h

:

:0

ThK

p

x⎟⎟⎞

⎜⎜⎛−=

pressure of the atmosphere

curvature surface tension:1 2

Th

Kxx⎟⎠

⎜⎝ +

=

1 2 TKdf

curvature surface tension

)(21 2

xhyonconstm

TKgydzdf

==++

dfLcL )(,22

),(0 −∞→→±=±=== ycdzdfLxoncLUxhyonV

)(xh

y

UV

)(

x

xh

22cLcL

−

22LxLx =−= ⎯→⎯ f

⎟⎠⎞

⎜⎝⎛−=

cLifπς 2exp

If we introduce new variable ⎠⎝

2exp,1log =⎟⎠⎞

⎜⎝⎛−=+=⎟

⎠⎞

⎜⎝⎛= ρπςτθω σe

cLifi

dzdf

ci i

If we introduce new variable

tan1

==

⎠⎝⎠⎝

θρ

h

cLdzcthen, when

and

0sin

tan

2 =⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+−∂∂ −

σθ

σθ

στ

θ

τττ eqpee

hx and

⎠⎝ ∂∂∂ σσσ

Levi-Civita equation

0sin2

2

=−⎟⎟⎠

⎞⎜⎜⎝

⎛+ − θθ θθ

θHH

H

pedsdqee

dsd

2 ⎟⎠

⎜⎝ dsds

)(2)2/(: 22 LmcTqcgLpH ππ ==

Levi-Civita (1925)

Hilb t t f )(2 )2/( : LmcTqcgLpH ππ ==

L : wave length c : speed of the wave m : densityf

Hilbert transform

T : surface tension T : gravitational acceleration

LdyLdx )(sin2

),(cos2

)()( seLdsdyseL

dsdx sHsH θ

πθ

πθθ −− −=−=

)( sθ

x

Gravity waveWhen q=0 （surface tension is neglected）,

)(3 )(sin)(θ θθsH spsHdsde =

0

)0(3)(3 )(sin3θθ θsHsH dwwpee

ds

=− ∫

0

)0(3 )(sin3log)(3 θ θθsH dwwpesH ⎟

⎠⎞⎜

⎝⎛ += ∫

0

1 )(sin3log31)( θμθ

sdwwpHs − ⎟

⎠⎞⎜

⎝⎛ += ∫

( ))0(333

θμ Hep −=⎠⎝

Nekrasov’s equation

⎧ 2

Levi-Civita equation is equivalent to

⎪⎪⎧

⎟⎠⎞⎜

⎝⎛ +⋅

−= −∫ ∫ )(sinlog

2cot

61)(

0

12

0θμ

πθ

π tdtdwwsts

⎪

⎪⎪⎨ −=−

⎠⎝∫)()(

26 0

θθ

π

ss

⎪⎪⎪

⎩==+ − ).3(,)()2( )0(3 θμθπθ Hpess

A I N k (1921)⎪⎩ A. I. Nekrasov (1921)

When for , we call this solution aspositive solution

),0(0)( πθ ∈> sspositive solution.

Positive solution expresses water wave which has only one peak and one trough per periodone peak and one trough per period.

Shape of the positive solution

)(sθ

6π 100=μ

)(sθ

6 100μ

s0 π s

Wh i ll dWhen , water wave is called

“Stokes’ extreme wave”

∞=μ

（or called “Stokes’ highest wave”）

Stokes’ wave of extreme formStokes wave of extreme form

)(sθ

6π

)(sθ

sπ0

s

Solution is no longer continuous at s = 0

Known results for existenceKnown results for existence

Nekrasov’s eq. (the case μ<∞) has a positive solution.q ( μ ) p

G.Keady and J.Norbury (1978)

As μ→∞, the positive solution of Nekrasov’s eq. have a convergent subsequence, and Stokes’ wave of extreme form is coincide with the limit of the subsequence.

J.F.Toland (1978)

Known results for uniquenessKnown results for uniqueness

For 3<μ≦170, the positive solution of Nekrasov’s eq. is μ≦ , p qglobally unique (the numerical verification method is employed).K K b hi N i l ifi ti f th l b l iK. Kobayashi, Numerical verification of the global uniqueness of a positive solution for Nekrasov's equation,Japan Journal of Industrial and Applied Mathematics, Vol.21 [2] (2004) 181 218(2004), pp.181-218.

We also proved the uniqueness ofp qStokes’ wave of extreme formby the numerical verification method.K. Kobayashi, On the global uniqueness of Stokes' wave of extreme form, to appear in IMA Journal of Applied Mathematics.pp

Evaluating the solutionEvaluating the solution

∫ ⎟⎠⎞⎜

⎝⎛⋅= ∫

πθθ )(sinlogsin21)( dtdwwss

t

∫ ⎟⎠⎞⎜

⎝⎛ +⋅

−= ∫−

πθμθ

21 )(sinlogcot1)( dtdwwsts

t∫ ⎟⎠⎞⎜

⎝⎛⋅

−= ∫

πθθ

2)(sinlogcot1)( dtdwwsts

t∫ ⎟⎠

⎜⎝− ∫π 0 0

)(gcoscos6

)(ts

∫ ⎞⎛π i21

∫ ⎟⎠

⎜⎝

+ ∫ θμπ

θ0 0

)(sinlog2

cot6

)( dtdwws ∫ ⎟⎠

⎜⎝ ∫ θ

πθ

0 0)(sinlog

2cot

6)( dtdwws

( )∫ ⎟⎠⎞⎜

⎝⎛ ⋅+⋅⋅

−≤ ∫ ≥<

πθθ

π 0 0)(1)(1sinlog

coscossin2

61 dtdwww

tss t

swsw

⎟⎞

⎜⎛ ∫

−+− ππθ

||)(i d

ts( )∫ ⎟

⎟⎞

⎜⎜⎛ +∫

−+−

− ≥<πππ

θθ||

||)(sin1)(sin11 dwwwt

ts

ts swsw

∫ ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

++⋅≤

∫∫∫

−

−

−−

π

θθ

θ

π 0||

)||min(

)||,min(||

)(sin)(sin

)(sin1log

2cot

61

0

dtdwwdww

dwwtts

tss

tssts( )∫ ⎟

⎟

⎠⎜⎜

⎝+

+⋅=∫

∫−

≥<

−

θθπ 0||

||

0)(sin1)(sin1

1log2

cot61 dt

dwww

tts

swsw

ts

⎠⎝ ∫∫ )||,min(0 tss

are bounded by lower and upper bound as)(sθ

)0()()()( πθθθ ≤<≤≤ ssss

∫ ⎟⎟⎞

⎜⎜⎛+≤

∫−+−

−π

ππθ

θ

||

||)(sin

1logcot1)( dtdwwts

ts

ts∫ ⎟⎟

⎠⎜⎜

⎝+

+⋅≤∫∫

−

−

−θθπ

θ0

||

)||,min(

)||,min()(sin)(sin

1log2

cot6

)(

0

dtdwwdww

s ts

tss

tss

We write right side of this expression as ( ) )(, sJ θθ( )

( ) )(θθ is a renewed upper bound of the solution.

( ) )(, sJ θθ

I th i h t b( ) )(J θθIn the same way, is shown to be a renewed lower bound of the solution.

( ) )(, sJ θθ

Known results about bounds

Spielvogel(1970)

Known results about bounds

2/)( πθ <s

It means the fact that water waves do not have vertical tangents.

/20100005.0)(θ ≤<⋅≥s K Kobayashi/2s0100005.0)( πθ ≤<≥s K.Kobayashi

we can compute more accurate upper and lower bound( ) ( )by and repeatedly.( ) )(, sJ θθ ( ) )(, sJ θθ

An outline of proof of the uniquenessAn outline of proof of the uniqueness

Compute upper and lower bound of theFirst Step Compute upper and lower bound of the solution repeatedly.

Second Step Proving uniqueness by contraction mapping principle

Functions are evaluated by step functions

mapping principle.

Functions are evaluated by step functions.

We obtain mathematically rigorous result by t lli fl ti i t di dcontrolling floating point rounding mode.

Evaluated by step functionsy p

)(sθ ―― θ（ｓ）―― exact sol

―

6π exact sol.

―― θ（ｓ）

π0

s

)(sθ

6π

s

The condition of the solution capturedn : 20000 CPU : Pentium4 2.0GHz

π2

3π

6π

0 π

Aft 40 ti f it ti ld th iAfter 40 times of iteration, we could prove the uniqueness.

Proof of the uniqueness using contraction mapping.

We make sequences of functions that restricts the solution.

⎪⎩

⎪⎨⎧

⋅==

≤< /2s00

0

100005.0)(2/)(

πθπθ

ss( )( )

( )( )⎪⎩

⎪⎨⎧

=

=

+

+

)( , )(, max)( )( , )(, min)(

1

1

ssJsssJs

kkkk

kkkk

θθθθ

θθθθ

⎩( )( )⎩ +1 kkkk

Now, we take positive function .)(sg

nk ≥n

p )(g

If there exists such that, for any ,nk ≥

1supsup ,)(

)()( )(

)()(

00

11 <−

⋅−

≤<≤<≤++ λλ θθθθ

ππ sgss

sgss kkkk

ss

nIf there exists such that, for any ,

)()( 00 ≤<≤< ππ gg ss

holds, then the global uniqueness is proved.

Evaluating the difference between upper and lower bound

( ) ( ) )(, )(, )()( 11 sJsJss kkkkkk θθθθθθ −− ≤++

( ) )(,, sgG nn θθ

)||min(|| tssts ππ⎛ ∫∫−−+−

∫

)(

)( sgsg≤

( )∫∫+⎛

∫)||min(|| ππ tssts

0 )(sin

)( )(sin

2cot

)(61 0

||

0

)||,min(||

||

dww

dwwgdwwtsg

n

n

ts

tssts

tsπ

θ

θ

π

ππ

⎜⎜⎜

⎝

⎛⋅

∫∫∫≤ −

−−+−

−∫( )

∫∫∫≤ −

−−+−

−

⎜⎜⎜

⎝

⎛ −⋅∫ ||

0

)||,min(||

||

0 )(sin

)()( )(sin

2cot

)(61 0 k

πππ

θ

θθθ

π ts

tssts

ts

dww

dwwwdwwtsg

k

kk

)()()(

)(i

dt )( k

0sup||

||

||

0

sg

ss

ddwwg k

tsts

tsθθ

θππ

ππ −•⋅

⎟⎟⎟⎞

⎝

∫∫

∫

−+−

−+−

−+

∫( )

∫∫

∫

−+−

−+−

−⋅⎟⎟⎟⎞

−

⎝

+

∫||

||

||

0

)(i

dt )()( k ππππ

θθθ ts

ts

ts ddwww k )()(sin 0

0 || sgdww snθ π⎟

⎠ ≤<∫ ∫⎟⎠ 0

|| )(sinθ dwwk

( ))(

)()( )(

)(,, ksup sgss

sgsgG knn θθθθ −

•

Then, we write last expression as

)()( 0p sgsg s π≤<

( ))(

)()( )(

)(,, )(

)()( supsupsup 11 sssgGss kknnkk θθθθθθ −⋅

− ≤++

)()()( supsupsup000 sgsgsg sss πππ ≤<≤<≤<

)(( ) 1

)()(,,sup

0<

sgsgG nn θθ

If we can find certain which satisfy

the global uniqueness has proved.

)(sg

)(0 ≤< sgs π

We made such by iteration as follows,)(sg

( )⎪⎨

⎧ =

)(,,)(

,1)(0

sgGsg

sg

knn θθ( )( )⎪

⎩

⎨ =

≤<

+ )(,,sup)()(

0

1 sgGgsg

knns

knnk θθ

π

Actually, we use and then)()(40 2 sgsgn ==( ) 10 99298)(,,sup <≤sgG nn θθ( ) 10.99298

)(sup0

<≤≤< sg

nn

s π

Possibility for the application to the equation in mathematical financemathematical finance

To establish benchmark for the numerical scheme in the mathematical finance e need e plicitl erified res ltsmathematical finance, we need explicitly verified results.

Example: Exercise boundary for the American put option with geometric Brownian motionwith geometric Brownian motion.

)()( ufKeuTB −=−

( ) ( ))2/()()()2/()( 22

+ ++−−−−− NeNeu

urufufu

urufruσ

σσ

σ

( ) 10

)2/()()( 2

=+ ∫ −−−−− dvNeru

vvrvufufru

σσ

0

Since explicit form of the solution is not known for this equation it is important to obtain verified resultsequation, it is important to obtain verified results.

Concluding remarks

・ Numerical verification method has been generallyconstructed for elliptic PDE’s.

・ We have proved the uniqueness of the Stokes’ wave・ We have proved the uniqueness of the Stokes wave of extreme form which was open for many decades.

・ It seems that there is a large possibility to be applied to the equation in mathematical finance.

Thank you very much!