FOR C ONSERVATION L AWS : N UMERICAL ANALYSIS Margarete O. Domingues - - PowerPoint PPT Presentation

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FOR C ONSERVATION L AWS : N UMERICAL ANALYSIS Margarete O. Domingues - - PowerPoint PPT Presentation

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A DAPTIVE M ULTIRESOLUTION M ETHODS FOR C ONSERVATION L AWS : N UMERICAL ANALYSIS Margarete O. Domingues 1 , S onia M. Gomes 2 , Olivier Roussel 3 , and Kai Schneider 3 , 4 1 Instituto Nacional de Pesquisas Espaciais - LAC, Brazil 2 Universidade

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ADAPTIVE MULTIRESOLUTION METHODS

FOR CONSERVATION LAWS :

NUMERICAL ANALYSIS

Margarete O. Domingues1, Sˆ

  • nia M. Gomes2,

Olivier Roussel3, and Kai Schneider3,4

1 Instituto Nacional de Pesquisas Espaciais - LAC, Brazil 2 Universidade Estadual de Campinas - IMECC, Brazil 3 M2P2, Universit´

es d’Aix-Marseille, France

4 ´

Ecole Centrale de Marseille, France

Adaptive Multiresolution Methods for Conservation Laws – p.1/80

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Summary

Motivation First part: MR analyses for point values and cell averages Harten’ approach Discrete and functional aspects Local regularity indicator and data compression. Stability. Lifting methodology Second part: applications to adaptive methods for PDE The SPR method The FV/MR method Examples

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Basic References

Books

  • A. Cohen. Wavelet Methods in Numerical Analysis.

Handbook of Numerical Analysis, Elsevier, 2000, Ciarlet, P.

  • G. and Lions, J. L. (eds),

Papers

  • A. Harten, Multiresolution representation of data: a general

framework, SIAM J. Numer. Anal. (1996),33 (3): 385-394,

  • W. Sweldens, The lifting scheme: a construction of second

generation wavelets, SIAM J. Math. Anal., (1996), 29: 511-543.

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Motivation

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Multiresolutions analysis - WT

fJ ← → fMR

J

= (f0, d0, . . . dJ−1) fJ → information of f at the nest scale level J f0 → information of f at coarsest level dj → difference of information (details) between levels j and j + 1 fMR

J

→ multiresolution representation

Adaptive Multiresolution Methods for Conservation Laws – p.5/80

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Illustration: direct WT

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Multiresolution approximations: inverse WT

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Multiresolution approximations: cont..

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Multiresolution approximations: cont..

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Fundamental properties

Perfect reconstruction Efcient algorithms Localization Polynomial cancellation Stability Multiresolution representation

VJ = V0 +

J

  • j=0

Wj

  • k

fJ,kΦJ,k =

  • k

f0,kΦ0,k +

  • j
  • k

dj,kΨj,k

Adaptive Multiresolution Methods for Conservation Laws – p.10/80

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SLIDE 11

Brief history

1911: Haar; 1930: Littlewood-Paley; 1940: Gabor; 1960: Calderón-Zigmund 1984: Morlet e Grossmann (continuous WT ) 1985: Meyer, Mallat (multiresolution analysis) 1988: Daubechies (orthogonal wavelets) 1992: Cohen, Daubechies e Feauveau

(biorthogonal wavelets )

1992–1993: Harten; Donoho (interpolating and cell-averages

MR )

1994: Sweldens (Second generation wavelets) . . .

Adaptive Multiresolution Methods for Conservation Laws – p.11/80

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Goals

Introduction of the principal concepts in MRA Applications in the design of adaptive schemes for PDE wavelet coefcients as regularity indicators construction of adaptive grids: rened close to singularities × coarse in smooth regions

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initial condition grid solution at t = 15 grid

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Wavelet ideas appear in several contexts ⇓ and there are different approaches for their constructions and for the interpretation of their properties.

Harten’s approach and Lifting schemes

Fourier techniques are not necessary Appropriate for non-cartesian geometries Main focus: characterization of local regularity stability of the algorithms

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Harten’s approach

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MR analysis for point values

Dyadic grids on the interval

X3 X2 X1 X4 X0

Xj = {xj,k = k2−j, 0 ≤ k ≤ 2j} ⊂ Ω = [0, 1]

Adaptive Multiresolution Methods for Conservation Laws – p.17/80

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Xj ⊂ Xj+1 xj,k = xj+1,2k – old points Xj+1 renement of Xj xj+1,2k+1 ∈ Xj+1 \ Xj – new points Discretization by point values at Xj: Dj : f → fj fj,k = f(xj,k)

Adaptive Multiresolution Methods for Conservation Laws – p.18/80

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Goal: To construct a multiresolution analysis for fJ. fJ

WT

− → fMR

J

= (f0, d0, . . . dJ−1) Principal tool: interpolating predicton operator to approximate the point values of f at level j + 1 from the information at the coarser level j. Pj→j+1 : fj → ˜ fj+1 ˜ fj+1,2k = fj,k ˜ fj+1,2k+1 ≈ fj+1,2k+1 Wavelets are prediction errors dj,k = fj+1,2k+1 − ˜ fj+1,2k+1

Adaptive Multiresolution Methods for Conservation Laws – p.19/80

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Simple Example: linear interpolation

˜ fj+1,2k

− 1 = fj,k + fj,k

2

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Cubic interpolation

˜ fj+1,2k−1 = 9 16 [fj,k + fj,k+1] − 1 16 [fj,k−1 + fj,k

+ 2]

Adaptive Multiresolution Methods for Conservation Laws – p.21/80

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SLIDE 41

Interpolation errors

dj,k = fj+1,2k+1 − ˜ fj+1,2k+1

xj+1,2k+1 dj,k

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Wavelet coefcients

Prediction errors dj,k = fj+1,2k+1 − ˜ fj+1,2k+1 dj = (dj,k) measures the difference of information between discretization levels j and j + 1. Polynomial cancellation: if f is a polynomial of degree ≤ r, then using interpolation of degree r dj,k = 0

Adaptive Multiresolution Methods for Conservation Laws – p.23/80

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Analysis algorithm

f J

fJ!1 fJ!2 f0 ...

d d d

J!1 J!2

For j = J − 1, . . . , 0, given fj+1 Do decimation: fj,k = fj+1,2k+1 Do prediction: ˜ fj+1,2k+1 = [Pj→j+1fj](2k + 1) Compute: dj,k = fj+1,2k+1 − ˜ fj+1,2k+1

Adaptive Multiresolution Methods for Conservation Laws – p.24/80

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Synthesis algorithm

... f f

f

J!1

f1

J

d d d

1 J!1

For j = 0, . . . , J − 1, given fj and dj Do prediction: ˜ fj+1,2k+1 = [Pj→j+1fj](2k + 1) Compute: fj+1,2k = fj,k fj+1,2k+1 = dj,k + ˜ fj+1,2k+1

Adaptive Multiresolution Methods for Conservation Laws – p.25/80

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Local regularity analysis

Polynomial Lagrange interpolation error For f ∈ Cs(Ij,k), 1 ≤ s ≤ r, with bounded derivative f(s+1), |dj,k| = |f(xj+1,2k+1) − p(xj+1,2k+1)| ≤ K2−(s+1)j max

ξ∈Ij,k

|f(s+1)(ξ)| where r is the polynomial degree, Ij,k is the interval containing the interpolation stencil, K = K(s, r).

Adaptive Multiresolution Methods for Conservation Laws – p.26/80

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Consequence

There is a relation between the decay rate of the wavelet coefcients and the local regularity of the function. For high interpolation accuracy, a fast decay occurs on smooth regions Wavelet coefcients dj,k can be used as local regularity indicators

Adaptive Multiresolution Methods for Conservation Laws – p.27/80

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Example

f(x) =                    8.1 e1/4 e−|x−1/2|, x ≤ 1/4, 9 e−|x−1/2|, 1/4 ≤ x ≤ 3/4, e−|x−1/2| (16x2 − 24x + 18), x ≥ 3/4, ξ = 1/4 → f has a jump discontinuity ξ = 1/2 → f (1) has a jump discontinuity ξ = 3/4 → f (2) has a jump discontinuity

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f(x)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5.5 6 6.5 7 7.5 8 8.5 9 9.5

|dj,k| > 5 × 10−4

Prediction by cubic interpolation

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Decay rate of |dj,k| in the vicinity of ξ

Aj(ξ)

def

= max{|dj,k| : |xj+1,2k+1 − ξ| < δ} Aj(ξ) = O(2γj), γ = 0, −1, −2, for ξ = 1/4, 1/2, 3/4

5 6 7 8 9 10 11 12 13 −25 −20 −15 −10 −5 l !(l)

Estimation of γ          γ ∼ −0.0004 → ξ = 1/4 (o) γ ∼ −1.0149 → ξ = 1/2 (∗) γ ∼ −1.9538 → ξ = 3/4 (+)

Adaptive Multiresolution Methods for Conservation Laws – p.30/80

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Data compression

Input: fJ, = (j) Step 1: Analysis fJ

WT

− → fMR

J

= (f0, d0, . . . dJ−1). Step 2: Thresholding d

j,k =

   dj,k if |dj,k| > j if |dj,k| ≤ j Output: f ,MR

J

= (f0, d

0, . . . d j).

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Important aspect: stability

Compressed data f,MR

J

= (f0, d

0, . . . d J−1)

Perturbation: |d

j,k − dj,k| ≤ j

Inverse WT: f,MR

J

→ f

J

Wavelet perturbations are transmitted to higher levels by successive predictions. Will they be amplied or can they be controlled? ||fJ − f

J|| = O(||||)?

Yes, if the prediction scheme is convergent. convergence → stability

Adaptive Multiresolution Methods for Conservation Laws – p.32/80

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The concept of convergence

Prediction operator: Pj→j+1 Given any initial data: sj = (sj,k) Iterative prediction: s+1 = Pj→j+1s, ≥ j The predition operator is convergent if there exist a continuous function sj(x) such that sj(x,k) = s,k, for ≥ j The predictions by polynomial Lagrange interpolation of degree M − 1 are convergent.

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Φ5,k(x) ( far from the boundaries ) M = 2 M = 4

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Φ5,k(x) (Interaction with the left boundary, M = 4)

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Functional context

Scaling functions: Φj,k(x) – iteractive predition of delta-sequences sj,m = δ(k − m) Interpolation: Φj,k(xj,m) = δ(k − m) Approximation spaces: Vj = span{Φj,k(x), k = 0, · · · , 2j} Interpolation operator: fj(x) =

2j

  • k=0

fj,kΦj,k(x)

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Properties of Φj,k(x) (Donoho, 1992)

Scaling relation: ck

m = Φj,k(xj+1,m)

Φj,k(x) =

  • m

ck

mΦj+1,m(x)

Regularity: increases with interpolation degree M − 1 Φj,k(x) ∈ Cs , s = 0, 1, 2, for M = 2, 4, 6 Simmetry: Far from the boundaries, Φj,k(x) = φ(2jx − k), where φ = ϕM is the Dubuc-Delauriers fundamental function Compact support: |supp(Φj,k)| = O(2−j) (increases with M) Polinomial reproduction: xn =

2

  • k=0

(x

k)nΦj,k(x), n ≤ M − 1.

Adaptive Multiresolution Methods for Conservation Laws – p.37/80

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Complementary spaces

Hierarchy of spaces: Vj ⊂ Vj+1 Wj: functions in Vj+1 that vanish in Xj Wavelet functions: Ψj,k(x) = Φj+1,2k+1(x) Wj = span{Ψj,k(x), k = 0, · · · 2j − 1} Direct summation: Vj+1 = Vj ⊕ Wj

2j+1

  • k=0

fj+1,kΦj+1,k(x) =

2j

  • k=0

fj,kΦj,k(x) +

2j−1

  • k=0

dj,kΨj,k(x)

Adaptive Multiresolution Methods for Conservation Laws – p.38/80

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MR transforms ↔ change of basis

Vj+1 = Vj + Wj

  • k

fj+1,kϕj+1,k(x) =

  • k

fj,kΦj,k(x) +

  • k

dj,kΨj,k(x)

WT IWT

"j,k

V j Wj V j+1

#j+1,k #j,k

  • A. Cohen - Wavelet Methods in Numerical Analysis. In: P. G. Ciarlet and J. L.Lions (eds),

Handbook of Numerical Analysis,Elsevier, 2000.

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Change of Basis

V J = V0 + W0 + . . . + WJ−1 {ΦJ,k(x)} ↔ {Φ0,k(x)} ∪J−1

j=0 {Ψj,k(x)} 2J

  • k=0

fJ,kΦJ,k(x) =

1

  • k=0

f0,kΦ0,k(x) +

J−1

  • j=0

2j−1

  • k=0

dj,kΨj,k(x) fJ ↔ (f0, d0, . . . dJ−1)

Adaptive Multiresolution Methods for Conservation Laws – p.40/80

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Stability analysis

Compressed data: f,MR

J

= (f0, d

0, . . . d J−1)

Perturbation: |d

j,k − dj,k| ≤ j

Inverse WT: f,MR

J

→ f

J

Thresholding error: fJ(x) − f

J(x) = J−1

  • j=0
  • k

(dj,k − d

j,k)Ψj,k(x)

||fJ − f

J||∞ ≤ J−1

  • j=0

j||Ψj,k||∞ ≤ C

J−1

  • j=0

j

Adaptive Multiresolution Methods for Conservation Laws – p.41/80

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MR for point values in higher dimensions

  • → points of Xj
  • → points of Xj+1 \ Xj

fj,(k,m) = fj+1,(2k,2m) Prediction tensor product

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Wavelet coefcients are prediction (interpolation) errors

d(1)

j,(k,m) = fj+1,(2k,2m+1) − ˜

fj+1,(2k,2m+1) d(2)

j,(k,m) = fj+1,(2k+1,2m) − ˜

fj+1,(2k+1,2m) d(3)

j,(k,m) = fj+1,(2k+1,2m+1) − ˜

fj+1,(2k+1,2m+1)

  • d(1)

j,(k,m)

  • d(3)

j,(k,m)

  • fj,(k,m)
  • d(2)

j,(k,m)

  • Adaptive Multiresolution Methods for Conservation Laws – p.43/80
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Sparse Point Representation (SPR)

Wavelet coefcients ↔ polynomial interpolation error. d(i)

j,(k,m) → local regularity indicators

Thresholding the wavelet coefcients → 2D SPR of functions. Only the point values corresponnding to signicant wavelet coefcients are kept in the SPR grids (M. Homström 1997) Convergence of the interpolating prediction → stability (control

  • f perturbation errors)

Adaptive Multiresolution Methods for Conservation Laws – p.44/80

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Examples

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MR analysis for cell averages

Dyadic cells on the interval

$ 0,0 $ = $ 1,0 $

1 , 1

$ 2,0 $ 2,1 $ 2 2

,

$ 2,3 $ 3,0

3,

$

7

... ...

j=3 j=2 j=1 j=0

Ωj,k = (xj,k, xj,k+1), [0, 1] =

2j−1

  • k=0

Ωj,k Hierarchy Ωj,k = Ωj+1,2k ∪ Ωj+1,2k+1

Adaptive Multiresolution Methods for Conservation Laws – p.46/80

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Principal aspects

Discretization: Dj : f → fj fj,k = 2j

  • Ωj,k

f(x)dx Restriction: Pj+1→j : fj+1 → fj fj,k = fj+1,2k + fj+1,2k+1 2 Prediction: Pj→j+1 : fj → ˜ fj+1 Consistency: [Pj+1→jPj→j+1]fj = fj Localization + Reproduction of polynomyals + Stability

Adaptive Multiresolution Methods for Conservation Laws – p.47/80

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How to construct cell-average predicitons?

Cell-average polynomial interpolation: Choose a stencil of the N + 1 cells Ωj,k+m closest to Ωj,k Find a polynomial p(x) of degree N whose cell averages on Ωj,k+m coincide with fj,k+m Dene ˜ fj+1,2k+1 = 2j+1

  • Ωj+1,2k+1

p(x)dx ˜ fj+1,2k = 2j+1

  • Ωj+1,2k

p(x)dx

Adaptive Multiresolution Methods for Conservation Laws – p.48/80

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f

j,k

f

j,k!1

fj,k f

j,k+1

f

~

j+1,2k

f

j+1,2k f j+1,2k+1

level j

Restriction Prediction

level j+1

Adaptive Multiresolution Methods for Conservation Laws – p.49/80

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Examples

Haar prediction: exact for constant functions (N = 0) ˜ fj+1,2k = ˜ fj+1,2k+1 = fj,k Quadratic cell-average interpolation: (N = 2) ˜ fj+1,2k+1 = fj,k + 1 8[fj,k+1 − fj,k−1] ˜ fj+1,2k = fj,k − 1 8[fj,k+1 − fj,k−1] (Far from the boundaries, for 1 ≤ 2j − 2)

Adaptive Multiresolution Methods for Conservation Laws – p.50/80

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Wavelet coefcients

Prediction errors: dj,k = fj+1,2k+1 − ˜ fj+1,2k+1 Multiresolution transformations:

fL

1:1

← → ( u0, d0, · · · , dL−1)

Polynomial cancellation: if f(x) is a polynomial of degree ≤ N, then dj,k = 0 Thresholding: data compression d

j,k =

   dj,k if |dj,k| > j if |dj,k| ≤ j Stability? Yes, for convergent cell-average predictions.

Adaptive Multiresolution Methods for Conservation Laws – p.51/80

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Analysis algorithm

fJ

WT

− → (f0, d0, . . . dJ−1) For j = J − 1, . . . , 0, given fj+1 Do restriction: fj,k = [Pj+1→jfj+1](k) = [fj+1,2k + fj+1,2k+1]/2 Do prediction: ˜ fj+1,2k+1 = [Pj→j+1fj](2k + 1) Compute: dj,k = fj+1,2k+1 − ˜ fj+1,2k+1

Adaptive Multiresolution Methods for Conservation Laws – p.52/80

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Synthesis algorithm

(f0, d0, . . . dJ−1) IWT − → fJ For j = 0, . . . , J − 1, given fj and dj Do prediction: ˜ fj+1,2k+1 = [Pj→j+1fj](2k + 1) Compute: fj+1,2k+1 = dj,k + ˜ fj+1,2k+1 fj+1,2k = 2fj,k − fj+1,2k+1

Adaptive Multiresolution Methods for Conservation Laws – p.53/80

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The concept of convergence

Convergent predictions

Given any starting sequence sj = (sj,k), consider the iteractive cell-average subdivision s = P−1→s−1 A cell-average prediction is convergent if there exists an integrable function sj(x) such that its cell averages at levels ≥ j coincide with s The predictions by cell-average polynomial interpolation are convergent.

Adaptive Multiresolution Methods for Conservation Laws – p.54/80

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Functional context

Dual scaling functions:

Φ∗

j,k(x) = 2jχΩj,k(x)

fj,k = 2j

Ωj,k f(x)dx =< f, Φ∗ j,k >

Primal scaling functions:

Φj,k(x) – iteractive cell-average predition of delta-sequences [DjΦj,k]m = δ(k − m)

Approximation spaces:

Vj = span{Φj,k(x), k = 0, · · · , 2j}

Reconstruction operator:

fj(x) = 2j

k=0 < f, Φ∗ j,k > Φj,k(x)

Adaptive Multiresolution Methods for Conservation Laws – p.55/80

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SLIDE 75

Properties of Φj,k(x) (Donoho, 1993)

Scaling relation: ck(m) cell averages do Φj,k at level j + 1 Φj,k(x) =

  • m

ck(m)Φj+1,m(x) Regularity: increases with interpolation degree N Φj,k(x) ∈ Cs ; s = −1, 0, 1 for N = 0, 2, 4 Simmetry: Far from the boundaries Φj,k(x) = φ(2jx − k), φ = ϕ1,N+1 → Cohen-Daubechies-Feauveau spline scaling function. Compact support: |supp(Φj,k)| = O(2−j) (increases with N) Polinomial reproduction:

xn =

2−1

X

k=0

< xn, Φ∗

j,k > Φj,k(x), n ≤ N

Adaptive Multiresolution Methods for Conservation Laws – p.56/80

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SLIDE 76

Φ5,k(x)

Adaptive Multiresolution Methods for Conservation Laws – p.57/80

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SLIDE 77

Complementary spaces

Vj+1 = Vj + Wj

  • k

fj+1,kΦj,k(x) =

  • k

fj,kΦj,k(x) +

  • k

dj,kΨj,k(x) Dual wavelets:

dj,k =< f, Ψ∗

j,k >

Ψ∗

j,k = 2j+1χΩj+1,2k+1 − 2j m

hk

mχΩj,m

Primal wavelets: Ψj,k(x) = Φj+1,2k+1(x) − Φj+1,2k(x)

Wj = span{Ψj,k(x), k = 0, · · · , 2j − 1}

Adaptive Multiresolution Methods for Conservation Laws – p.58/80

slide-78
SLIDE 78

Primal wavelets Ψj,k(x)

Adaptive Multiresolution Methods for Conservation Laws – p.59/80

slide-79
SLIDE 79

Local regularity wavelet indicator

Polynomial cancellation property + classical local polynomial approximation results ⇓ For f ∈ Cs , s ≤ N + 1, |dj,k| =

  • < f − q, Ψ∗

j,k >

  • ≤

infq∈ΠN||f − q||L∞(Ij,k)||Ψ∗

j,k||L1

≤ C2−sj|f|Cs(Ij,k) Ij,k = supp(Ψ∗

j,k)

Adaptive Multiresolution Methods for Conservation Laws – p.60/80

slide-80
SLIDE 80

Data compression

Input: fJ, = (j) Step 1: Analysis fJ

WT

− → fMR

J

= (f0, d0, . . . dJ−1). Step 2: Thresholding d

j,k =

   dj,k if |dj,k| > j if |dj,k| ≤ j Output: f ,MR

J

= (f0, d

0, . . . d j).

Adaptive Multiresolution Methods for Conservation Laws – p.61/80

slide-81
SLIDE 81

Example

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Adaptive Multiresolution Methods for Conservation Laws – p.62/80

slide-82
SLIDE 82

Stability

Compressed data f ,MR

J

= (f0, d

0, . . . d J−1)

Perturbation: |d

j,k − dj,k| ≤ j

Inverse WT: f ,MR

J

→ f

J

For convergent predictions, since ||Ψj,k||L1 ≤ C2−j, then ||fJ − f

J||L1

= 1 |

J−1

  • j=0

2j−1

  • k=0

(dj,k − d

j,k)Ψj,k(x) dx|

≤

J−1

  • j=0

j

2j−1

  • k=0

||Ψj,k||L1 ≤ C

J−1

  • j=0

j

Adaptive Multiresolution Methods for Conservation Laws – p.63/80

slide-83
SLIDE 83

Cartesian grids in higher dimensions

Hierarchy of Meshes

G G G G

1 2 3

$ $

0,0,0

$1,0,0 $1,0,1 $1,1,1 $1,1,0

Gj = {Ωj,(k,m)}, Ω = ∪(k,m)Ωj,(k,m), |Ωj,(k,m)| ∼ 2−2j, Ωj,(k,m) ∈ Gj is the union of four children cells in Gj+1 Ωj,(k,m) = [

µ∈Cj,(k,m)

Ωj+1,µ

Adaptive Multiresolution Methods for Conservation Laws – p.64/80

slide-84
SLIDE 84

Discretization: Cell averages fj = (fj,(k,m)) fj,(k,m) = 1 | Ωj,(k,m) |

  • Ωj,(k,m)

u(x, y)dxdy Restriction Pj+1→j : fj+1 → fj fj,(k,m) = 1 4{fj+1,(2k,2m) + fj+1,(2k,2m+1) + fj+1,(2k+,2m) + fj+1,(2k+1,2m+1)} Tensor product prediction Pj→j+1 : fj → ˜ fj+1 two-level transformations fj+1 ↔ {fj, d(1)

j , d(2) j , d(3) j } d(1)

j,(k,m)

= fj+1,(2k,2m+1) − ˜ fj+1,(2k,2m+1) d(2)

j,(k,m)

= fj+1,(2k+1,2m) − ˜ fj+1,(2k+1,2m) d(3)

j,(k,m)

= fj+1,(2k+1,2m+1) − ˜ fj+1,(2k+1,2m+1)

Adaptive Multiresolution Methods for Conservation Laws – p.65/80

slide-85
SLIDE 85

Solution of Lax-Liu Riemman problem

Adaptive Multiresolution Methods for Conservation Laws – p.66/80

slide-86
SLIDE 86

Non-cartesian grids

  • R. Abgrall and A. Harten, Multiresolution representation in

unstructured meshes. SIAM J. Numer. Anal. (1998), 35: 2128-2146 Hierarchy of triangular meshes

T T T T T T

j+1 j+1 j+1

Tj+1

j j j 01 02 03 00 0(1) 0(2) 0(3)

Gj = {T j

γ}, Ω = ∪γ∈Sj T j µ,

|T j

γ| ∼ 2−2j,

T

j γ =

[

µ∈Cj

γ

T

j+1 µ

Cj

γ = {T j+1 µ

, µ = (γ, i), i = 0, 1, 2, 3} childrem at level j + 1

Adaptive Multiresolution Methods for Conservation Laws – p.67/80

slide-87
SLIDE 87

Discretization: Cell averages fj = (fj,γ) fj,γ = 1 | T j

γ |

  • T j

γ

u(x)dx Restriction Pj+1→j : fj+1 → fj fj,γ = 1 | T j

γ |

  • µ∈Cj

γ

| T j+1

µ

| fj+1,µ Prediction Pj→j+1 : fj → ˜ fj+1 Wavelet coefcients: d(i)

j,γ = fj+1,µ − ˜

fj+1,µ, µ = (γ, i) ∈ Cj

γ, i = 1, 2, 3

Adaptive Multiresolution Methods for Conservation Laws – p.68/80

slide-88
SLIDE 88

Haar wavelet coefcients: prediction by piecewise constants d(i)

j,µ = fj+1,µ − fj,γ, µ = (γ, i) ∈ Cj γ, i = 1, 2, 3

uj+1

4k+1

uj+1

4k

uj

k

uj+1

4k+3

u4k+2

j+1

C j+1

4k+1 4k

C C j+1

4k+3 j+1

C 4k+2

j+1

C j

k

dj

k3

dj

k2

dk1

j

Adaptive Multiresolution Methods for Conservation Laws – p.69/80

slide-89
SLIDE 89

Higer order prediction (Cohen-Dyn-Kaber-Postel, 2000) ˜ fj+1,µ =              fj,γ µ = (γ, 0) fj,γ + 1

6[fj,γ(2) − fj,γ(1)] + 1 6[fj,γ(3) − fj,γ(1)]

µ = (γ, 1) fj,γ + 1

6[fj,γ(1) − fj,γ(2)] + 1 6[fj,γ(3) − fj,γ(2)]

µ = (γ, 2) fj,γ + 1

6[fj,γ(1) − fj,γ(3)] + 1 6[fj,γ(2) − fj,γ(3)]

µ = (γ, 3) Holds for equilateral triangular partitions, T j

γ(i), i = 1, 2, 3 denote the three neighbours of T j γ

This prediction is convergent, and exact for rst order polynomials

  • A. Cohen and N. Dyn and S. M. Kaber and M.Postel. Multiresolution

nite volume schemes on triangles, J. Comput. Phys. (2000), 161: 264-286

Adaptive Multiresolution Methods for Conservation Laws – p.70/80

slide-90
SLIDE 90

Lifting Methodology

Adaptive Multiresolution Methods for Conservation Laws – p.71/80

slide-91
SLIDE 91

Principles of lifting schemes (W. Sweldens)

Given an original MR framework: uj+1 ↔ {uj, dj} Construct a new MR setting: uj+1 ↔ {unew

j

, dnew

j

}

+ +

! !

P P

f dj

j+1

fj fj d j

U U

f d

j j

fj+1

new new

Prediction: to modify the wavelet coefcients

dnew

j

= dj − Puj

Updating: to modify the scaling coefcients

unew

j

= uj + Udnew

j

Simple inversion and computations in situ (without auxiliary memory locations)

Adaptive Multiresolution Methods for Conservation Laws – p.72/80

slide-92
SLIDE 92

Interpolating MR = lifting the lazy MR

Original MR : Lazy MR

(evenj, oddj) := Split(uj+1) uj,k = uj+1,2k dj,k = uj+1,2k+1 uj+1 := Merge(evenj, oddj) uj+1,2k = uj,k uj+1,2k+1 = dj,k

Prediction, to modify the lazy wavelet coefcients dnew

j

=

  • ddj − P(evenj)

fj+1 fj+1

!! split + merge

P P

f d

j j

even

  • dd

even

  • dd

Adaptive Multiresolution Methods for Conservation Laws – p.73/80

slide-93
SLIDE 93

Updating the interpolating MR

Original MR: uj+1 ↔ {uj, dj} (interpolating MR) Update: to modify the scaling coefcients

+

!

fj+1 fj+1 dj fj fj d j

U U

f d

j j new

Goal: to preserve averages 1 2j

  • k

unew

j,k =

1 2j+1

  • k

uj+1,k

Adaptive Multiresolution Methods for Conservation Laws – p.74/80

slide-94
SLIDE 94

Example

dj,k = uj+1,2k+1 − uj,k + uj,k+1 2 unew

j,k = uj,k + dj,k−1 + dj,k

4

  • k

dj,k =

  • k

uj+1,2k+1 −

  • k

uj+1,2k + uj+1,2k+2 2

  • k

unew

j,k

=

  • k

uj+1,2k + dj,k−1 + dj,k 4 =

  • k

uj+1,2k + 1 2

  • k

uj+1,2k+1 − 1 2

  • k

uj+1,2k = 1 2

  • k

uj+1,k

Adaptive Multiresolution Methods for Conservation Laws – p.75/80

slide-95
SLIDE 95

The effect of updating

The primal scaling functions are preserved: Φnew

j,k (x) = Φj,k(x)

V new

j

= Vj Since the wavelet coefcients are not modied, the dual wavelets Ψ∗

j,k(x) are preserved dj,k =< u, Ψ∗ j,k >

The modication of the scaling coefcients ⇒ new dual scaling functions Φ∗,new

j,k

(x) such that unew

j,k =< u, Φ∗,new j,k

> and new primal wavelets Ψnew

j,k (x)

i.e. new complementary spaces V V + W new

Adaptive Multiresolution Methods for Conservation Laws – p.76/80

slide-96
SLIDE 96

The effect of prediction

Since the scaling coefcients are not modied, the dual scaling functions are preserved, such that unew

j,k =< u, Φ∗ j,k >

The modication of the wavelet coefcients ⇒ new dual wavelets Ψ∗,new

j,k

(x) (dual lifting) dnew

j,k =< u, Ψ∗,new j,k

> new primal scaling functions Φnew

j,k and new primal wavelets

Ψnew

j,k , spanning new spaces

V new

j+j = V new j

+ W new

j

Adaptive Multiresolution Methods for Conservation Laws – p.77/80

slide-97
SLIDE 97

Lifting Haar wavelets

For discretizations by cell averages, Haar wavelet coefcients vanish only for constant polynomials Lifting can be used to modify the wavelet coefcients, to have new zero wavelet coefcients for higher degree polynomials dnew

j,k = dHaar j,k

−

  • m∈S(k)

ck

muj,m

Example: cancelling quadratic polynomials dnew

j,γ

= uj+1,2k+1 − uj,k − 1 8[uj,k+1 − uj,k−1] = dHaar

j,k

− 1 8[uj,k+1 − uj,k−1] Ψ∗new

j,γ

= Ψ∗,Haar

j,γ

− 1 8[Φ∗

j,k+1 − Φ∗ j,k−1]

Adaptive Multiresolution Methods for Conservation Laws – p.78/80

slide-98
SLIDE 98

Lifting Haar-wavelets in general geometry

d(i)

j,µ = d(i),Haar j,µ

−         

1 6[fj,γ(2) − fj,γ(1)] + 1 6[fj,γ(3) − fj,γ(1)]

µ = (γ, 1)

1 6[fj,γ(1) − fj,γ(2)] + 1 6[fj,γ(3) − fj,γ(2)]

µ = (γ, 2)

1 6[fj,γ(1) − fj,γ(3)] + 1 6[fj,γ(2) − fj,γ(3)]

µ = (γ, 3)

T T T T T T

j+1 j+1 j+1

Tj+1

j j j 01 02 03 00 0(1) 0(2) 0(3)

Adaptive Multiresolution Methods for Conservation Laws – p.79/80

slide-99
SLIDE 99

Concluding Remarks

Two methodologies for the construction of MR transforms (without using Fourier techniques) Harten’s approach Lifting schemes Important aspects Wavelet coefcients: local regularity indicators Data compression: thresholding wavelet coefcients Convergent prediction operators ⇒ Stability Functional context: WT and IWT as change of bases Next topic: applications to adaptive methods for PDE

Adaptive Multiresolution Methods for Conservation Laws – p.80/80