Wavelet collocation for fourth order problems

7/27/2019 Wavelet collocation for fourth order problems

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Wavelet collocation for fourth order problems

Silvia Bertoluzza

IMATI Enrico Magenes - CNR, Pavia

Joint work with Valerie Perrier, Lab. Jean Kuntzmann, INP Grenoble

Silvia Bertoluzza (IMATI-CNR) Wavelet collocation for fourth order problems 1 / 24
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Wavelet Collocation

Introduced O. Vasilyev & S. Paolucci [1995] and by S.B. & G. Naldi[1996]

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Wavelet Collocation


Two ingredients

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Wavelet Collocation


Two ingredients

Use of Deslaurier-Dubuc interpolating functions


W l C ll i
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Wavelet Collocation


Two ingredients

Use of Deslaurier-Dubuc interpolating functionsUse of a collocation approach


W l t C ll ti
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Wavelet Collocation


Two ingredients


Pointwise nonlinerities are easily treated (no integral computation)


Wavelet Collocation
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Wavelet Collocation


Two ingredients



Very simple yet effective refining/derefining strategy


Wavelet Collocation
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Wavelet Collocation


Two ingredients




Applied to a wide class of problems


Wavelet Collocation


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Wavelet Collocation


Two ingredients





Fluid dynamics


Wavelet Collocation
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Wavelet Collocation


Two ingredients





Fluid dynamicsElasticity


Wavelet Collocation
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Two ingredients





Fluid dynamicsElasticityFluid structure interaction


Wavelet Collocation
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Two ingredients





Fluid dynamicsElasticityFluid structure interactionSemiconductors



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Wavelet Collocation


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The Schauder basis

Vj: p.w. linears on uniform grid, step h = 2j

E

Tff

ffff

ff

ffff

ff

ffff

ff

ffff

ff

ffff

ff

ffff

ff

ffff

ff

ffff

Basis: translated of

f

fff


Wavelet Collocation
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The Schauder basis

Vj: p.w. linears on uniform grid, step h = 2j

E

Tff

ffff

ff

ffff

ff

ffff

ff

ffff

ff

ffff

ff

ffff

ff

ffff

ff

ffff


f

fff

coarser space Vj1Ttttttt

t

ttttt

t

ttttt

t

ttttt


t

ttt


Wavelet Collocation


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What is the difference

Tttttt

tt

t

tttt

tt

t

tttt

tt

t

tttt

tt


Wavelet Collocation
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Tttttt

tt

t

tttt

tt

t

tttt

tt

t

tttt

tt

fffff

ff

fffff

ff

fffff

ff

fffff

ff


Wavelet Collocation


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Tttttt

tt

t

tttt

tt

t

tttt

tt

t

tttt

tt

Difference

T

ffffffffffffffffffffffffffff


Wavelet Collocation


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Deslaurier-Dubuc [DD] Interpolating functions

Same structure but with more regular functions

1 0 10

0.2

0.4

0.6

0.8

1

5 0 50.2

0

0.2

0.4

0.6

0.8

1

1.2

5 0 50.2

0

0.2

0.4

0.6

0.8

1

1.2

10 0 100.2

0

0.2

0.4

0.6

0.8

1

1.2

10 0 100.2

0

0.2

0.4

0.6

0.8

1

1.2

20 0 200.2

0

0.2

0.4

0.6

0.8

1

1.2


Wavelet Collocation
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Two ways of constructing :

limit of a polynomial interpolatory subdivision scheme:


Wavelet Collocation
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limit of a polynomial interpolatory subdivision scheme:(k) = k0, k Z


Wavelet Collocation
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limit of a polynomial interpolatory subdivision scheme:(k) = k0, k Z{(k/2j)} {(k/2j+1)}

((2n 1)/2j+1) = jn((2n 1)/2j+1)

with jn polynomial of degree 2L + 1 defined as

jn(k) = (k/2j), k = n L, ., n + L


Wavelet Collocation
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((2n 1)/2j+1) = jn((2n 1)/2j+1)


jn(k) = (k/2j), k = n L, ., n + L

is the autocorrelation function of the Daubechies scaling functionwith L 1 zero moments


Wavelet Collocation
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((2n 1)/2j+1) = jn((2n 1)/2j+1)


jn(k) = (k/2j), k = n L, ., n + L

is the autocorrelation function of the Daubechies scaling functionwith L 1 zero moments

is interpolatory ((k) = k0)


Wavelet Collocation


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DD interpolating wavelets

Vj

: span of {jk

, k N}


Wavelet Collocation


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Vj

: span of {jk

, k N}

Ij : C0(R) Vj Langrangian interpolation operator


Wavelet Collocation
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Vj: span of {jk, k N}


Vj Vj+1


Wavelet Collocation
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Vj Vj+1

detail space: Wj = (Ij+1 Ij)Vj+1


Wavelet Collocation
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Vj Vj+1


Wj: span of {j,k = (2jx k) = (2(2jx k) 1), k N}


Wavelet Collocation

DD i l i l


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Vj Vj+1



Well defined correspondence between dyadic points and basisfunctions:


Wavelet Collocation

DD i l i l


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Vj Vj+1




at fixed coarse level j0 x = k2

j0 j0,k


Wavelet Collocation

DD i t l ti l t
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Vj Vj+1





j0 j0,kfor j > j0 and odd k = 2n + 1 x = k2

j j1,n


Wavelet Collocation

DD i t l ti l t
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Vj Vj+1





j0 j0,kfor j > j0 and odd k = 2n + 1 x = k2

j j1,n

bases for Rd by tensor product.

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Wavelet Collocation

DD interpolating wavelets on [0 1]


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DD interpolating wavelets on [0,1]

Basis on [0,1] built by polynomial extrapolation [Donoho, 92]

given values at xj,k = k2j, k = 0, . . . , 2j you want to uniquely select

one function in Vj


Wavelet Collocation

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one function in Vjobtain values at xj,k outside [0, 1] by extrapolating values in [0, 1]


Wavelet Collocation

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one function in Vjobtain values at xj,k outside [0, 1] by extrapolating values in [0, 1]this yields subspace Ej Vj


Wavelet Collocation

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one function in Vjobtain values at xj,k outside [0, 1] by extrapolating values in [0, 1]this yields subspace Ej VjVj[0, 1] = Ej|[0,1]


Wavelet Collocation



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Basis of Vj[0, 1]: {j,k, k = 0, . . . , 2j} with

j,k(xj,n) = n,k, n = 0, . . . , 2j

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Wavelet Collocation



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Basis of Vj[0, 1]: {j,k, k = 0, . . . , 2j} with

j,k(xj,n) = n,k, n = 0, . . . , 2j

Vj[0, 1] Vj+1[0, 1]

detail space and its basis built as for R


Wavelet Collocation

Collocation method on a uniform grid
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Second order problem

Au = f in ]0, 1[d, Bu = g on ]0, 1[d


Wavelet Collocation

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Au = f in ]0, 1[d, Bu = g on ]0, 1[d

Gj: grid of dyadic points of [0, 1]

Gj = {k/2j : k Zd [0, 1]d}


Wavelet Collocation

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g


Au = f in ]0, 1[d, Bu = g on ]0, 1[d

Gj: grid of dyadic points of [0, 1]

Gj = {k/2j : k Zd [0, 1]d}

ProblemLook for uh Vj[0, 1]

d such that

Auh() = f() Gj]0, 1[, Bu() = g() Gj ]0, 1[d

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Wavelet Collocation

Collocation method on a non-uniform grid


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g

detail grids: Jj = Gj+1 \ Gj = Gj0 jj0 Jj: set of all dyadic points


Wavelet Collocation



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g


interpolating wavelet


Wavelet Collocation

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h Vh Vh = span < , h >


Wavelet Collocation

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h Vh Vh = span < , h >

Problem

Look for uh Vh such that

Auh() = f() h]0, 1[, Bu() = f() h ]0, 1[d


Wavelet Collocation

Adaptivity
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Simple adaptive strategy: look at coefficients of tentative solution todesign the next grid


Wavelet Collocation

Adaptivity
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if u small (|u| < r): delete it from the grid


Wavelet Collocation

Adaptivity
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if u small (|u| < r): delete it from the gridif u big (|u| > a > r): add neighbouring points at higher level


Wavelet Collocation

Adaptivity
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solve a sequence of problems with greed designed by looking at the

solution at the previous step


Wavelet Collocation

Adaptivity
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other strategies might be applied but no theory available


Wavelet Collocation

Adaptivity
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other strategies might be applied but no theory available

strategy effectively applied to different equations (Burgers equation,Convection-Diffusion problems, Euler-Poisson system forsemiconductors,. . . )


Wavelet Collocation

Fourth order problems
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Problem

Find u : [0, 1]d R such that

Au = f in ]0, 1[d, Bu = g on ]0, 1[d

where

A: fourth order operator

B takes values in R2 (two boundary conditions)

Examples

Structural mechanics (beams, plates, . . . )Fluid dynamics (ice formations, fluids in the lungs)

Nanotechnologies

Denoising


Wavelet Collocation

Fourth order problems on [0, 1]
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Extension to 4th order problems: two things need to be facedtwo boundary condition at each extrema, only one degree of freedom:if we impose the equation at all interior points we have two moreequations than d.o.f.



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Wavelet Collocation



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Extension to 4th order problems: two things need to be faced

two boundary condition at each extrema, only one degree of freedom:if we impose the equation at all interior points we have two moreequations than d.o.f.

possible solutions

do not impose equation at all interior points



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Wavelet Collocation



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possible solutions

do not impose equation at all interior pointsadd one d.o.f. from a higher levelmodify the construction of DD wavelets on [0, 1] so that we have twod.o.f. at each extrema

conditioning of the matrices is so bad that round-off is killer


Wavelet Collocation

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possible solutions

do not impose equation at all interior pointsadd one d.o.f. from a higher levelmodify the construction of DD wavelets on [0, 1] so that we have twod.o.f. at each extrema

conditioning of the matrices is so bad that round-off is killer

solution: increase the precision of (selected) computations


Wavelet Collocation

Hermite boundary functions
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New basis on [0,1] built by modified polynomial extrapolation

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Wavelet Collocation



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Lagrangian at interior points / Hermite at boundary points

Basis of Vj[0, 1]: {j,k, k = 1, 0, . . . , 2j, 2j + 1, } with


Wavelet Collocation

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j,k(xj,n) = n,k, k = 1, . . . , 2j + 1, n = 0, . . . , 2j


Wavelet Collocation

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j,k(xj,n) = n,k, k = 1, . . . , 2j + 1, n = 0, . . . , 2j

j,k(0) = j,k(1) = 0, k = 0, . . . , 2

j


Wavelet Collocation

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j,k(xj,n) = n,k, k = 1, . . . , 2j + 1, n = 0, . . . , 2j

j,k(0) = j,k(1) = 0, k = 0, . . . , 2

j

j,1(0) =

j,2j+1(1) = 1,

j,1(1) =

j,2j+1(0) = 0

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Wavelet Collocation

Multiresolution analysis


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Vj[0, 1] defined as the span of such functions

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Wavelet Collocation



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Vj[0, 1] defined as the span of such functions

Interpolation Ij : C1(0, 1) Vj[0, 1]

Ijf = f

(0)j,1 +

2j

k=0

f(xk)j,k + f

(1)j,2j+1

Vj[0, 1] Vj+1[0, 1] [S.B., Perrier, 2012]


Wavelet Collocation

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Vj

[0, 1] defined as the span of such functions


Ijf = f

(0)j,1 +

2j

k=0

f(xk)j,k + f

(1)j,2j+1

Vj[0, 1] Vj+1[0, 1] [S.B., Perrier, 2012]

Wj[0, 1] = (Ij+1 Ij)Vj+1[0, 1]


Wavelet Collocation



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Vj

[0, 1] defined as the span of such functions


Ijf = f

(0)j,1 +

2j

k=0

f(xk)j,k + f

(1)j,2j+1

Vj[0, 1] Vj+1[0, 1] [S.B., Perrier, 2012]

Wj[0, 1] = (Ij+1 Ij)Vj+1[0, 1]

Wj[0, 1] span of {j,k = j+1,2k1, k = 1, . . . , 2j}


Wavelet Collocation

Smoothness charachterization
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TheoremLet

f =

2j0 +1

k=1

fkj0,k +

j

2j

k=1

dj,k2j/2j,k

and let 1 + 1/p < < R. Then f B,pq if and only if

(2j0 +1

k=1

|fk|p)q/p +

j

2jsq/p(2j

k=1

|dj,k|p)q/p < +,

with s = + 1/2 1/p.


Wavelet Collocation

Killer round-off: example

test problem:
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test problem:

u(iv)

= 0, u(0) = u(1) = 1, u

(0) = u

(1) = 0


Wavelet Collocation


test problem:
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test problem:

u(iv)

= 0, u(0) = u(1) = 1, u

(0) = u

(1) = 0

errors for solution in Vj[0, 1] (scaling function basis)

h L2 error L error

.0625 9.696e-06 1.5677e-05


Wavelet Collocation


test problem:
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test problem:

u(iv)

= 0, u(0) = u(1) = 1, u

(0) = u

(1) = 0


h L2 error L error

.0625 9.696e-06 1.5677e-05.0313 .00015748 .00025088

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Wavelet Collocation


test problem:


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test problem:

u(iv)

= 0, u(0) = u(1) = 1, u

(0) = u

(1) = 0


h L2 error L error

.0625 9.696e-06 1.5677e-05.0313 .00015748 .00025088.0156 .0025464 .0040258.0078 .042886 .067578

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Wavelet Collocation


test problem:


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test problem:

u(iv)

= 0, u(0) = u(1) = 1, u

(0) = u

(1) = 0errors for solution in Vj[0, 1] (scaling function basis)

h L2 error L error

.0625 9.696e-06 1.5677e-05.0313 .00015748 .00025088.0156 .0025464 .0040258.0078 .042886 .067578.0039 3.0851 4.8962

try with the multiscale basis

Silvia Bertoluzza (IMATI-CNR) Wavelet collocation for fourth order problems 17 / 24 Wavelet Collocation


test problem:
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test problem:

u(iv)

= 0, u(0) = u(1) = 1, u

(0) = u

(1) = 0errors for solution in Vj[0, 1] (scaling function basis)

h L2 error L error

.0625 9.696e-06 1.5677e-05.0313 .00015748 .00025088.0156 .0025464 .0040258.0078 .042886 .067578.0039 3.0851 4.8962

L2 error L error

9.696e-06 1.5677e-059.8408e-06 1.5677e-059.9162e-06 1.5677e-059.9545e-06 1.5677e-059.9734e-06 1.5677e-05

try with the multiscale basis

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Wavelet Collocation

Computation with 30 exact digits

M l b M l i i i lb


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Matlab Multiprecision toolbox [Advanpix]



M tl b M lti i i t lb
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Same computation with 30 exact digits



M tl b M lti i i t lb
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Same computation with 30 exact digitserrors

h L2 error L error

.0625 2.3871e-25 3.8769e-25

.0313 3.8769e-24 6.1761e-24

.0156 6.2506e-23 9.8818e-23

.0078 1.004e-21 1.5811e-21

.0039 1.6095e-20 2.5297e-20



Matlab Multiprecision toolbox
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h L2 error L error

.0625 2.3871e-25 3.8769e-25

.0313 3.8769e-24 6.1761e-24

.0156 6.2506e-23 9.8818e-23

.0078 1.004e-21 1.5811e-21

.0039 1.6095e-20 2.5297e-20

good! but very expensive solve linear system in double precision



Matlab Multiprecision toolbox [Ad i ]
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h L2 error L error

.0625 2.3871e-25 3.8769e-25

.0313 3.8769e-24 6.1761e-24

.0156 6.2506e-23 9.8818e-23

.0078 1.004e-21 1.5811e-21

.0039 1.6095e-20 2.5297e-20

L2 error L error

3.3959e-13 9.952e-13

3.1068e-13 4.6829e-136.1219e-13 8.5487e-131.6684e-13 5.0271e-131.0915e-13 2.4181e-13


Silvia Bertoluzza (IMATI CNR) Wavelet collocation for fourth order problems 18 / 24 Wavelet Collocation


Matlab Multiprecision toolbox [Ad i ]
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h L2 error L error

.0625 2.3871e-25 3.8769e-25

.0313 3.8769e-24 6.1761e-24

.0156 6.2506e-23 9.8818e-23

.0078 1.004e-21 1.5811e-21

.0039 1.6095e-20 2.5297e-20

L2 error L error

3.3959e-13 9.952e-13

3.1068e-13 4.6829e-136.1219e-13 8.5487e-131.6684e-13 5.0271e-131.0915e-13 2.4181e-13


much more satisfactory!!


Multiple precision computation
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What exactly is computed in multiple precision:




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initialization is performed in multiple precision




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eigenvalues/eigenvectors needed for computing derivatives of atintegers


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refinement to compute values of at dyadic points


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refinement to compute values of at dyadic pointsextrapolation to compute basis for [0, 1]


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matrix and right hand side assembled in double precision

Sil i B t l (IMATI CNR) W l t ll ti f f th d bl s 19 / 24 Wavelet Collocation

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matrix and right hand side assembled in double precision

linear system solved in double precision

Sil i B t l (IMATI CNR) W l t ll ti f f th d bl 19 / 24 Wavelet Collocation

Test 1. [Twizell & Tirmizi, 86]
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Equation: w(iv) + 4w = 1, in (1, 1)

Boundary conditions

w(1) = w(1) = 0, w

(1) = w

(1) =

sinh(2) sin(2)

4(cosh(2) + cos(2))

Exact solution

w =1 2(sin(1)sinh(1)sin(x)sinh(x) + cos(1)cosh(1)cos(x)cosh(x)

4(cosh(2) + cos(2))

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Errors

h L2 error L error

.0625 4.8859e-10 8.071e-10

.0313 2.087e-12 4.6301e-12

.0156 1.2167e-14 1.9113e-14

.0078 3.2334e-17 9.2613e-17

.0039 4.5559e-17 9.395e-17

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Errors

h L2 error L error

.0625 4.8859e-10 8.071e-10

.0313 2.087e-12 4.6301e-12

.0156 1.2167e-14 1.9113e-14

.0078 3.2334e-17 9.2613e-17

.0039 4.5559e-17 9.395e-17

L2 error L error

4.8859e-10 8.0717e-102.8972e-12 4.6153e-12

5.7798e-09 2.0393e-084.4425e-09 2.0393e-083.1474e-08 2.0393e-08

Sil i B l (IMATI CNR) W l ll i f f h d bl 21 / 24 Wavelet Collocation

Test 2. Clamped-Clamped beam, half-loaded
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Equation:EIw(iv) = q

Boundary conditions

f, w(0) = w(2L) = w(0) = w(2L) = 0

Exact solution is known

h L2 error.0625 0.0173

.0313 0.0039.0156 9.8449 e-04

.0078 2.4704 e-04

.0039 6.1876 e-05

Sil i B l (IMATI CNR) W l ll i f f h d bl 22 / 24 Wavelet Collocation

Test 3. Cantilever beam, tip loaded
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Equation:EIw(iv) = 0

Boundary conditions

w(0) = w(0) = w(L) = 0, EIw

(L) = P

h L2 error.0625 8.5334e-10.0313 1.6134e-12

.0156 6.3875e-11

.0078 7.0735e-11

.0039 8.205e-11


Wavelet Collocation

Conclusions

Modified interpolating wavelets on [0, 1] well suited for 4th orderproblems


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problems


Wavelet Collocation

Conclusions



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problems

Computations must be (partially) carried out in multiple precisions


Wavelet Collocation

Conclusions

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p


First tests show that the method behaves as expected


Wavelet Collocation

Conclusions

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p



Plenty of open problems!


Wavelet Collocation

Conclusions

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p




Adaptivity needs to be studied


Wavelet Collocation

Conclusions

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Adaptivity needs to be studiedPreconditioning is of paramount importance


Wavelet Collocation

Conclusions

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No theory!


Wavelet Collocation

Conclusions

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No theory!

For whom might be interested: the Advanpix Multiple PrecisionToolbox for Matlab works beautifully!!http://www.advanpix.com

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Wavelet collocation for fourth order problems

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