text: Table 1: Summary statistics from model ﬁts relating to ﬁgure 1. The values of xˆ and αˆ min were obtained CSN method. The parameter θ¯ is the average posterior value of θ in expression 5. The column, % < xˆ , gives the proportion of data that are less the min xˆ . min Data set xˆ % < xˆ αˆ θ¯ min min Github 49 97 2.74 3.21 Moby Dick 7 84 1.95 1.95 Petster 29 93 2.90 3.80 Swiss Prot 47 95 2.07 2.07 In each example g(x; θ) is a power law distribution (with x = 1), i.e. min − x θ g(x, θ) = for x = 1, . . . (5) ζ(θ) (cid:88) where θ is the power law scaling parameter and ∞ 1 ζ(θ) = iθ i=1 is Riemann zeta function. To model the deviation away from the power law distribution, we use the unit exponential cumulative density function − − − − − − D(x; φ) = 1 e φ0 φ1(x 1) φ2(x 1)2 (6) { } ≥ − − ∞ where φ = φ0, φ1, φ2 0. So D(x = 1; φ) = 1 e φ0 and g(x = ; φ) = 1. Figure 1a gives the empirical CDF of each data set and the ﬁtted function f(x; θ, φ). Also shown is a ﬁtted power law distribution where x and the scaling parameter θ were estimated min using the CSN method from Clauset et al.[6]. Figure 1b plots the diﬀerence function D(x; φ) along with the estimated x value obtained from the CSN method. min For each data set in ﬁgure 1a, f(x; θ, φ) provides an excellent ﬁt. However, the key beneﬁt is that all data is used (see table 1 for an exact breakdown). For example, in the Moby Dick data set since x = 7, ﬁtting a power law using CSN would result in discarding around 84% min of the collected data. For the network data sets, the standard power law ﬁt is relatively poor, however, our ﬂexible formulation still provides an excellent ﬁt. Inference can also be made about plausible values x by calculating min arg min D(x; φ) > 0.95 , x where φ are samples from the posterior. The results given in ﬁgure 1 are broadly consistent with the CSN method. However, the diﬀerence method also provides credible regions to assess parameter uncertainty. 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Summary

label: caption

text: statistics

text: from

text: model

text: ﬁts

text: relating

text: to

text: ﬁgure

text: The

text: values

text: of

text: xˆ

text: and

text: αˆ

text: min

text: were

text: obtained

text: CSN

text: method.

text: The

text: parameter

text: θ¯

text: is

text: the

text: average

text: posterior

text: value

text: of

text: θ

text: in

text: expression

text: The

text: column,

text: xˆ

text: gives

text: the

text: proportion

text: of

text: data

text: that

text: are

text: less

text: the

text: min

text: xˆ

text: min

text: Data

label: table

text: set

text: xˆ

text: xˆ

text: αˆ

text: θ¯

text: min

text: min

text: Github

text: Moby

text: Dick

text: Petster

text: Swiss

text: Prot

text: In

text: each

text: example

text: g(x;

text: θ)

text: is

text: a

text: power

text: law

text: distribution

text: (with

text: x

text: i.e.

text: min

text: x

text: θ

text: g(x,

text: θ)

text: for

text: x

text: ζ(θ)

text: where

text: θ

text: is

text: the

text: power

text: law

text: scaling

text: parameter

text: and

text: ζ(θ)

text: iθ

text: is

text: Riemann

text: zeta

text: function.

text: To

text: model

text: the

text: deviation

text: away

text: from

text: the

text: power

text: law

text: distribution,

text: we

text: use

text: the

text: unit

text: exponential

text: cumulative

text: density

text: function

text: D(x;

text: φ)

text: e

text: φ0

text: where

text: φ

text: φ2

text: So

text: D(x

text: φ)

text: e

text: φ0

text: and

text: g(x

text: φ)

text: Figure

text: gives

text: the

text: empirical

text: CDF

text: of

text: each

text: data

text: set

text: and

text: the

text: ﬁtted

text: function

text: f(x;

text: θ,

text: φ).

text: Also

text: shown

text: is

text: a

text: ﬁtted

text: power

text: law

text: distribution

text: where

text: x

text: and

text: the

text: scaling

text: parameter

text: θ

text: were

text: estimated

text: min

text: using

text: the

text: CSN

text: method

text: from

text: Clauset

text: et

text: Figure

text: plots

text: the

text: diﬀerence

text: function

text: D(x;

text: φ)

text: along

text: with

text: the

text: estimated

text: x

text: value

text: obtained

text: from

text: the

text: CSN

text: method.

text: min

text: For

text: each

text: data

text: set

text: in

text: ﬁgure

text: f(x;

text: θ,

text: φ)

text: provides

text: an

text: excellent

text: ﬁt.

text: However,

text: the

text: key

text: beneﬁt

text: is

text: that

text: all

text: data

text: is

text: used

text: (see

text: table

text: for

text: an

text: exact

text: breakdown).

text: For

text: example,

text: in

text: the

text: Moby

text: Dick

text: data

text: set

text: since

text: x

text: ﬁtting

text: a

text: power

text: law

text: using

text: CSN

text: would

text: result

text: in

text: discarding

text: around

text: min

text: of

text: the

text: collected

text: data.

text: For

text: the

text: network

text: data

text: sets,

text: the

text: standard

text: power

text: law

text: ﬁt

text: is

text: relatively

text: poor,

text: however,

text: our

text: ﬂexible

text: formulation

text: still

text: provides

text: an

text: excellent

text: ﬁt.

text: Inference

text: can

text: also

text: be

text: made

text: about

text: plausible

text: values

text: x

text: by

text: calculating

text: min

text: arg

text: min

text: D(x;

text: φ)

text: x

text: where

text: φ

text: are

text: samples

text: from

text: the

text: posterior.

text: The

text: results

text: given

text: in

text: ﬁgure

text: are

text: broadly

text: consistent

text: with

text: the

text: CSN

text: method.

text: However,

text: the

text: diﬀerence

text: method

text: also

text: provides

text: credible

text: regions

text: to

text: assess

text: parameter

text: uncertainty.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≤ ≤ Let l (0 j n ) be the Hamming weight of the part in the jth level of a vector j j v, and (l , l , ..., l ) be the Hamming weight spectrum vector of v. Then, we can express 1 2 s the last equality in an equivalent but diﬀerent form which can be directly expressed by the parameters l = (l (cid:0), l , ..., l ) and A = A (cid:1) as (cid:88)follows(cid:89): 1 2 s l (l1,l2,...,ls) (cid:80)| ∈ ≤ ≤ s C C z : a Rni, 1 i s = A z W i:w(a) l j:lj l i=1 ≤ ≤ where the summation takes over all l satisfying 0 l n for each j and A j j l l denotes the number of codewords in C having the Hamming weight spectrum vector l = (l , l , ..., l ). 1 2 s In the following theorem, we derive a MacWilliams-type identity for the complete − poset level weight enumerator of a linear P code. R − Theorem 4.1. Let C be a linear P code over a poset of size N and s levels. Then, R the re(cid:0)lation(cid:12) between the complete po(cid:1)set level weight enumerator of C and its dual is (cid:12) given by     ⊥ (cid:88)∈ (cid:89)≤ (cid:88)≤ (cid:88) (cid:0) (cid:1) (cid:0) (cid:1)  CW C zi:w(a) : a Rni, 1 i s    1 s nj pj − − − − = | | A ( 1)aj(q 1)pj aj lj (cid:80)nj−lj z , C l aj pj aj j:pj l j=1 pj=0 aj=0 where χ is a nontrivial additive character of R and the summ(cid:0)ati(cid:1)on takes over all l (cid:0) (cid:1) ≤ ≤ l satisfying 0 l n for each j and A denotes the number of codewords in C having j j l the Hamming weight spectrum vector l = (l , l , ..., l ). Here, lj = 0 for l < a and (cid:81) 1 2 s aj j j 0 = 1. 0 n Proof. Set f (v) = z , by Lemma 2.2, we obtain w(vj(cid:88)) (cid:88) (cid:89) (cid:101) j=1 (cid:104) (cid:105) (cid:104) (cid:105) s f (u) = (cid:88) χ ( u(cid:88), v ) f (v) = χ ( u, v (cid:89)) z j:w(vj) ∈ ∈ v RN v RN j=1 s (cid:32) (cid:33) = (cid:88) ... (cid:88) χ ((cid:89)u1v1 + ... + usvs) zj:w(vj) ∈ ∈ v1 Rn1 v2 Rns j=1  s (cid:104) (cid:105) = ... χ ( u , v ) z (cid:89) (cid:88) j j j:wH(vj) ∈ ∈ v1 Rn1 v2 Rns j=1  s (cid:104) (cid:105) = χ ( u , v ) z . j j j:w(vj) ∈ j=1 vj Rnj 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Let

text: l

text: j

text: n

text: be

text: the

text: Hamming

text: weight

text: of

text: the

text: part

text: in

text: the

text: jth

text: level

text: of

text: a

text: vector

text: j

text: j

text: v,

text: and

text: (l

text: l

text: l

text: be

text: the

text: Hamming

text: weight

text: spectrum

text: vector

text: of

text: v.

text: Then,

text: we

text: can

text: express

text: s

text: the

text: last

text: equality

text: in

text: an

text: equivalent

text: but

text: diﬀerent

text: form

text: which

text: can

text: be

text: directly

text: expressed

text: by

text: the

text: parameters

text: l

text: (l

text: l

text: l

text: and

text: A

text: A

text: as

text: s

text: l

text: s

text: C

text: C

text: z

text: a

text: Rni,

text: i

text: s

text: A

text: z

text: W

text: i:w(a)

text: l

text: j:lj

text: l

text: where

text: the

text: summation

text: takes

text: over

text: all

text: l

text: satisfying

text: l

text: n

text: for

text: each

text: j

text: and

text: A

text: j

text: j

text: l

text: l

text: denotes

text: the

text: number

text: of

text: codewords

text: in

text: C

text: having

text: the

text: Hamming

text: weight

text: spectrum

text: vector

text: l

text: (l

text: l

text: l

text: s

text: In

text: the

text: following

text: theorem,

text: we

text: derive

text: a

text: MacWilliams-type

text: identity

text: for

text: the

text: complete

text: poset

text: level

text: weight

text: enumerator

text: of

text: a

text: linear

text: P

text: code.

text: R

text: Theorem

text: Let

text: C

text: be

text: a

text: linear

text: P

text: code

text: over

text: a

text: poset

text: of

text: size

text: N

text: and

text: s

text: levels.

text: Then,

text: R

text: the

text: between

text: the

text: complete

text: level

text: weight

text: enumerator

text: of

text: C

text: and

text: its

text: dual

text: is

text: given

text: by

text: CW

text: C

text: zi:w(a)

text: a

text: Rni,

text: i

text: s

text: s

text: nj

text: pj

text: A

text: aj

text: lj

text: z

text: C

text: l

text: aj

text: pj

text: aj

text: j:pj

text: l

text: where

text: χ

text: is

text: a

text: nontrivial

text: additive

text: character

text: of

text: R

text: and

text: the

text: takes

text: over

text: all

text: l

text: l

text: satisfying

text: l

text: n

text: for

text: each

text: j

text: and

text: A

text: denotes

text: the

text: number

text: of

text: codewords

text: in

text: C

text: having

text: j

text: j

text: l

text: the

text: Hamming

text: weight

text: spectrum

text: vector

text: l

text: (l

text: l

text: l

text: Here,

text: lj

text: for

text: l

text: a

text: and

text: s

text: aj

text: j

text: j

text: n

text: Proof.

text: Set

text: f

text: (v)

text: z

text: by

text: Lemma

text: we

text: obtain

text: s

text: f

text: (u)

text: χ

text: v

text: f

text: (v)

text: χ

text: u,

text: v

text: z

text: j:w(vj)

text: v

text: RN

text: v

text: RN

text: s

text: χ

text: usvs)

text: zj:w(vj)

text: Rns

text: s

text: χ

text: u

text: v

text: z

text: j

text: j

text: j:wH(vj)

text: Rns

text: s

text: χ

text: u

text: v

text: z

text: j

text: j

text: j:w(vj)

text: vj

text: Rnj

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [13] J. N. Gutierrez, H. Tapia-Recillas, A MacWilliams Identity for Poset-Codes, Congr. Numer. 133 (1998) 63–73. [14] K. Huang, C. Liang, X. Ma, B. Bai, Unequal error protection by partial superpo- sition transmission using LDPC Codes. Available in arXiv:1309.3864 [cs.IT], 2013. [15] H. K.Kim, D. Y. Oh , A Classiﬁcation of Posets Admitting the MacWilliams Identity, IEEE Transactions on Information Theory, 51 (4) (2005) 1424–1431. [16] J. Korhonen, P. Frossard, Bit-error resilient packetization for streaming H.264/AVC video, Proc. of the 1st ACM International Workshop on Mobile Video, Germany, 2007, pp. 25-30. [17] E. Kuriata, Creation of unequal error protection codes for two groups of sym- bols,Int. J. Appl. Math. Comput. Sci. 18 (2008) 251-257. [18] H. K.Kim, D. Y. Oh, A classiﬁcation of posets admitting the MacWilliams identity, IEEE Trans. Inform. Theory. 51 (2005) 1424-1431. [19] P. Larsson, Codes for Correction of Localized Errors (Linkoping Studies in Science and Technology, dissertations, no. 374). Linkoping Sweden, (1995). [20] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error Correcting Codes, North- Holland Pub. Co., Amsterdam, 1977. [21] A. Mardjuadi, J. H. Weber, Codes for multiple localized burst error correction, IEEE Trans. Inform. Theory. 44 (1998) 2020-2024. [22] B. Masnik , J. Wolf, On linear unequal error protection codes, IEEE Trans. on Information Theory, IT- 13(4) (1967) 600–607. [23] H. Niederreiter, Point sets and sequences with small discrepancy, Mh. Math. 104 (1987) 273-337. [24] H. Niederreiter, A combinatorial problem for vector spaces over ﬁnite ﬁelds, Dis- crete Math. 96 (1991) 221-228. [25] H. Niederreiter, Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes, Discrete Math. 106/107 (1992) 361-367. [26] M. Ozen, V. Siap, The MacWilliams identity for m-spotty weight enumerators of linear codes over ﬁnite ﬁelds, Comput. Math. Appl. 61 (2011) 1000-1004. [27] M. Ozen, V. Siap, The MacWilliams identity for m-spotty Rosenbloom-Tsfasman weight enumerator, J. Franklin Inst. doi:10.1016/j.franklin.2012.06.002. 19

label: reference

text: J.

text: N.

text: Gutierrez,

text: H.

text: Tapia-Recillas,

text: A

text: MacWilliams

text: Identity

text: for

text: Poset-Codes,

text: Congr.

text: Numer.

text: K.

text: Huang,

text: C.

text: Liang,

text: X.

text: Ma,

text: B.

text: Bai,

text: Unequal

text: error

text: protection

text: by

text: partial

text: superpo-

text: sition

text: transmission

text: using

text: LDPC

text: Codes.

text: Available

text: in

text: [cs.IT],

text: H.

text: K.Kim,

text: D.

text: Y.

text: Oh

text: A

text: Classiﬁcation

text: of

text: Posets

text: Admitting

text: the

text: MacWilliams

text: Identity,

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: J.

text: Korhonen,

text: P.

text: Frossard,

text: Bit-error

text: resilient

text: packetization

text: for

text: streaming

text: video,

text: Proc.

text: of

text: the

text: ACM

text: International

text: Workshop

text: on

text: Mobile

text: Video,

text: Germany,

text: pp.

text: E.

text: Kuriata,

text: Creation

text: of

text: unequal

text: error

text: protection

text: codes

text: for

text: two

text: groups

text: of

text: sym-

text: bols,Int.

text: J.

text: Appl.

text: Math.

text: Comput.

text: Sci.

text: H.

text: K.Kim,

text: D.

text: Y.

text: Oh,

text: A

text: classiﬁcation

text: of

text: posets

text: admitting

text: the

text: MacWilliams

text: identity,

text: IEEE

text: Trans.

text: Inform.

text: Theory.

text: P.

text: Larsson,

text: Codes

text: for

text: Correction

text: of

text: Localized

text: Errors

text: (Linkoping

text: Studies

text: in

text: Science

text: and

text: Technology,

text: dissertations,

text: no.

text: Linkoping

text: Sweden,

text: F.

text: J.

text: MacWilliams,

text: N.

text: J.

text: A.

text: Sloane,

text: The

text: Theory

text: of

text: Error

text: Correcting

text: Codes,

text: North-

text: Holland

text: Pub.

text: Co.,

text: Amsterdam,

text: A.

text: Mardjuadi,

text: J.

text: H.

text: Weber,

text: Codes

text: for

text: multiple

text: localized

text: burst

text: error

text: correction,

text: IEEE

text: Trans.

text: Inform.

text: Theory.

text: B.

text: Masnik

text: J.

text: Wolf,

text: On

text: linear

text: unequal

text: error

text: protection

text: codes,

text: IEEE

text: Trans.

text: on

text: Information

text: Theory,

text: IT-

text: H.

text: Niederreiter,

text: Point

text: sets

text: and

text: sequences

text: with

text: small

text: discrepancy,

text: Mh.

text: Math.

text: H.

text: Niederreiter,

text: A

text: combinatorial

text: problem

text: for

text: vector

text: spaces

text: over

text: ﬁnite

text: ﬁelds,

text: Dis-

text: crete

text: Math.

text: H.

text: Niederreiter,

text: Orthogonal

text: arrays

text: and

text: other

text: combinatorial

text: aspects

text: in

text: the

text: theory

text: of

text: uniform

text: point

text: distributions

text: in

text: unit

text: cubes,

text: Discrete

text: Math.

text: M.

text: Ozen,

text: V.

text: Siap,

text: The

text: MacWilliams

text: identity

text: for

text: m-spotty

text: weight

text: enumerators

text: of

text: linear

text: codes

text: over

text: ﬁnite

text: ﬁelds,

text: Comput.

text: Math.

text: Appl.

text: M.

text: Ozen,

text: V.

text: Siap,

text: The

text: MacWilliams

text: identity

text: for

text: m-spotty

text: Rosenbloom-Tsfasman

text: weight

text: enumerator,

text: J.

text: Franklin

text: Inst.


text: (cid:26) 1, if (i , i , ..., i ) = (j , j , ..., j ) (ii) if S = k, then η (β , ..., β ) = 1 2 nk 1 2 nk S:i1i2...ink j1 jnk 0, otherwise ∈ { } where S, k 1, 2, ..., s . − Example 3.1. Let C be a binary PF code with N = 6 and s = 3. Consider the vector 2 u in Example 2.1. Then, η (β , β ) = η (β ) = η (β , β , β ) = 1. Otherwise 1:10 1 0 2:1 1 3:100 1 0 0 η (β , ..., β ) = 0. S:i1i2...ink j1 jnk We are now ready to give the deﬁnition of a byte poset level weight enumerator for − a linear P code. R − Deﬁnition 3.2. Let C be a linear P code over a poset of size N and s levels. Then R the byte poset level weight enumerat(cid:88)or of(cid:89)C (cid:89)is deﬁned as follows: B (C|z : a ∈ Rni, 1 ≤ i ≤ s) = b s zµS:00...0(u)...zµS:i1...inj (u)...zµS:−q−1...q−−1(u), W i:a ∈ S:00...0 S:i1...inj S:q 1...q 1 u C nj=a S=1 { } { } (cid:88) where a = min n : j = 1, ..., s , b = max n : j = 1, ..., s , and j j s µ (u) = η (uk). (3.1) S:i1...inj S:i1i2...inj k=1 − Now, we state the MacWilliams identity for a linear P code. R − Theorem 3.1. Let C be a linear P code over a poset of size N and s levels. Then, R the relation between the byte poset level weight enu(cid:88)mer(cid:89)ator(cid:89)of C (cid:89)and its dual is given by ⊥| ∈ ≤ ≤ 1 b s B (C z : a Rni, 1 i s) = | | A, W i:a C ∈ ∈ u C nj=a S=1 i1,i2,...,inj Q   where χ is a nontrivial additive character of R, (cid:88) (cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17)     µS:i1,i2,...,inj (u) A = χ β , β , ..., β β , β , ..., β z , ∈ j1 j2 jnj i1 i2 inj S:j1,j2,...,jnj βj1,βj2,...,βjnj Rnj { − } and Q = 0, 1, ..., q 1 . 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: if

text: (i

text: i

text: i

text: (j

text: j

text: j

text: (ii)

text: if

text: S

text: k,

text: then

text: η

text: (β

text: β

text: nk

text: nk

text: jnk

text: otherwise

text: where

text: S,

text: k

text: s

text: Example

text: Let

text: C

text: be

text: a

text: binary

text: PF

text: code

text: with

text: N

text: and

text: s

text: Consider

text: the

text: vector

text: u

text: in

text: Example

text: Then,

text: η

text: (β

text: β

text: η

text: (β

text: η

text: (β

text: β

text: β

text: Otherwise

text: η

text: (β

text: β

text: jnk

text: We

text: are

text: now

text: ready

text: to

text: give

text: the

text: deﬁnition

text: of

text: a

text: byte

text: poset

text: level

text: weight

text: enumerator

text: for

text: a

text: linear

text: P

text: code.

text: R

text: Deﬁnition

text: Let

text: C

text: be

text: a

text: linear

text: P

text: code

text: over

text: a

text: poset

text: of

text: size

text: N

text: and

text: s

text: levels.

text: Then

text: R

text: the

text: byte

text: poset

text: level

text: weight

text: deﬁned

text: as

text: follows:

text: B

text: (C|z

text: a

text: Rni,

text: i

text: s)

text: b

text: s

text: W

text: i:a

text: S:q

text: u

text: C

text: nj=a

text: where

text: a

text: min

text: n

text: j

text: s

text: b

text: max

text: n

text: j

text: s

text: and

text: j

text: j

text: s

text: µ

text: (u)

text: η

text: (uk).

text: Now,

text: we

text: state

text: the

text: MacWilliams

text: identity

text: for

text: a

text: linear

text: P

text: code.

text: R

text: Theorem

text: Let

text: C

text: be

text: a

text: linear

text: P

text: code

text: over

text: a

text: poset

text: of

text: size

text: N

text: and

text: s

text: levels.

text: Then,

text: R

text: the

text: relation

text: between

text: the

text: byte

text: poset

text: level

text: weight

text: C

text: its

text: dual

text: is

text: given

text: by

text: b

text: s

text: B

text: (C

text: z

text: a

text: Rni,

text: i

text: s)

text: A,

text: W

text: i:a

text: C

text: u

text: C

text: nj=a

text: Q

text: where

text: χ

text: is

text: a

text: nontrivial

text: additive

text: character

text: of

text: R,

text: (u)

text: A

text: χ

text: β

text: β

text: β

text: β

text: β

text: β

text: z

text: jnj

text: inj

text: Rnj

text: and

text: Q

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: with a convergence radius determined by the stability constant [7, 8]. The recovery of polyhedral interfaces then becomes a shape optimization. The analysis in [7] makes explicit use of a Landwe- ber iteration. Via successive approximations, and making use of estimates for the corresponding growth of the stability constant, the reconstruction can be cast into a multi-level scheme [8] ef- fectively enlarging the radius of convergence. As an important application, we mention so-called time-harmonic full waveform inversion (FWI) developed in reﬂection seismology [14,15,19,21] with the goal to image wavespeed variations in Earth’s interior. The data, here, are essentially the single- layer potential operator. However, stability estimates for the Dirichlet-to-Neumann map directly carry over to stability estimates for this operator. We give an outline of the paper. We ﬁrst state the main result and the main assumptions (Section ). Then we establish a rough stability estimate for the potentials using complex geometrical optics (CGO) solutions following the outline of an estimate in Beretta et al. [5] (Section ). The CGO solutions were introduced by Sylvester and Uhlmann [20] in their proof of uniqueness of this inverse boundary value problem. The CGO solutions in our analysis diﬀer slightly from theirs to obtain better constants in the stability estimates as proposed in [17]. We proceed with establishing the recovery of the number of tetrahedra in the mesh from the potential, and with expressing the Hausdorﬀ distance between meshes in terms of the diﬀerence of piecewise constant potentials deﬁned on these meshes. Naturally, the information on the Hausdorﬀ distance between meshes can be transformed to information on the vertices of the tetrahedra forming the meshes (Section ). The main part of the proof of our result pertains to obtaining a lower bound for the Gateaux derivative of the Dirichlet-to-Neumann map under mesh deformation (Section ). (cid:90) We use the Fourier transform convention, · fˆ(ξ) = f(x)eix ξdx. R 3 If the function f is deﬁned on a subset of R3, it is extended to R3 attaining the value zero. We denote by fˇ the inverse Fourier transform of f, (cid:90) 1 − · (1.1) fˇ(x) = f(ξ)e ix ξdξ. (2π)3 R 3 We introduce coordinates, x = (x(cid:48), x ), in R3, where x(cid:48) ∈ R2 and x ∈ R. We denote the open ball in R3 centered at x of radius r by3 B (x), and the open ball in R23centered at x(cid:48) of radius r by (cid:48) (cid:48) r B (x ). r We let Ω be a bounded domain in R3 such that R3 \ Ω is connected, ⊂ (2.1) Ω B (0) for some R > 0, R and (2.2) Ω has a Lipschitz boundary with constants r and K , 0 0 2 ##LTLine##

text: with

text: a

text: convergence

text: radius

text: determined

text: by

text: the

text: stability

text: constant

text: The

text: recovery

text: of

text: polyhedral

text: interfaces

text: then

text: becomes

text: a

text: shape

text: optimization.

text: The

text: analysis

text: in

text: makes

text: explicit

text: use

text: of

text: a

text: Landwe-

text: ber

text: iteration.

text: Via

text: successive

text: approximations,

text: and

text: making

text: use

text: of

text: estimates

text: for

text: the

text: corresponding

text: growth

text: of

text: the

text: stability

text: constant,

text: the

text: reconstruction

text: can

text: be

text: cast

text: into

text: a

text: multi-level

text: scheme

text: ef-

text: fectively

text: enlarging

text: the

text: radius

text: of

text: convergence.

text: As

text: an

text: important

text: application,

text: we

text: mention

text: so-called

text: time-harmonic

text: full

text: waveform

text: inversion

text: (FWI)

text: developed

text: in

text: reﬂection

text: seismology

text: with

text: the

text: goal

text: to

text: image

text: wavespeed

text: variations

text: in

text: Earth’s

text: interior.

text: The

text: data,

text: here,

text: are

text: essentially

text: the

text: single-

text: layer

text: potential

text: operator.

text: However,

text: stability

text: estimates

text: for

text: the

text: Dirichlet-to-Neumann

text: map

text: directly

text: carry

text: over

text: to

text: stability

text: estimates

text: for

text: this

text: operator.

text: We

text: give

text: an

text: outline

text: of

text: the

text: paper.

text: We

text: ﬁrst

text: state

text: the

text: main

text: result

text: and

text: the

text: main

text: assumptions

text: (Section

text: Then

text: we

text: establish

text: a

text: rough

text: stability

text: estimate

text: for

text: the

text: potentials

text: using

text: complex

text: geometrical

text: optics

text: (CGO)

text: solutions

text: following

text: the

text: outline

text: of

text: an

text: estimate

text: in

text: Beretta

text: et

text: al.

text: (Section

text: The

text: CGO

text: solutions

text: were

text: introduced

text: by

text: Sylvester

text: and

text: Uhlmann

text: in

text: their

text: proof

text: of

text: uniqueness

text: of

text: this

text: inverse

text: boundary

text: value

text: problem.

text: The

text: CGO

text: solutions

text: in

text: our

text: analysis

text: diﬀer

text: slightly

text: from

text: theirs

text: to

text: obtain

text: better

text: constants

text: in

text: the

text: stability

text: estimates

text: as

text: proposed

text: in

text: We

text: proceed

text: with

text: establishing

text: the

text: recovery

text: of

text: the

text: number

text: of

text: tetrahedra

text: in

text: the

text: mesh

text: from

text: the

text: potential,

text: and

text: with

text: expressing

text: the

text: Hausdorﬀ

text: distance

text: between

text: meshes

text: in

text: terms

text: of

text: the

text: diﬀerence

text: of

text: piecewise

text: constant

text: potentials

text: deﬁned

text: on

text: these

text: meshes.

text: Naturally,

text: the

text: information

text: on

text: the

text: Hausdorﬀ

text: distance

text: between

text: meshes

text: can

text: be

text: transformed

text: to

text: information

text: on

text: the

text: vertices

text: of

text: the

text: tetrahedra

text: forming

text: the

text: meshes

text: (Section

text: The

text: main

text: part

text: of

text: the

text: proof

text: of

text: our

text: result

text: pertains

text: to

text: obtaining

text: a

text: lower

text: bound

text: for

text: the

text: Gateaux

text: derivative

text: of

text: the

text: Dirichlet-to-Neumann

text: map

text: under

text: mesh

text: deformation

text: (Section

text: We

text: use

text: the

text: Fourier

text: transform

text: convention,

text: fˆ(ξ)

text: f(x)eix

text: ξdx.

text: R

text: If

text: the

text: function

text: f

text: is

text: deﬁned

text: on

text: a

text: subset

text: of

text: it

text: is

text: extended

text: to

text: attaining

text: the

text: value

text: zero.

text: We

text: denote

text: by

text: fˇ

text: the

text: inverse

text: Fourier

text: transform

text: of

text: f,

text: fˇ(x)

text: f(ξ)e

text: ix

text: ξdξ.

text: R

text: We

text: introduce

text: coordinates,

text: x

text: x

text: in

text: where

text: and

text: x

text: R.

text: We

text: denote

text: the

text: open

text: ball

text: in

text: centered

text: at

text: x

text: of

text: radius

text: r

text: B

text: (x),

text: and

text: the

text: open

text: ball

text: in

text: at

text: of

text: radius

text: r

text: by

text: r

text: B

text: (x

text: r

text: We

text: let

text: Ω

text: be

text: a

text: bounded

text: domain

text: in

text: such

text: that

text: Ω

text: is

text: connected,

text: Ω

text: B

text: for

text: some

text: R

text: R

text: and

text: Ω

text: has

text: a

text: Lipschitz

text: boundary

text: with

text: constants

text: r

text: and

text: K

text: ##LTLine##


text: ∩ ∅ (3.22) T(1) T(0) = k j,δσ with (cid:18) (cid:19) σ2 1/3 (3.23) δ = . σ c2C 0 1 ∈ B ∈ { } ∈ B ∈ Indeed, assume that j (0) for some l 1, . . . , N , k / (1) and that there is a point P ∩ l l T(0) T(1) such that j k ≥ (3.24) d(P, ∂T(0)) δ, j ⊂ that is, Bδ(P) Tj(0). Using as(cid:12)sumption ((cid:12)2.11(cid:12)) and (2.12) in(cid:12) Remark 1, it then follows that (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ∩ ≥ ∩ ≥ (3.25) T(0) T(1) B (P) T(1) C δ3 j k δ k 1 if δ < r . By (3.21) 1 ≤ σ2 (3.26) C δ3 . 1 c2 0 (cid:18) (cid:19) Thus (3.22) holds provided that σ2 1/3 ≤ δ = r , σ c2C 1 0 1 that is, (cid:113) ≤ (3.27) σ σ = r3c2C . 1 1 0 1 ≤ ∈ B { } Now we consider T(0) for σ σ and j (0) for some l. Since T(1) is a partition of Ω, we can write j,δσ 1 l k k (cid:32) (cid:33) (cid:91) ∩ M Tj(,0δ)σ = T(cid:91)j(,0δ)σ(cid:16) Tk(1)(cid:17) k=1 M ∩ = T(0) T(1) . j,δσ k k=1 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: k

text: j,δσ

text: with

text: σ2

text: δ

text: σ

text: B

text: B

text: Indeed,

text: assume

text: that

text: j

text: for

text: some

text: l

text: N

text: k

text: and

text: that

text: there

text: is

text: a

text: point

text: P

text: l

text: l

text: such

text: that

text: j

text: k

text: d(P,

text: δ,

text: j

text: that

text: is,

text: Bδ(P)

text: Using

text: and

text: Remark

text: it

text: then

text: follows

text: that

text: B

text: (P)

text: C

text: δ3

text: j

text: k

text: δ

text: k

text: if

text: δ

text: r

text: By

text: σ2

text: C

text: δ3

text: Thus

text: holds

text: provided

text: that

text: σ2

text: δ

text: r

text: σ

text: that

text: is,

text: σ

text: σ

text: B

text: Now

text: we

text: consider

text: for

text: σ

text: σ

text: and

text: j

text: for

text: some

text: l.

text: Since

text: is

text: a

text: partition

text: of

text: Ω,

text: we

text: can

text: write

text: j,δσ

text: l

text: k

text: k

text: M

text: M

text: j,δσ

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:12) ≤ (cid:12) ∈ with q˜0 = Q0 (cf. (2.13)). For(cid:12)(cid:12) ω ω(cid:12)(cid:12)1 and t [0, 1], we have ≤ ≤ 1 2 ω2q˜(t) ω 2Q λ (B ) = λ (Ω˜), 1 0 2 1 R 3 1 cf. (4.7) and (2.6), whence the Dirichlet problem (cid:26) ∆u + ω2q˜(t)u = 0 in Ω˜, (4.10) u = φ on ∂Ω˜, ∈ ∈ has a unique solution u H1(Ω˜) for every φ H1/2(∂Ω˜). Thus the one-parameter family of Dirichlet-to-Neumann maps, ∈ (4.11) Λ˜ = Λ , for t [0, 1] t q˜(t) L − (cid:107) (cid:107) is well deﬁned in (H1/2(∂Ω˜), H 1/2(∂Ω˜)). We denote the norm in this space by T . (cid:63)˜ (cid:26) ∈ (cid:26) To proceed, we take φ, ψ H1/2(∂Ω˜) and let u˜ and u˜ be the solutions to 0 1 ∆u˜ + ω2q˜(0)u˜ = 0 in Ω˜, ∆u˜ + ω2q˜(1)u˜ = 0 in Ω˜, 0 0 and 1 1 u˜ = φ on ∂Ω˜, u˜ = ψ on ∂Ω˜. 0 1 We then use Alessandrini’s identity and write (cid:90) (cid:90) (cid:104) − (cid:105) − − (Λ˜ Λ˜ )(φ), ψ = (q˜(1) q˜(0))u˜ u˜ dx = (q(1) q(0))u˜ u˜ dx 1 0 0 1 0 1 (cid:104) − | Ω˜ | (cid:105) ≤ (cid:107) − (cid:107) (cid:107) (cid:107)Ω (cid:107) (cid:107) = (Λ Λ )(u˜ ), u˜ Λ Λ u˜ u˜ . 1 0 0 ∂Ω 1 ∂Ω 1 0 (cid:63) 0 H1/2(∂Ω) 1 H1/2(∂Ω) Moreover, by trace and regularity estimates, we have (cid:107) (cid:107) ≤ (cid:107) (cid:107) ≤ (cid:107) (cid:107) u˜ C u˜ C u˜ for k = 0, 1, k H1/2(∂Ω) k H1(Ω˜) k H1/2(∂Ω˜) where C depends on the a priori data. We have then shown that (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) (4.12) Λ˜ Λ˜ C Λ Λ . 1 0 (cid:63)˜ 3 1 0 (cid:63) In this section, we give the proof of Lipschitz stability starting from the logarithmic estimate ob- tained in Corollary 3.3. We split the proof into three steps: First step. We show that for any pair of functions φ and ψ in H1/2(∂Ω˜), the function F (cid:104) (cid:105) (t, φ, ψ) = Λ˜ (φ), ψ t is diﬀerentiable. 16 ##LTLine## ##LTLine##

text: with

text: (cf.

text: ω

text: and

text: t

text: we

text: have

text: ω

text: λ

text: (B

text: λ

text: (Ω˜),

text: R

text: cf.

text: and

text: whence

text: the

text: Dirichlet

text: problem

text: ∆u

text: in

text: Ω˜,

text: u

text: φ

text: on

text: ∂Ω˜,

text: has

text: a

text: unique

text: solution

text: u

text: for

text: every

text: φ

text: Thus

text: the

text: one-parameter

text: family

text: of

text: Dirichlet-to-Neumann

text: maps,

text: Λ˜

text: Λ

text: for

text: t

text: t

text: q˜(t)

text: L

text: is

text: well

text: deﬁned

text: in

text: H

text: We

text: denote

text: the

text: norm

text: in

text: this

text: space

text: by

text: T

text: To

text: proceed,

text: we

text: take

text: φ,

text: ψ

text: and

text: let

text: u˜

text: and

text: u˜

text: be

text: the

text: solutions

text: to

text: ∆u˜

text: in

text: Ω˜,

text: ∆u˜

text: in

text: Ω˜,

text: and

text: u˜

text: φ

text: on

text: ∂Ω˜,

text: u˜

text: ψ

text: on

text: ∂Ω˜.

text: We

text: then

text: use

text: Alessandrini’s

text: identity

text: and

text: write

text: (Λ˜

text: Λ˜

text: )(φ),

text: ψ

text: u˜

text: dx

text: u˜

text: dx

text: Ω˜

text: (Λ

text: Λ

text: )(u˜

text: u˜

text: Λ

text: Λ

text: u˜

text: u˜

text: ∂Ω

text: ∂Ω

text: Moreover,

text: by

text: trace

text: and

text: regularity

text: estimates,

text: we

text: have

text: u˜

text: C

text: u˜

text: C

text: u˜

text: for

text: k

text: k

text: k

text: k

text: where

text: C

text: depends

text: on

text: the

text: a

text: priori

text: data.

text: We

text: have

text: then

text: shown

text: that

text: Λ˜

text: Λ˜

text: C

text: Λ

text: Λ

text: In

text: this

text: section,

text: we

text: give

text: the

text: proof

text: of

text: Lipschitz

text: stability

text: starting

text: from

text: the

text: logarithmic

text: estimate

text: ob-

text: tained

text: in

text: Corollary

text: We

text: split

text: the

text: proof

text: into

text: three

text: steps:

text: First

text: step.

text: We

text: show

text: that

text: for

text: any

text: pair

text: of

text: functions

text: φ

text: and

text: ψ

text: in

text: the

text: function

text: F

text: (t,

text: φ,

text: ψ)

text: Λ˜

text: (φ),

text: ψ

text: t

text: is

text: diﬀerentiable.

text: ##LTLine##

text: ##LTLine##


text: ∈ that is, for any point P ∂Ω, there exists a rigid transformation of coordinates under which P = 0 and ∩ { (cid:48) ∈ R | (cid:48)| | | } { (cid:48) | (cid:48)| (cid:48) } Ω (x , x ) 3 : x < r , x < K r = (x , x ) : x < r , x > ψ(x ) , 3 0 3 0 0 3 0 3 (cid:48) where ψ is a Lipschitz continuous (level set) function in B such that r0 (cid:107)∇ (cid:107) ≤ ψ(0) = 0 and ψ ∞ (cid:48) K . L (B ) 0 r0 We consider the boundary value problem for the Helmholtz equation, (cid:26) ∆u + ω2qu = 0 in Ω, (2.3) u = φ on ∂Ω ∈ for φ H1/2(∂Ω), and introduce the Dirichlet-to-Neumann map → − (2.4) Λ : H1/2 (∂Ω) H 1/2 (∂Ω) q according to (cid:12) (cid:12) (cid:12) (cid:12) → ∂u (2.5) φ Λ (φ) := . q ∂ν ∂Ω (cid:28) (cid:29) (cid:90) The normal derivative is deﬁned in the weak sense as ∂u | ∇ · ∇ − , ψ = ( u ψ ω2quψ)dx ∂ν ∂Ω Ω ∈ ∈ ∞ − for every ψ H1(Ω). In the above, q L (∂Ω) is identiﬁed with c 2 where c denotes the wavespeed. − The solution of (2.3) exists in H1 (Ω) and is unique if ω is not in the Dirichlet spectrum of q 1∆ on Ω. We introduce ω , ω such that 0 < ω < ω and 0 1 0 1 (cid:115) ≤ λ (B ) (2.6) ω 1 R , 1 2Q 0 − − where λ (B ) is the ﬁrst eigenvalue of ∆ on B . We recall that λ (B ) = λ (B )R 2. (If we 1 R R 1 R 1 1 detect the spectrum, we substitute the true ﬁrst eigenfrequency for ω .) We then assume that 1 ≤ ≤ (2.7) ω ω ω . 0 1 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: that

text: is,

text: for

text: any

text: point

text: P

text: ∂Ω,

text: there

text: exists

text: a

text: rigid

text: transformation

text: of

text: coordinates

text: under

text: which

text: P

text: and

text: R

text: Ω

text: (x

text: x

text: x

text: r

text: x

text: K

text: r

text: (x

text: x

text: x

text: r

text: x

text: ψ(x

text: where

text: ψ

text: is

text: a

text: Lipschitz

text: continuous

text: (level

text: set)

text: function

text: in

text: B

text: such

text: that

text: and

text: ψ

text: K

text: L

text: (B

text: We

text: consider

text: the

text: boundary

text: value

text: problem

text: for

text: the

text: Helmholtz

text: equation,

text: ∆u

text: in

text: Ω,

text: u

text: φ

text: on

text: ∂Ω

text: for

text: φ

text: and

text: introduce

text: the

text: Dirichlet-to-Neumann

text: map

text: Λ

text: (∂Ω)

text: H

text: (∂Ω)

text: q

text: according

text: to

text: ∂u

text: φ

text: Λ

text: (φ)

text: q

text: ∂ν

text: ∂Ω

text: The

text: normal

text: derivative

text: is

text: deﬁned

text: in

text: the

text: weak

text: sense

text: as

text: ∂u

text: ψ

text: u

text: ψ

text: ∂ν

text: ∂Ω

text: Ω

text: for

text: every

text: ψ

text: In

text: the

text: above,

text: q

text: L

text: (∂Ω)

text: is

text: identiﬁed

text: with

text: c

text: where

text: c

text: denotes

text: the

text: wavespeed.

text: The

text: solution

text: of

text: exists

text: in

text: (Ω)

text: and

text: is

text: unique

text: if

text: ω

text: is

text: not

text: in

text: the

text: Dirichlet

text: spectrum

text: of

text: q

text: 1∆

text: on

text: Ω.

text: We

text: introduce

text: ω

text: ω

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text: that

text: ω

text: ω

text: and

text: λ

text: (B

text: ω

text: R

text: where

text: λ

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text: is

text: the

text: ﬁrst

text: eigenvalue

text: of

text: on

text: B

text: We

text: recall

text: that

text: λ

text: (B

text: λ

text: (B

text: )R

text: (If

text: we

text: R

text: R

text: R

text: detect

text: the

text: spectrum,

text: we

text: substitute

text: the

text: true

text: ﬁrst

text: eigenfrequency

text: for

text: ω

text: We

text: then

text: assume

text: that

text: ω

text: ω

text: ω

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: ≥ where C = C(R, ω , ω , Q ). But then, for µ max(c , c ), 0 1 0 (cid:90) (cid:12) (cid:12)1 2 (cid:90) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:98) (cid:98) (cid:12) (cid:12) (cid:98) (cid:98) (cid:12) (cid:107) − (cid:107) (cid:18) − 2 (cid:19) − 2 q(0) q(1) 2 = (q(0) q(1))(ξ) dξ + (q(0) q(1))(ξ) dξ L2(Ω) | |≤ | | ξ ρ ξ >ρ ≤ (cid:90) (cid:12) (cid:107) − (cid:107)(cid:12) 1 (3.9) Cρ3 e8R(cid:12) (µ+ρ) Λ Λ (cid:12) 2 + (cid:12) (cid:98) (cid:98)0 1(cid:12) (cid:63) µ2 − 2 + (q(0) q(1))(ξ) dξ. | | ξ >ρ ∈ To estimate the integral in (3.9) we show that for every s (0, 1/2) (cid:112) (cid:107) − (cid:107) ≤ (3.10) q(0) q(1) 2 C N , Hs(Ω) 0 where C = C(R, r , Q ). Indeed, by [11] we have 0 0 (cid:16) (cid:17)   (cid:107) − (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) q(0) (cid:88)q(1) 2 2 q(0) 2 +(cid:88)q(1) 2  Hs(Ω) Hs(Ω) Hs(Ω)  ≤ N | | | | − | | M | | | | − | | 2 q(0) 2 T(0) 1 2s ∂T(0) 2s + q(1) 2 T(1) 1 2s ∂T(1) 2s j j j k k k j=1 k=1 ≤ CN , 0 where C = C(R, r , Q ). 0 0 Using (3.10), (cid:90) (cid:12)(cid:16) (cid:17) (cid:12) (cid:90) (cid:12)(cid:16) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:98) (cid:98) (cid:12) (cid:12) (cid:98) (cid:98) (cid:12) − 2 ≤ 1 | | − 2 q(0) q(1) (ξ) dξ (1 + ξ s)2 q(0) q(1) (ξ) dξ | | ρ2s | | ξ >ρ ξ >ρ ≤ 1 (cid:107) − (cid:107) ≤ CN (3.11) q(0) q(1) 2 0 . ρ2s Hs(Ω) ρ2s (cid:26) (cid:18) (cid:19) (cid:27) Finally, by inserting (3.11) into (3.9), we get that (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) 1 1 q(0) q(1) 2 CN ρ3 e8R(µ+ρ) Λ Λ 2 + + , L2(Ω) 0 0 1 (cid:63) µ2 ρ2s where C = C(R, r , ω , ω , Q ). We then choose 0 0 1 0 ρ = µ3+22s , ≥ and observe that there is a constant c depending only on R such that, for µ c , 3 3 ≤ ρ3e8R(µ+ρ) e18Rµ 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: C

text: C(R,

text: ω

text: ω

text: Q

text: But

text: then,

text: for

text: µ

text: max(c

text: c

text: dξ

text: dξ

text: ξ

text: ρ

text: ξ

text: >ρ

text: (µ+ρ)

text: Λ

text: Λ

text: µ2

text: dξ.

text: ξ

text: >ρ

text: To

text: estimate

text: the

text: integral

text: in

text: we

text: show

text: that

text: for

text: every

text: s

text: C

text: N

text: Hs(Ω)

text: where

text: C

text: C(R,

text: r

text: Q

text: Indeed,

text: by

text: we

text: have

text: Hs(Ω)

text: Hs(Ω)

text: Hs(Ω)

text: N

text: M

text: j

text: j

text: j

text: k

text: k

text: k

text: CN

text: where

text: C

text: C(R,

text: r

text: Q

text: Using

text: (ξ)

text: dξ

text: ξ

text: (ξ)

text: dξ

text: ξ

text: >ρ

text: ξ

text: >ρ

text: CN

text: Hs(Ω)

text: Finally,

text: by

text: inserting

text: into

text: we

text: get

text: that

text: CN

text: ρ3

text: Λ

text: Λ

text: µ2

text: where

text: C

text: C(R,

text: r

text: ω

text: ω

text: Q

text: We

text: then

text: choose

text: ρ

text: and

text: observe

text: that

text: there

text: is

text: a

text: constant

text: c

text: depending

text: only

text: on

text: R

text: such

text: that,

text: for

text: µ

text: c

text: ##LTLine##

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text: ##LTLine##


text: 10 0.3 100 0.1 MA/m Theory 25 m 0.3 MA/m µ Measurement 25 m 0.2 1.15 MMAA//mm Hz] Theory 50 µµm m] 0.1 uency [ 10 Measurement 50 µm mNTorque [ -0. 10 um rotation freq 1 m xi -0.2 Ma 0.1 0 100 200 300 400 500 600 -0.3 -80 -60 -40 -20 0 20 40 60 80 Object length [ m] Field angle [o] µ FIG. 11: Measured out-of-plane torque curves as a FIG. 13: Maximum rotation frequency as a function function of the ﬁeld angle for varying magnetic ﬁeld of object length for rectangles of 25 (red squares) µ strength. and 50 m (black circles) width. (Model points are connected by a line to guide the eye, see table I). 0.2 Theory Measured Torque 0.15 not caused by a diﬀerence in drag, which is domi- µ m] nated by the length of the element. The 25 m el- N/ ements however have magnetic strips with a much me [ 0.1 smaller width, leading to higher saturation ﬁelds and qu relatively low eﬀective ﬁelds (h in table I). We spec- or T ulate that the assumption of uniform magnetisation 0.05 might not hold for ﬁelds that only 15% of saturation ﬁeld. Comparing with ﬁelds applied out of the plane, we indeed observe hysteresis eﬀects for ﬁelds below 0 0 0.002 0.004 0.006 0.008 0.01 300 kA/m, which is 25% of the saturation ﬁeld. It 1/H [m/kA] is possible that the magnetisation in the strip splits up in domains, leading to a reduction in torque and increase in scatter due to hysteresis eﬀects. The elec- FIG. 12: Measured maximum out-of-plane torque as tromagnet in our setup was unfortunately not capable a function of magnetic ﬁeld strength compared to of generating higher rotating ﬁelds, so this hypothesis the modeled expected behaviour (solid line), which could not be veriﬁed. does not include any ﬁt parameter. C. Maximum torque of magnetotactic bacteria Figure 14 shows a typical trajectory of a magneto- tion, using drag coeﬃcient and magnetic torque as tactic bacterium under application of a rotating mag- listed in table I. Since the dimensions of the magnetic netic ﬁeld. In the sequence of images, the frequency elements is varying with object length, the theoreti- of rotation is increased at constant rate until the bac- cal maximum rotation frequency is only calculated for teria no longer follows the ﬁeld. In this way we es- speciﬁc elements, and connected by a line to guide the timate that that maximum rotation frequency lies in eye. the range fo 7 to 10 rad/s. For all measurements, the observed maximum ro- From SEM and TEM images of ﬁfteen magneto- tation frequency lies below the theoretical prediction. tactic bacteria (ﬁgure 9), we determined their dimen- This can only partly be explained by the measure- sions and the size, spacing and number of crystals ment procedure, in which the maximum frequency is in the magnetosomes. The minimum, maximum and approached from below. The diﬀerence between the average values are listed in table III. Applying the µ 25 and 50 m elements is however pronounced. The chain-of-spheres model to the magnetosome data re- µ 50 m elements show small scatter between experi- sults in ∆N values of 0.07 to 0.13. As a result torque ments, and for the larger element the diﬀerence with values ranging from 0.02 to 4 aNm, with an average µ theory is small. The 25 m elements however show of about 1 aNm at 7.9 mT. much more scatter, and diﬀer from theory by a ap- For the calculation of the ﬁeld dependent torque, proximately a factor of three. we assume that the magnetisation of the crystals is The diﬀerence between the two series is probably that of bulk magnetite (Fe O , 480 kA/m [22]). Us- 3 4 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: MA/m

text: Theory

text: m

text: MA/m

text: µ

text: Measurement

text: m

text: MMAA//mm

text: Hz]

text: Theory

text: µµm

text: m]

text: uency

text: Measurement

text: µm

text: mNTorque

text: um

text: rotation

text: freq

text: m

text: xi

text: Ma

text: Object

text: length

text: m]

text: Field

text: angle

text: [o]

text: µ

text: FIG.

text: Measured

text: out-of-plane

text: torque

text: curves

text: as

text: a

text: FIG.

text: Maximum

text: rotation

text: frequency

text: as

text: a

text: function

text: function

text: of

text: the

text: ﬁeld

text: angle

text: for

text: varying

text: magnetic

text: ﬁeld

text: of

text: object

text: length

text: for

text: rectangles

text: of

text: (red

text: squares)

text: µ

text: strength.

text: and

text: m

text: (black

text: circles)

text: width.

text: (Model

text: points

text: are

text: connected

text: by

text: a

text: line

text: to

text: guide

text: the

text: eye,

text: see

text: table

text: I).

text: Theory

text: Measured

text: Torque

text: not

text: caused

text: by

text: a

text: diﬀerence

text: in

text: drag,

text: which

text: is

text: domi-

text: µ

text: m]

text: nated

text: by

text: the

text: length

text: of

text: the

text: element.

text: The

text: m

text: el-

text: N/

text: ements

text: however

text: have

text: magnetic

text: strips

text: with

text: a

text: much

text: me

text: smaller

text: width,

text: leading

text: to

text: higher

text: saturation

text: ﬁelds

text: and

text: qu

text: relatively

text: low

text: eﬀective

text: ﬁelds

text: (h

text: in

text: table

text: I).

text: We

text: spec-

text: or

text: T

text: ulate

text: that

text: the

text: assumption

text: of

text: uniform

text: magnetisation

text: might

text: not

text: hold

text: for

text: ﬁelds

text: that

text: only

text: of

text: saturation

text: ﬁeld.

text: Comparing

text: with

text: ﬁelds

text: applied

text: out

text: of

text: the

text: plane,

text: we

text: indeed

text: observe

text: hysteresis

text: eﬀects

text: for

text: ﬁelds

text: below

text: kA/m,

text: which

text: is

text: of

text: the

text: saturation

text: ﬁeld.

text: It

text: [m/kA]

text: is

text: possible

text: that

text: the

text: magnetisation

text: in

text: the

text: strip

text: splits

text: up

text: in

text: domains,

text: leading

text: to

text: a

text: reduction

text: in

text: torque

text: and

text: increase

text: in

text: scatter

text: due

text: to

text: hysteresis

text: eﬀects.

text: The

text: elec-

text: FIG.

text: Measured

text: maximum

text: out-of-plane

text: torque

text: as

text: tromagnet

text: in

text: our

text: setup

text: was

text: unfortunately

text: not

text: capable

text: a

text: function

text: of

text: magnetic

text: ﬁeld

text: strength

text: compared

text: to

text: of

text: generating

text: higher

text: rotating

text: ﬁelds,

text: so

text: this

text: hypothesis

text: the

text: modeled

text: expected

text: behaviour

text: (solid

text: line),

text: which

text: could

text: not

text: be

text: veriﬁed.

text: does

text: not

text: include

text: any

text: ﬁt

text: parameter.

text: C.

text: Maximum

text: torque

text: of

text: magnetotactic

text: bacteria

text: Figure

text: shows

text: a

text: typical

text: trajectory

text: of

text: a

text: magneto-

text: tion,

text: using

text: drag

text: coeﬃcient

text: and

text: magnetic

text: torque

text: as

text: tactic

text: bacterium

text: under

text: application

text: of

text: a

text: rotating

text: mag-

text: listed

text: in

text: table

text: I.

text: Since

text: the

text: dimensions

text: of

text: the

text: magnetic

text: netic

text: ﬁeld.

text: In

text: the

text: sequence

text: of

text: images,

text: the

text: frequency

text: elements

text: is

text: varying

text: with

text: object

text: length,

text: the

text: theoreti-

text: of

text: rotation

text: is

text: increased

text: at

text: constant

text: rate

text: until

text: the

text: bac-

text: cal

text: maximum

text: rotation

text: frequency

text: is

text: only

text: calculated

text: for

text: teria

text: no

text: longer

text: follows

text: the

text: ﬁeld.

text: In

text: this

text: way

text: we

text: es-

text: speciﬁc

text: elements,

text: and

text: connected

text: by

text: a

text: line

text: to

text: guide

text: the

text: timate

text: that

text: that

text: maximum

text: rotation

text: frequency

text: lies

text: in

text: eye.

text: the

text: range

text: fo

text: to

text: rad/s.

text: For

text: all

text: measurements,

text: the

text: observed

text: maximum

text: ro-

text: From

text: SEM

text: and

text: TEM

text: images

text: of

text: ﬁfteen

text: magneto-

text: tation

text: frequency

text: lies

text: below

text: the

text: theoretical

text: prediction.

text: tactic

text: bacteria

text: (ﬁgure

text: we

text: determined

text: their

text: dimen-

text: This

text: can

text: only

text: partly

text: be

text: explained

text: by

text: the

text: measure-

text: sions

text: and

text: the

text: size,

text: spacing

text: and

text: number

text: of

text: crystals

text: ment

text: procedure,

text: in

text: which

text: the

text: maximum

text: frequency

text: is

text: in

text: the

text: magnetosomes.

text: The

text: minimum,

text: maximum

text: and

text: approached

text: from

text: below.

text: The

text: diﬀerence

text: between

text: the

text: average

text: values

text: are

text: listed

text: in

text: table

text: III.

text: Applying

text: the

text: µ

text: and

text: m

text: elements

text: is

text: however

text: pronounced.

text: The

text: chain-of-spheres

text: model

text: to

text: the

text: magnetosome

text: data

text: re-

text: µ

text: m

text: elements

text: show

text: small

text: scatter

text: between

text: experi-

text: sults

text: in

text: ∆N

text: values

text: of

text: to

text: As

text: a

text: result

text: torque

text: ments,

text: and

text: for

text: the

text: larger

text: element

text: the

text: diﬀerence

text: with

text: values

text: ranging

text: from

text: to

text: aNm,

text: with

text: an

text: average

text: µ

text: theory

text: is

text: small.

text: The

text: m

text: elements

text: however

text: show

text: of

text: about

text: aNm

text: at

text: mT.

text: much

text: more

text: scatter,

text: and

text: diﬀer

text: from

text: theory

text: by

text: a

text: ap-

text: For

text: the

text: calculation

text: of

text: the

text: ﬁeld

text: dependent

text: torque,

text: proximately

text: a

text: factor

text: of

text: three.

text: we

text: assume

text: that

text: the

text: magnetisation

text: of

text: the

text: crystals

text: is

text: The

text: diﬀerence

text: between

text: the

text: two

text: series

text: is

text: probably

text: that

text: of

text: bulk

text: magnetite

text: (Fe

text: O

text: kA/m

text: Us-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTLine##

text: ##LTLine##


text: DIRECTIONAL PRECONDITIONER FOR HIGH FREQUENCY OBSTACLE SCATTERING LEXING YING Abstract. The boundary integral method is an eﬃcient approach for solving time-harmonic obstacle scattering problems by a bounded scat- terer. This paper presents the directional preconditioner for the itera- tive solution of linear systems of the boundary integral method. This new preconditioner builds a data-sparse approximation of the integral operator, transforms it into a sparse linear system, and computes an approximate inverse with eﬃcient sparse and hierarchical linear algebra algorithms. This preconditioner is eﬃcient and results in small and al- most frequency-independent iteration counts when combined with stan- dard iterative solvers. Numerical results are provided to demonstrate the eﬀectiveness of the new preconditioner. 1. Introduction This paper is concerned with solving the time-harmonic acoustic obstacle ⊂ scattering problems in two dimensions. Let Ω R2 be a bounded scatterer with smooth boundary ∂Ω, ω be the frequency, and eiωtu (x) be the time- I harmonic incident wave. In the sound-soft scattering problem, the scattered ﬁeld u(x) satisﬁes the Helmholtz equation with the Dirichlet boundary con- dition ∈ \ ∆u(x) + ω2u(x) = 0, x R2 Ω, − ∈ u(x) = u (x), x ∂Ω. I In the sound-hard scattering problem, the scattered ﬁeld u(x) satisﬁes the Helmholtz equation with the Neumann boundary condition ∈ \ ∆u(x) + ω2u(x) = 0, x R2 Ω, ∂u(x) −∂u (x) ∈ I = , x ∂Ω. ∂n(x) ∂n(x) 2010 Mathematics Subject Classiﬁcation. 65N38, 65R20, 78A45. Key words and phrases. Boundary integral method, scattering, high-frequency waves, preconditioner, low-rank approximation, sparse linear algebra. This work was partially supported by the National Science Foundation under award DMS-0846501 and the U.S. Department of Energy’s Advanced Scientiﬁc Computing Re- search program under award DE-FC02-13ER26134/DE-SC0009409. The author thanks Anil Damle for comments and suggestions. 1 ##LTLine## ##LTLine## ##LTLine##

text: DIRECTIONAL

text: PRECONDITIONER

text: FOR

text: HIGH

text: FREQUENCY

text: OBSTACLE

text: SCATTERING

text: LEXING

text: YING

text: Abstract.

text: The

label: abstract

text: boundary

text: integral

text: method

text: is

text: an

text: eﬃcient

text: approach

text: for

text: solving

text: time-harmonic

text: obstacle

text: scattering

text: problems

text: by

text: a

text: bounded

text: scat-

text: terer.

text: This

text: paper

text: presents

text: the

text: directional

text: preconditioner

text: for

text: the

text: itera-

text: tive

text: solution

text: of

text: linear

text: systems

text: of

text: the

text: boundary

text: integral

text: method.

text: This

text: new

text: preconditioner

text: builds

text: a

text: data-sparse

text: approximation

text: of

text: the

text: integral

text: operator,

text: transforms

text: it

text: into

text: a

text: sparse

text: linear

text: system,

text: and

text: computes

text: an

text: approximate

text: inverse

text: with

text: eﬃcient

text: sparse

text: and

text: hierarchical

text: linear

text: algebra

text: algorithms.

text: This

text: preconditioner

text: is

text: eﬃcient

text: and

text: results

text: in

text: small

text: and

text: al-

text: most

text: frequency-independent

text: iteration

text: counts

text: when

text: combined

text: with

text: stan-

text: dard

text: iterative

text: solvers.

text: Numerical

text: results

text: are

text: provided

text: to

text: demonstrate

text: the

text: eﬀectiveness

text: of

text: the

text: new

text: preconditioner.

text: Introduction

text: This

text: paper

text: is

text: concerned

text: with

text: solving

text: the

text: time-harmonic

text: acoustic

text: obstacle

text: scattering

text: problems

text: in

text: two

text: dimensions.

text: Let

text: Ω

text: be

text: a

text: bounded

text: scatterer

text: with

text: smooth

text: boundary

text: ∂Ω,

text: ω

text: be

text: the

text: frequency,

text: and

text: eiωtu

text: (x)

text: be

text: the

text: time-

text: I

text: harmonic

text: incident

text: wave.

text: In

text: the

text: sound-soft

text: scattering

text: problem,

text: the

text: scattered

text: ﬁeld

text: u(x)

text: satisﬁes

text: the

text: Helmholtz

text: equation

text: with

text: the

text: Dirichlet

text: boundary

text: con-

text: dition

text: ∆u(x)

text: x

text: Ω,

text: u(x)

text: u

text: (x),

text: x

text: ∂Ω.

text: I

text: In

text: the

text: sound-hard

text: scattering

text: problem,

text: the

text: scattered

text: ﬁeld

text: u(x)

text: satisﬁes

text: the

text: Helmholtz

text: equation

text: with

text: the

text: Neumann

text: boundary

text: condition

text: ∆u(x)

text: x

text: Ω,

text: ∂u(x)

text: −∂u

text: (x)

text: I

text: x

text: ∂Ω.

text: ∂n(x)

text: ∂n(x)

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Key

text: words

text: and

text: phrases.

text: Boundary

text: integral

text: method,

text: scattering,

text: high-frequency

text: waves,

text: preconditioner,

text: low-rank

text: approximation,

text: sparse

text: linear

text: algebra.

text: This

text: work

text: was

text: partially

text: supported

text: by

text: the

text: National

text: Science

text: Foundation

text: under

text: award

text: and

text: the

text: U.S.

text: Department

text: of

text: Energy’s

text: Advanced

text: Scientiﬁc

text: Computing

text: Re-

text: search

text: program

text: under

text: award

text: The

text: author

text: thanks

text: Anil

text: Damle

text: for

text: comments

text: and

text: suggestions.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:78) { } Given (cid:96) in 1, . . . , p , Lemma 4.7 ensures that we may apply Lemma 4.6 to tthheis myiaepldsw and to the measures with density(cid:79)2i(cid:96)1Ai(cid:96) and (cid:96)(cid:48)(cid:54)=(cid:96) 2i(cid:96)(cid:48)1Ai(cid:48)(cid:96) and δ −d2+σ+O((cid:15)) (cid:28) (cid:107)2i(cid:96)1Ai(cid:96)(cid:107)(cid:89)2 · (cid:107) (cid:54) (cid:48) 2i(cid:96)(cid:48)1Ai(cid:96)(cid:48) (cid:107)1 (cid:96)=(cid:96) | | | | = 2i(cid:96) Ai(cid:96) 12 (cid:48)(cid:54) 2i(cid:96)(cid:48) Ai(cid:96)(cid:48) . (cid:96) =(cid:96) Using also that the deﬁnition of the A ’s implies that i | | (cid:28) | | (cid:28) (cid:107) (cid:107) 2i Ai 1 and 2i Ai 12 µδ 2, the above forces ≥ − ∀ (cid:48) (cid:54) | | ≥ 2i(cid:96)/2 δ d2+σ+O((cid:15)) and (cid:96) = (cid:96), 2i(cid:96)(cid:48) Ai (cid:48) δO((cid:15)). (cid:96) This must hold for each (cid:96), and therefore, for each (cid:96), − | | 2i(cid:96) = δ d+σ+O((cid:15)) and 2i(cid:96) Ai(cid:96) = δO((cid:15)). (11) As the set A is a union of ball of radius δ, this shows that i(cid:96) (cid:29) − | | ≥ − N(A , δ) δ d A δ σ+O((cid:15)). i(cid:96) i(cid:96) ≥ Moreover, as the measure µ is σ-Frostman, we have, for all ρ δ, | ∩ | (cid:28) (cid:28) 2i(cid:96) Ai(cid:96) B(x, ρ) µ(B(x, 4ρ)) ρσ, whence ∩ (cid:28) − | ∩ | ≤ − N(A B(x, ρ), δ) δ d A B(x, ρ) ρσδ O((cid:15))N(A , δ). i(cid:96) i(cid:96) i(cid:96) Thus, each A is a (σ, O((cid:15)))-set at scal(cid:91)e δ, and therefore, so is i(cid:96) p A := A . i(cid:96) (cid:96)=1 ⊗ · · · ⊗ NOonwonleethϕanbde,tbhye (d1e0n)s,itwyefhuanvcetion of the measure w∗(2i11Ai1 2ip1Aip). (cid:107) (cid:107) − ϕ 2 = δ d+σ+O((cid:15)). 2 ∗ On the other hand, µ is σ-Frostman and ν can be written ν µ for some 1 probability measure ν , so that ν is also σ-Frostman, which implies 1 (cid:107) (cid:107) ≤ − ϕ ∞ δ d+σ. 25 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Given

text: in

text: p

text: Lemma

text: ensures

text: that

text: we

text: may

text: apply

text: Lemma

text: to

text: tthheis

text: myiaepldsw

text: and

text: to

text: the

text: measures

text: with

text: and

text: and

text: δ

text: Using

text: also

text: that

text: the

text: deﬁnition

text: of

text: the

text: A

text: ’s

text: implies

text: that

text: i

text: Ai

text: and

text: Ai

text: µδ

text: the

text: above

text: forces

text: δ

text: and

text: Ai

text: This

text: must

text: hold

text: for

text: each

text: and

text: therefore,

text: for

text: each

text: δ

text: and

text: As

text: the

text: set

text: A

text: is

text: a

text: union

text: of

text: ball

text: of

text: radius

text: δ,

text: this

text: shows

text: that

text: N(A

text: δ)

text: δ

text: d

text: A

text: δ

text: Moreover,

text: as

text: the

text: measure

text: µ

text: is

text: σ-Frostman,

text: we

text: have,

text: for

text: all

text: ρ

text: δ,

text: B(x,

text: ρ)

text: µ(B(x,

text: ρσ,

text: whence

text: N(A

text: B(x,

text: ρ),

text: δ)

text: δ

text: d

text: A

text: B(x,

text: ρ)

text: ρσδ

text: δ).

text: Thus,

text: each

text: A

text: is

text: a

text: (σ,

text: at

text: δ,

text: and

text: therefore,

text: so

text: is

text: p

text: A

text: A

text: NOonwonleethϕanbde,tbhye

text: of

text: the

text: measure

text: ϕ

text: δ

text: On

text: the

text: other

text: hand,

text: µ

text: is

text: σ-Frostman

text: and

text: ν

text: can

text: be

text: written

text: ν

text: µ

text: for

text: some

text: probability

text: measure

text: ν

text: so

text: that

text: ν

text: is

text: also

text: σ-Frostman,

text: which

text: implies

text: ϕ

text: δ

text: d+σ.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 E. BOLKER, J. HATCH, AND C. ZARA Our model is particularly straightforward mathematically. For some questions all you need for good bounds or even exact answers (to the model) is a calculator and the number and size of turbines. For others an Excel spreadsheet (which we provide) does the job. One important insight to draw from our geometric model is that birds passing through the wind farm turbine height, may encounter surprisingly few turbines and that this number is probably greatly re- duced by avoidance. Chamberlain et. al [6] show that Band model predictions are much more sensitive to errors in estimating avoidance behavior than to equivalent errors in all other input parameters. Birds may act to avoid both whole turbines individual blades, so may lower both the average number of turbines encountered and the encounter mortality probability. Since we assume no active avoidance, our mor- tality estimates are likely to conservative – that is, too high. Our model is generic – it accepts turbine coordinates as input. In this paper we apply it to a simple example that makes the geometry and mathematics clear. In Section we report on several studies that use it in real situations. 2. The basic model We assume that each bird follows a path T when it ﬂies at a constant speed, height and heading (direction) across an area containing a wind farm. We want to compute the mortality probability M(T ) that the bird fails to survive its passage through the wind farm. To that end, let E(T ) be the number of turbines the bird encounters in its travel along T . Let p be the probability of safe passage through one turbine, and assume that surviving turbine encounters are independent events. Then the probability of surviving all the encounters is pE(T) so − M(T ) = 1 pE(T). The price for this simple computation is the unrealistic assumptions we need to justify it. The ﬁrst is our requirement that paths be straight lines. The second is our use of a single value p for the survival proba- bility for any single encounter. In fact the value of p depends not only on the geometry of the encounter, thus on the size of the bird, on the speed at which it crosses through a revolving rotor, on the angle the ﬂight path makes with the vertical plane of the turbine and the distance from the center of the turbine at which the encounter takes place, but also on the active avoidance behavior of the bird. So to use the model in any particular case you must decide on an appropriate average value for p or, to compute an upper bound on mortality, a minimum value.

text: E.

text: BOLKER,

text: J.

text: HATCH,

text: AND

text: C.

text: ZARA

text: Our

text: model

text: is

text: particularly

text: straightforward

text: mathematically.

text: For

text: some

text: questions

text: all

text: you

text: need

text: for

text: good

text: bounds

text: or

text: even

text: exact

text: answers

text: (to

text: the

text: model)

text: is

text: a

text: calculator

text: and

text: the

text: number

text: and

text: size

text: of

text: turbines.

text: For

text: others

text: an

text: Excel

text: spreadsheet

text: (which

text: we

text: provide)

text: does

text: the

text: job.

text: One

text: important

text: insight

text: to

text: draw

text: from

text: our

text: geometric

text: model

text: is

text: that

text: birds

text: passing

text: through

text: the

text: wind

text: farm

text: turbine

text: height,

text: may

text: encounter

text: surprisingly

text: few

text: turbines

text: and

text: that

text: this

text: number

text: is

text: probably

text: greatly

text: re-

text: duced

text: by

text: avoidance.

text: Chamberlain

text: et.

text: al

text: show

text: that

text: Band

text: model

text: predictions

text: are

text: much

text: more

text: sensitive

text: to

text: errors

text: in

text: estimating

text: avoidance

text: behavior

text: than

text: to

text: equivalent

text: errors

text: in

text: all

text: other

text: input

text: parameters.

text: Birds

text: may

text: act

text: to

text: avoid

text: both

text: whole

text: turbines

text: individual

text: blades,

text: so

text: may

text: lower

text: both

text: the

text: average

text: number

text: of

text: turbines

text: encountered

text: and

text: the

text: encounter

text: mortality

text: probability.

text: Since

text: we

text: assume

text: no

text: active

text: avoidance,

text: our

text: mor-

text: tality

text: estimates

text: are

text: likely

text: to

text: conservative

text: that

text: is,

text: too

text: high.

text: Our

text: model

text: is

text: generic

text: it

text: accepts

text: turbine

text: coordinates

text: as

text: input.

text: In

text: this

text: paper

text: we

text: apply

text: it

text: to

text: a

text: simple

text: example

text: that

text: makes

text: the

text: geometry

text: and

text: mathematics

text: clear.

text: In

text: Section

text: we

text: report

text: on

text: several

text: studies

text: that

text: use

text: it

text: in

text: real

text: situations.

text: The

text: basic

text: model

text: We

text: assume

text: that

text: each

text: bird

text: follows

text: a

text: path

text: T

text: when

text: it

text: ﬂies

text: at

text: a

text: constant

text: speed,

text: height

text: and

text: heading

text: (direction)

text: across

text: an

text: area

text: containing

text: a

text: wind

text: farm.

text: We

text: want

text: to

text: compute

text: the

text: mortality

text: probability

text: M(T

text: that

text: the

text: bird

text: fails

text: to

text: survive

text: its

text: passage

text: through

text: the

text: wind

text: farm.

text: To

text: that

text: end,

text: let

text: E(T

text: be

text: the

text: number

text: of

text: turbines

text: the

text: bird

text: encounters

text: in

text: its

text: travel

text: along

text: T

text: Let

text: p

text: be

text: the

text: probability

text: of

text: safe

text: passage

text: through

text: one

text: turbine,

text: and

text: assume

text: that

text: surviving

text: turbine

text: encounters

text: are

text: independent

text: events.

text: Then

text: the

text: probability

text: of

text: surviving

text: all

text: the

text: encounters

text: is

text: pE(T)

text: so

text: M(T

text: pE(T).

text: The

text: price

text: for

text: this

text: simple

text: computation

text: is

text: the

text: unrealistic

text: assumptions

text: we

text: need

text: to

text: justify

text: it.

text: The

text: ﬁrst

text: is

text: our

text: requirement

text: that

text: paths

text: be

text: straight

text: lines.

text: The

text: second

text: is

text: our

text: use

text: of

text: a

text: single

text: value

text: p

text: for

text: the

text: survival

text: proba-

text: bility

text: for

text: any

text: single

text: encounter.

text: In

text: fact

text: the

text: value

text: of

text: p

text: depends

text: not

text: only

text: on

text: the

text: geometry

text: of

text: the

text: encounter,

text: thus

text: on

text: the

text: size

text: of

text: the

text: bird,

text: on

text: the

text: speed

text: at

text: which

text: it

text: crosses

text: through

text: a

text: revolving

text: rotor,

text: on

text: the

text: angle

text: the

text: ﬂight

text: path

text: makes

text: with

text: the

text: vertical

text: plane

text: of

text: the

text: turbine

text: and

text: the

text: distance

text: from

text: the

text: center

text: of

text: the

text: turbine

text: at

text: which

text: the

text: encounter

text: takes

text: place,

text: but

text: also

text: on

text: the

text: active

text: avoidance

text: behavior

text: of

text: the

text: bird.

text: So

text: to

text: use

text: the

text: model

text: in

text: any

text: particular

text: case

text: you

text: must

text: decide

text: on

text: an

text: appropriate

text: average

text: value

text: for

text: p

text: or,

text: to

text: compute

text: an

text: upper

text: bound

text: on

text: mortality,

text: a

text: minimum

text: value.


text: 14 E. BOLKER, J. HATCH, AND C. ZARA based on the model outputs, including the use of a formal uncertainty analysis. Nicolas Vanermen and Eric W.M. Stienen [14] studied bird mortality for a proposed wind farm in the Belgian Part of the North Sea. They used our model to ﬁnd a worst case estimate of the number of turbines encountered. Appendix A. Algorithms Here we provide an algorithm to evaluate the integral in Equation 6, which is the formal statement of the numerator in Equation 1. It’s an analogue of Theorem 1, true for some arrangements of targets on a dart board – fortunately, the ones we are interested in. Let D , D , . . . , D be a sequence of disks in the plane with the same 1 2 N radius and collinear centers, arranged in numerical order along the line of centers. Then for each i, ⊇ ∩ ⊇ ∩ (7) D D D D D . . . . i i i+1 i i+2 This is just what you can see in Figure 2. It’s also true for the shaded regions in Figure 4. It’s exactly what we need for the next theorem. Theorem 6. Let D , D , . . . , D be a sequence of plane regions for 1 2 N which Equation 7 is true. Let χ be the characteristic function of D , ∪ ∪ · · · ∪ i i X = D D (cid:90) D and X = X Then i i i+1 N (cid:88) (cid:88)1 (cid:80) N N (8) p Ni=1 χi(x)dx = λj−i(p)Aij , X i=1 j=i ∩  where A = Area(D D ) and ij i j   p, if r = 0, − (9) λ = λ (p) = p2 2p, if r = 1, r r − − (cid:62) pr 1(p 1)2, if r 2 .

text: E.

text: BOLKER,

text: J.

text: HATCH,

text: AND

text: C.

text: ZARA

text: based

text: on

text: the

text: model

text: outputs,

text: including

text: the

text: use

text: of

text: a

text: formal

text: uncertainty

text: analysis.

text: Nicolas

text: Vanermen

text: and

text: Eric

text: W.M.

text: Stienen

text: studied

text: bird

text: mortality

text: for

text: a

text: proposed

text: wind

text: farm

text: in

text: the

text: Belgian

text: Part

text: of

text: the

text: North

text: Sea.

text: They

text: used

text: our

text: model

text: to

text: ﬁnd

text: a

text: worst

text: case

text: estimate

text: of

text: the

text: number

text: of

text: turbines

text: encountered.

text: Appendix

label: section

text: A.

text: Algorithms

text: Here

text: we

text: provide

text: an

text: algorithm

text: to

text: evaluate

text: the

text: integral

text: in

text: Equation

text: which

text: is

text: the

text: formal

text: statement

text: of

text: the

text: numerator

text: in

text: Equation

text: It’s

text: an

text: analogue

text: of

text: Theorem

text: true

text: for

text: some

text: arrangements

text: of

text: targets

text: on

text: a

text: dart

text: board

text: fortunately,

text: the

text: ones

text: we

text: are

text: interested

text: in.

text: Let

text: D

text: D

text: D

text: be

text: a

text: sequence

text: of

text: disks

text: in

text: the

text: plane

text: with

text: the

text: same

text: N

text: radius

text: and

text: collinear

text: centers,

text: arranged

text: in

text: numerical

text: order

text: along

text: the

text: line

text: of

text: centers.

text: Then

text: for

text: each

text: i,

text: D

text: D

text: D

text: D

text: D

text: i

text: i

text: i

text: This

text: is

text: just

text: what

text: you

text: can

text: see

text: in

text: Figure

text: It’s

text: also

text: true

text: for

text: the

text: shaded

text: regions

text: in

text: Figure

text: It’s

text: exactly

text: what

text: we

text: need

text: for

text: the

text: next

text: theorem.

text: Theorem

text: Let

text: D

text: D

text: D

text: be

text: a

text: sequence

text: of

text: plane

text: regions

text: for

text: N

text: which

text: Equation

text: is

text: true.

text: Let

text: χ

text: be

text: the

text: characteristic

text: function

text: of

text: D

text: i

text: i

text: X

text: D

text: D

text: D

text: and

text: X

text: X

text: Then

text: i

text: i

text: N

text: N

text: N

text: p

text: χi(x)dx

text: λj−i(p)Aij

text: X

text: j=i

text: where

text: A

text: Area(D

text: D

text: and

text: ij

text: i

text: j

text: p,

text: if

text: r

text: λ

text: λ

text: (p)

text: if

text: r

text: r

text: r

text: pr

text: if

text: r


text: WINDFARM GEOMETRY 15 (cid:90) (cid:90) Proof. (cid:80) (cid:80) − − p Ni=1 χi(x) dx p Ni=2 χi(x) dx = p(A11 A12) X −X2 − − − + (p2 p)(A A ) + (p3 p2)(A A ) 12 13 13 14 · · · − − − − − − + + (pk 1 pk 2)(A − A ) + (pN pN 1)A 1,k 1 1,k 1,N − − (cid:88)· · · = pA + (p2 2p)A + (p3 2p2 + p)A + 11 12 13 · · · N = λ A + λ A + λ A + + λ − A = λ − A . 0 11 1 12 2 13 k 1 1N j 1 1j j=1 (cid:90) (cid:90) (cid:88) Similarly (cid:80) (cid:80) − N (cid:90) p Ni=2 χi(x)dx (cid:90) p Ni=3 χi(x)dx = (cid:88) λj−2A2j X2 (cid:80) X3 (cid:80) j=2 − N p Ni=3 χi(x)dx p Ni=4 χi(x)dx = λj−3A3j X3 X4 j=3 (cid:90) . .(cid:88) . . . . (cid:80) N p Ni=N χi(x)dx = λj−NANj XN j=N (cid:3) Adding these telescoping equations leads to (8). The last piece is the computation of the area of the intersections of ∩ the regions D D . The D are circles when the ﬂight direction is i j i up or downwind. When the ﬂight is at an angle to the wind, the D i are ellipses obtained by compressing circles along a diameter, with the same compression factor for each. Then we can ﬁnd the areas of the intersections by decompressing to circles, ﬁnding those areas and then compressing the results. Suppose C and C are circles of radius R, with centers O and 1 2 (cid:54) 1 O situated at a distance O O = 2d 2R. Let D and D be the 2 1 2 1 2 segments of the disks bounded by lines parallel to the line of the centers, at distance a and b from the line of centers; these distances are positive if the line is above the line of centers and negative otherwise. The ∩ area of the intersection D D can be computed as follows using the 1 2 elementary geometry illustrated in Figure 9.

text: WINDFARM

text: GEOMETRY

text: Proof.

text: p

text: χi(x)

text: dx

text: p

text: χi(x)

text: dx

text: X

text: p)(A

text: A

text: A

text: (pk

text: pk

text: A

text: (pN

text: pN

text: pA

text: p)A

text: N

text: λ

text: A

text: λ

text: A

text: λ

text: A

text: λ

text: A

text: λ

text: A

text: k

text: j

text: Similarly

text: N

text: p

text: χi(x)dx

text: p

text: χi(x)dx

text: N

text: p

text: χi(x)dx

text: p

text: χi(x)dx

text: N

text: p

text: Ni=N

text: χi(x)dx

text: λj−NANj

text: XN

text: j=N

text: Adding

text: these

text: telescoping

text: equations

text: leads

text: to

text: The

text: last

text: piece

text: is

text: the

text: computation

text: of

text: the

text: area

text: of

text: the

text: intersections

text: of

text: the

text: regions

text: D

text: D

text: The

text: D

text: are

text: circles

text: when

text: the

text: ﬂight

text: direction

text: is

text: i

text: j

text: i

text: up

text: or

text: downwind.

text: When

text: the

text: ﬂight

text: is

text: at

text: an

text: angle

text: to

text: the

text: wind,

text: the

text: D

text: i

text: are

text: ellipses

text: obtained

text: by

text: compressing

text: circles

text: along

text: a

text: diameter,

text: with

text: the

text: same

text: compression

text: factor

text: for

text: each.

text: Then

text: we

text: can

text: ﬁnd

text: the

text: areas

text: of

text: the

text: intersections

text: by

text: decompressing

text: to

text: circles,

text: ﬁnding

text: those

text: areas

text: and

text: then

text: compressing

text: the

text: results.

text: Suppose

text: C

text: and

text: C

text: are

text: circles

text: of

text: radius

text: R,

text: with

text: centers

text: O

text: and

text: O

text: situated

text: at

text: a

text: distance

text: O

text: O

text: Let

text: D

text: and

text: D

text: be

text: the

text: segments

text: of

text: the

text: disks

text: bounded

text: by

text: lines

text: parallel

text: to

text: the

text: line

text: of

text: the

text: centers,

text: at

text: distance

text: a

text: and

text: b

text: from

text: the

text: line

text: of

text: centers;

text: these

text: distances

text: are

text: positive

text: if

text: the

text: line

text: is

text: above

text: the

text: line

text: of

text: centers

text: and

text: negative

text: otherwise.

text: The

text: area

text: of

text: the

text: intersection

text: D

text: D

text: can

text: be

text: computed

text: as

text: follows

text: using

text: the

text: elementary

text: geometry

text: illustrated

text: in

text: Figure


text: 6 E. BOLKER, J. HATCH, AND C. ZARA independent of turbine placement. For ﬂights that cross the small circle NπB2 NπB ¯ ¯ (3) E = E(θ) = = (2B)(2R) 4R independent both of θ and of turbine placement. Imagine an onshore windfarm: N turbines with centers D meters apart in a line along a ridge perpendicular to the prevailing wind. For ﬂights with or against the wind, each bird meets no turbines, or just ≈ ≈ one. Since 2R L(θ) ND, NπB ≈ NπB π × B ¯ E(θ) = = . 2L(θ) 2ND 2 D ¯ When B/D is on the order of 1/15 to 1/7 ([11]), E is on the order of 0.1 to 0.2. Between 10 and 20 percent of the birds encounter a turbine. ¯ The average E is likely to be less than one for oﬀshore windfarms as well, even though turbines tend to be bunched and birds can meet (cid:112) more than one. To see why, imagine N turbines arranged on a square grid and roughly ﬁlling a circle. Then the radius R of the circle will be approximately N/πD, where D is the distance between turbine centers along grid lines. Then Equation 3 implies (cid:112) √ ≈ NπB π3/2 N × B ¯ (4) E = . 4 N/πD 4 D ¯ For B/D = 1/10, E < 1 when N < 52. On average, a bird encoun- ¯ ters less than one turbine. For B/D = 1/20, E < 1 when N < 207. You can conﬁrm that using the Circle worksheet in the spreadsheet. The Cape Wind installation will have 130 turbines in an area roughly ≈ a rectangle twice as wide as high, with B/D 1/16. If they were ≈ ¯ packed in a circle Equation 4 would yield E 0.85. The spreadsheet ≈ ¯ calculations show E 0.6 for the actual positions of the turbines. ¯ Unfortunately, M is usually harder to come by. One case is easy. Theorem 3. When birds encounter at most one turbine on bearing θ − ¯ ¯ (5) M(θ) = (1 p)E(θ). ¯ Proof. The probability that a bird encounters a turbine is E(θ). If it − (cid:3) does, it dies with probability (1 p). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: E.

text: BOLKER,

text: J.

text: HATCH,

text: AND

text: C.

text: ZARA

text: independent

text: of

text: turbine

text: placement.

text: For

text: ﬂights

text: that

text: cross

text: the

text: small

text: circle

text: NπB

text: E

text: E(θ)

text: independent

text: both

text: of

text: θ

text: and

text: of

text: turbine

text: placement.

text: Imagine

text: an

text: onshore

text: windfarm:

text: N

text: turbines

text: with

text: centers

text: D

text: meters

text: apart

text: in

text: a

text: line

text: along

text: a

text: ridge

text: perpendicular

text: to

text: the

text: prevailing

text: wind.

text: For

text: ﬂights

text: with

text: or

text: against

text: the

text: wind,

text: each

text: bird

text: meets

text: no

text: turbines,

text: or

text: just

text: one.

text: Since

text: L(θ)

text: ND,

text: NπB

text: NπB

text: π

text: B

text: E(θ)

text: D

text: When

text: B/D

text: is

text: on

text: the

text: order

text: of

text: to

text: E

text: is

text: on

text: the

text: order

text: of

text: to

text: Between

text: and

text: percent

text: of

text: the

text: birds

text: encounter

text: a

text: turbine.

text: The

text: average

text: E

text: is

text: likely

text: to

text: be

text: less

text: than

text: one

text: for

text: oﬀshore

text: windfarms

text: as

text: well,

text: even

text: though

text: turbines

text: tend

text: to

text: be

text: bunched

text: and

text: birds

text: can

text: meet

text: more

text: than

text: one.

text: To

text: see

text: why,

text: imagine

text: N

text: turbines

text: arranged

text: on

text: a

text: square

text: grid

text: and

text: roughly

text: ﬁlling

text: a

text: circle.

text: Then

text: the

text: radius

text: R

text: of

text: the

text: circle

text: will

text: be

text: approximately

text: N/πD,

text: where

text: D

text: is

text: the

text: distance

text: between

text: turbine

text: centers

text: along

text: grid

text: lines.

text: Then

text: Equation

text: implies

text: NπB

text: N

text: B

text: E

text: N/πD

text: D

text: For

text: B/D

text: E

text: when

text: N

text: On

text: average,

text: a

text: bird

text: encoun-

text: ters

text: less

text: than

text: one

text: turbine.

text: For

text: B/D

text: E

text: when

text: N

text: You

text: can

text: conﬁrm

text: that

text: using

text: the

text: Circle

text: worksheet

text: in

text: the

text: spreadsheet.

text: The

text: Cape

text: Wind

text: installation

text: will

text: have

text: turbines

text: in

text: an

text: area

text: roughly

text: a

text: rectangle

text: twice

text: as

text: wide

text: as

text: high,

text: with

text: B/D

text: If

text: they

text: were

text: packed

text: in

text: a

text: circle

text: Equation

text: would

text: yield

text: E

text: The

text: spreadsheet

text: calculations

text: show

text: E

text: for

text: the

text: actual

text: positions

text: of

text: the

text: turbines.

text: Unfortunately,

text: M

text: is

text: usually

text: harder

text: to

text: come

text: by.

text: One

text: case

text: is

text: easy.

text: Theorem

text: When

text: birds

text: encounter

text: at

text: most

text: one

text: turbine

text: on

text: bearing

text: θ

text: M(θ)

text: p)E(θ).

text: Proof.

text: The

text: probability

text: that

text: a

text: bird

text: encounters

text: a

text: turbine

text: is

text: E(θ).

text: If

text: it

text: does,

text: it

text: dies

text: with

text: probability

text: p).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 E. BOLKER, J. HATCH, AND C. ZARA Cape Wind computed and estimated mortality probabilities as a function of flight bearing, survival probability p=0.95 for a single encounter 0.070 0.060 bility 0.050 a b 0.040 o pr ality 0.030 ort 0.020 m 0.010 0.000 0 20 40 60 80 100 120 140 160 180 flight bearing computed value (1-p)(E bar) Figure 3. Estimated and computed mortality probabilities. ¯ gray and dark areas are diﬀerent but the computations for E(θ) and ¯ M(θ) are the same. 2B L(θ) 2R Figure 4. Side view when birds tend to ﬂy above tur- bine centers. A second feature allows you to evaluate the model when birds ﬂy at an angle to the wind. In that case the birds see ellipses rather than circles. Their view is shown in Figure 5. 2B L(θ) 2R Figure 5. Side view when birds ﬂy at an angle to the wind. 5. The spreadsheet ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: E.

text: BOLKER,

text: J.

text: HATCH,

text: AND

text: C.

text: ZARA

text: Cape

text: Wind

text: computed

text: and

text: estimated

text: mortality

text: probabilities

text: as

text: a

text: function

text: of

text: flight

text: bearing,

text: survival

text: probability

text: for

text: a

text: single

text: encounter

text: bility

text: a

text: b

text: o

text: pr

text: ality

text: ort

text: m

text: flight

text: bearing

text: computed

text: value

text: bar)

text: Figure

text: Estimated

text: and

text: computed

text: mortality

text: probabilities.

text: gray

text: and

text: dark

text: areas

text: are

text: diﬀerent

text: but

text: the

text: computations

text: for

text: E(θ)

text: and

text: M(θ)

text: are

text: the

text: same.

text: L(θ)

text: Figure

text: Side

text: view

text: when

text: birds

text: tend

text: to

text: ﬂy

text: above

text: tur-

text: bine

text: centers.

text: A

text: second

text: feature

text: allows

text: you

text: to

text: evaluate

text: the

text: model

text: when

text: birds

text: ﬂy

text: at

text: an

text: angle

text: to

text: the

text: wind.

text: In

text: that

text: case

text: the

text: birds

text: see

text: ellipses

text: rather

text: than

text: circles.

text: Their

text: view

text: is

text: shown

text: in

text: Figure

text: L(θ)

text: Figure

text: Side

text: view

text: when

text: birds

text: ﬂy

text: at

text: an

text: angle

text: to

text: the

text: wind.

text: The

text: spreadsheet

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: THE PRESCRIBED JACOBIAN INEQUALITY WITH Lp DATA 19 (cid:101) (cid:101) (52) φ = id on ∂[0, 1]2, (cid:107) (cid:101)− (cid:107) ≤ | − | ≤ ≤ ∞ (53) φ id 2Cτ ψ 1(M) 1/(2p) for all 1 p , W1,p([0,1]2) (cid:107) − (cid:107) (cid:101) ≤ | −(cid:101) | (54) φ id ∞ 2Cτ ψ 1(M) 1/2, L ([0,1]2) |∇ (cid:101)| ≤ ∇ (55) φ C det φ a. e. in (0, 1)2, ∇(cid:101) ≥ − (56) det φ 1 + 2τ a. e. in ψ 1(M), ∇ ≥ − | − | (57) det φ 1 2Cτ ψ 1(M) 1/2 a. e. in (0, 1)2. (cid:101) We claim that ◦ ◦ − φ := ψ φ ψ 1 satisﬁes (5)-(10). Indeed ﬁrst (5) follows trivially from (52). In what follows C (cid:107) (cid:107) (cid:107) − (cid:107) ψ will denote a generic constant depending only on ψ and ψ 1 (and hence C1,1 C1,1 only on Ω) which may change from appearance to appearance. Using (51) and (54) (cid:101) we get (cid:107) − (cid:107) ≤ (cid:107) ◦ − − − (cid:107) ≤ | | φ id ∞ C φ ψ 1 ψ 1 ∞ C M 1/2τ, L (Ω) ψ L (Ω) ψ (cid:101) (cid:101) (cid:101) which proves (7). Next note that ∇ − ∇ ◦ ◦ − − ∇ ◦ (cid:101)− · ∇ − − · ∇ − φ Id = (ψ φ ψ 1) Id = ψ(φ ψ 1) ( φ(ψ 1) Id) ψ 1 ∇ ◦ − − ∇ − · ∇ − + ( ψ(φ ψ 1) ψ(ψ 1)) ψ 1 (cid:101) (cid:101) which yields, using (51), (53), and (54), (cid:107)∇ − (cid:107) ≤ (cid:107)∇ − (cid:107) (cid:107) − (cid:107) φ Id C φ Id + C φ id ∞ Lp(Ω) ψ Lp([0,1]2) ψ L ([0,1]2) ≤ | | C M 1/(2p)τ, ψ (cid:101) (cid:101) proving (6). Next, using (55), we deduce, a. e. in Ω, |∇ | ≤ |∇ − | ≤ ∇ − ≤ ∇ φ C φ(ψ 1) C det φ(ψ 1) C det φ, ψ ψ ψ (cid:101) (cid:101) (cid:101) which proves (8). Then (cid:16) (cid:17) ∇ ◦(cid:101) ◦ − ∇ ◦ − (cid:101)∇ − ∇ − (cid:101) det (ψ φ ψ 1) = det ψ(φ ψ 1) det φ(ψ 1) det ψ 1 ∇(cid:101) ◦ − − ∇ (cid:101)− ∇ − ∇ − ∇ − = det ψ(φ ψ 1) det ψ(ψ 1) det φ(ψ 1) det ψ 1 + det φ(ψ 1) ≥ − | ◦ − − − | (cid:101) ∇ − C φ ψ 1 ψ 1 + det φ(ψ 1) ψ ≥ − | | ∇ − C M 1/2τ + det φ(ψ 1), ψ where we have used (54) f(cid:26)or the last inequality. Hence using (56) and (57) we get from the last inequality − | | ∇ ≥ 1 + τ(2 C M 1/2) a. e. in M det φ − | ψ| \ 1 C M 1/2τ a. e. in Ω M. ψ | | | | ≤ Taking M small enough so that C M 1/2 1 gives (9) and (10). ψ Step 2. We now prove the lemma in the general case.

text: THE

text: PRESCRIBED

text: JACOBIAN

text: INEQUALITY

text: WITH

text: Lp

text: DATA

text: φ

text: id

text: on

text: φ

text: id

text: ψ

text: for

text: all

text: p

text: φ

text: id

text: ψ

text: L

text: φ

text: C

text: det

text: φ

text: a.

text: e.

text: in

text: det

text: φ

text: 2τ

text: a.

text: e.

text: in

text: ψ

text: det

text: φ

text: ψ

text: a.

text: e.

text: in

text: We

text: claim

text: that

text: φ

text: ψ

text: φ

text: ψ

text: satisﬁes

text: Indeed

text: ﬁrst

text: follows

text: trivially

text: from

text: In

text: what

text: follows

text: C

text: ψ

text: will

text: denote

text: a

text: generic

text: constant

text: depending

text: only

text: on

text: ψ

text: and

text: ψ

text: (and

text: hence

text: only

text: on

text: Ω)

text: which

text: may

text: change

text: from

text: appearance

text: to

text: appearance.

text: Using

text: and

text: we

text: get

text: φ

text: id

text: C

text: φ

text: ψ

text: ψ

text: C

text: M

text: L

text: (Ω)

text: ψ

text: L

text: (Ω)

text: ψ

text: which

text: proves

text: Next

text: note

text: that

text: φ

text: Id

text: (ψ

text: φ

text: ψ

text: Id

text: ψ(φ

text: ψ

text: φ(ψ

text: Id)

text: ψ

text: ψ(φ

text: ψ

text: ψ(ψ

text: ψ

text: which

text: yields,

text: using

text: and

text: φ

text: Id

text: C

text: φ

text: Id

text: C

text: φ

text: id

text: Lp(Ω)

text: ψ

text: ψ

text: L

text: C

text: M

text: ψ

text: proving

text: Next,

text: using

text: we

text: deduce,

text: a.

text: e.

text: in

text: Ω,

text: φ

text: C

text: φ(ψ

text: C

text: det

text: φ(ψ

text: C

text: det

text: φ,

text: ψ

text: ψ

text: ψ

text: which

text: proves

text: Then

text: det

text: (ψ

text: φ

text: ψ

text: det

text: ψ(φ

text: ψ

text: det

text: φ(ψ

text: det

text: ψ

text: det

text: ψ(φ

text: ψ

text: det

text: ψ(ψ

text: det

text: φ(ψ

text: det

text: ψ

text: det

text: φ(ψ

text: C

text: φ

text: ψ

text: ψ

text: det

text: φ(ψ

text: ψ

text: C

text: M

text: det

text: φ(ψ

text: ψ

text: where

text: we

text: have

text: used

text: the

text: last

text: inequality.

text: Hence

text: using

text: and

text: we

text: get

text: from

text: the

text: last

text: inequality

text: C

text: M

text: a.

text: e.

text: in

text: M

text: det

text: φ

text: ψ|

text: C

text: M

text: a.

text: e.

text: in

text: Ω

text: M.

text: ψ

text: Taking

text: M

text: small

text: enough

text: so

text: that

text: C

text: M

text: gives

text: and

text: ψ

text: Step

text: We

text: now

text: prove

text: the

text: lemma

text: in

text: the

text: general

text: case.