Improvements to the Stable Completion

In view of the almost fully populated part of the right half of ˜Mj for j = 3 in Table 3.4 and the large absolute values in the denominators of the nonzero entries, we construct an improved stable completion.

To this end, we inspect the relation (2.3.9) and explicitly compare the involved matrices on the levels j = j0 and j = j0+ 1. We find that the number of nonzero entries of I − M^j,0M˜^T_j,0 can be significantly reduced by taking appropriate linear combinations of columns. This corresponds to a multiplication from the right with a matrix ˇLj, yielding the modified expression

Mj,1= (I − M^j,0M˜^T_j,0)ˇLjMˇj,1, (3.2.18) which corresponds to the substitution

Mˇj,17→ ˇM⁰_j,1:= ˇLjMˇj,1. (3.2.19) To show that this additional matrix fits into the framework of Section 2.3.1, we formulate the following Lemma 3.3. Given an initial stable completion ˇMj,1 and a target completion ˇM⁰_j,1, the transformation matrices from (2.3.1) are determined by

L⁽¹⁾_j := ˇGj,0Mˇ⁰_j,1, K⁽¹⁾_j := ˇGj,1Mˇ⁰_j,1. (3.2.20) Mˇ⁰_j,1 is again a stable completion if and only if L⁽¹⁾_j and K⁽¹⁾_j fulfil the assumptions of Theorem 2.13.

Proof. According to (2.3.2) and the definitions of Theorem 2.13, we have Mˇ⁰_j,1= Mj,0L⁽¹⁾_j + ˇMj,1K⁽¹⁾_j = (Mj,0, ˇMj,1) L⁽¹⁾_j

K⁽¹⁾_j

!

= ˇMj L⁽¹⁾_j K⁽¹⁾_j

!

. (3.2.21)

Multiplication with ˇGj = ˇM⁻¹_j leads to

L⁽¹⁾_j K⁽¹⁾_j

!

= ˇGjMˇ⁰_j,1, (3.2.22)

which is equivalent to (3.2.20).

The improved initial stable completion ˇM⁰_j,1 is presented in Table 3.5, together with the matrices K⁽¹⁾_j and L⁽¹⁾_j from (3.2.20). These transformations are applied prior to the biorthogonalisation, which is then performed by the matrices from (2.3.10),

L⁽²⁾_j := − ˜M^T_j,0Mˇ⁰_j,1, K⁽²⁾_j := I . (3.2.23) Thus, we deal with a two-stage process here. To formalise the concatenation of two such transformations, we derive the following

35

Chapter3.TwoConstructionsontheInterval

1 0 0 0 0 0 0 0 0 − ¹³⁹_{256 256}⁵¹ − ₂₅₆⁹ ₂₅₆¹ 0 0 0 0

2611 3750

1 2

3

100 0 −₁₂₅₀¹³ 0 0 0 0 _1920000⁶⁹⁹⁴³⁹ −₆₄₀₀₀₀⁹⁵¹⁷ ₆₄₀₀₀₀¹²⁶³ −_1920000¹⁶⁰²¹ ₆₄₀₀₀₀¹⁸⁰⁷ −₆₄₀₀₀₀⁶⁶³ ₆₄₀₀₀₀¹¹⁷ −₆₄₀₀₀₀¹³

−³¹⁵¹₅₆₂₅ 1 −₁₅₀²³ 0 −₅₆₂₅³⁰¹ 0 0 0 0 −_2880000⁷²⁸⁶⁹⁹ −³³⁴⁷⁰³_{960000 320000}⁴⁴⁴³⁹ −_2880000¹⁶⁴⁸³⁹ _2880000⁴¹⁸³⁹ −₉₆₀₀₀₀⁵¹¹⁷ ₃₂₀₀₀₀³⁰¹ −_2880000³⁰¹

−₂₂₅₀^{59 1}₂ ⁵³₆₀ 0 ₂₂₅₀²⁷¹ 0 0 0 0 −_1152000^{87511 215333}₃₈₄₀₀₀ −₁₂₈₀₀₀³⁰⁴²⁹ _1152000¹³⁸⁸²⁹ −_1152000³⁷⁶⁶⁹ ₃₈₄₀₀₀⁴⁶⁰⁷ −₁₂₈₀₀₀²⁷¹ _1152000²⁷¹

23

150 0 1 0 ₁₅₀²³ 0 0 0 0 ₇₆₈₀₀⁸⁶⁷⁷ −₂₅₆₀₀⁷⁶³¹ −₂₅₆₀₀^{8091 11897}₇₆₈₀₀ − ₇₆₈₀₀³¹⁹⁷ ₂₅₆₀₀³⁹¹ −₂₅₆₀₀⁶⁹ ₇₆₈₀₀²³

271

2250 0 ⁵³₆₀ ¹₂ −₂₂₅₀⁵⁹ 0 0 0 0 _1152000¹⁰¹²¹⁹ −₃₈₄₀₀₀⁸⁶⁸⁵⁷ ₁₂₈₀₀₀⁷¹⁸⁴¹ −_1152000⁹⁵⁴⁴¹ _1152000⁸²⁰¹ −₃₈₄₀₀₀¹⁰⁰³ ₁₂₈₀₀₀⁵⁹ −_1152000⁵⁹

−₅₆₂₅³⁰¹ 0 −₁₅₀²³ 1 −³¹⁵¹₅₆₂₅ 0 0 0 0 −_2880000^{119849 118747}₉₆₀₀₀₀ −₃₂₀₀₀₀⁹³⁴¹¹ −^1166989_{2880000 2880000}⁴³⁷⁹⁸⁹ −₉₆₀₀₀₀⁵³⁵⁶⁷ ₃₂₀₀₀₀³¹⁵¹ −_2880000³¹⁵¹

−₁₂₅₀¹³ 0 ₁₀₀^{3 1}₂ ²⁶¹¹₃₇₅₀ 0 0 0 0 −_1920000¹³²¹¹ ₆₄₀₀₀₀⁸⁴³³ −₆₄₀₀₀₀^{53787 1062329}_1920000 −_1920000³⁶²⁹²⁹ ₆₄₀₀₀₀⁴⁴³⁸⁷ −₆₄₀₀₀₀⁷⁸³³ _1920000²⁶¹¹

0 0 0 0 1 0 0 0 0 ₅₁₂¹ − ₅₁₂⁹ ₅₁₂⁵¹ − ¹³⁹₅₁₂ − ¹³⁹_{512 512}⁵¹ − ₅₁₂⁹ ₅₁₂¹

0 0 0 0 ²⁶¹¹₃₇₅₀ ¹_{2 100}³ 0 −₁₂₅₀¹³ _1920000²⁶¹¹ −₆₄₀₀₀₀⁷⁸³³ ₆₄₀₀₀₀⁴⁴³⁸⁷ −_1920000^{362929 1062329}_1920000 −₆₄₀₀₀₀⁵³⁷⁸⁷ ₆₄₀₀₀₀⁸⁴³³ −_1920000¹³²¹¹ 0 0 0 0 −³¹⁵¹₅₆₂₅ 1 −₁₅₀²³ 0 −₅₆₂₅³⁰¹ −_2880000³¹⁵¹ ₃₂₀₀₀₀³¹⁵¹ −₉₆₀₀₀₀⁵³⁵⁶⁷ _2880000⁴³⁷⁹⁸⁹ −^1166989_2880000 −₃₂₀₀₀₀^{93411 118747}₉₆₀₀₀₀ −_2880000¹¹⁹⁸⁴⁹ 0 0 0 0 −2250⁵⁹

1 2

53

60 0 ₂₂₅₀²⁷¹ −1152000⁵⁹

59

128000 −384000¹⁰⁰³

8201

1152000 −1152000⁹⁵⁴⁴¹

71841

128000 −384000⁸⁶⁸⁵⁷

101219 1152000

0 0 0 0 ₁₅₀²³ 0 1 0 ₁₅₀²³ ₇₆₈₀₀²³ −₂₅₆₀₀⁶⁹ ₂₅₆₀₀³⁹¹ − ₇₆₈₀₀^{3197 11897}₇₆₈₀₀ −₂₅₆₀₀⁸⁰⁹¹ −₂₅₆₀₀⁷⁶³¹ ₇₆₈₀₀⁸⁶⁷⁷ 0 0 0 0 ₂₂₅₀²⁷¹ 0 ⁵³₆₀ ¹₂ −₂₂₅₀⁵⁹ _1152000²⁷¹ −₁₂₈₀₀₀²⁷¹ ₃₈₄₀₀₀⁴⁶⁰⁷ −_1152000³⁷⁶⁶⁹ _1152000¹³⁸⁸²⁹ −₁₂₈₀₀₀^{30429 215333}₃₈₄₀₀₀ −_1152000⁸⁷⁵¹¹ 0 0 0 0 −₅₆₂₅³⁰¹ 0 −₁₅₀²³ 1 −³¹⁵¹₅₆₂₅ −_2880000³⁰¹ ₃₂₀₀₀₀³⁰¹ −₉₆₀₀₀₀⁵¹¹⁷ _2880000⁴¹⁸³⁹ −_2880000¹⁶⁴⁸³⁹ ₃₂₀₀₀₀⁴⁴⁴³⁹ −³³⁴⁷⁰³₉₆₀₀₀₀ −_2880000⁷²⁸⁶⁹⁹ 0 0 0 0 −₁₂₅₀¹³ 0 ₁₀₀^{3 1}₂ ²⁶¹¹₃₇₅₀ −₆₄₀₀₀₀¹³ ₆₄₀₀₀₀¹¹⁷ −₆₄₀₀₀₀⁶⁶³ ₆₄₀₀₀₀¹⁸⁰⁷ −_1920000¹⁶⁰²¹ ₆₄₀₀₀₀¹²⁶³ −₆₄₀₀₀₀⁹⁵¹⁷ _1920000⁶⁹⁹⁴³⁹

0 0 0 0 0 0 0 0 1 0 0 0 0 ₂₅₆¹ − ₂₅₆⁹ ₂₅₆⁵¹ − ¹³⁹₂₅₆

Table 3.4: The primal two-level transform √

2Mj at level j = 3, for the simple stable completion.

36

3.2. Finite Element Wavelets

Table 3.5: The left table shows the new proposal for a modified stable completion √

2 ˇM_j,1 on the coarsest level j = 2. It contains off-diagonal entries which induce linear combinations of the columns of Mj,1. It is trivially extended to higher levels by repetition. In the middle and on the right we display the associated transformation matrices K⁽¹⁾_j and L⁽¹⁾_j .

Lemma 3.4. Let K⁽ⁱ⁾_j and L⁽ⁱ⁾_j , i = 1, 2, denote the transformation matrices for two successive stable completions. These two steps can be written as a single step with the matrices

Lj = L⁽²⁾_j + L⁽¹⁾_j K⁽²⁾_j , Kj = K⁽¹⁾_j K⁽²⁾_j . (3.2.24) Proof. We combine (2.3.2) and the left part of (3.2.21) and obtain

Mj,1= Mj,0L⁽²⁾_j + ˇM⁰_j,1K⁽²⁾_j = Mj,0L⁽²⁾_j +

Mj,0L⁽¹⁾_j + ˇMj,1K⁽¹⁾_j 

K⁽²⁾_j . (3.2.25) The claim is verified by comparison with (3.2.24).

Using this result, we can now determine the structure of the complete transformation.

Theorem 3.5. With the definitions of Theorem 2.13, the transformation matrices for the stable comple-tion modified according to (3.2.19) are given by

Lj=

− ˜M^T_j,0+ ˇGj,0 ˇLjMˇj,1, Kj = ˇGj,1LˇjMˇj,1. (3.2.26)

By carrying out these calculations in our example we indeed achieve a reduction of the quantity of nonzero entries in the right part of Mj by about a factor of 2. The improved transformation matrices on level j = 2 are shown in Table 3.6. In contrast to the simple stable completion from Table 3.3, the block-banded structure can be seen in all four parts of the matrices. The full effect only becomes apparent on the next higher level j = 3. The corresponding primal matrix Mj is shown in Table 3.7, and the dual matrix ˜Mj in Table 3.8. There are several benefits of the new construction.

• The number of arithmetic operations in the forward transformation (which is given by the number of nonzero entries in the matrix) is reduced by a factor of almost two.

• The size of the support of the wavelets is reduced by a factor of two to four.

• The denominators in the fractions are significantly smaller, indicating less irrational numbers.

• The transformation matrices at level j = 2 contain the same entries as the matrices on higher levels which facilitates the implementation.

• The pattern of nonzero entries is similar for the primal and dual matrices.

37

Chapter 3. Two Constructions on the Interval

Table 3.6: We show the new proposal for the primal and dual two-level transformation matrices on the coarsest level, √

2Mj and√

2 ˜Mj, derived from the modified stable completion from Table 3.5.

38

3.2.FiniteElementWavelets

1 0 0 0 0 0 0 0 0 −₁₂₈⁷⁵ 0 0 0 0 0 0 0

2611 3750

1 2

3

100 0 −₁₂₅₀¹³ 0 0 0 0 ²⁶¹¹_{6400 3200}⁸³⁹ ₃₂₀₀³⁹ −₁₂₈₀₀¹¹⁷ ₁₂₈₀₀³⁹ 0 0 0

−³¹⁵¹₅₆₂₅ 1 −₁₅₀²³ 0 −₅₆₂₅³⁰¹ 0 0 0 0 −³¹⁵¹₉₆₀₀ −⁴⁴⁹⁹_{4800 4800}³⁰¹ −₆₄₀₀³⁰¹ ₁₉₂₀₀³⁰¹ 0 0 0

−₂₂₅₀^{59 1}₂ ⁵³₆₀ 0 ₂₂₅₀²⁷¹ 0 0 0 0 −₃₈₄₀^{59 2129}₁₉₂₀ −₁₉₂₀²⁷¹ ₂₅₆₀²⁷¹ −₇₆₈₀²⁷¹ 0 0 0

23

150 0 1 0 ₁₅₀²³ 0 0 0 0 ₂₅₆²³ − ⁴⁹₆₄ − ⁴⁹_{64 512}⁶⁹ − ₅₁₂²³ 0 0 0

271

2250 0 ⁵³₆₀ ¹₂ −₂₂₅₀⁵⁹ 0 0 0 0 ₃₈₄₀²⁷¹ −₁₉₂₀^{271 2129}₁₉₂₀ −₂₅₆₀⁵⁹ ₇₆₈₀⁵⁹ 0 0 0

−₅₆₂₅³⁰¹ 0 −₁₅₀²³ 1 −³¹⁵¹₅₆₂₅ 0 0 0 0 −₉₆₀₀³⁰¹ ₄₈₀₀³⁰¹ −⁴⁴⁹⁹₄₈₀₀ −³¹⁵¹_{6400 19200}³¹⁵¹ 0 0 0

−₁₂₅₀¹³ 0 ₁₀₀^{3 1}₂ ²⁶¹¹₃₇₅₀ 0 0 0 0 −₆₄₀₀³⁹ ₃₂₀₀³⁹ ₃₂₀₀⁸³⁹ ₁₂₈₀₀⁷⁸³³ −₁₂₈₀₀²⁶¹¹ 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 − ₂₅₆⁷⁵ − ₂₅₆⁷⁵ 0 0 0

0 0 0 0 ²⁶¹¹₃₇₅₀ ¹_{2 100}³ 0 −₁₂₅₀¹³ 0 0 0 −₁₂₈₀₀²⁶¹¹ ₁₂₈₀₀⁷⁸³³ ₃₂₀₀⁸³⁹ ₃₂₀₀³⁹ −₆₄₀₀³⁹ 0 0 0 0 −³¹⁵¹5625 1 −150²³ 0 −5625³⁰¹ 0 0 0 ₁₉₂₀₀³¹⁵¹ −³¹⁵¹6400 −⁴⁴⁹⁹4800

301

4800 −9600³⁰¹

0 0 0 0 −₂₂₅₀^{59 1}₂ ⁵³₆₀ 0 ₂₂₅₀²⁷¹ 0 0 0 ₇₆₈₀⁵⁹ −₂₅₆₀^{59 2129}₁₉₂₀ −₁₉₂₀²⁷¹ ₃₈₄₀²⁷¹

0 0 0 0 ₁₅₀²³ 0 1 0 ₁₅₀²³ 0 0 0 − ₅₁₂²³ ₅₁₂⁶⁹ − ⁴⁹₆₄ − ⁴⁹_{64 256}²³

0 0 0 0 ₂₂₅₀²⁷¹ 0 ⁵³₆₀ ¹₂ −₂₂₅₀⁵⁹ 0 0 0 −₇₆₈₀²⁷¹ ₂₅₆₀²⁷¹ −₁₉₂₀^{271 2129}₁₉₂₀ −₃₈₄₀⁵⁹ 0 0 0 0 −₅₆₂₅³⁰¹ 0 −₁₅₀²³ 1 −³¹⁵¹₅₆₂₅ 0 0 0 ₁₉₂₀₀³⁰¹ −₆₄₀₀³⁰¹ ₄₈₀₀³⁰¹ −⁴⁴⁹⁹₄₈₀₀ −³¹⁵¹₉₆₀₀ 0 0 0 0 −₁₂₅₀¹³ 0 ₁₀₀^{3 1}₂ ²⁶¹¹₃₇₅₀ 0 0 0 ₁₂₈₀₀³⁹ −₁₂₈₀₀¹¹⁷ ₃₂₀₀³⁹ ₃₂₀₀^{839 2611}₆₄₀₀

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 −₁₂₈⁷⁵

Table 3.7: The primal two-level transform √

2Mj at level j = 3 constructed with our modified stable completion.

39

Chapter3.TwoConstructionsontheInterval

1 0 0 0 0 0 0 0 0 −¹²⁸₇₅ 0 0 0 0 0 0 0

139 128

3373

3200 −₁₆₀₀⁶²⁷ ₃₂₀₀²⁵³ −₂₅₆¹ 0 0 0 0 ¹³⁹₇₅ −₇₅⁸ ₇₅² −₇₅¹ 0 0 0 0

−¹¹₃₂ ²³⁷₂₀₀ −₁₀₀¹³ ₂₀₀⁷ −₆₄¹ 0 0 0 0 −⁴⁴₇₅ −³²_{75 75}⁸ −₇₅⁴ 0 0 0 0

−₁₂₈^{51 1843}₃₂₀₀ ¹⁰⁴³₁₆₀₀ −₃₂₀₀⁴⁷⁷ ₂₅₆⁹ 0 0 0 0 −¹⁷₂₅ ²⁴₂₅ −₂₅⁶ ₂₅³ 0 0 0 0

1

5 −₁₂₅^{28 104}₁₂₅ −₁₂₅²⁸ ₁₀¹ 0 0 0 0 ¹²⁸₃₇₅ −²⁵⁶₃₇₅ −²⁵⁶₃₇₅ ¹²⁸₃₇₅ 0 0 0 0

9

128 −₃₂₀₀^{477 1043}₁₆₀₀ ¹⁸⁴³₃₂₀₀ −₂₅₆⁵¹ 0 0 0 0 ₂₅³ −₂₅^{6 24}₂₅ −¹⁷₂₅ 0 0 0 0

−₃₂¹ ₂₀₀⁷ −₁₀₀^{13 237}₂₀₀ −¹¹₆₄ 0 0 0 0 −₇₅⁴ ₇₅⁸ −³²₇₅ −⁴⁴₇₅ 0 0 0 0

−₁₂₈¹ ₃₂₀₀²⁵³ −₁₆₀₀^{627 3373}₃₂₀₀ ¹³⁹₂₅₆ 0 0 0 0 −₇₅¹ ₇₅² −₇₅^{8 139}₇₅ 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 −¹²⁸₇₅ −¹²⁸₇₅ 0 0 0

0 0 0 0 ¹³⁹₂₅₆ ³³⁷³₃₂₀₀ −₁₆₀₀⁶²⁷ ₃₂₀₀²⁵³ −₁₂₈¹ 0 0 0 0 ¹³⁹₇₅ −₇₅⁸ ₇₅² −₇₅¹ 0 0 0 0 −¹¹₆₄ ²³⁷₂₀₀ −₁₀₀¹³ ₂₀₀⁷ −₃₂¹ 0 0 0 0 −⁴⁴₇₅ −³²_{75 75}⁸ −₇₅⁴ 0 0 0 0 −₂₅₆^{51 1843}₃₂₀₀ ¹⁰⁴³₁₆₀₀ −₃₂₀₀⁴⁷⁷ ₁₂₈⁹ 0 0 0 0 −¹⁷₂₅ ²⁴₂₅ −₂₅⁶ ₂₅³ 0 0 0 0 ₁₀¹ −₁₂₅^{28 104}₁₂₅ −₁₂₅^{28 1}₅ 0 0 0 0 ¹²⁸₃₇₅ −²⁵⁶₃₇₅ −²⁵⁶₃₇₅ ¹²⁸₃₇₅ 0 0 0 0 ₂₅₆⁹ −₃₂₀₀^{477 1043}₁₆₀₀ ¹⁸⁴³₃₂₀₀ −₁₂₈⁵¹ 0 0 0 0 ₂₅³ −₂₅^{6 24}₂₅ −¹⁷₂₅

0 0 0 0 −64¹

7

200 −100¹³

237

200 −¹¹32 0 0 0 0 −75⁴

8

75 −³²75 −⁴⁴75

0 0 0 0 −₂₅₆¹ ₃₂₀₀²⁵³ −₁₆₀₀^{627 3373}₃₂₀₀ ¹³⁹₁₂₈ 0 0 0 0 −₇₅¹ ₇₅² −₇₅^{8 139}₇₅

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 −¹²⁸₇₅

Table 3.8: The dual two-level transform √

2 ˜Mj at level j = 3 constructed with our modified stable completion.

40

3.2. Finite Element Wavelets

Figure 3.4: These graphs show the primal and dual wavelets for the coarsest complement space Wj (thus with j = 2), obtained by the modified stable completion. Only two of the four functions in each set are displayed, as the missing ones result by mirroring around x = 1/2. The primal wavelets on this level are piecewise linear with mesh size 1/8, the dual wavelets are piecewise cubic with mesh size 1/4.

1 0 0 0 0

Table 3.9: This table shows the forward and backward transformation matrices between the single-scale basis Φj

and the nodal basis for j = j0 = 2. C⁻¹_j is shown on the left (note that the values in the first and last columns correspond to those from Table 3.1). Cj is shown on the right. Due to the special structure of these matrices, the inversion reduces to a change in sign of the off-diagonal entries.

The only disadvantage lies in the fact that the dual transformation has gained in the number of nonzero entries. Since the dual transform is rarely used in the context of partial differential equations, this is only a minor issue which is outweighed by far by the positive effects. The wavelets for the primal and dual spaces are finally shown in Figure 3.4.

The construction has so far been carried out for free boundary conditions. However, it is a straightforward procedure to obtain a biorthogonal wavelet basis with homogeneous boundary conditions on both the primal and dual side by simply deleting the two functions with nonzero value on the boundary. The corresponding rows and columns are then removed from the transformation matrices, which conserves biorthogonality.

In document Fast Optimised Wavelet Methods for Control Problems Constrained by Elliptic PDEs (Page 47-53)