text: THE D-BAR METHOD FOR DIFFUSE OPTICAL TOMOGRAPHY 13 | | q ( z ) Test potentials, Case 1 Case 2 0 6 3 0 0 0.5 1 0 0.5 |1| z Figure 3. Proﬁle plots of Case 1 and Case 2 radial potentials | | q (z) = q ( z ). 0 0 − Let us ﬁx E = 1. In the numerical tests that follow, for the DN- and S - λ matrices we use N = 16, see the deﬁnition (20). We add gaussian noise to each element using equation (21) so that the relative matrix norm between the original DN-matrix and the noisy DN-matrix is 0.005%. In the mesh for the FEM we have 1048576 triangles. We reconstruct two radially symmetric potentials (Case 1 and Case 2) of ﬁgure | | 3. For radially symmetric potentials we have t(λ) = t( λ ) and Im (t(λ)) = 0. In ﬁgure 4 we have the radially symmetric scattering transform computed in three ways; black solid line indicates computation directly from (7) using the knowledge of q , blue dashed line indicates computation using the DN-matrices L and L− 0 q E and using (22) and (23), red dashed line indicates the same but with noisy DN -matrix L(cid:15). On the left we used Case 1 potential and on the right we used Case 2 q potential. In ﬁgure ?? we plot the reconstructions of Cases 1 and 2 using the three diﬀerent scattering transforms. Black solid line indicates the original potential, black dashed line indicates the reconstruction using the knowledge of the potential via the scat- tering transform (10), blue dashed line indicates reconstruction using (27) without noise, and red dashed line indicates reconstruction using (27) with added noise. 3. Numerical investigation of exceptional points at negative energy − Again, ﬁx E = 1. We use exactly the same radially symmetric potentials as in the numerical part of [?] and [?]: ##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##

text: THE

text: D-BAR

text: METHOD

text: FOR

text: DIFFUSE

text: OPTICAL

text: TOMOGRAPHY

text: q

text: z

text: Test

text: potentials,

text: Case

text: Case

text: z

text: Figure

text: Proﬁle

text: plots

text: of

text: Case

text: and

text: Case

text: radial

text: potentials

text: q

text: (z)

text: q

text: z

text: Let

text: us

text: ﬁx

text: E

text: In

text: the

text: numerical

text: tests

text: that

text: follow,

text: for

text: the

text: DN-

text: and

text: S

text: λ

text: matrices

text: we

text: use

text: N

text: see

text: the

text: deﬁnition

text: We

text: add

text: gaussian

text: noise

text: to

text: each

text: element

text: using

text: equation

text: so

text: that

text: the

text: relative

text: matrix

text: norm

text: between

text: the

text: original

text: DN-matrix

text: and

text: the

text: noisy

text: DN-matrix

text: is

text: In

text: the

text: mesh

text: for

text: the

text: FEM

text: we

text: have

text: triangles.

text: We

text: reconstruct

text: two

text: radially

text: symmetric

text: potentials

text: (Case

text: and

text: Case

text: of

text: ﬁgure

text: For

text: radially

text: symmetric

text: potentials

text: we

text: have

text: t(λ)

text: t(

text: λ

text: and

text: Im

text: (t(λ))

text: In

text: ﬁgure

text: we

text: have

text: the

text: radially

text: symmetric

text: scattering

text: transform

text: computed

text: in

text: three

text: ways;

text: black

text: solid

text: line

text: indicates

text: computation

text: directly

text: from

text: using

text: the

text: knowledge

text: of

text: q

text: blue

text: dashed

text: line

text: indicates

text: computation

text: using

text: the

text: DN-matrices

text: L

text: and

text: L−

text: q

text: E

text: and

text: using

text: and

text: red

text: dashed

text: line

text: indicates

text: the

text: same

text: but

text: with

text: noisy

text: DN

text: -matrix

text: On

text: the

text: left

text: we

text: used

text: Case

text: potential

text: and

text: on

text: the

text: right

text: we

text: used

text: Case

text: q

text: potential.

text: In

text: ﬁgure

text: we

text: plot

text: the

text: reconstructions

text: of

text: Cases

text: and

text: using

text: the

text: three

text: diﬀerent

text: scattering

text: transforms.

text: Black

text: solid

text: line

text: indicates

text: the

text: original

text: potential,

text: black

text: dashed

text: line

text: indicates

text: the

text: reconstruction

text: using

text: the

text: knowledge

text: of

text: the

text: potential

text: via

text: the

text: scat-

text: tering

text: transform

text: blue

text: dashed

text: line

text: indicates

text: reconstruction

text: using

text: without

text: noise,

text: and

text: red

text: dashed

text: line

text: indicates

text: reconstruction

text: using

text: with

text: added

text: noise.

text: Numerical

text: investigation

text: of

text: exceptional

text: points

text: at

text: negative

text: energy

text: Again,

text: ﬁx

text: E

text: We

text: use

text: exactly

text: the

text: same

text: radially

text: symmetric

text: potentials

text: as

text: in

text: the

text: numerical

text: part

text: of

text: and

text: ##LTFigure##

text: ##LTFigure##

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: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: THE D-BAR METHOD FOR DIFFUSE OPTICAL TOMOGRAPHY 19 Case 3 Case 4 | | t( λ ) 15 10 5 0 −5 −10 −151 10 20 30 1 10 20 30| | λ | | Figure 10. The radial scattering transform t( λ ) for the conduc- − tivities of Case 3 and Case 4, at energy E = 1, computed in three ways; black solid line indicates computation directly from (7) using the knowledge of q , blue dashed line indicates computation using 0 the DN-matrices L , L− and using equations (22) and (23), red q E dashed line indicates the same but with noisy DN-matrix L(cid:15). q Case 3 Case 4 | | σ( z ) 2.5 2 1.5 1 0 0.5 1 0 0.5 1| | z Figure 11. Reconstructions of test conductivities Case 3 and Case − 4, at energy E = 1. Black solid line: the original conductivity. Black dashed line: reconstruction using the knowledge of the po- tential via the scattering transform (10). Blue dashed line: recon- struction using the new computational method and (??) without noise. Red dashed line: the same but with added noise. We developed a numerical method to compute the Faddeev Green’s function g (z) for negative energy and thus extended the numerical D-bar method from the λ ##LTFigure## ##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##

text: THE

label: paragraph

text: D-BAR

text: METHOD

text: FOR

text: DIFFUSE

text: OPTICAL

text: TOMOGRAPHY

text: Case

text: Case

text: t(

text: λ

font: default

text: −5

text: λ

label: caption

text: Figure

text: The

text: radial

text: scattering

text: transform

text: t(

text: λ

text: for

text: the

text: conduc-

text: tivities

text: of

text: Case

text: and

text: Case

text: at

text: energy

text: E

text: computed

text: in

text: three

text: ways;

text: black

text: solid

text: line

text: indicates

text: computation

text: directly

text: from

text: using

text: the

text: knowledge

text: of

text: q

text: blue

text: dashed

text: line

text: indicates

text: computation

text: using

text: the

text: DN-matrices

text: L

text: L−

text: and

text: using

text: equations

text: and

text: red

text: q

text: E

text: dashed

text: line

text: indicates

text: the

text: same

text: but

text: with

text: noisy

text: DN-matrix

text: q

text: Case

text: Case

text: σ(

text: z

text: z

text: Figure

text: Reconstructions

text: of

text: test

text: conductivities

text: Case

text: and

text: Case

text: at

text: energy

text: E

text: Black

text: solid

text: line:

text: the

text: original

text: conductivity.

text: Black

text: dashed

text: line:

text: reconstruction

text: using

text: the

text: knowledge

text: of

text: the

text: po-

text: tential

text: via

text: the

text: scattering

text: transform

text: Blue

text: dashed

text: line:

text: recon-

text: struction

text: using

text: the

text: new

text: computational

text: method

text: and

text: without

text: noise.

text: Red

text: dashed

text: line:

text: the

text: same

text: but

text: with

text: added

text: noise.

text: We

text: developed

text: a

text: numerical

text: method

text: to

text: compute

text: the

text: Faddeev

text: Green’s

text: function

text: g

text: (z)

text: for

text: negative

text: energy

text: and

text: thus

text: extended

text: the

text: numerical

text: D-bar

text: method

text: from

text: the

text: λ

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: THE D-BAR METHOD FOR DIFFUSE OPTICAL TOMOGRAPHY 23 References [1] M. J. Ablowitz and A. I. Nachman. Multidimensional nonlinear evolution equations and in- verse scattering. Physica D, 18:223–241, 1986. [2] S. R. Arridge. Optical tomography in medical imaging. Inverse Problems, 15(2):R41–R93, 1999. [3] S.R. Arridge and J.C. Schotland. Optical tomography: forward and inverse problems. Inverse Problems, 25:123010 (59pp), 2009. [4] Richard Beals and Ronald R. Coifman. Scattering, transformations spectrales et ´equations d’´evolution non lin´eaires. In Goulaouic-Meyer-Schwartz Seminar, 1980–1981, pages Exp. No. XXII,10. E´cole Polytech., Palaiseau, 1981. [5] A.-P. Calder´on. On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pages 65–73. Soc. Brasil. Mat., Rio de Janeiro, 1980. [6] Maarten V. de Hoop, Matti Lassas, Matteo Santacesaria, Samuli Siltanen, and Janne. P Tamminen. D-bar method and exceptional points at positive energy: a computational study. Inverse Problems, 32(2):025003, 2016. [7] L. D. Faddeev. Increasing solutions of the Schro¨dinger equation. Soviet Physics Doklady, 10:1033–1035, 1966. [8] I. M. Gelfand. Some problems of functional analysis and algebra. International Mathematical Congress in Amsterdam (in Russian), Nauka, Moscow, pages 49–74, 1961. [9] A. P. Gibson, J. C. Hebden, and S. R. Arridge. Recent advances in diﬀuse optical imaging. Physics in Medicine and Biology, 50:R1–R43, 2005. [10] P. G. Grinevich. Scattering transformation at ﬁxed non-zero energy for the two-dimensional Schr¨odinger operator with potential decaying at inﬁnity. Russian Mathematical Surveys, 55(2):1015–1083, 2000. [11] P. G. Grinevich and S. P. Novikov. Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions: I. energies below the ground state. Funct. Anal. Appl., 11(1):19–27, 1988. [12] K. Knudsen, M. Lassas, J.L. Mueller, and S. Siltanen. Regularized D-bar method for the inverse conductivity problem. Inverse Problems and Imaging, 3(4):599–624, 2009. [13] K. Knudsen, J.L. Mueller, and S. Siltanen. Numerical solution method for the dbar-equation in the plane. Journal of Computational Physics, 198:500–517, 2004. [14] E. Lakshtanov, R. G. Novikov, and B. R. Vainberg. A global riemann-hilbert problem for two-dimensional inverse scattering at ﬁxed energy. arXiv:1509.06495v1, 2015. [15] E. Lakshtanov and B. Vainberg. Exceptional points in faddeev scattering problem. arXiv:1407.1548v4, 2015. [16] E. Lakshtanov and B. Vainberg. On reconstruction of complex-valued once diﬀerentiable con- ductivities. arXiv:1511.08780v2, 2015. [17] S. Moskow and J.C. Schotland. Convergence and stability of the inverse scattering series for diﬀuse waves. Inverse Problems, 24:065005, 2008. [18] S. Moskow and J.C. Schotland. Numerical studies of the inverse born series for diﬀuse waves. Inverse Problems, 25:095007, 2009. [19] J.L. Mueller and S. Siltanen. Linear and Nonlinear Inverse Problems with Practical Applica- tions. SIAM, 2012. [20] M Music, P Perry, and S Siltanen. Exceptional circles of radial potentials. Inverse Problems, 29(4):045004, 2013. [21] A. I. Nachman. Global uniqueness for a two-dimensional inverse boundary value problem. Annals of Mathematics, 143:71–96, 1996. [22] R. G. Novikov. The inverse scattering problem on a ﬁxed energy level for the two-dimensional Schr¨odinger operator. Journal of Functional Analysis, 103(2):409–463, 1992. − − [23] R.G. Novikov. A multidimensional inverse spectral problem for the equation δψ + (v(x) eu(x))ψ = 0. Functional Analysis and Its Applications, 22(4):263–272, 1988. [24] J. Ripoll and V. Ntziachristos. From ﬁnite to inﬁnite volumes: Removal of boundaries in diﬀuse wave imaging. Physical Review Letters, 96(17):173903, 2006. [25] Matteo Santacesaria. Stability estimates for an inverse problem for the schrodinger equation at negative energy in two dimensions. Applicable Analysis, 92(8):1666–1681, 2013.

text: THE

text: D-BAR

text: METHOD

text: FOR

text: DIFFUSE

text: OPTICAL

text: TOMOGRAPHY

text: References

label: reference

text: M.

text: J.

text: Ablowitz

text: and

text: A.

text: I.

text: Nachman.

text: Multidimensional

text: nonlinear

text: evolution

text: equations

text: and

text: in-

text: verse

text: scattering.

text: Physica

text: D,

text: S.

text: R.

text: Arridge.

text: Optical

text: tomography

text: in

text: medical

text: imaging.

text: Inverse

text: Problems,

text: S.R.

text: Arridge

text: and

text: J.C.

text: Schotland.

text: Optical

text: tomography:

text: forward

text: and

text: inverse

text: problems.

text: Inverse

text: Problems,

text: Richard

text: Beals

text: and

text: Ronald

text: R.

text: Coifman.

text: Scattering,

text: transformations

text: spectrales

text: et

text: ´equations

text: d’´evolution

text: non

text: lin´eaires.

text: In

text: Goulaouic-Meyer-Schwartz

text: Seminar,

text: pages

text: Exp.

text: No.

text: E´cole

text: Polytech.,

text: Palaiseau,

text: A.-P.

text: Calder´on.

text: On

text: an

text: inverse

text: boundary

text: value

text: problem.

text: In

text: Seminar

text: on

text: Numerical

text: Analysis

text: and

text: its

text: Applications

text: to

text: Continuum

text: Physics

text: (Rio

text: de

text: Janeiro,

text: pages

text: Soc.

text: Brasil.

text: Mat.,

text: Rio

text: de

text: Janeiro,

text: Maarten

text: V.

text: de

text: Hoop,

text: Matti

text: Lassas,

text: Matteo

text: Santacesaria,

text: Samuli

text: Siltanen,

text: and

text: Janne.

text: P

text: Tamminen.

text: D-bar

text: method

text: and

text: exceptional

text: points

text: at

text: positive

text: energy:

text: a

text: computational

text: study.

text: Inverse

text: Problems,

text: L.

text: D.

text: Faddeev.

text: Increasing

text: solutions

text: of

text: the

text: Schro¨dinger

text: equation.

text: Soviet

text: Physics

text: Doklady,

text: I.

text: M.

text: Gelfand.

text: Some

text: problems

text: of

text: functional

text: analysis

text: and

text: algebra.

text: International

text: Mathematical

text: Congress

text: in

text: Amsterdam

text: (in

text: Russian),

text: Nauka,

text: Moscow,

text: pages

text: A.

text: P.

text: Gibson,

text: J.

text: C.

text: Hebden,

text: and

text: S.

text: R.

text: Arridge.

text: Recent

text: advances

text: in

text: diﬀuse

text: optical

text: imaging.

text: Physics

text: in

text: Medicine

text: and

text: Biology,

text: P.

text: G.

text: Grinevich.

text: Scattering

text: transformation

text: at

text: ﬁxed

text: non-zero

text: energy

text: for

text: the

text: two-dimensional

text: Schr¨odinger

text: operator

text: with

text: potential

text: decaying

text: at

text: inﬁnity.

text: Russian

text: Mathematical

text: Surveys,

text: P.

text: G.

text: Grinevich

text: and

text: S.

text: P.

text: Novikov.

text: Two-dimensional

text: “inverse

text: scattering

text: problem”

text: for

text: negative

text: energies

text: and

text: generalized-analytic

text: functions:

text: I.

text: energies

text: below

text: the

text: ground

text: state.

text: Funct.

text: Anal.

text: Appl.,

text: K.

text: Knudsen,

text: M.

text: Lassas,

text: J.L.

text: Mueller,

text: and

text: S.

text: Siltanen.

text: Regularized

text: D-bar

text: method

text: for

text: the

text: inverse

text: conductivity

text: problem.

text: Inverse

text: Problems

text: and

text: Imaging,

text: K.

text: Knudsen,

text: J.L.

text: Mueller,

text: and

text: S.

text: Siltanen.

text: Numerical

text: solution

text: method

text: for

text: the

text: dbar-equation

text: in

text: the

text: plane.

text: Journal

text: of

text: Computational

text: Physics,

text: E.

text: Lakshtanov,

text: R.

text: G.

text: Novikov,

text: and

text: B.

text: R.

text: Vainberg.

text: A

text: global

text: riemann-hilbert

text: problem

text: for

text: two-dimensional

text: inverse

text: scattering

text: at

text: ﬁxed

text: energy.

text: E.

text: Lakshtanov

text: and

text: B.

text: Vainberg.

text: Exceptional

text: points

text: in

text: faddeev

text: scattering

text: problem.

text: E.

text: Lakshtanov

text: and

text: B.

text: Vainberg.

text: On

text: reconstruction

text: of

text: complex-valued

text: once

text: diﬀerentiable

text: con-

text: ductivities.

text: S.

text: Moskow

text: and

text: J.C.

text: Schotland.

text: Convergence

text: and

text: stability

text: of

text: the

text: inverse

text: scattering

text: series

text: for

text: diﬀuse

text: waves.

text: Inverse

text: Problems,

text: S.

text: Moskow

text: and

text: J.C.

text: Schotland.

text: Numerical

text: studies

text: of

text: the

text: inverse

text: born

text: series

text: for

text: diﬀuse

text: waves.

text: Inverse

text: Problems,

text: J.L.

text: Mueller

text: and

text: S.

text: Siltanen.

text: Linear

text: and

text: Nonlinear

text: Inverse

text: Problems

text: with

text: Practical

text: Applica-

text: tions.

text: SIAM,

text: M

text: Music,

text: P

text: Perry,

text: and

text: S

text: Siltanen.

text: Exceptional

text: circles

text: of

text: radial

text: potentials.

text: Inverse

text: Problems,

text: A.

text: I.

text: Nachman.

text: Global

text: uniqueness

text: for

text: a

text: two-dimensional

text: inverse

text: boundary

text: value

text: problem.

text: Annals

text: of

text: Mathematics,

text: R.

text: G.

text: Novikov.

text: The

text: inverse

text: scattering

text: problem

text: on

text: a

text: ﬁxed

text: energy

text: level

text: for

text: the

text: two-dimensional

text: Schr¨odinger

text: operator.

text: Journal

text: of

text: Functional

text: Analysis,

text: R.G.

text: Novikov.

text: A

text: multidimensional

text: inverse

text: spectral

text: problem

text: for

text: the

text: equation

text: δψ

text: (v(x)

text: eu(x))ψ

text: Functional

text: Analysis

text: and

text: Its

text: Applications,

text: J.

text: Ripoll

text: and

text: V.

text: Ntziachristos.

text: From

text: ﬁnite

text: to

text: inﬁnite

text: volumes:

text: Removal

text: of

text: boundaries

text: in

text: diﬀuse

text: wave

text: imaging.

text: Physical

text: Review

text: Letters,

text: Matteo

text: Santacesaria.

text: Stability

text: estimates

text: for

text: an

text: inverse

text: problem

text: for

text: the

text: schrodinger

text: equation

text: at

text: negative

text: energy

text: in

text: two

text: dimensions.

text: Applicable

text: Analysis,


text: 24 J. P. TAMMINEN, T. TARVAINEN AND S. SILTANEN [26] S. Siltanen, J. Mueller, and D. Isaacson. An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem. Inverse Problems, 16:681–699, 2000. [27] S. Siltanen, J. Mueller, and D. Isaacson. Errata: An implementation of the reconstruction al- gorithm of A. Nachman for the 2-D inverse conductivity problem. Inverse Problems, 17:1561– 1563, 2001. [28] Samuli Siltanen. Electrical impedance tomography and Faddeev Green’s functions. Annales Academiae Scientiarum Fennicae Mathematica Dissertationes, 121(121):56, 1999. Disserta- tion, Helsinki University of Technology, Espoo, 1999. [29] Gennadi Vainikko. Fast solvers of the Lippmann-Schwinger equation. In Direct and inverse problems of mathematical physics (Newark, DE, 1997), volume 5 of Int. Soc. Anal. Appl. Comput., pages 423–440. Kluwer Acad. Publ., Dordrecht, 2000. Email address: janne.tamminen@ttu.ee Email address: tanja.tarvainen@uef.fi Email address: samuli.siltanen@helsinki.fi

text: J.

text: P.

text: TAMMINEN,

text: T.

text: TARVAINEN

text: AND

text: S.

text: SILTANEN

text: S.

text: Siltanen,

text: J.

text: Mueller,

text: and

text: D.

text: Isaacson.

text: An

text: implementation

text: of

text: the

text: reconstruction

text: algorithm

text: of

text: A.

text: Nachman

text: for

text: the

text: inverse

text: conductivity

text: problem.

text: Inverse

text: Problems,

text: S.

text: Siltanen,

text: J.

text: Mueller,

text: and

text: D.

text: Isaacson.

text: Errata:

text: An

text: implementation

text: of

text: the

text: reconstruction

text: al-

text: gorithm

text: of

text: A.

text: Nachman

text: for

text: the

text: inverse

text: conductivity

text: problem.

text: Inverse

text: Problems,

text: Samuli

text: Siltanen.

text: Electrical

text: impedance

text: tomography

text: and

text: Faddeev

text: Green’s

text: functions.

text: Annales

text: Academiae

text: Scientiarum

text: Fennicae

text: Mathematica

text: Dissertationes,

text: Disserta-

text: tion,

text: Helsinki

text: University

text: of

text: Technology,

text: Espoo,

text: Gennadi

text: Vainikko.

text: Fast

text: solvers

text: of

text: the

text: Lippmann-Schwinger

text: equation.

text: In

text: Direct

text: and

text: inverse

text: problems

text: of

text: mathematical

text: physics

text: (Newark,

text: DE,

text: volume

text: of

text: Int.

text: Soc.

text: Anal.

text: Appl.

text: Comput.,

text: pages

text: Kluwer

text: Acad.

text: Publ.,

text: Dordrecht,

text: Email

text: address:

text: janne.tamminen@ttu.ee

text: Email

text: address:

text: tanja.tarvainen@uef.fi

text: Email

text: address:

text: samuli.siltanen@helsinki.fi


text: 12 F. LINDNER, N. MARHEINEKE, H. STROOT, A. VIBE, AND R. WEGENER ∈ M Proof. Let (r, v, µ) be a local solution to (3.1) up to time σ with initial conditions r and ∈ M 0 P v T . Furthermore let σ˜ be any ﬁxed stopping time fulﬁlling σ˜ < σ. Note that, -almost 0 r0 ◦ surely, g r is diﬀerentiable and constantly zero on [0, σ˜]. Thus, applying the chain rule on the P constraint yields, -almost surely, d ∇ (cid:62) ∈ 0 = g(r(t)) = ( g(r(t))) v(t) for all t [0, σ˜]. dt ∧ ∧ Recall r˜ = (r(t σ˜)) ≥ and v˜ = (v(t σ˜)) ≥ from Deﬁnition 8. Since r, r˜ as well as v, v˜ coincide Pt 0 t 0 on [[0, σ˜]], it holds, -almost surely, that ∇ (cid:62) ∈ ∞ 0 = ( g(r˜(t))) v˜(t) for all t [0, ). (3.7) R × R → R ∇ (cid:62) ∈ R ∇ Let f : dN dN be deﬁned by f(x, y) = ( g(x)) y for any x, y dN, and let and ∇ ∇ ∇ · x be the gradients w.r.t. the variables x and y, respectively, i.e., f(x, y) = (f( , y))(x). We y x want to apply Itˆo’s formula to f(r˜, v˜). Note that the process r˜ has bounded variation and therefore regarded as a semimartingale its ma(cid:90)rtingale part is zero. Then,(cid:90)by Itˆo’s formula, − t ∇ t ∇ f(r˜(t), v˜(t)) f(r , v ) = f(r˜(u), v˜(u))dr˜(u) + f(r˜(u), v˜(u))dv˜(u) (3.8) 0 0 x y 0 0 where all second order terms vanish since (r˜(t)) ≥ has bounded variation and f is linear in y. t 0 Combining (3.7) and(cid:90) (3.8) and (cid:26)inserting the d(cid:104)eﬁnition of f, r˜ and v˜ yields t ∇ (cid:62)(cid:105) ∇ (cid:27) 0 = 1[0,σ˜](u) ( g(r˜(u))) ( g(r˜(u)))dµ˜(u) + a(u, r˜(u), v˜(u))du 0 (3.9) + B(u, r˜(u), v˜(u))dw(u) + D2g(r˜(u))(v˜(u), v˜(u))du . − · Integrating G 1(r( )) w.r.t. the semimartingale deﬁned by the right hand side of (3.9) results in (3.6). Let (r, v) be any local solution to the explicit equation (3.5) up to time σ with initial ∈ M ∈ M conditions r and v T . Further let σ˜ be again any ﬁxed stopping time fulﬁlling σ˜ < σ. P0 0 ◦r0 Note that, -almost surely, g r is diﬀerentiable on [0, σ˜]. Recall that g(r ) = 0. Therefore (r, v) → R 0 combined with µ : [[0, σ)) N deﬁned by (3.6) is a local solution to (3.1) up to time σ with initial P conditions r , v , if, -almost surely, 0 0 d ∇ (cid:62) ∈ 0 = g(r(t)) = ( g(r(t))) v(t) for all t [0, σ˜]. dt (cid:3) This equality can be veriﬁed by using Itˆo’s formula. The unique local solvability of the manifold-valued explicit SDE (3.5) can be shown by using the general (abstract) solution theory in [18]. For the sake of completeness and in order to gain more M insight in our speciﬁc problem, we state here a concise proof that makes use of the structure of and the explicit equation (3.5). Lemma 11. The SDE (3.5) has a unique local solution (r, v) up to its explosion time σ. Proof. We consider a locally Lipschitz continuous extension of the coeﬃcients in the explicit SDE R (3.5) towards the space dN. Recall that the drift and diﬀusion functions a and B are locally R ∇ R Lipschitz continuous and already deﬁned on dN. Moreover, g and D2g are deﬁned on dN and − are smooth. According to Lemma 9, G(x) has an inverse G 1(x) whose matrix norm is bounded ∈ M for all x ε. We ﬁx ε > 0 and introduce the extension ##LTLine## ##LTLine##

text: F.

text: LINDNER,

text: N.

text: MARHEINEKE,

text: H.

text: STROOT,

text: A.

text: VIBE,

text: AND

text: R.

text: WEGENER

text: M

text: Proof.

text: Let

text: (r,

text: v,

text: µ)

text: be

text: a

text: local

text: solution

text: to

text: up

text: to

text: time

text: σ

text: with

text: initial

text: conditions

text: r

text: and

text: M

text: P

text: v

text: T

text: Furthermore

text: let

text: σ˜

text: be

text: any

text: ﬁxed

text: stopping

text: time

text: fulﬁlling

text: σ˜

text: σ.

text: Note

text: that,

text: -almost

text: surely,

text: g

text: r

text: is

text: diﬀerentiable

text: and

text: constantly

text: zero

text: on

text: σ˜].

text: Thus,

text: applying

text: the

text: chain

text: rule

text: on

text: the

text: P

text: constraint

text: yields,

text: -almost

text: surely,

text: d

text: g(r(t))

text: g(r(t)))

text: v(t)

text: for

text: all

text: t

text: σ˜].

text: dt

text: Recall

text: r˜

text: (r(t

text: σ˜))

text: and

text: v˜

text: (v(t

text: σ˜))

text: from

text: Deﬁnition

text: Since

text: r,

text: r˜

text: as

text: well

text: as

text: v,

text: v˜

text: coincide

text: Pt

text: t

text: on

text: σ˜]],

text: it

text: holds,

text: -almost

text: surely,

text: that

text: g(r˜(t)))

text: v˜(t)

text: for

text: all

text: t

text: R

text: R

text: R

text: R

text: Let

text: f

text: dN

text: dN

text: be

text: deﬁned

text: by

text: f(x,

text: y)

text: g(x))

text: y

text: for

text: any

text: x,

text: y

text: dN,

text: and

text: let

text: and

text: x

text: be

text: the

text: gradients

text: w.r.t.

text: the

text: variables

text: x

text: and

text: y,

text: respectively,

text: i.e.,

text: f(x,

text: y)

text: (f(

text: y))(x).

text: We

text: y

text: x

text: want

text: to

text: apply

text: Itˆo’s

text: formula

text: to

text: f(r˜,

text: v˜).

text: Note

text: that

text: the

text: process

text: r˜

text: has

text: bounded

text: variation

text: and

text: therefore

text: regarded

text: as

text: a

text: semimartingale

text: its

text: part

text: is

text: zero.

text: Itˆo’s

text: formula,

text: t

text: t

text: f(r˜(t),

text: v˜(t))

text: f(r

text: v

text: f(r˜(u),

text: v˜(u))dr˜(u)

text: f(r˜(u),

text: v˜(u))dv˜(u)

text: x

text: y

text: where

text: all

text: second

text: order

text: terms

text: vanish

text: since

text: (r˜(t))

text: has

text: bounded

text: variation

text: and

text: f

text: is

text: linear

text: in

text: y.

text: t

text: Combining

text: and

text: the

text: of

text: f,

text: r˜

text: and

text: v˜

text: yields

text: t

text: g(r˜(u)))

text: g(r˜(u)))dµ˜(u)

text: a(u,

text: r˜(u),

text: v˜(u))du

text: B(u,

text: r˜(u),

text: v˜(u))dw(u)

text: v˜(u))du

text: Integrating

text: G

text: w.r.t.

text: the

text: semimartingale

text: deﬁned

text: by

text: the

text: right

text: hand

text: side

text: of

text: results

text: in

text: Let

text: (r,

text: v)

text: be

text: any

text: local

text: solution

text: to

text: the

text: explicit

text: equation

text: up

text: to

text: time

text: σ

text: with

text: initial

text: M

text: M

text: conditions

text: r

text: and

text: v

text: T

text: Further

text: let

text: σ˜

text: be

text: again

text: any

text: ﬁxed

text: stopping

text: time

text: fulﬁlling

text: σ˜

text: σ.

text: Note

text: that,

text: -almost

text: surely,

text: g

text: r

text: is

text: diﬀerentiable

text: on

text: σ˜].

text: Recall

text: that

text: g(r

text: Therefore

text: (r,

text: v)

text: R

text: combined

text: with

text: µ

text: σ))

text: N

text: deﬁned

text: by

text: is

text: a

text: local

text: solution

text: to

text: up

text: to

text: time

text: σ

text: with

text: initial

text: P

text: conditions

text: r

text: v

text: if,

text: -almost

text: surely,

text: d

text: g(r(t))

text: g(r(t)))

text: v(t)

text: for

text: all

text: t

text: σ˜].

text: dt

text: This

text: equality

text: can

text: be

text: veriﬁed

text: by

text: using

text: Itˆo’s

text: formula.

text: The

text: unique

text: local

text: solvability

text: of

text: the

text: manifold-valued

text: explicit

text: SDE

text: can

text: be

text: shown

text: by

text: using

text: the

text: general

text: (abstract)

text: solution

text: theory

text: in

text: For

text: the

text: sake

text: of

text: completeness

text: and

text: in

text: order

text: to

text: gain

text: more

text: M

text: insight

text: in

text: our

text: speciﬁc

text: problem,

text: we

text: state

text: here

text: a

text: concise

text: proof

text: that

text: makes

text: use

text: of

text: the

text: structure

text: of

text: and

text: the

text: explicit

text: equation

text: Lemma

text: The

text: SDE

text: has

text: a

text: unique

text: local

text: solution

text: (r,

text: v)

text: up

text: to

text: its

text: explosion

text: time

text: σ.

text: Proof.

text: We

text: consider

text: a

text: locally

text: Lipschitz

text: continuous

text: extension

text: of

text: the

text: coeﬃcients

text: in

text: the

text: explicit

text: SDE

text: R

text: towards

text: the

text: space

text: dN.

text: Recall

text: that

text: the

text: drift

text: and

text: diﬀusion

text: functions

text: a

text: and

text: B

text: are

text: locally

text: R

text: R

text: Lipschitz

text: continuous

text: and

text: already

text: deﬁned

text: on

text: dN.

text: Moreover,

text: g

text: and

text: are

text: deﬁned

text: on

text: dN

text: and

text: are

text: smooth.

text: According

text: to

text: Lemma

text: G(x)

text: has

text: an

text: inverse

text: G

text: whose

text: matrix

text: norm

text: is

text: bounded

text: M

text: for

text: all

text: x

text: ε.

text: We

text: ﬁx

text: ε

text: and

text: introduce

text: the

text: extension

text: ##LTLine##

text: ##LTLine##


text: 4 F. LINDNER, N. MARHEINEKE, H. STROOT, A. VIBE, AND R. WEGENER dynamics is driven by the inner forces due to traction and bending and by the acting outer forces due to gravity and drag. The drag forces arising in a turbulent ﬂow are modeled as superposition of a deterministic and a stochastic part, [30, 31]. Whereas the deterministic force is determined by the relative velocity between the mean ﬂow velocity u (evaluated at (r(s, t), t)) and the ﬁber velocity, the stochastic force takes into account the impact of the turbulent ﬂuctuations. It is modeled as three-dimensional space-time white noise ξ with ﬂow-dependent amplitude D. We use a linear R × (Stokes-type) drag model where the d d-valued drag operators C and D depend explicitly on the ﬁber velocity and/or orientation; the further dependence on the ﬂow ﬁeld enters via the evaluation of the ﬂow quantities at (r, t), cf., u(r, t). The stochastic ﬁber system (2.1) is characterized by four dimensionless numbers: the bending number α (ratio of inertial and bending forces), the Froude (cid:107) (cid:107) number Fr (ratio of inertial and gravitational forces with gravitational direction e , e = 1), the g g drag number Dr (ratio of inertial and mean drag forces) and the turbulent ﬂuctuation number β. Assumption 1 (Drag force model). For the subsequent investigations we assume that the ﬂow R × R → R velocity u : d + d is continuously diﬀerentiable with at most linear growth in the space 0 R × argument. Moreover, the drag operators are continuously diﬀerentiable and bounded, i.e., C : 2d R → R × R × R → R × + d d and D : 3d + d d. Note that the consideration of nonlinear drag models as, 0 0 e.g., given in [31] is possible, presupposing respective regularity and growth conditions. The ﬁber model (2.1) is a wave-like system with elliptic regularization. For the spatial dis- cretization of the deterministic version diﬀerent approaches can be found in literature, such as, e.g., geometric Lagrangian methods [25], ﬁnite element schemes [1, 2, 16] or ﬁnite volume approaches [40]. We apply a ﬁnite volume method in combination with a ﬁnite diﬀerence approximation for the constraint. The usage of a staggered grid allows particularly for small discretization stencils. In the deterministic case, it is a conservative ﬁrst-order scheme. However, note that the derivation of the spatially semi-discrete stochastic system is here purely formal, a rigorous interpretation follows in Sec. . We consider a ﬁnite volume discretization that is based on a conforming partition of [0, (cid:96)] into subintervals (control cells) I of length ∆s. The partition is identiﬁed with the sequence of nodes { } i (cid:82) s . The idea is to formally integrate the evolution equations in (2.1) over the control cells I = i ± i [s − , s ], s ± = s ∆s/2 for i = 1, ..., N, and to set up a stochastic diﬀerential system in i 1/2 i+1/2 i 1/2 i ∈ { } time for the cell averages ϕ (t) = ϕ(s, t)ds/∆s of the unknowns ϕ r, v . In this way the ﬁber i Ii position and velocity are assigned to the cell nodes, r (t), v (t), whereas we assign the inner traction i i –and consequently also the constraint– to the cell edges, λ ± (t) = λ(s ± , t). Proceeding from i 1/2 i 1/2 the integral equations (cid:18) (cid:90) (cid:90) (cid:19) d r (t) = v (t), dt i i d 1 − (2.3) v (t) = φ(s , t) φ(s − , t) + f(s, t) ds + A(s, t)ξ(s, t) ds , dt i ∆s i+1/2 i 1/2 (cid:107) (cid:107) Ii Ii ∂ r(s − , t) = 1, s i 1/2 − − where we abbreviate the ﬂux function by φ = λ∂ r α 2∂ r, the deterministic source terms by − − − s sss − f = Fr 2e +Dr 2C(r, ∂(cid:90)r, .)(u(r, .) v) and the noise amplitude by A = Dr 2βC(r, ∂ r, .)D(r, v, ∂ r, .), g s s s we evaluate the integrals by means of trapezoidal quadrature rules, (cid:90) ≈ 1 (cid:90) f(s, t) ds (f(s , t) + f(s − , t))∆s, 2 i+1/2 i 1/2 Ii ≈ 1 A(s, t)ξ(s, t) ds (A(s , t) + A(s − , t)) ξ(s, t) ds. 2 i+1/2 i 1/2 Ii Ii ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F.

text: LINDNER,

text: N.

text: MARHEINEKE,

text: H.

text: STROOT,

text: A.

text: VIBE,

text: AND

text: R.

text: WEGENER

text: dynamics

text: is

text: driven

text: by

text: the

text: inner

text: forces

text: due

text: to

text: traction

text: and

text: bending

text: and

text: by

text: the

text: acting

text: outer

text: forces

text: due

text: to

text: gravity

text: and

text: drag.

text: The

text: drag

text: forces

text: arising

text: in

text: a

text: turbulent

text: ﬂow

text: are

text: modeled

text: as

text: superposition

text: of

text: a

text: deterministic

text: and

text: a

text: stochastic

text: part,

text: Whereas

text: the

text: deterministic

text: force

text: is

text: determined

text: by

text: the

text: relative

text: velocity

text: between

text: the

text: mean

text: ﬂow

text: velocity

text: u

text: (evaluated

text: at

text: (r(s,

text: t),

text: t))

text: and

text: the

text: ﬁber

text: velocity,

text: the

text: stochastic

text: force

text: takes

text: into

text: account

text: the

text: impact

text: of

text: the

text: turbulent

text: ﬂuctuations.

text: It

text: is

text: modeled

text: as

text: three-dimensional

text: space-time

text: white

text: noise

text: ξ

text: with

text: ﬂow-dependent

text: amplitude

text: D.

text: We

text: use

text: a

text: linear

text: R

text: (Stokes-type)

text: drag

text: model

text: where

text: the

text: d

text: d-valued

text: drag

text: operators

text: C

text: and

text: D

text: depend

text: explicitly

text: on

text: the

text: ﬁber

text: velocity

text: and/or

text: orientation;

text: the

text: further

text: dependence

text: on

text: the

text: ﬂow

text: ﬁeld

text: enters

text: via

text: the

text: evaluation

text: of

text: the

text: ﬂow

text: quantities

text: at

text: (r,

text: t),

text: cf.,

text: u(r,

text: t).

text: The

text: stochastic

text: ﬁber

text: system

text: is

text: characterized

text: by

text: four

text: dimensionless

text: numbers:

text: the

text: bending

text: number

text: α

text: (ratio

text: of

text: inertial

text: and

text: bending

text: forces),

text: the

text: Froude

text: number

text: Fr

text: (ratio

text: of

text: inertial

text: and

text: gravitational

text: forces

text: with

text: gravitational

text: direction

text: e

text: e

text: the

text: g

text: g

text: drag

text: number

text: Dr

text: (ratio

text: of

text: inertial

text: and

text: mean

text: drag

text: forces)

text: and

text: the

text: turbulent

text: ﬂuctuation

text: number

text: β.

text: Assumption

text: (Drag

text: force

text: model).

text: For

text: the

text: subsequent

text: investigations

text: we

text: assume

text: that

text: the

text: ﬂow

text: R

text: R

text: R

text: velocity

text: u

text: d

text: d

text: is

text: continuously

text: diﬀerentiable

text: with

text: at

text: most

text: linear

text: growth

text: in

text: the

text: space

text: R

text: argument.

text: Moreover,

text: the

text: drag

text: operators

text: are

text: continuously

text: diﬀerentiable

text: and

text: bounded,

text: i.e.,

text: C

text: R

text: R

text: R

text: R

text: R

text: d

text: d

text: and

text: D

text: d

text: d.

text: Note

text: that

text: the

text: consideration

text: of

text: nonlinear

text: drag

text: models

text: as,

text: e.g.,

text: given

text: in

text: is

text: possible,

text: presupposing

text: respective

text: regularity

text: and

text: growth

text: conditions.

text: The

text: ﬁber

text: model

text: is

text: a

text: wave-like

text: system

text: with

text: elliptic

text: regularization.

text: For

text: the

text: spatial

text: dis-

text: cretization

text: of

text: the

text: deterministic

text: version

text: diﬀerent

text: approaches

text: can

text: be

text: found

text: in

text: literature,

text: such

text: as,

text: e.g.,

text: geometric

text: Lagrangian

text: methods

text: ﬁnite

text: element

text: schemes

text: or

text: ﬁnite

text: volume

text: approaches

text: We

text: apply

text: a

text: ﬁnite

text: volume

text: method

text: in

text: combination

text: with

text: a

text: ﬁnite

text: diﬀerence

text: approximation

text: for

text: the

text: constraint.

text: The

text: usage

text: of

text: a

text: staggered

text: grid

text: allows

text: particularly

text: for

text: small

text: discretization

text: stencils.

text: In

text: the

text: deterministic

text: case,

text: it

text: is

text: a

text: conservative

text: ﬁrst-order

text: scheme.

text: However,

text: note

text: that

text: the

text: derivation

text: of

text: the

text: spatially

text: semi-discrete

text: stochastic

text: system

text: is

text: here

text: purely

text: formal,

text: a

text: rigorous

text: interpretation

text: follows

text: in

text: Sec.

text: We

text: consider

text: a

text: ﬁnite

text: volume

text: discretization

text: that

text: is

text: based

text: on

text: a

text: conforming

text: partition

text: of

text: into

text: subintervals

text: (control

text: cells)

text: I

text: of

text: length

text: ∆s.

text: The

text: partition

text: is

text: identiﬁed

text: with

text: the

text: sequence

text: of

text: nodes

text: i

text: s

text: The

text: idea

text: is

text: to

text: formally

text: integrate

text: the

text: evolution

text: equations

text: in

text: over

text: the

text: control

text: cells

text: I

text: i

text: i

text: [s

text: s

text: s

text: s

text: for

text: i

text: N,

text: and

text: to

text: set

text: up

text: a

text: stochastic

text: diﬀerential

text: system

text: in

text: i

text: i

text: i

text: time

text: for

text: the

text: cell

text: averages

text: ϕ

text: (t)

text: ϕ(s,

text: t)ds/∆s

text: of

text: the

text: unknowns

text: ϕ

text: r,

text: v

text: In

text: this

text: way

text: the

text: ﬁber

text: i

text: Ii

text: position

text: and

text: velocity

text: are

text: assigned

text: to

text: the

text: cell

text: nodes,

text: r

text: (t),

text: v

text: (t),

text: whereas

text: we

text: assign

text: the

text: inner

text: traction

text: i

text: i

text: –and

text: consequently

text: also

text: the

text: constraint–

text: to

text: the

text: cell

text: edges,

text: λ

text: (t)

text: λ(s

text: t).

text: Proceeding

text: from

text: i

text: i

text: the

text: integral

text: equations

text: d

text: r

text: (t)

text: v

text: (t),

text: dt

text: i

text: i

text: d

text: v

text: (t)

text: φ(s

text: t)

text: φ(s

text: t)

text: f(s,

text: t)

text: ds

text: A(s,

text: t)ξ(s,

text: t)

text: ds

text: dt

text: i

text: ∆s

text: i

text: Ii

text: Ii

text: r(s

text: t)

text: s

text: i

text: where

text: we

text: abbreviate

text: the

text: ﬂux

text: function

text: by

text: φ

text: λ∂

text: r

text: α

text: 2∂

text: r,

text: the

text: deterministic

text: source

text: terms

text: by

text: s

text: sss

text: f

text: Fr

text: +Dr

text: .)(u(r,

text: v)

text: and

text: the

text: noise

text: amplitude

text: by

text: A

text: Dr

text: r,

text: .)D(r,

text: v,

text: r,

text: g

text: s

text: s

text: s

text: we

text: evaluate

text: the

text: integrals

text: by

text: means

text: of

text: trapezoidal

text: quadrature

text: rules,

text: f(s,

text: t)

text: ds

text: (f(s

text: t)

text: f(s

text: t))∆s,

text: i

text: Ii

text: A(s,

text: t)ξ(s,

text: t)

text: ds

text: (A(s

text: t)

text: A(s

text: t))

text: ξ(s,

text: t)

text: ds.

text: i

text: Ii

text: Ii

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Azimuthal diﬀusion of the large-scale-circulation plane 3 close to West. Solving a Fokker-Planck equation with this model potential and a noise intensity derived from the measured D yielded a p(θ ) which agreed quantitatively with θ 0 experiment. For that case the Prandtl number was 4.38 and Γ = 1.00. Two samples were (cid:39) investigated, one with L = 24.8 and the other with L 50 cm. The Rayleigh number × ∼ ∼ was in the range 3 108 < Ra < 1011. The present study is for a physically larger sample, with L = D = 112 cm, and with (cid:39) × × the smaller Pr 0.8. Rayleigh numbers ranged from 2 1012 to 1.5 1014 and thus were (cid:112) much larger than in the previous work. Again we found that the preferred θ was close 0 to West, consistent with Earth’s Coriolis force. We also measured D and determined θ the corresponding Reynolds number Re = L D /ν. Comparison with an extrapolation θ θ ∝ of the earlier data indicates consistency with Re Pr1.2, a Prandtl dependence found θ before for Re (He et al. 2015). Quite remarkably, the diﬀusivity measurements conﬁrmed V ∗ (cid:39) × that the ultimate-state transition occurs over a range of Ra, from Ra 2 1013 to ∗ (cid:39) × 1 Ra 8 1013, as was found earlier from measurements of Nu(Ra) and Re (Ra). 2 V We also present new measurements of Nu which are for the same sample as that used for the study of the azimuthal diﬀusion. They, as well as Re measurements reported V before (He et al. 2015) for a diﬀerent sample with Γ = 1.00, show the ultimate-state ∗ ∗ transition and all three physical properties yield consistent results for Ra and Re as 1 2 shown below in Fig. 4. Recently it was claimed by Skrbek & Urban (2015) that the temperature dependence of ﬂuid properties (non-OB conditions) may be responsible for the transitions seen in the Grenoble and the G¨ottingen data and that hypothetical corresponding data for an § OB system may not show any transitions. In we show two reasons why this claim is ﬂawed. 2. Apparatus and procedure The apparatus was described elsewhere (Ahlers et al. 2009b, 2012b, 2014). It consisted of a sample cell located inside a vessel known as the “Uboot of G¨ottingen” which was ℃ pressurized with sulfur hexaﬂuoride (SF ) at an average temperature T = 21.5 and 6 m at pressures that ranged from 2.0 to 17.7 bars, resulting in the relatively narrow Prandtl- × × number range from 0.78 to 0.86 over the Ra range from 2 1012 to 1.5 1014 (see Fig. 1b of Ahlers et al. (2014)). The Uboot had a volume of about 25 m3, and about 2000 kg of SF were required to ﬁll it to the maximum pressure. The sample cell had aluminium 6 plates at the top and bottom that were separated by a distance L = 112 cm and had an ± − aspect ratio Γ = 1.00. The plates were leveled relative to gravity to 10 4 rad. Three sets of eight thermistors, one each at the heights z/L = 0.25, 0.50, and 0.75, were imbedded in the sidewall (Brown et al. 2005). Since the LSC carries warm ﬂuid upward at one side of the cell and cold ﬂuid downward at the opposite, one can detect the azimuthal orientation of the LSC by ﬁtting − T = T + δ cos (iπ/4 θ ) (2.1) f w,k k k,0 to all eight thermistors at one speciﬁc height and time. The index “i” stands for the azimuthal location of the thermistors and takes values i = 0 . . . 7, with the location of i = 0 located at a point directly East of the sample center and i increasing in the counter- clockwise direction when viewed from above. The index k denotes the vertical location of the thermistor and will in the following have letters b (z = L/4), m (z = L/2) and t (z = 3L/4). The orientation of the up-ﬂow is given by the phase θ . k,0 From the eight temperatures at a given k we computed the coeﬃcients of the lowest four Fourier modes (Stevens et al. 2011; Weiss & Ahlers 2011). This yielded all eight Fourier ##LTLine##

text: Azimuthal

text: diﬀusion

text: of

text: the

text: large-scale-circulation

text: plane

text: close

text: to

text: West.

text: Solving

text: a

text: Fokker-Planck

text: equation

text: with

text: this

text: model

text: potential

text: and

text: a

text: noise

text: intensity

text: derived

text: from

text: the

text: measured

text: D

text: yielded

text: a

text: p(θ

text: which

text: agreed

text: quantitatively

text: with

text: θ

text: experiment.

text: For

text: that

text: case

text: the

text: Prandtl

text: number

text: was

text: and

text: Γ

text: Two

text: samples

text: were

text: investigated,

text: one

text: with

text: L

text: and

text: the

text: other

text: with

text: L

text: cm.

text: The

text: Rayleigh

text: number

text: was

text: in

text: the

text: range

text: Ra

text: The

text: present

text: study

text: is

text: for

text: a

text: physically

text: larger

text: sample,

text: with

text: L

text: D

text: cm,

text: and

text: with

text: the

text: smaller

text: Pr

text: Rayleigh

text: numbers

text: ranged

text: from

text: to

text: and

text: thus

text: were

text: much

text: larger

text: than

text: in

text: the

text: previous

text: work.

text: Again

text: we

text: found

text: that

text: the

text: preferred

text: θ

text: was

text: close

text: to

text: West,

text: consistent

text: with

text: Earth’s

text: Coriolis

text: force.

text: We

text: also

text: measured

text: D

text: and

text: determined

text: θ

text: the

text: corresponding

text: Reynolds

text: number

text: Re

text: L

text: D

text: /ν.

text: Comparison

text: with

text: an

text: extrapolation

text: θ

text: θ

text: of

text: the

text: earlier

text: data

text: indicates

text: consistency

text: with

text: Re

text: a

text: Prandtl

text: dependence

text: found

text: θ

text: before

text: for

text: Re

text: (He

text: et

text: al.

text: Quite

text: remarkably,

text: the

text: diﬀusivity

text: measurements

text: conﬁrmed

text: V

text: that

text: the

text: ultimate-state

text: transition

text: occurs

text: over

text: a

text: range

text: of

text: Ra,

text: from

text: Ra

text: to

text: Ra

text: as

text: was

text: found

text: earlier

text: from

text: measurements

text: of

text: Nu(Ra)

text: and

text: Re

text: (Ra).

text: V

text: We

text: also

text: present

text: new

text: measurements

text: of

text: Nu

text: which

text: are

text: for

text: the

text: same

text: sample

text: as

text: that

text: used

text: for

text: the

text: study

text: of

text: the

text: azimuthal

text: diﬀusion.

text: They,

text: as

text: well

text: as

text: Re

text: measurements

text: reported

text: V

text: before

text: (He

text: et

text: al.

text: for

text: a

text: diﬀerent

text: sample

text: with

text: Γ

text: show

text: the

text: ultimate-state

text: transition

text: and

text: all

text: three

text: physical

text: properties

text: yield

text: consistent

text: results

text: for

text: Ra

text: and

text: Re

text: as

text: shown

text: below

text: in

text: Fig.

text: Recently

text: it

text: was

text: claimed

text: by

text: Skrbek

text: Urban

text: that

text: the

text: temperature

text: dependence

text: of

text: ﬂuid

text: properties

text: (non-OB

text: conditions)

text: may

text: be

text: responsible

text: for

text: the

text: transitions

text: seen

text: in

text: the

text: Grenoble

text: and

text: the

text: G¨ottingen

text: data

text: and

text: that

text: hypothetical

text: corresponding

text: data

text: for

text: an

text: OB

text: system

text: may

text: not

text: show

text: any

text: transitions.

text: In

text: we

text: show

text: two

text: reasons

text: why

text: this

text: claim

text: is

text: ﬂawed.

text: Apparatus

text: and

text: procedure

text: The

text: apparatus

text: was

text: described

text: elsewhere

text: (Ahlers

text: et

text: al.

text: It

text: consisted

text: of

text: a

text: sample

text: cell

text: located

text: inside

text: a

text: vessel

text: known

text: as

text: the

text: “Uboot

text: of

text: G¨ottingen”

text: which

text: was

text: pressurized

text: with

text: sulfur

text: hexaﬂuoride

text: (SF

text: at

text: an

text: average

text: temperature

text: T

text: and

text: m

text: at

text: pressures

text: that

text: ranged

text: from

text: to

text: bars,

text: resulting

text: in

text: the

text: relatively

text: narrow

text: Prandtl-

text: number

text: range

text: from

text: to

text: over

text: the

text: Ra

text: range

text: from

text: to

text: (see

text: Fig.

text: of

text: Ahlers

text: et

text: al.

text: The

text: Uboot

text: had

text: a

text: volume

text: of

text: about

text: and

text: about

text: kg

text: of

text: SF

text: were

text: required

text: to

text: ﬁll

text: it

text: to

text: the

text: maximum

text: pressure.

text: The

text: sample

text: cell

text: had

text: aluminium

text: plates

text: at

text: the

text: top

text: and

text: bottom

text: that

text: were

text: separated

text: by

text: a

text: distance

text: L

text: cm

text: and

text: had

text: an

text: aspect

text: ratio

text: Γ

text: The

text: plates

text: were

text: leveled

text: relative

text: to

text: gravity

text: to

text: rad.

text: Three

text: sets

text: of

text: eight

text: thermistors,

text: one

text: each

text: at

text: the

text: heights

text: z/L

text: and

text: were

text: imbedded

text: in

text: the

text: sidewall

text: (Brown

text: et

text: al.

text: Since

text: the

text: LSC

text: carries

text: warm

text: ﬂuid

text: upward

text: at

text: one

text: side

text: of

text: the

text: cell

text: and

text: cold

text: ﬂuid

text: downward

text: at

text: the

text: opposite,

text: one

text: can

text: detect

text: the

text: azimuthal

text: orientation

text: of

text: the

text: LSC

text: by

text: ﬁtting

text: T

text: T

text: δ

text: cos

text: θ

text: f

text: w,k

text: k

text: to

text: all

text: eight

text: thermistors

text: at

text: one

text: speciﬁc

text: height

text: and

text: time.

text: The

text: index

text: “i”

text: stands

text: for

text: the

text: azimuthal

text: location

text: of

text: the

text: thermistors

text: and

text: takes

text: values

text: i

text: with

text: the

text: location

text: of

text: i

text: located

text: at

text: a

text: point

text: directly

text: East

text: of

text: the

text: sample

text: center

text: and

text: i

text: increasing

text: in

text: the

text: counter-

text: clockwise

text: direction

text: when

text: viewed

text: from

text: above.

text: The

text: index

text: k

text: denotes

text: the

text: vertical

text: location

text: of

text: the

text: thermistor

text: and

text: will

text: in

text: the

text: following

text: have

text: letters

text: b

text: (z

text: m

text: (z

text: and

text: t

text: (z

text: The

text: orientation

text: of

text: the

text: up-ﬂow

text: is

text: given

text: by

text: the

text: phase

text: θ

text: From

text: the

text: eight

text: temperatures

text: at

text: a

text: given

text: k

text: we

text: computed

text: the

text: coeﬃcients

text: of

text: the

text: lowest

text: four

text: Fourier

text: modes

text: (Stevens

text: et

text: al.

text: Weiss

text: Ahlers

text: This

text: yielded

text: all

text: eight

text: Fourier

text: ##LTLine##


text: 6 Xiaozhou He1,2,6, Eberhard Bodenschatz1,3,4,6, and Guenter Ahlers1,5,6 0.004 (a) 1000 (b) 400 0.079 0.003 (d) 0.1 (c) 0.075 0.08 1012 1013 1014 1012 1013 1014 Ra Ra (cid:39) Figure 4. (a): Re Pr1.2 as a function of Ra for z/L = 0.50 and Pr 0.8. Solid line: a θm power-law ﬁt with an exponent of 0.40 to the circles with red dots. Dashed line: 0.124Ra0.278 as determined using the data from (Brown & Ahlers 2006b) for Pr = 4.38. (b): Reduced Reynolds (cid:39) number Re Pr1.2/Ra0.4 as a function of Ra for Pr 0.8. Dashed line: an extrapolation of θm the results for Pr = 4.38 of Brown & Ahlers (2006b). The horizontal solid bar corresponds to 0.00364. (c): The reduced velocity-ﬂuctuation Reynolds number Re /Ra0.5 as a function of Ra V ≤ ∗ × from He et al. (2015). The solid line is the power-law ﬁt to the data with Ra Ra = 2 1013, 1 ∗ which gave an exponent of 0.43. The dashed line is the extrapolation of that ﬁt beyond Ra . 1 The horizontal solid bar corresponds to 0.0863. (d): The reduced Nusselt number Nu/Ra0.321 as a function of Ra on logarithmic scales. The solid line is the power-law ﬁt Nu = 0.0159Ra0.370 to the red squares, and the horizontal dashed line is at Nu/Ra0.321 = 0.0763. In all panels vertical ∗ (cid:39) × ∗ (cid:39) × dotted lines represent Ra 2 1013 and Ra 8 1013. 1 2 values of D from a ﬁt of Eq. 3.3 to data at small nδt such as those shown as solid θm symbols in Fig. 3(a). The corresponding results for Re are shown in Fig. 3(b). θm The dashed line in Fig. 3(b) is an extrapolation of the result obtained by Brown & ∼ Ahlers (2006b) for Ra < 1011 and Pr = 4.38. Although our data are at much larger Ra, we note that this extrapolation can be made to pass through our data for classical RBC × ∼ (Ra < 2 1013) when Re Pr1.2 is plotted vs. Ra, as in Fig. 4(a). We note that the θm same dependence on Pr1.2 was found also for the velocity-ﬂuctuation Reynolds number ∼ Re in the range Pr < 7 (He et al. 2015). V ∗ ∗ 3.3. Nu, Re , and Re , and the ultimate-state transition-range from Ra to Ra V θ 1 2 In Fig. 4(a) one sees that there is a change in the Ra dependence of Re Pr1.2 near × θm Ra = 2 1013 which we identify with the beginning of the transition range to the ∗ ultimate state of RBC at Ra (left vertical dotted line). This becomes more obvious 1 when the reduced data for Re Pr1.2/Ra0.4 are plotted, as in Fig. 4(b). There one also θm ∗ (cid:39) × sees that the scatter in the data suddenly decreases as Ra exceeds Ra 8 1013, as (cid:104) (cid:105) 2 was the case for Re [see e.g. He et al. (2012b)], and for E /E in Fig. 2(c) shown V k,1 k,tot above. The origin of the scatter in the transition region is unclear. One reason could be that between the classical and ultimate states the system can choose from two or more states with diﬀerent boundary-layer conﬁgurations where the BLs are, spatially, neither entirely laminar nor entirely turbulent. Above the transition the ﬂuctuations in the data are small and the state seems uniquely deﬁned. Further experiments are needed to better understand the transition region. 2.1rP mθeR 123.0aR / uN 4.0aR / 2.1rP mθeR 2/1aR / VeR ##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##

text: Xiaozhou

label: author

text: Eberhard

text: and

text: Guenter

text: (a)

text: (b)

text: (d)

text: (c)

text: Ra

text: Ra

text: Figure

text: (a):

text: Re

text: as

text: a

text: function

text: of

text: Ra

text: for

text: z/L

text: and

text: Pr

text: Solid

text: line:

text: a

text: θm

text: power-law

text: ﬁt

text: with

text: an

text: exponent

text: of

text: to

text: the

text: circles

text: with

text: red

text: dots.

text: Dashed

text: line:

text: as

text: determined

text: using

text: the

text: data

text: from

text: (Brown

text: Ahlers

text: for

text: Pr

text: (b):

text: Reduced

text: Reynolds

text: number

text: Re

text: as

text: a

text: function

text: of

text: Ra

text: for

text: Pr

text: Dashed

text: line:

text: an

text: extrapolation

text: of

text: θm

text: the

text: results

text: for

text: Pr

text: of

text: Brown

text: Ahlers

text: The

text: horizontal

text: solid

text: bar

text: corresponds

text: to

text: (c):

text: The

text: reduced

text: velocity-ﬂuctuation

text: Reynolds

text: number

text: Re

text: as

text: a

text: function

text: of

text: Ra

text: V

text: from

text: He

text: et

text: al.

text: The

text: solid

text: line

text: is

text: the

text: power-law

text: ﬁt

text: to

text: the

text: data

text: with

text: Ra

text: Ra

text: which

text: gave

text: an

text: exponent

text: of

text: The

text: dashed

text: line

text: is

text: the

text: extrapolation

text: of

text: that

text: ﬁt

text: beyond

text: Ra

text: The

text: horizontal

text: solid

text: bar

text: corresponds

text: to

text: (d):

text: The

text: reduced

text: Nusselt

text: number

text: as

text: a

text: function

text: of

text: Ra

text: on

text: logarithmic

text: scales.

text: The

text: solid

text: line

text: is

text: the

text: power-law

text: ﬁt

text: Nu

text: to

text: the

text: red

text: squares,

text: and

text: the

text: horizontal

text: dashed

text: line

text: is

text: at

text: In

text: all

text: panels

text: vertical

text: dotted

text: lines

text: represent

text: Ra

text: and

text: Ra

text: values

text: of

text: D

text: from

text: a

text: ﬁt

text: of

text: Eq.

text: to

text: data

text: at

text: small

text: nδt

text: such

text: as

text: those

text: shown

text: as

text: solid

text: θm

text: symbols

text: in

text: Fig.

text: The

text: corresponding

text: results

text: for

text: Re

text: are

text: shown

text: in

text: Fig.

text: θm

text: The

text: dashed

text: line

text: in

text: Fig.

text: is

text: an

text: extrapolation

text: of

text: the

text: result

text: obtained

text: by

text: Brown

text: Ahlers

text: for

text: Ra

text: and

text: Pr

text: Although

text: our

text: data

text: are

text: at

text: much

text: larger

text: Ra,

text: we

text: note

text: that

text: this

text: extrapolation

text: can

text: be

text: made

text: to

text: pass

text: through

text: our

text: data

text: for

text: classical

text: RBC

text: (Ra

text: when

text: Re

text: is

text: plotted

text: vs.

text: Ra,

text: as

text: in

text: Fig.

text: We

text: note

text: that

text: the

text: θm

text: same

text: dependence

text: on

text: was

text: found

text: also

text: for

text: the

text: velocity-ﬂuctuation

text: Reynolds

text: number

text: Re

text: in

text: the

text: range

text: Pr

text: (He

text: et

text: al.

text: V

text: Nu,

text: Re

text: and

text: Re

text: and

text: the

text: ultimate-state

text: transition-range

text: from

text: Ra

text: to

text: Ra

text: V

text: θ

text: In

text: Fig.

text: one

text: sees

text: that

text: there

text: is

text: a

text: change

text: in

text: the

text: Ra

text: dependence

text: of

text: Re

text: near

text: θm

text: Ra

text: which

text: we

text: identify

text: with

text: the

text: beginning

text: of

text: the

text: transition

text: range

text: to

text: the

text: ultimate

text: state

text: of

text: RBC

text: at

text: Ra

text: (left

text: vertical

text: dotted

text: line).

text: This

text: becomes

text: more

text: obvious

text: when

text: the

text: reduced

text: data

text: for

text: Re

text: are

text: plotted,

text: as

text: in

text: Fig.

text: There

text: one

text: also

text: θm

text: sees

text: that

text: the

text: scatter

text: in

text: the

text: data

text: suddenly

text: decreases

text: as

text: Ra

text: exceeds

text: Ra

text: as

text: was

text: the

text: case

text: for

text: Re

text: [see

text: e.g.

text: He

text: et

text: al.

text: and

text: for

text: E

text: /E

text: in

text: Fig.

text: shown

text: V

text: k,tot

text: above.

text: The

text: origin

text: of

text: the

text: scatter

text: in

text: the

text: transition

text: region

text: is

text: unclear.

text: One

text: reason

text: could

text: be

text: that

text: between

text: the

text: classical

text: and

text: ultimate

text: states

text: the

text: system

text: can

text: choose

text: from

text: two

text: or

text: more

text: states

text: with

text: diﬀerent

text: boundary-layer

text: conﬁgurations

text: where

text: the

text: BLs

text: are,

text: spatially,

text: neither

text: entirely

text: laminar

text: nor

text: entirely

text: turbulent.

text: Above

text: the

text: transition

text: the

text: ﬂuctuations

text: in

text: the

text: data

text: are

text: small

text: and

text: the

text: state

text: seems

text: uniquely

text: deﬁned.

text: Further

text: experiments

text: are

text: needed

text: to

text: better

text: understand

text: the

text: transition

text: region.

text: mθeR

text: uN

text: mθeR

text: VeR

text: ##LTFigure##

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: ##LTLine##

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: ##LTLine##


text: 12 G. SIMONS, C. TARDIF, D. WEHLAU   (cid:88)   − · D (4) (X X ) (u, v) /θ( ) = 0G , u,v v,u D(H) ∈ (u,v) A(H) D and the -signature system is the system consisting of the ﬂow constraints (1), D the parity constraint (2) and the -signature constraint (4). (cid:48) Proposition 11. Let H, H be graphs such that there exists a homomorphism → (cid:48) D ψ : H H . If the -signature system has integer solutions on H, then it has (cid:48) integer solutions on H . → (cid:48) Proof. First note that ψˆ : A(H) A(H ) deﬁned by ψˆ(u, v) = (ψ(u), ψ(v)) extends to a group homomo(cid:80)rphism of ZA(H) to ZA(H(cid:48)), which we also denote ψˆ. We will show that ψˆ maps 0Z /θ(D) to 0Z (cid:48) /θ(D). Inde(cid:80)ed any generator of D A(H) A(·H ) D 0Z /θ( ) is of the form ∈ φ(u, v) (f(u), f(v)), where (D, φ) is in A(H) (u,v) A(D) · and f is a homomorphism of D in H. It is mapped by ψ to ∈ φ(u, v) ◦ ◦ D ◦ (u,v) A(D) (ψ f(u), ψ f(v)). The latter is in 0Z (cid:48) /θ( ), since ψ f is a homomorphism of D to H(cid:48). Therefore ψˆ induces a grouAp(Hh)omomorphism of ZA(H)/θ(D) = GD(H) to ZA(H(cid:48))/θ(D) = GD (cid:48) , which we again call ψˆ. (H ) ∈ D Now suppose that X (cid:88), (u, v) A(H) and N are solutions to the -signature (cid:48) (cid:48)u,∈v (cid:48) system of H. For (u , v ) A(H ), put (cid:48) { (cid:48) (cid:48)} X (cid:48) (cid:48) = X : ψ(u, v) = u and ψ(v) = v . u ,v u,v (cid:48) (cid:48) (cid:48) ∈ (cid:48) (cid:48) Claim 1. X (cid:88)(cid:48) (cid:48), (u , v ) A(H ) and N s(cid:88)atisfy th(cid:88)e ﬂow constraint (1) on H . u ,v(cid:48) ∈ (cid:48) Indeed for u V (H ) we have (cid:48) − (cid:48) − (X (cid:48) (cid:48) X (cid:48) (cid:48)) = (X X ) = 0. u ,v v ,u u,v v,u v(cid:48)∈NH(cid:48)(u(cid:48)) u∈ψ−1(u(cid:48)) v∈NH(u) (cid:48) (cid:48) (cid:48) ∈ (cid:48) (cid:48) Claim 2. X (cid:48) (cid:48), (u (cid:88), v ) A(H ) and N satisf(cid:88)y the parity constraint (2) on H . u ,v Indeed we have (cid:48) − − X (cid:48) (cid:48) 2N = X 2N = 1. u ,v u,v (cid:48) (cid:48) ∈ (cid:48) ∈ (u ,v ) A(H ) (u,v) A(H) (cid:48) (cid:48) (cid:48) ∈ (cid:48) D Claim 3. X (cid:48) (cid:48), (u , v ) A(H ) and N satisfy the -signature constraint (4) on (cid:48) u ,v H .   (cid:88) Indeed since   − · D (X X ) (u, v) /θ( ) = 0G u,v v,u D(H) ∈ (u,v) A(H)   and ψˆ is a group (cid:88)homomorphism, we have   − · D (Xu,v Xv,u) (ψ(u), ψ(v)) /θ( ) = 0GD(H(cid:48)). ∈ (u,v) A(H)

text: G.

text: SIMONS,

text: C.

text: TARDIF,

text: D.

text: WEHLAU

label: equation

text: D

text: (X

text: X

text: (u,

text: v)

text: /θ(

text: u,v

text: v,u

text: D(H)

text: (u,v)

text: A(H)

text: D

text: and

text: the

text: -signature

text: system

text: is

text: the

text: system

text: consisting

text: of

text: the

text: ﬂow

text: constraints

text: D

text: the

text: parity

text: constraint

text: and

text: the

text: -signature

text: constraint

text: Proposition

text: Let

text: H,

text: H

text: be

text: graphs

text: such

text: that

text: there

text: exists

text: a

text: homomorphism

text: D

text: ψ

text: H

text: H

text: If

text: the

text: -signature

text: system

text: has

text: integer

text: solutions

text: on

text: H,

text: then

text: it

text: has

text: integer

text: solutions

text: on

text: H

text: Proof.

text: First

text: note

text: that

text: ψˆ

text: A(H)

text: A(H

text: deﬁned

text: by

text: ψˆ(u,

text: v)

text: (ψ(u),

text: ψ(v))

text: extends

text: to

text: a

text: group

text: of

text: ZA(H)

text: to

text: which

text: we

text: also

text: denote

text: ψˆ.

text: We

text: will

text: show

text: that

text: ψˆ

text: maps

text: /θ(D)

text: to

text: /θ(D).

text: any

text: generator

text: of

text: D

text: A(H)

text: A(·H

text: D

text: /θ(

text: is

text: of

text: the

text: form

text: φ(u,

text: v)

text: (f(u),

text: f(v)),

text: where

text: (D,

text: φ)

text: is

text: in

text: A(H)

text: (u,v)

text: A(D)

text: and

text: f

text: is

text: a

text: homomorphism

text: of

text: D

text: in

text: H.

text: It

text: is

text: mapped

text: by

text: ψ

text: to

text: φ(u,

text: v)

text: D

text: (u,v)

text: A(D)

text: (ψ

text: f(u),

text: ψ

text: f(v)).

text: The

text: latter

text: is

text: in

text: /θ(

text: since

text: ψ

text: f

text: is

text: a

text: homomorphism

text: of

text: D

text: to

text: Therefore

text: ψˆ

text: induces

text: a

text: grouAp(Hh)omomorphism

text: of

text: ZA(H)/θ(D)

text: GD(H)

text: to

text: GD

text: which

text: we

text: again

text: call

text: ψˆ.

text: (H

text: D

text: Now

text: suppose

text: that

text: X

text: (u,

text: v)

text: A(H)

text: and

text: N

text: are

text: solutions

text: to

text: the

text: -signature

text: system

text: of

text: H.

text: For

text: (u

text: v

text: A(H

text: put

text: X

text: X

text: ψ(u,

text: v)

text: u

text: and

text: ψ(v)

text: v

text: u

text: ,v

text: u,v

text: Claim

text: X

text: (u

text: v

text: A(H

text: and

text: N

text: ﬂow

text: constraint

text: on

text: H

text: u

text: Indeed

text: for

text: u

text: V

text: (H

text: we

text: have

text: (X

text: X

text: (X

text: X

text: u

text: ,v

text: v

text: ,u

text: u,v

text: v,u

text: v∈NH(u)

text: Claim

text: X

text: (u

text: v

text: A(H

text: and

text: N

text: the

text: parity

text: constraint

text: on

text: H

text: u

text: ,v

text: Indeed

text: we

text: have

text: X

text: X

text: u

text: ,v

text: u,v

text: (u

text: ,v

text: A(H

text: (u,v)

text: A(H)

text: D

text: Claim

text: X

text: (u

text: v

text: A(H

text: and

text: N

text: satisfy

text: the

text: -signature

text: constraint

text: on

text: u

text: ,v

text: H

text: Indeed

text: since

text: D

text: (X

text: X

text: (u,

text: v)

text: /θ(

text: u,v

text: v,u

text: D(H)

text: (u,v)

text: A(H)

text: and

text: ψˆ

text: is

text: a

text: group

text: we

text: have

text: D

text: (Xu,v

text: Xv,u)

text: (ψ(u),

text: ψ(v))

text: /θ(

text: (u,v)

text: A(H)


text: 4 G. SIMONS, C. TARDIF, D. WEHLAU We will suppose that H is connected, thus all the constant maps are in the same connected component of HG, which we call the connected component of the ≥ constants. Suppose that H satisﬁes coind(B(H)) + 2 4. Then by Corollary 3, ∈ K there exists an odd cycle C such that for some integer m, M (C) admits a m 4 homomorphism to H. By the above remark, this is equivalent to the existence of a loop in the connected component of the constants in HC. Since HC is ﬁnite, such a loop could be found in ﬁnite time. Thus for a ﬁxed C, the question of the existence of a m such that M (C) admits a homomorphism to H is decidable. However to m ≥ settle the question as to whether coind(B(H)) + 2 4, we need to look at the inﬁnitely many possible choices for C, so the search is still inﬁnite. Our next step is to look for easily computable invariants that are constant on the components of HC. For a graph H, let A(H) denote the set of its arcs. That is, for ∈ F∗ [u, v] E(H), A(H) contains the two arcs (u, v) and (v, u). Let (A(H)) be the free group generated by the elements of A(H). Let C be an odd cycle with vertex-set Z { } = 0, . . . , 2n . The elements of HC are not necessarily homomorphisms of C 2n+1 to H. However, if (f, g) is an arc of HC, then f(0), g(1), f(2), . . . , g(2n) is a closed F∗ walk of length 4n + 2 in H. We deﬁne the (A(H))-signature σF∗ (f, g) of (A(H)) (f, g) as · − σF∗ (f, g) = Π2n (f(2i), g(2i + 1)) (f(2i + 2), g(2i + 1)) 1. (A(H)) i=0 The indices are of course taken modulo 2n + 1. Also since the product is non- commutative, it is necessary to specify that the product is developed left to right: · · · · · · Πk x = x x x rather than x x − x . Thus σF∗ (f, g) is essentially i=0 i 0 1 k k k 1 0 (A(H)) a list of the arcs in the closed walk generated by f and g, with trivial simpliﬁcations. ∗ F∗ We deﬁne a congruence θ on (A(H)) as follows: If u, v, w, x is a 4-cycle of H, we put · − ∗ · − (u, v) (w, v) 1 θ (u, x) (w, x) 1, that is, · − · · − ∗ (u, v) (w, v) 1 (w, x) (u, x) 1 θ 1F . (A(H)) G∗ The group (H) is deﬁned by G∗ F∗ ∗ (H) := (A(H))/θ . (cid:48) Now if (f, g) and (f, g ) are two arcs of HC, then the corresponding terms in the product deﬁning their signatures are congruent: · − ∗ (cid:48) · (cid:48) − (f(2i), g(2i+1)) (f(2i+2), g(2i+1)) 1θ (f(2i), g (2i+1)) (f(2i+2), g (2i+1)) 1 ∗ (cid:48) ∗ for i = 0, . . . , 2n. Therefore σF∗ (f, g)/θ = σF∗ (f, g )/θ . We deﬁne G∗ (A(H)) (A(H)) the (H)-signature σG∗ (f) of a non-isolated vertex f in HC by (H) ∗ σG∗ (f) = σF∗ (f, g)/θ (H) (A(H)) for any neighbour g of f. → Proposition 5. Let C be an odd cycle and f : C H a homomorphism. If f belongs to the connected component of the constants in HC, then σG∗ (f) = 1G∗ . (H) (H) Proof. If f is a constant map in HC, then for any neighbour g of f we have

text: G.

text: SIMONS,

text: C.

text: TARDIF,

text: D.

text: WEHLAU

text: We

text: will

text: suppose

text: that

text: H

text: is

text: connected,

text: thus

text: all

text: the

text: constant

text: maps

text: are

text: in

text: the

text: same

text: connected

text: component

text: of

text: HG,

text: which

text: we

text: call

text: the

text: connected

text: component

text: of

text: the

text: constants.

text: Suppose

text: that

text: H

text: satisﬁes

text: coind(B(H))

text: Then

text: by

text: Corollary

text: K

text: there

text: exists

text: an

text: odd

text: cycle

text: C

text: such

text: that

text: for

text: some

text: integer

text: m,

text: M

text: (C)

text: admits

text: a

text: m

text: homomorphism

text: to

text: H.

text: By

text: the

text: above

text: remark,

text: this

text: is

text: equivalent

text: to

text: the

text: existence

text: of

text: a

text: loop

text: in

text: the

text: connected

text: component

text: of

text: the

text: constants

text: in

text: HC.

text: Since

text: HC

text: is

text: ﬁnite,

text: such

text: a

text: loop

text: could

text: be

text: found

text: in

text: ﬁnite

text: time.

text: Thus

text: for

text: a

text: ﬁxed

text: C,

text: the

text: question

text: of

text: the

text: existence

text: of

text: a

text: m

text: such

text: that

text: M

text: (C)

text: admits

text: a

text: homomorphism

text: to

text: H

text: is

text: decidable.

text: However

text: to

text: m

text: settle

text: the

text: question

text: as

text: to

text: whether

text: coind(B(H))

text: we

text: need

text: to

text: look

text: at

text: the

text: inﬁnitely

text: many

text: possible

text: choices

text: for

text: C,

text: so

text: the

text: search

text: is

text: still

text: inﬁnite.

text: Our

text: next

text: step

text: is

text: to

text: look

text: for

text: easily

text: computable

text: invariants

text: that

text: are

text: constant

text: on

text: the

text: components

text: of

text: HC.

text: For

text: a

text: graph

text: H,

text: let

text: A(H)

text: denote

text: the

text: set

text: of

text: its

text: arcs.

text: That

text: is,

text: for

text: F∗

text: [u,

text: v]

text: E(H),

text: A(H)

text: contains

text: the

text: two

text: arcs

text: (u,

text: v)

text: and

text: (v,

text: u).

text: Let

text: (A(H))

text: be

text: the

text: free

text: group

text: generated

text: by

text: the

text: elements

text: of

text: A(H).

text: Let

text: C

text: be

text: an

text: odd

text: cycle

text: with

text: vertex-set

text: Z

text: The

text: elements

text: of

text: HC

text: are

text: not

text: necessarily

text: homomorphisms

text: of

text: C

text: to

text: H.

text: However,

text: if

text: (f,

text: g)

text: is

text: an

text: arc

text: of

text: HC,

text: then

text: is

text: a

text: closed

text: F∗

text: walk

text: of

text: length

text: in

text: H.

text: We

text: deﬁne

text: the

text: (A(H))-signature

text: σF∗

text: (f,

text: g)

text: of

text: (A(H))

text: (f,

text: g)

text: as

text: σF∗

text: (f,

text: g)

text: (A(H))

text: The

text: indices

text: are

text: of

text: course

text: taken

text: modulo

text: Also

text: since

text: the

text: product

text: is

text: non-

text: commutative,

text: it

text: is

text: necessary

text: to

text: specify

text: that

text: the

text: product

text: is

text: developed

text: left

text: to

text: right:

text: Πk

text: x

text: x

text: x

text: x

text: rather

text: than

text: x

text: x

text: x

text: Thus

text: σF∗

text: (f,

text: g)

text: is

text: essentially

text: i

text: k

text: k

text: k

text: (A(H))

text: a

text: list

text: of

text: the

text: arcs

text: in

text: the

text: closed

text: walk

text: generated

text: by

text: f

text: and

text: g,

text: with

text: trivial

text: simpliﬁcations.

text: F∗

text: We

text: deﬁne

text: a

text: congruence

text: θ

text: on

text: (A(H))

text: as

text: follows:

text: If

text: u,

text: v,

text: w,

text: x

text: is

text: a

text: of

text: H,

text: we

text: put

text: (u,

text: v)

text: (w,

text: v)

text: θ

text: (u,

text: x)

text: (w,

text: x)

text: that

text: is,

text: (u,

text: v)

text: (w,

text: v)

text: (w,

text: x)

text: (u,

text: x)

text: θ

text: (A(H))

text: G∗

text: The

text: group

text: (H)

text: is

text: deﬁned

text: by

text: G∗

text: F∗

text: (H)

text: (A(H))/θ

text: Now

text: if

text: (f,

text: g)

text: and

text: (f,

text: g

text: are

text: two

text: arcs

text: of

text: HC,

text: then

text: the

text: corresponding

text: terms

text: in

text: the

text: product

text: deﬁning

text: their

text: signatures

text: are

text: congruent:

text: 1θ

text: g

text: g

text: for

text: i

text: Therefore

text: σF∗

text: (f,

text: g)/θ

text: σF∗

text: (f,

text: g

text: )/θ

text: We

text: deﬁne

text: G∗

text: (A(H))

text: (A(H))

text: the

text: (H)-signature

text: σG∗

text: (f)

text: of

text: a

text: non-isolated

text: vertex

text: f

text: in

text: HC

text: by

text: (H)

text: σG∗

text: (f)

text: σF∗

text: (f,

text: g)/θ

text: (H)

text: (A(H))

text: for

text: any

text: neighbour

text: g

text: of

text: f.

text: Proposition

text: Let

text: C

text: be

text: an

text: odd

text: cycle

text: and

text: f

text: C

text: H

text: a

text: homomorphism.

text: If

text: f

text: belongs

text: to

text: the

text: connected

text: component

text: of

text: the

text: constants

text: in

text: HC,

text: then

text: σG∗

text: (f)

text: (H)

text: (H)

text: Proof.

text: If

text: f

text: is

text: a

text: constant

text: map

text: in

text: HC,

text: then

text: for

text: any

text: neighbour

text: g

text: of

text: f

text: we

text: have


text: 9 q over both ρ and σ, subject to constraints (17) and (18), (10), and prove a lower bound on the inference error q s s which yields ([18], sec. 5.2), for any convex (ρ, σ): (cid:2) (cid:3) σρoopptt((xx)) == −−MMqsopt[[ −−EE(cid:15)qsopt ]]((xx)),, ((2289)) qs ≥ αJ [ (cid:15)qs ]1+ J s0 . (31) qdopt sqdopt This results reﬂects a fundamental generalization of the high-dimensional CR bound (19) that includes infor- where qopt and qopt satisfy s d mation about the signal distribution P that can be opti- s (cid:104) (cid:105) mally exploited by a regularizer σ. Since J [ (cid:15) ] < J [ (cid:15) ], qs 1 by the data processing inequality for Fisher information, qopt = , qopt = qMMSE(qopt), (30) d αJ (cid:15) s s d this high dimensional lower bound is larger than th→e low- qsopt dimensional one (10) under the replacement α N. Thus, as in the unregularized case (19), the price paid and the function qMMSE is deﬁned in (9). Again, the for even optimal high-dimensional regularized inference s validity of (28)-(29) under the RS assumption requires at ﬁnite measurement density, relative to scalar Bayesian convexity of ρopt and σopt. Convexity of the signal and inference at asymptotically inﬁnite density, is increased noise energies, E and E are suﬃcient to guarantee con- error due to the presence of additional gaussian noise s (cid:15) vexity of ρopt and σopt, and so for this class of signal with dimensionality dependent variance qsopt. and noise, with log concave distributions, (28)-(29) yields 4. Optimal high dimensional inference smoothly interpolates between MAP and quadratic inference an optimal inference procedure. However, by judicious applications of the Cauchy-Schwarz inequality, we prove ([18], sec. 4.1) that even for non-convex E and E , the The optimal inference procedure (28)-(29) is a smoothed s (cid:15) inference error q for any convex procedure (ρ, σ) must version of MAP inference (see Fig. 4C for an example of s exceed qsopt in (30). This result yields a fundamental limit smoothing), where the MAP choices ρ = E(cid:15) and σ = Es on the performance of any convex inference procedure of are smoothed over scales qopt and qopt respectively to s → ∞ d the form (1) in high dimensions. obtain ρopt and σopt. As α , both qopt and qopt s → d Intriguingly, by comparing the optimal achievable high approach 0 at the same rate, implying ρopt E and → (cid:15) dimensional M-estimation performance qopt in (30) to σopt E . Thus at high measurement density, MAP s s the asymptotic performance of low dimensional scalar inference is the optimal M-estimator. This conclusion Bayesian inference in (8) and (9), we ﬁnd a striking par- is intuitively reasonable because at high measurement allel. In particular, qopt corresponds to the low dimen- densities, the mode of the posterior distribution over s sional asymptotic MMSE in a scal(cid:112)ar estimation problem the signal, returned by the MAP estimate, is typically where the eﬀective number of measurements N = α and close to the mean of the posterior distribution, which the noise (cid:15) is further corrupted by additional Gaussian is the optimal MMSE estimate amongst all inference → noise of variance qopt ((cid:15) (cid:15) + qoptz). The correc- procedures. s s → → (cid:104) (cid:105) tion to the low dimensional scalar asymptotics (9), valid Alternatively, as α 0, qopt s2 from below, while s only at large N, in the high dimensional regime at ﬁnite qopt diverges as 1 . The divergence of qopt implies that d α d measurement density α, is obtained by self-consistently σ in (29) approaches a quadratic. Thus, remarkably, opt solving for qopt in (30). In essence, at ﬁnite measure- at low measurement density, simple quadratic regular- s ment density, there is irreducible error in estimating the ization, independent of the signal distribution, becomes signal, qopt. This error contributes to the eﬀective Gaus- an optimal inference procedure. Furthermore, in the low s (cid:104) (cid:105) (cid:28) (cid:104) (cid:105) sian noise qopt in the scalar MFT estimation problem for density plus high SNR limit, where (cid:15)2 s2 , ρopt d the signal, shown in Fig. 3B, where the proximal map also approaches a quadratic. Thus overall, optimal high becomes the Bayesian posterior mean map in the optimal dimensional inference at high SNR interpolates between case. On the otherhand, this irreducible, extra gaussian MAP and quadratic inference as the measurement den- noise is absent in low dimensions (compare LHS of (30) sity decreases. In Figure 6 we demonstrate, for Laplacian to (8)). This irreducible error qopt can be found by self- signal and noise, that optimal inference outperforms both s consistently solving for it in the RHS of (30). Finally, MAP and quadratic inference at all α, approaching the as a simple point, we note that direct calculation reveals former at large α and the latter at small α. that (30) reduces to (21) when the signal and noise are 5. A relation between optimal high dimensional inference of both Gaussian distributed, as expected, since optimal signal, and low-dimensional Bayesian inference of noise quadratic inference is the best procedure for Gaussian signal and noise. There is an interesting connection between optimal Furthermore, using the fact that the equalities in (30) high dimensional inference, and low-dimensional scalar become inequalities for non-optimal procedures ([18], sec- Bayesian inference. Indeed, when ρ and σ take their tion 4.2), we can derive a high dimensional analogue of optimal forms in (28) and (29), then the proximal de- ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: q

text: over

text: both

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text: σ,

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text: ]]((xx)),,

text: qs

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text: price

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text: function

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text: the

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text: A

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text: optimal

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text: optimal

text: signal,

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text: low-dimensional

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text: the

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text: optimal

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text: using

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text: fact

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text: dimensional

text: inference,

text: and

text: low-dimensional

text: scalar

text: become

text: inequalities

text: for

text: non-optimal

text: procedures

text: sec-

text: Bayesian

text: inference.

text: Indeed,

text: when

text: ρ

text: and

text: σ

text: take

text: their

text: tion

text: we

text: can

text: derive

text: a

text: high

text: dimensional

text: analogue

text: of

text: optimal

text: forms

text: in

text: and

text: then

text: the

text: proximal

text: de-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##