text: Modiﬁed Schr¨odinger-Newton equation probe space-time below the string length scale. In loop quantum gravity, the existence of a mini- mum length has a very interesting consequence, as it turns the big bang into a big bounce [14,15]. Furthermore, the arguments from black hole physics suggest that any theory of quantum gravity must be equipped with a minimum length scale [16, 17], due to the fact that the energy required to probe any region of space below the Plank length is greater than the energy required to create a mini black hole in that region of space. The minimal length exists in other subjects too; such as, path integral quantum gravity [18, 19], special relativity [20], doubly special relativity [21–23], etc. In short, the existence of minimal measurable length, by now, has become a universal feature in almost all approaches of quantum gravity. Moreover, some Gedankenexperiments and thought experiments [8, 24, 25] in the spirit of black hole physics have also supported the idea. For fur- ther informations on the subject one may follow some review articles devoted to the subject, for instance, [26–28]. In spite of having several serious proposals for quantizing the general relativity, unfortunately we do not have a fully consistent quantum theory of gravity yet. This has motivated the study of semi-classical quantum gravity (SCQG), where the gravitational ﬁled is treated as a background classical ﬁeld, and the matter ﬁelds are treated quantum mechanically. Under such approximation, | (cid:105) if ψ is the wave function of the matter ﬁeld, the Einstein tensor G can be obtained in terms of the µν (cid:104) | | (cid:105) quantum mechanical energy-momentum tensor for matter ﬁelds T µν as G = 8πG ψ T µν ψ /c4. µν However, Newtonian gravity has been observed to be the correct approximation to general relativ- ity till the smallest length scale (0.4mm) to which general relativity has been tested [29]. Thus, at small distances, it is expected that the semi-classical approximation can be described by a Schr¨odinger-Newton equation [30–34], which is the nonrelativistic limit of the Dirac equation and the Klein-Gordon equation with a classical Newtonian potential [35]. It has been proposed that the Schr¨odinger-Newton equation can be utilized to explore various interesting properties of gravita- tional systems at the given length scale [36–38]. Some interesting consequences may follow from the combination of the above two frameworks, namely, the GUP and SCQG, and it is allowed since both of the eﬀects occur at small scales. Technically, this can be achieved by imposing the minimal length structure into the semi-classical scheme of gravity by means of deforming the Schr¨odinger-Newton equation in accordance with the laws of GUP. The most interesting fact is that the new theory resulting from the combination of GUP and SCQG may provide an intermediate theory between a full quantum theory of gravity and the SCQG and, this is precisely the issue that we address in the present manuscript. Recently, it has been suggested that both the Schr¨odinger-Newton equation [39, 40] and the deformation of quantum mechanical structure by the GUP [41] can be tested experimentally by using the opto- mechanical systems. Therefore, it becomes important to understand the eﬀects of the combined theory, especially when there is a strong viability that the theory may be tested by using a similar type of experimental set-up in near future. 2 GUP and minimal length Let us commence by introducing a particular version of modiﬁed commutation relation between the position and momentum operators X, P [3,4] (cid:126) [Xi, Pj] = i (δij + βδijP 2 + 2βPiPj), [Xi, Xj] = [Pi, P(cid:80)j] = 0, (2.1) with β having the dimension of inverse squared momentum and, P 2 = 3 P 2. As obviously, → j=1 j in the limit β 0, the deformed relations (2.1) reduce to the standard canonical commutation (cid:126) relations [x , p ] = i δ . However, there exist many other similar type of deformations in the i j ij literature, which have been used to investigated many interesting phenomena in diﬀerent contexts; 2 ##LTLine##

text: Modiﬁed

text: Schr¨odinger-Newton

text: equation

text: probe

text: space-time

text: below

text: the

text: string

text: length

text: scale.

text: In

text: loop

text: quantum

text: gravity,

text: the

text: existence

text: of

text: a

text: mini-

text: mum

text: length

text: has

text: a

text: very

text: interesting

text: consequence,

text: as

text: it

text: turns

text: the

text: big

text: bang

text: into

text: a

text: big

text: bounce

text: Furthermore,

text: the

text: arguments

text: from

text: black

text: hole

text: physics

text: suggest

text: that

text: any

text: theory

text: of

text: quantum

text: gravity

text: must

text: be

text: equipped

text: with

text: a

text: minimum

text: length

text: scale

text: due

text: to

text: the

text: fact

text: that

text: the

text: energy

text: required

text: to

text: probe

text: any

text: region

text: of

text: space

text: below

text: the

text: Plank

text: length

text: is

text: greater

text: than

text: the

text: energy

text: required

text: to

text: create

text: a

text: mini

text: black

text: hole

text: in

text: that

text: region

text: of

text: space.

text: The

text: minimal

text: length

text: exists

text: in

text: other

text: subjects

text: too;

text: such

text: as,

text: path

text: integral

text: quantum

text: gravity

text: special

text: relativity

text: doubly

text: special

text: relativity

text: etc.

text: In

text: short,

text: the

text: existence

text: of

text: minimal

text: measurable

text: length,

text: by

text: now,

text: has

text: become

text: a

text: universal

text: feature

text: in

text: almost

text: all

text: approaches

text: of

text: quantum

text: gravity.

text: Moreover,

text: some

text: Gedankenexperiments

text: and

text: thought

text: experiments

text: in

text: the

text: spirit

text: of

text: black

text: hole

text: physics

text: have

text: also

text: supported

text: the

text: idea.

text: For

text: fur-

text: ther

text: informations

text: on

text: the

text: subject

text: one

text: may

text: follow

text: some

text: review

text: articles

text: devoted

text: to

text: the

text: subject,

text: for

text: instance,

text: In

text: spite

text: of

text: having

text: several

text: serious

text: proposals

text: for

text: quantizing

text: the

text: general

text: relativity,

text: unfortunately

text: we

text: do

text: not

text: have

text: a

text: fully

text: consistent

text: quantum

text: theory

text: of

text: gravity

text: yet.

text: This

text: has

text: motivated

text: the

text: study

text: of

text: semi-classical

text: quantum

text: gravity

text: (SCQG),

text: where

text: the

text: gravitational

text: ﬁled

text: is

text: treated

text: as

text: a

text: background

text: classical

text: ﬁeld,

text: and

text: the

text: matter

text: ﬁelds

text: are

text: treated

text: quantum

text: mechanically.

text: Under

text: such

text: approximation,

text: if

text: ψ

text: is

text: the

text: wave

text: function

text: of

text: the

text: matter

text: ﬁeld,

text: the

text: Einstein

text: tensor

text: G

text: can

text: be

text: obtained

text: in

text: terms

text: of

text: the

text: µν

text: quantum

text: mechanical

text: energy-momentum

text: tensor

text: for

text: matter

text: ﬁelds

text: T

text: µν

text: as

text: G

text: ψ

text: T

text: µν

text: ψ

text: µν

text: However,

text: Newtonian

text: gravity

text: has

text: been

text: observed

text: to

text: be

text: the

text: correct

text: approximation

text: to

text: general

text: relativ-

text: ity

text: till

text: the

text: smallest

text: length

text: scale

text: to

text: which

text: general

text: relativity

text: has

text: been

text: tested

text: Thus,

text: at

text: small

text: distances,

text: it

text: is

text: expected

text: that

text: the

text: semi-classical

text: approximation

text: can

text: be

text: described

text: by

text: a

text: Schr¨odinger-Newton

text: equation

text: which

text: is

text: the

text: nonrelativistic

text: limit

text: of

text: the

text: Dirac

text: equation

text: and

text: the

text: Klein-Gordon

text: equation

text: with

text: a

text: classical

text: Newtonian

text: potential

text: It

text: has

text: been

text: proposed

text: that

text: the

text: Schr¨odinger-Newton

text: equation

text: can

text: be

text: utilized

text: to

text: explore

text: various

text: interesting

text: properties

text: of

text: gravita-

text: tional

text: systems

text: at

text: the

text: given

text: length

text: scale

text: Some

text: interesting

text: consequences

text: may

text: follow

text: from

text: the

text: combination

text: of

text: the

text: above

text: two

text: frameworks,

text: namely,

text: the

text: GUP

text: and

text: SCQG,

text: and

text: it

text: is

text: allowed

text: since

text: both

text: of

text: the

text: eﬀects

text: occur

text: at

text: small

text: scales.

text: Technically,

text: this

text: can

text: be

text: achieved

text: by

text: imposing

text: the

text: minimal

text: length

text: structure

text: into

text: the

text: semi-classical

text: scheme

text: of

text: gravity

text: by

text: means

text: of

text: deforming

text: the

text: Schr¨odinger-Newton

text: equation

text: in

text: accordance

text: with

text: the

text: laws

text: of

text: GUP.

text: The

text: most

text: interesting

text: fact

text: is

text: that

text: the

text: new

text: theory

text: resulting

text: from

text: the

text: combination

text: of

text: GUP

text: and

text: SCQG

text: may

text: provide

text: an

text: intermediate

text: theory

text: between

text: a

text: full

text: quantum

text: theory

text: of

text: gravity

text: and

text: the

text: SCQG

text: and,

text: this

text: is

text: precisely

text: the

text: issue

text: that

text: we

text: address

text: in

text: the

text: present

text: manuscript.

text: Recently,

text: it

text: has

text: been

text: suggested

text: that

text: both

text: the

text: Schr¨odinger-Newton

text: equation

text: and

text: the

text: deformation

text: of

text: quantum

text: mechanical

text: structure

text: by

text: the

text: GUP

text: can

text: be

text: tested

text: experimentally

text: by

text: using

text: the

text: opto-

text: mechanical

text: systems.

text: Therefore,

text: it

text: becomes

text: important

text: to

text: understand

text: the

text: eﬀects

text: of

text: the

text: combined

text: theory,

text: especially

text: when

text: there

text: is

text: a

text: strong

text: viability

text: that

text: the

text: theory

text: may

text: be

text: tested

text: by

text: using

text: a

text: similar

text: type

text: of

text: experimental

text: set-up

text: in

text: near

text: future.

text: GUP

text: and

text: minimal

text: length

text: Let

text: us

text: commence

text: by

text: introducing

text: a

text: particular

text: version

text: of

text: modiﬁed

text: commutation

text: relation

text: between

text: the

text: position

text: and

text: momentum

text: operators

text: X,

text: P

text: [Xi,

text: Pj]

text: i

text: (δij

text: βδijP

text: [Xi,

text: Xj]

text: [Pi,

text: with

text: β

text: having

text: the

text: dimension

text: of

text: inverse

text: squared

text: momentum

text: and,

text: P

text: P

text: As

text: obviously,

text: j

text: in

text: the

text: limit

text: β

text: the

text: deformed

text: relations

text: reduce

text: to

text: the

text: standard

text: canonical

text: commutation

text: relations

text: [x

text: p

text: i

text: δ

text: However,

text: there

text: exist

text: many

text: other

text: similar

text: type

text: of

text: deformations

text: in

text: the

text: i

text: j

text: ij

text: literature,

text: which

text: have

text: been

text: used

text: to

text: investigated

text: many

text: interesting

text: phenomena

text: in

text: diﬀerent

text: contexts;

text: ##LTLine##


text: Modiﬁed Schr¨odinger-Newton equation [40] C. C. Gan, C. M. Savage and S. Z. Scully, Optomechanical tests of a Schr¨odinger-Newton equation for gravitational quantum mechanics, Phys. Rev. D 93, 124049 (2016). [41] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. Kim and C. Brukner, Probing Planck-scale physics with quantum optics, Nat. Phys. 8, 393–397 (2012). [42] P. Pedram, K. Nozari and S. H. Taheri, The eﬀects of minimal length and maximal momentum on the transition rate of ultra cold neutrons in gravitational ﬁeld, JHEP 2011, 1–11 (2011). PT [43] S. Dey, A. Fring and L. Gouba, -symmetric non-commutative spaces with minimal volume uncertainty relations, J. Phys. A: Math. Theor. 45, 385302 (2012). [44] S. Dey and V. Hussin, Entangled squeezed states in noncommutative spaces with minimal length uncertainty relations, Phys. Rev. D 91, 124017 (2015). [45] S. Dey, A. Fring and B. Khantoul, Hermitian versus non-Hermitian representations for mini- mal length uncertainty relations, J. Phys. A: Math. Theor. 46, 335304 (2013). [46] V. V. Nesvizhevsky et al, Quantum states of neutrons in the Earth’s gravitational ﬁeld, Nature 415, 297–299 (2002). [47] S. Flu¨gge, Practical quantum mechanics, Springer Verlag: Berlin (1999). [48] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, Dover Publications: New York (1964). [49] V. V. Nesvizhevsky et al, Study of the neutron quantum states in the gravity ﬁeld, Euro. Phys. J. C 40, 479–491 (2005). [50] O. Bertolami, J. G. Rosa, C. M. L. de Arag˜ao, P. Castorina and D. Zappal`a, Noncommutative gravitational quantum well, Phys. Rev. D 72, 025010 (2005). [51] F. Brau and F. Buisseret, Minimal length uncertainty relation and gravitational quantum well, Phys. Rev. D 74, 036002 (2006). [52] R. Banerjee, B. D. Roy and S. Samanta, Remarks on the noncommutative gravitational quantum well, Phys. Rev. D 74, 045015 (2006). [53] F. Buisseret, B. Silvestre-Brac and V. Mathieu, Modiﬁed Newton’s law, braneworlds, and the gravitational quantum well, Class. Quant. Grav. 24, 855 (2007). [54] L. Randall and R. Sundrum, An alternative to compactiﬁcation, Phys. Rev. Lett. 83, 4690 (1999). [55] F. Scardigli and R. Casadio, Gravitational tests of the generalized uncertainty principle, Euro. Phys. J. C 75, 425 (2015). [56] M. Bawaj et al., Probing deformed commutators with macroscopic harmonic oscillators, Nature Commun. 6, 7503 (2015). [57] F. Scardigli, G. Lambiase and E. C. Vagenas, GUP parameter from quantum corrections to the Newtonian potential, Phys. Lett. B 767, 242–246 (2017). [58] T. J. Kippenberg and K. J. Vahala, Cavity optomechanics: back-action at the mesoscale, Science 321, 1172–1176 (2008). 12 ##LTLine##

text: Modiﬁed

label: paragraph

text: Schr¨odinger-Newton

text: equation

label: reference

text: C.

text: C.

text: Gan,

text: C.

text: M.

text: Savage

text: and

text: S.

text: Z.

text: Scully,

text: Optomechanical

text: tests

text: of

text: a

text: Schr¨odinger-Newton

text: equation

text: for

text: gravitational

text: quantum

text: mechanics,

text: Phys.

text: Rev.

text: D

text: I.

text: Pikovski,

text: M.

text: R.

text: Vanner,

text: M.

text: Aspelmeyer,

text: M.

text: Kim

text: and

text: C.

text: Brukner,

text: Probing

text: Planck-scale

text: physics

text: with

text: quantum

text: optics,

text: Nat.

text: Phys.

text: P.

text: Pedram,

text: K.

text: Nozari

text: and

text: S.

text: H.

text: Taheri,

text: The

text: eﬀects

text: of

text: minimal

text: length

text: and

text: maximal

text: momentum

text: on

text: the

text: transition

text: rate

text: of

text: ultra

text: cold

text: neutrons

text: in

text: gravitational

text: ﬁeld,

text: JHEP

text: PT

text: S.

text: Dey,

text: A.

text: Fring

text: and

text: L.

text: Gouba,

text: -symmetric

text: non-commutative

text: spaces

text: with

text: minimal

text: volume

text: uncertainty

text: relations,

text: J.

text: Phys.

text: A:

text: Math.

text: Theor.

text: S.

text: Dey

text: and

text: V.

text: Hussin,

text: Entangled

text: squeezed

text: states

text: in

text: noncommutative

text: spaces

text: with

text: minimal

text: length

text: uncertainty

text: relations,

text: Phys.

text: Rev.

text: D

text: S.

text: Dey,

text: A.

text: Fring

text: and

text: B.

text: Khantoul,

text: Hermitian

text: versus

text: non-Hermitian

text: representations

text: for

text: mini-

text: mal

text: length

text: uncertainty

text: relations,

text: J.

text: Phys.

text: A:

text: Math.

text: Theor.

text: V.

text: V.

text: Nesvizhevsky

text: et

text: al,

text: Quantum

text: states

text: of

text: neutrons

text: in

text: the

text: Earth’s

text: gravitational

text: ﬁeld,

text: Nature

text: S.

text: Flu¨gge,

text: Practical

text: quantum

text: mechanics,

text: Springer

text: Verlag:

text: Berlin

text: M.

text: Abramowitz

text: and

text: I.

text: A.

text: Stegun,

text: Handbook

text: of

text: mathematical

text: functions:

text: with

text: formulas,

text: graphs,

text: and

text: mathematical

text: tables,

text: Dover

text: Publications:

text: New

text: York

text: V.

text: V.

text: Nesvizhevsky

text: et

text: al,

text: Study

text: of

text: the

text: neutron

text: quantum

text: states

text: in

text: the

text: gravity

text: ﬁeld,

text: Euro.

text: Phys.

text: J.

text: C

text: O.

text: Bertolami,

text: J.

text: G.

text: Rosa,

text: C.

text: M.

text: L.

text: de

text: Arag˜ao,

text: P.

text: Castorina

text: and

text: D.

text: Zappal`a,

text: Noncommutative

text: gravitational

text: quantum

text: well,

text: Phys.

text: Rev.

text: D

text: F.

text: Brau

text: and

text: F.

text: Buisseret,

text: Minimal

text: length

text: uncertainty

text: relation

text: and

text: gravitational

text: quantum

text: well,

text: Phys.

text: Rev.

text: D

text: R.

text: Banerjee,

text: B.

text: D.

text: Roy

text: and

text: S.

text: Samanta,

text: Remarks

text: on

text: the

text: noncommutative

text: gravitational

text: quantum

text: well,

text: Phys.

text: Rev.

text: D

text: F.

text: Buisseret,

text: B.

text: Silvestre-Brac

text: and

text: V.

text: Mathieu,

text: Modiﬁed

text: Newton’s

text: law,

text: braneworlds,

text: and

text: the

text: gravitational

text: quantum

text: well,

text: Class.

text: Quant.

text: Grav.

text: L.

text: Randall

text: and

text: R.

text: Sundrum,

text: An

text: alternative

text: to

text: compactiﬁcation,

text: Phys.

text: Rev.

text: Lett.

text: F.

text: Scardigli

text: and

text: R.

text: Casadio,

text: Gravitational

text: tests

text: of

text: the

text: generalized

text: uncertainty

text: principle,

text: Euro.

text: Phys.

text: J.

text: C

text: M.

text: Bawaj

text: et

text: al.,

text: Probing

text: deformed

text: commutators

text: with

text: macroscopic

text: harmonic

text: oscillators,

text: Nature

text: Commun.

text: F.

text: Scardigli,

text: G.

text: Lambiase

text: and

text: E.

text: C.

text: Vagenas,

text: GUP

text: parameter

text: from

text: quantum

text: corrections

text: to

text: the

text: Newtonian

text: potential,

text: Phys.

text: Lett.

text: B

text: T.

text: J.

text: Kippenberg

text: and

text: K.

text: J.

text: Vahala,

text: Cavity

text: optomechanics:

text: back-action

text: at

text: the

text: mesoscale,

text: Science

text: ##LTLine##


text: Modiﬁed Schr¨odinger-Newton equation supported by a CARMIN postdoctoral fellowship by IHES and IHP. The work of QZ is supported by NUS Tier 1 FRC Grant R-144-000-360-112. References [1] H. S. Snyder, Quantized space-time, Phys. Rev. 71, 38 (1947). [2] N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 1999, 032 (1999). [3] A. Kempf, G. Mangano and R. B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52, 1108 (1995). [4] S. Das and E. C. Vagenas, Universality of quantum gravity corrections, Phys. Rev. Lett. 101, 221301 (2008). [5] M. Gomes and V. G. Kupriyanov, Position-dependent noncommutativity in quantum me- chanics, Phys. Rev. D 79, 125011 (2009). [6] B. Bagchi and A. Fring, Minimal length in quantum mechanics and non-Hermitian Hamilto- nian systems, Phys. Lett. A 373, 4307–4310 (2009). [7] C. Quesne and V. M. Tkachuk, Composite system in deformed space with minimal length, Phys. Rev. A 81, 012106 (2010). [8] C. A. Mead, Possible connection between gravitation and fundamental length, Phys. Rev. 135, B849 (1964). [9] G. Veneziano, A stringy nature needs just two constants, Europhys. Lett. 2, 199 (1986). [10] D. J. Gross and P. F. Mende, String theory beyond the Planck scale, Nuc. Phys. B 303, 407–454 (1988). [11] D. Amati, M. Ciafaloni and G. Veneziano, Can spacetime be probed below the string size? Phys. Lett. B 216, 41–47 (1989). [12] T. Yoneya, On the interpretation of minimal length in string theories, Mod. Phys. Lett. A 4, 1587–1595 (1989). [13] K. Konishi, G. Paﬀuti and P. Provero, Minimum physical length and the generalized uncer- tainty principle in string theory, Phys. Lett. B 234, 276–284 (1990). [14] C. Rovelli, Loop quantum gravity, Living Rev. Rel. 1, 1 (1998). [15] P. Dzierzak, J. Jezierski, P. Malkiewicz and W. Piechocki, The minimum length problem of loop quantum cosmology, Acta Phys. Pol. B 41, 717–726 (2010). [16] M. Maggiore, A generalized uncertainty principle in quantum gravity, Phys. Lett. B 304, 65–69 (1993). [17] M. Park, The generalized uncertainty principle in (A)dS space and the modiﬁcation of Hawk- ing temperature from the minimal length, Phys. Lett. B 659, 698–702 (2008). [18] T. Padmanabhan, Physical signiﬁcance of Planck length, Ann. Phys. 165, 38–58 (1985). [19] J. Greensite, Is there a minimum length in d = 4 lattice quantum gravity? Phys. Lett. B 255, 375–380 (1991). 10 ##LTLine##

text: Modiﬁed

text: Schr¨odinger-Newton

text: equation

text: supported

text: by

text: a

text: CARMIN

text: postdoctoral

text: fellowship

text: by

text: IHES

text: and

text: IHP.

text: The

text: work

text: of

text: QZ

text: is

text: supported

text: by

text: NUS

text: Tier

text: FRC

text: Grant

text: References

text: H.

text: S.

text: Snyder,

text: Quantized

text: space-time,

text: Phys.

text: Rev.

text: N.

text: Seiberg

text: and

text: E.

text: Witten,

text: String

text: theory

text: and

text: noncommutative

text: geometry,

text: JHEP

text: A.

text: Kempf,

text: G.

text: Mangano

text: and

text: R.

text: B.

text: Mann,

text: Hilbert

text: space

text: representation

text: of

text: the

text: minimal

text: length

text: uncertainty

text: relation,

text: Phys.

text: Rev.

text: D

text: S.

text: Das

text: and

text: E.

text: C.

text: Vagenas,

text: Universality

text: of

text: quantum

text: gravity

text: corrections,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: Gomes

text: and

text: V.

text: G.

text: Kupriyanov,

text: Position-dependent

text: noncommutativity

text: in

text: quantum

text: me-

text: chanics,

text: Phys.

text: Rev.

text: D

text: B.

text: Bagchi

text: and

text: A.

text: Fring,

text: Minimal

text: length

text: in

text: quantum

text: mechanics

text: and

text: non-Hermitian

text: Hamilto-

text: nian

text: systems,

text: Phys.

text: Lett.

text: A

text: C.

text: Quesne

text: and

text: V.

text: M.

text: Tkachuk,

text: Composite

text: system

text: in

text: deformed

text: space

text: with

text: minimal

text: length,

text: Phys.

text: Rev.

text: A

text: C.

text: A.

text: Mead,

text: Possible

text: connection

text: between

text: gravitation

text: and

text: fundamental

text: length,

text: Phys.

text: Rev.

text: G.

text: Veneziano,

text: A

text: stringy

text: nature

text: needs

text: just

text: two

text: constants,

text: Europhys.

text: Lett.

text: D.

text: J.

text: Gross

text: and

text: P.

text: F.

text: Mende,

text: String

text: theory

text: beyond

text: the

text: Planck

text: scale,

text: Nuc.

text: Phys.

text: B

text: D.

text: Amati,

text: M.

text: Ciafaloni

text: and

text: G.

text: Veneziano,

text: Can

text: spacetime

text: be

text: probed

text: below

text: the

text: string

text: size?

text: Phys.

text: Lett.

text: B

text: T.

text: Yoneya,

text: On

text: the

text: interpretation

text: of

text: minimal

text: length

text: in

text: string

text: theories,

text: Mod.

text: Phys.

text: Lett.

text: A

text: K.

text: Konishi,

text: G.

text: Paﬀuti

text: and

text: P.

text: Provero,

text: Minimum

text: physical

text: length

text: and

text: the

text: generalized

text: uncer-

text: tainty

text: principle

text: in

text: string

text: theory,

text: Phys.

text: Lett.

text: B

text: C.

text: Rovelli,

text: Loop

text: quantum

text: gravity,

text: Living

text: Rev.

text: Rel.

text: P.

text: Dzierzak,

text: J.

text: Jezierski,

text: P.

text: Malkiewicz

text: and

text: W.

text: Piechocki,

text: The

text: minimum

text: length

text: problem

text: of

text: loop

text: quantum

text: cosmology,

text: Acta

text: Phys.

text: Pol.

text: B

text: M.

text: Maggiore,

text: A

text: generalized

text: uncertainty

text: principle

text: in

text: quantum

text: gravity,

text: Phys.

text: Lett.

text: B

text: M.

text: Park,

text: The

text: generalized

text: uncertainty

text: principle

text: in

text: (A)dS

text: space

text: and

text: the

text: modiﬁcation

text: of

text: Hawk-

text: ing

text: temperature

text: from

text: the

text: minimal

text: length,

text: Phys.

text: Lett.

text: B

text: T.

text: Padmanabhan,

text: Physical

text: signiﬁcance

text: of

text: Planck

text: length,

text: Ann.

text: Phys.

text: J.

text: Greensite,

text: Is

text: there

text: a

text: minimum

text: length

text: in

text: d

text: lattice

text: quantum

text: gravity?

text: Phys.

text: Lett.

text: B

text: ##LTLine##


text: assume that h (t) includes a low-pass baseband ﬁlter with an T equivalent amplitude characteristic of the spectral mask. B. Channel and receiver For the entire duration of the SCPHY packet, the channel is treated as linear and time-invariant [17]. Assuming the received signal contains echoes from L paths, the continuous channel response is Fig. 1. Composition of IEEE 802.11ad SCPHY frame. (cid:88) aperiodic autocorrelation functions have sidelobes equal in magnitude L − but opposite in sign. The sum of the two autocorrelations, therefore, hCH(t) = αl.δ(t τl), (8) ∈ l=1 has a peak of 2N and a sidelobe level of zero: where αl C and τl are the path losses and delays, respectively. The received signal is then given by ∗ − ∗ − GaN[n] GaN[ n] + GbN[n] GbN[ n] = 2Nδ[n], (1) ∗ ∗ ∗ ∗ where denotes linear convolution. To observe this property in the xR(t) = (xT hCH)(t) + η(t) = (sT hT hCH)(t) + η(t), (9) frequency domain, consider Fa and Fb as the Discrete Fourier N N where η(t) is additive white Gaussian noise (AWGN). Prior to Transforms (DFTs) of the sequences Ga and Gb , respectively. N N sampling at the receiver, an anti-aliasing continuous-time ﬁlter h (t) Then, the sum of the spectral densities of the two sequences is ﬂat, R (usually a milled analog ﬁlter at 60 GHz) is used to remove i.e., for each DFT index k, high-frequency noise from the signal x (t) and also to attenuate R adjacent channels. The signal processed by the ﬁlter h (t) is R | | | | FaN[k] 2 + FbN[k] 2 = 2N. (2) The convolution in (1) is linear while the DFT has the circular ∗ ∗ ∗ ∗ ∗ sR(t) = (xR hR)(t) = (sT hT hCH hR)(t) + (η hR)(t) convolution property. Therefore, for these relations to hold, we must ∗ ∗ choose a DFT of length NDFT ≥ 2N −1. Other perfect autocorrelation = (sT ∗hTR hCH)(t) + z(t), (10) sequences [22] and complementary pairs [23] also exist and have been where hTR(t) = (hT h∗R)(t) represents the overall analog ﬁlter impulse used for radar and communication applications. response and z(t) = (η hR)(t) is the ﬁltered noise. The received signal sR(t) is sampled at rate Fc = 2.64 GHz to obtain the discrete-time The 802.11ad CEF transmit signal is a concatenated sequence signal s [n] = s (nT ). R R c C. Channel estimation − ··· In a Nyquist receiver, the received signal samples are rCESC[n] = Gu512[n] + Gv512[n 512], n = 0,1, ,1151. (3≤) downsampled by a factor of 1.5. Since the constant spectrum property He≤re, the sequences Gu512[n] and Gv512[n] are deﬁned for 0 (2) of Golay sequences is not aﬀected due to upsampling, here we n 511; for other values of n, they are set to zero. Typical skip the downsampling stage. The samples s [n] are passed through R 802.11ad transmitters employ DACs that operate at 2.64 GHz to also four correlators, each corresponding to one of the Golay sequences of enable OFDM signals. Therefore, the discrete-time signal rCESC[n] is length 256. For example, if we assume that the overall analog ﬁlter upsampled by a factor of 1.5 to obtain a sequence sT[n] just before has a ﬂat unity frequency response over the band of interest, then the sending it to a digital-to-analog-converter (DAC) module (Fig. 2): signal trail of one of the correlators is (cid:40) r1CESC [n] = r0C,EoStCh[emr]w,inse=, 3m, (4) hˆ1au[n] = sR[n] ∗ Gau256[−n] = (hCH(nTc) ∗ sT[n]) ∗ Gau256[−n]. (11) By correlating the received samples with Gbu , Gav , and Gbv , sT[n] = r1CESC [2n]. (5) we similarly obtain hˆ [n], hˆ [n], and hˆ [n25]6. 256 256 The corresponding upsampled constituting sequences of s [n] are 1bu 1av 1bv T denoted as Gu [n], Gv [n], and so on. We represent the Since we intend to exploit the property in (1), we delay the 512up 512up discrete-time sequence s [n] as a weighted sum of Dirac impulses correlator outputs: hˆ [n] = hˆ [n],hˆ [n] = hˆ [n + 256], hˆ [n] = T 1a 1au 1b 1bu 2a (cid:88) hˆ2av[n+512], hˆ2b[n] = hˆ2bv[n+768]. The outputs of the complementary pairs are then summed up. As shown in Fig. 2, this results in two 1535 − estimates of the channel, hˆ1[n] and hˆ2[n]. For instance, the ﬁrst sT(t) = sT[m]δ(t mTc), (6) estimate of the channel is m=0 where Fc = 2.64GHz = 1/Tc. This signal is passed through the transmit shaping ﬁlter h (t) (Fig. 2) to obtain: T hˆ [n] = hˆ [n] + hˆ [n] 1 1a 1b ∗ ∗ − ∗ − = h [n] (Gau [n] Gau [ n] + Gbu [n] Gbu [ n]) CH 256 256 256 256 ∗ · xT(t) = (sT hT)(t). (7) = 512 hCH[n]. (12) The 802.11ad protocol speciﬁes a spectral mask for the transmit We average to obtain the ﬁnal estimate hˆ[n] = 1 1(hˆ [n] + hˆ [n]). 512 2 1 2 signal to limit inter-symbol interference [2, section 21.3.2]. We In the absence of noise, hˆ[n] recovers the true channel perfectly. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: assume

text: that

text: h

text: (t)

text: includes

text: a

text: low-pass

text: baseband

text: ﬁlter

text: with

text: an

text: T

text: equivalent

text: amplitude

text: characteristic

text: of

text: the

text: spectral

text: mask.

text: B.

label: section

text: Channel

text: and

text: receiver

text: For

text: the

text: entire

text: duration

text: of

text: the

text: SCPHY

text: packet,

text: the

text: channel

text: is

text: treated

text: as

text: linear

text: and

text: time-invariant

text: Assuming

text: the

text: received

text: signal

text: contains

text: echoes

text: from

text: L

text: paths,

text: the

text: continuous

text: channel

text: response

text: is

text: Fig.

text: Composition

label: caption

text: of

text: IEEE

text: SCPHY

text: frame.

label: equation

text: aperiodic

text: autocorrelation

text: functions

text: have

text: sidelobes

text: equal

text: in

text: magnitude

text: L

font: NGMZKW+txsy

text: but

text: opposite

text: in

text: sign.

text: The

text: sum

text: of

text: the

text: two

text: autocorrelations,

text: therefore,

text: hCH(t)

font: OOTUFF+rtxr

text: αl.δ(t

font: WPKQIU+rtxmi

text: τl),

text: has

text: a

text: peak

text: of

text: and

text: a

text: sidelobe

text: level

text: of

text: zero:

text: where

text: αl

text: C

text: and

text: τl

text: are

text: the

text: path

text: losses

text: and

text: delays,

text: respectively.

text: The

text: received

text: signal

text: is

text: then

text: given

text: by

text: GaN[n]

text: GaN[

text: n]

text: GbN[n]

text: GbN[

text: n]

text: where

text: denotes

text: linear

text: convolution.

text: To

text: observe

text: this

text: property

text: in

text: the

text: xR(t)

text: (xT

text: hCH)(t)

text: η(t)

text: (sT

text: hT

text: hCH)(t)

text: η(t),

text: frequency

text: domain,

text: consider

text: Fa

text: and

text: Fb

text: as

text: the

text: Discrete

text: Fourier

text: N

text: N

text: where

text: η(t)

text: is

text: additive

text: white

text: Gaussian

text: noise

text: (AWGN).

text: Prior

text: to

text: Transforms

text: (DFTs)

text: of

text: the

text: sequences

text: Ga

text: and

text: Gb

text: respectively.

text: N

text: N

text: sampling

text: at

text: the

text: receiver,

text: an

text: anti-aliasing

text: continuous-time

text: ﬁlter

text: h

text: (t)

text: Then,

text: the

text: sum

text: of

text: the

text: spectral

text: densities

text: of

text: the

text: two

text: sequences

text: is

text: ﬂat,

text: R

text: (usually

text: a

text: milled

text: analog

text: ﬁlter

text: at

text: GHz)

text: is

text: used

text: to

text: remove

text: i.e.,

text: for

text: each

text: DFT

text: index

text: k,

text: high-frequency

text: noise

text: from

text: the

text: signal

text: x

text: (t)

text: and

text: also

text: to

text: attenuate

text: R

text: adjacent

text: channels.

text: The

text: signal

text: processed

text: by

text: the

text: ﬁlter

text: h

text: (t)

text: is

text: R

text: FaN[k]

text: FbN[k]

text: The

text: convolution

text: in

text: is

text: linear

text: while

text: the

text: DFT

text: has

text: the

text: circular

text: sR(t)

text: (xR

text: hR)(t)

text: (sT

text: hT

text: hCH

text: hR)(t)

text: (η

text: hR)(t)

text: convolution

text: property.

text: Therefore,

text: for

text: these

text: relations

text: to

text: hold,

text: we

text: must

text: choose

text: a

text: DFT

text: of

text: length

text: NDFT

text: Other

text: perfect

text: autocorrelation

text: (sT

text: ∗hTR

text: hCH)(t)

text: z(t),

text: sequences

text: and

text: complementary

text: pairs

text: also

text: exist

text: and

text: have

text: been

text: where

text: hTR(t)

text: (hT

text: h∗R)(t)

text: represents

text: the

text: overall

text: analog

text: ﬁlter

text: impulse

text: used

text: for

text: radar

text: and

text: communication

text: applications.

text: response

text: and

text: z(t)

text: (η

text: hR)(t)

text: is

text: the

text: ﬁltered

text: noise.

text: The

text: received

text: signal

text: sR(t)

text: is

text: sampled

text: at

text: rate

text: Fc

text: GHz

text: to

text: obtain

text: the

text: discrete-time

text: The

text: CEF

text: transmit

text: signal

text: is

text: a

text: concatenated

text: sequence

text: signal

text: s

text: [n]

text: s

text: (nT

text: R

text: R

text: c

text: C.

text: Channel

text: estimation

text: In

text: a

text: Nyquist

text: receiver,

text: the

text: received

text: signal

text: samples

text: are

text: rCESC[n]

text: n

text: downsampled

text: by

text: a

text: factor

text: of

text: Since

text: the

text: constant

text: spectrum

text: property

text: He≤re,

text: the

text: sequences

text: and

text: are

text: deﬁned

text: for

text: of

text: Golay

text: sequences

text: is

text: not

text: aﬀected

text: due

text: to

text: upsampling,

text: here

text: we

text: n

text: for

text: other

text: values

text: of

text: n,

text: they

text: are

text: set

text: to

text: zero.

text: Typical

text: skip

text: the

text: downsampling

text: stage.

text: The

text: samples

text: s

text: [n]

text: are

text: passed

text: through

text: R

text: transmitters

text: employ

text: DACs

text: that

text: operate

text: at

text: GHz

text: to

text: also

text: four

text: correlators,

text: each

text: corresponding

text: to

text: one

text: of

text: the

text: Golay

text: sequences

text: of

text: enable

text: OFDM

text: signals.

text: Therefore,

text: the

text: discrete-time

text: signal

text: rCESC[n]

text: is

text: length

text: For

text: example,

text: if

text: we

text: assume

text: that

text: the

text: overall

text: analog

text: ﬁlter

text: upsampled

text: by

text: a

text: factor

text: of

text: to

text: obtain

text: a

text: sequence

text: sT[n]

text: just

text: before

text: has

text: a

text: ﬂat

text: unity

text: frequency

text: response

text: over

text: the

text: band

text: of

text: interest,

text: then

text: the

text: sending

text: it

text: to

text: a

text: digital-to-analog-converter

text: (DAC)

text: module

text: (Fig.

text: signal

text: trail

text: of

text: one

text: of

text: the

text: correlators

text: is

text: [n]

text: sR[n]

text: (hCH(nTc)

text: sT[n])

text: By

text: correlating

text: the

text: received

text: samples

text: with

text: Gbu

text: Gav

text: and

text: Gbv

text: sT[n]

text: we

text: similarly

text: obtain

text: hˆ

text: [n],

text: hˆ

text: [n],

text: and

text: hˆ

text: The

text: corresponding

text: upsampled

text: constituting

text: sequences

text: of

text: s

text: [n]

text: are

text: T

text: denoted

text: as

text: Gu

text: [n],

text: Gv

text: [n],

text: and

text: so

text: on.

text: We

text: represent

text: the

text: Since

text: we

text: intend

text: to

text: exploit

text: the

text: property

text: in

text: we

text: delay

text: the

text: discrete-time

text: sequence

text: s

text: [n]

text: as

text: a

text: weighted

text: sum

text: of

text: Dirac

text: impulses

text: correlator

text: outputs:

text: hˆ

text: [n]

text: hˆ

text: [n],hˆ

text: [n]

text: hˆ

text: [n

text: hˆ

text: [n]

text: T

text: The

text: outputs

text: of

text: the

text: complementary

text: pairs

text: are

text: then

text: summed

text: up.

text: As

text: shown

text: in

text: Fig.

text: this

text: results

text: in

text: two

text: estimates

text: of

text: the

text: channel,

text: and

text: For

text: instance,

text: the

text: ﬁrst

text: sT(t)

text: sT[m]δ(t

text: mTc),

text: estimate

text: of

text: the

text: channel

text: is

text: where

text: Fc

text: This

text: signal

text: is

text: passed

text: through

text: the

text: transmit

text: shaping

text: ﬁlter

text: h

text: (t)

text: (Fig.

text: to

text: obtain:

text: T

text: hˆ

text: [n]

text: hˆ

text: [n]

text: hˆ

text: [n]

text: h

text: [n]

text: (Gau

text: [n]

text: Gau

text: n]

text: Gbu

text: [n]

text: Gbu

text: n])

text: CH

text: xT(t)

text: (sT

text: hT)(t).

text: hCH[n].

text: The

text: protocol

text: speciﬁes

text: a

text: spectral

text: mask

text: for

text: the

text: transmit

text: We

text: average

text: to

text: obtain

text: the

text: ﬁnal

text: estimate

text: hˆ[n]

text: [n]

text: hˆ

text: [n]).

text: signal

text: to

text: limit

text: inter-symbol

text: interference

text: section

text: We

text: In

text: the

text: absence

text: of

text: noise,

text: hˆ[n]

text: recovers

text: the

text: true

text: channel

text: perfectly.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Waveform Diversity and Design, M. Wicks, E. Makole, S. Blunt, R. Schneible, and V. Amuso, Eds. SciTech Publishing, 2011. [14] K. M. Cohen, C. Attias, B. Farbman, I. Tselniker, and Y. C. Eldar, “Channel estimation in UWB channels using compressed sensing,” in IEEE Int. Conf. on Acoustics, Speech and Signal Process., 2014, pp. 1966–1970. [15] Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications. Cambridge University Press, 2012. [16] Y. C. Eldar, Sampling Theory: Beyond Bandlimited Systems. Cambridge University Press, 2015. [17] T. S. Rappaport, R. W. Heath Jr., R. C. Daniels, and J. N. Murdock, Millimeter Wave Wireless Communications. Prentice Hall, 2014. [18] R. T. Suryaprakash, M. Pajovic, K. J. Kim, and P. Orlik, “Millimeter wave communications channel estimation via bayesian group sparse recovery,” in IEEE Int. Conf. on Acoustics, Speech and Signal Process., 2016, pp. 3406–3410. [19] W.-C. Liu, T.-C. Wei, Y.-S. Huang, C.-D. Chan, and S.-J. Jou, “All-digital synchronization for SC/OFDM mode of IEEE 802.15. 3c and IEEE 802.11 ad,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 62, no. 2, pp. 545–553, 2015. [20] B. Ye and Z. Zhang, “Improved pilot design and channel estimation for 60 Ghz OFDM based on IEEE 802.11ad,” in IEEE Wireless Commun. Netw. Conf., 2013, pp. 4129–4133. [21] P. Kumari, N. Gonza´lez-Prelcic, and R. W. Heath Jr., “Investigating the IEEE 802.11ad standard for millimeter wave automotive radar,” in Vehicular Technology Conf., 2015, pp. 1–5. [22] H. D. Luke, “Sequences and arrays with perfect periodic correlation,” IEEE Trans. Aerosp. Electron. Syst., vol. 24, no. 3, pp. 287–294, 1988. [23] E. E. Hollis, “Another type of complementary sequence,” IEEE Trans. Aerosp. Electron. Syst., vol. 5, pp. 916–920, 1975. [24] M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with ﬁnite rate of innovation,” IEEE Trans. Signal Process., vol. 50, no. 6, pp. 1417–1428, 2002. [25] E. Baransky, G. Itzhak, I. Shmuel, N. Wagner, E. Shoshan, and Y. C. Eldar, “A sub-Nyquist radar prototype: Hardware and applications,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 2, pp. 809–822, 2014. [26] K. V. Mishra, E. Shoshan, M. Namer, M. Meltsin, D. Cohen, R. Madmoni, S. Dror, R. Ifraimov, and Y. C. Eldar, “Cognitive sub-Nyquist hardware prototype of a collocated MIMO radar,” in Compressed Sens. Theory Appl. Radar, Sonar Remote Sens., 2016, pp. 56–60.

text: Waveform

text: Diversity

text: and

text: Design,

text: M.

text: Wicks,

text: E.

text: Makole,

text: S.

text: Blunt,

text: R.

text: Schneible,

text: and

text: V.

text: Amuso,

text: Eds.

text: SciTech

text: Publishing,

text: K.

text: M.

text: Cohen,

text: C.

text: Attias,

text: B.

text: Farbman,

text: I.

text: Tselniker,

text: and

text: Y.

text: C.

text: Eldar,

text: “Channel

text: estimation

text: in

text: UWB

text: channels

text: using

text: compressed

text: sensing,”

text: in

text: IEEE

text: Int.

text: Conf.

text: on

text: Acoustics,

text: Speech

text: and

text: Signal

text: Process.,

text: pp.

text: Y.

text: C.

text: Eldar

text: and

text: G.

text: Kutyniok,

text: Compressed

text: Sensing:

text: Theory

text: and

text: Applications.

text: Cambridge

text: University

text: Press,

text: Y.

text: C.

text: Eldar,

text: Sampling

text: Theory:

text: Beyond

text: Bandlimited

text: Systems.

text: Cambridge

text: University

text: Press,

text: T.

text: S.

text: Rappaport,

text: R.

text: W.

text: Heath

text: Jr.,

text: R.

text: C.

text: Daniels,

text: and

text: J.

text: N.

text: Murdock,

text: Millimeter

text: Wave

text: Wireless

text: Communications.

text: Prentice

text: Hall,

text: R.

text: T.

text: Suryaprakash,

text: M.

text: Pajovic,

text: K.

text: J.

text: Kim,

text: and

text: P.

text: Orlik,

text: “Millimeter

text: wave

text: communications

text: channel

text: estimation

text: via

text: bayesian

text: group

text: sparse

text: recovery,”

text: in

text: IEEE

text: Int.

text: Conf.

text: on

text: Acoustics,

text: Speech

text: and

text: Signal

text: Process.,

text: pp.

text: W.-C.

text: Liu,

text: T.-C.

text: Wei,

text: Y.-S.

text: Huang,

text: C.-D.

text: Chan,

text: and

text: S.-J.

text: Jou,

text: “All-digital

text: synchronization

text: for

text: SC/OFDM

text: mode

text: of

text: IEEE

text: and

text: IEEE

text: ad,”

text: IEEE

text: Trans.

text: Circuits

text: Syst.

text: I,

text: Reg.

text: Papers,

text: vol.

text: no.

text: pp.

text: B.

text: Ye

text: and

text: Z.

text: Zhang,

text: “Improved

text: pilot

text: design

text: and

text: channel

text: estimation

text: for

text: Ghz

text: OFDM

text: based

text: on

text: IEEE

text: in

text: IEEE

text: Wireless

text: Commun.

text: Netw.

text: Conf.,

text: pp.

text: P.

text: Kumari,

text: N.

text: Gonza´lez-Prelcic,

text: and

text: R.

text: W.

text: Heath

text: Jr.,

text: “Investigating

text: the

text: IEEE

text: standard

text: for

text: millimeter

text: wave

text: automotive

text: radar,”

text: in

text: Vehicular

text: Technology

text: Conf.,

text: pp.

text: H.

text: D.

text: Luke,

text: “Sequences

text: and

text: arrays

text: with

text: perfect

text: periodic

text: correlation,”

text: IEEE

text: Trans.

text: Aerosp.

text: Electron.

text: Syst.,

text: vol.

text: no.

text: pp.

text: E.

text: E.

text: Hollis,

text: “Another

text: type

text: of

text: complementary

text: sequence,”

text: IEEE

text: Trans.

text: Aerosp.

text: Electron.

text: Syst.,

text: vol.

text: pp.

text: M.

text: Vetterli,

text: P.

text: Marziliano,

text: and

text: T.

text: Blu,

text: “Sampling

text: signals

text: with

text: ﬁnite

text: rate

text: of

text: innovation,”

text: IEEE

text: Trans.

text: Signal

text: Process.,

text: vol.

text: no.

text: pp.

text: E.

text: Baransky,

text: G.

text: Itzhak,

text: I.

text: Shmuel,

text: N.

text: Wagner,

text: E.

text: Shoshan,

text: and

text: Y.

text: C.

text: Eldar,

text: “A

text: sub-Nyquist

text: radar

text: prototype:

text: Hardware

text: and

text: applications,”

text: IEEE

text: Trans.

text: Aerosp.

text: Electron.

text: Syst.,

text: vol.

text: no.

text: pp.

text: K.

text: V.

text: Mishra,

text: E.

text: Shoshan,

text: M.

text: Namer,

text: M.

text: Meltsin,

text: D.

text: Cohen,

text: R.

text: Madmoni,

text: S.

text: Dror,

text: R.

text: Ifraimov,

text: and

text: Y.

text: C.

text: Eldar,

text: “Cognitive

text: sub-Nyquist

text: hardware

text: prototype

text: of

text: a

text: collocated

text: MIMO

text: radar,”

text: in

text: Compressed

text: Sens.

text: Theory

text: Appl.

text: Radar,

text: Sonar

text: Remote

text: Sens.,

text: pp.


text: (a) Noise std: 0 (b) Noise std: 0.2 (c) Noise std: 1.4 Figure 4: Sobol function. 3-dimensional input. 17 ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: Noise

text: std:

text: (b)

text: Noise

text: std:

text: (c)

text: Noise

text: std:

text: Figure

text: Sobol

text: function.

text: input.

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##

text: ##LTFigure##


text: Figure 5: Ishigami function. 3-dimensional input. Figure 6: Environmental function. 4-dimensional input. 18 ##LTFigure## ##LTFigure##

text: Figure

text: Ishigami

text: function.

text: input.

text: Figure

text: Environmental

text: function.

text: input.

text: ##LTFigure##

text: ##LTFigure##


text: (a) Noise std: 0 (b) Noise std: 5 Figure 8: WingWeight function. 10-dimensional input Figure 9: Truss model. 10-dimensional input 20 ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: Noise

text: std:

text: (b)

text: Noise

text: std:

text: Figure

text: WingWeight

text: function.

text: input

text: Figure

text: Truss

text: model.

text: input

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: probabilistically. However, a common situation in practice is to have a limited budget for data acqui- sition and to end up with scarce datasets. This introduces epistemic uncertainty (lack of knowledge) alongside aleatory uncertainty (natural variability). Then, a more general framework is required to characterize the uncertainty in model parameters beyond probability theory. Imprecise probabilities describes a set of methodologies which are more general than the well- established probability theory. They include Dempster-Shafer’s theory of evidence (Dempster, 1967; Shafer, 1976), Bayesian hierarchical models (Gelman, 2006; Gelman et al., 2009), possibility theory (De Cooman et al., 1995; Dubois and Prade, 1988), probability-boxes (Ferson and Ginzburg, 1996; Ferson and Hajagos, 2004), info-gap models (Ben-Haim, 2006; Kanno and Takewaki, 2006), and fuzzy sets (Zadeh, 1978; M¨oller and Beer, 2004). Among the various methods, probability-boxes provide a simple structure to distinguish aleatory and epistemic uncertainty naturally. In this context, engineers are concerned with the reliability of their designs, which is assessed by structural reliability analysis. The goal of such an analysis is the estimation of the failure probability, which describes the probability that the systems performs below a minimal acceptable threshold value. − − A typical value for the failure probability in engineering is in the range of 10 3 to 10 6. Altogether, the complexity of structural reliability analysis comprises three main factors: (i) a high-ﬁdelity computer simulation, (ii) uncertainty in the parameters, and (iii) the estimation of rare events (i.e. the estimation of small failure probabilities). In the presence of probabilistic variables, a number of methods are known to estimate failure probabilities (Lemaire, 2009; Morio et al., 2014), including Monte Carlo simulation, subset simulation (Au and Beck, 2001, 2003), importance sampling, ﬁrst/second order reliability method (FORM/SORM) (Breitung, 1989; Hohenbichler and Rackwitz, 1988), and line sampling (Koutsourelakis et al., 2004; De Angelis et al., 2015). In the presence of imprecise probabilities, however, fewer publications are available, some of which are mentioned here. (Eldred and Swiler, 2009; Hurtado, 2013) use nested Monte Carlo simulations to cope with imprecise probabilities. (Alvarez and Hurtado, 2014) combine subset simulation with random sets. (Zhang et al., 2015) generalize FORM/SORM for Dempster- Shafer’s evidence theory. A combination of line sampling and imprecise probabilities is discussed in (De Angelis et al., 2015). These publications typically require a large number of model evaluations and hence rely on an inexpensive-to-evaluate performance function to make the analysis tractable. A way to reduce the number of model evaluations and hence allowing expensive-to-evaluate limit- state functions are meta-models. Meta-modelling techniques approximate the expensive-to-evaluate limit-state function by a fast-to-evaluate surrogate based on a relatively small number of runs of the original one. The meta-model is then used in the structural reliability analysis. In particular, meta- models with adaptive experimental designs have been proven to be eﬃcient in the estimation of failure probabilities. As an example, (Bichon et al., 2008; Echard et al., 2011; Dubourg et al., 2013) combine Kriging (Gaussian process models) (Santner et al., 2003) and structural reliability analysis in the context of probability theory. However, only a few authors combine meta-modelling techniques with imprecise probabilities for structural reliability analysis. (Balesdent et al., 2014; Morio and Balesdent, 2016) use an adaptive- Kriging importance sampling algorithm for distributions with interval-valued parameters. A combi- nation of Karhunen-Lo`eve expansions and evidence theory is presented in (Oberguggenberger, 2014). (Zhang et al., 2014) use radial basis expansions in the presence of evidence theory. Based on these developments, a set of algorithms is proposed in this paper to further increase the eﬃciency of structural reliability analysis in the presence of epistemic uncertainties. In particular, free and parametric p-boxes are presented separately and compared later in this paper. The application 2

text: probabilistically.

text: However,

text: a

text: common

text: situation

text: in

text: practice

text: is

text: to

text: have

text: a

text: limited

text: budget

text: for

text: data

text: acqui-

text: sition

text: and

text: to

text: end

text: up

text: with

text: scarce

text: datasets.

text: This

text: introduces

text: epistemic

text: uncertainty

text: (lack

text: of

text: knowledge)

text: alongside

text: aleatory

text: uncertainty

text: (natural

text: variability).

text: Then,

text: a

text: more

text: general

text: framework

text: is

text: required

text: to

text: characterize

text: the

text: uncertainty

text: in

text: model

text: parameters

text: beyond

text: probability

text: theory.

text: Imprecise

text: probabilities

text: describes

text: a

text: set

text: of

text: methodologies

text: which

text: are

text: more

text: general

text: than

text: the

text: well-

text: established

text: probability

text: theory.

text: They

text: include

text: Dempster-Shafer’s

text: theory

text: of

text: evidence

text: (Dempster,

text: Shafer,

text: Bayesian

text: hierarchical

text: models

text: (Gelman,

text: Gelman

text: et

text: al.,

text: possibility

text: theory

text: (De

text: Cooman

text: et

text: al.,

text: Dubois

text: and

text: Prade,

text: probability-boxes

text: (Ferson

text: and

text: Ginzburg,

text: Ferson

text: and

text: Hajagos,

text: info-gap

text: models

text: (Ben-Haim,

text: Kanno

text: and

text: Takewaki,

text: and

text: fuzzy

text: sets

text: (Zadeh,

text: M¨oller

text: and

text: Beer,

text: Among

text: the

text: various

text: methods,

text: probability-boxes

text: provide

text: a

text: simple

text: structure

text: to

text: distinguish

text: aleatory

text: and

text: epistemic

text: uncertainty

text: naturally.

text: In

text: this

text: context,

text: engineers

text: are

text: concerned

text: with

text: the

text: reliability

text: of

text: their

text: designs,

text: which

text: is

text: assessed

text: by

text: structural

text: reliability

text: analysis.

text: The

text: goal

text: of

text: such

text: an

text: analysis

text: is

text: the

text: estimation

text: of

text: the

text: failure

text: probability,

text: which

text: describes

text: the

text: probability

text: that

text: the

text: systems

text: performs

text: below

text: a

text: minimal

text: acceptable

text: threshold

text: value.

text: A

text: typical

text: value

text: for

text: the

text: failure

text: probability

text: in

text: engineering

text: is

text: in

text: the

text: range

text: of

text: to

text: Altogether,

text: the

text: complexity

text: of

text: structural

text: reliability

text: analysis

text: comprises

text: three

text: main

text: factors:

text: (i)

text: a

text: high-ﬁdelity

text: computer

text: simulation,

text: (ii)

text: uncertainty

text: in

text: the

text: parameters,

text: and

text: (iii)

text: the

text: estimation

text: of

text: rare

text: events

text: (i.e.

text: the

text: estimation

text: of

text: small

text: failure

text: probabilities).

text: In

text: the

text: presence

text: of

text: probabilistic

text: variables,

text: a

text: number

text: of

text: methods

text: are

text: known

text: to

text: estimate

text: failure

text: probabilities

text: (Lemaire,

text: Morio

text: et

text: al.,

text: including

text: Monte

text: Carlo

text: simulation,

text: subset

text: simulation

text: (Au

text: and

text: Beck,

text: importance

text: sampling,

text: ﬁrst/second

text: order

text: reliability

text: method

text: (FORM/SORM)

text: (Breitung,

text: Hohenbichler

text: and

text: Rackwitz,

text: and

text: line

text: sampling

text: (Koutsourelakis

text: et

text: al.,

text: De

text: Angelis

text: et

text: al.,

text: In

text: the

text: presence

text: of

text: imprecise

text: probabilities,

text: however,

text: fewer

text: publications

text: are

text: available,

text: some

text: of

text: which

text: are

text: mentioned

text: here.

text: (Eldred

text: and

text: Swiler,

text: Hurtado,

text: use

text: nested

text: Monte

text: Carlo

text: simulations

text: to

text: cope

text: with

text: imprecise

text: probabilities.

text: (Alvarez

text: and

text: Hurtado,

text: combine

text: subset

text: simulation

text: with

text: random

text: sets.

text: (Zhang

text: et

text: al.,

text: generalize

text: FORM/SORM

text: for

text: Dempster-

text: Shafer’s

text: evidence

text: theory.

text: A

text: combination

text: of

text: line

text: sampling

text: and

text: imprecise

text: probabilities

text: is

text: discussed

text: in

text: (De

text: Angelis

text: et

text: al.,

text: These

text: publications

text: typically

text: require

text: a

text: large

text: number

text: of

text: model

text: evaluations

text: and

text: hence

text: rely

text: on

text: an

text: inexpensive-to-evaluate

text: performance

text: function

text: to

text: make

text: the

text: analysis

text: tractable.

text: A

text: way

text: to

text: reduce

text: the

text: number

text: of

text: model

text: evaluations

text: and

text: hence

text: allowing

text: expensive-to-evaluate

text: limit-

text: state

text: functions

text: are

text: meta-models.

text: Meta-modelling

text: techniques

text: approximate

text: the

text: expensive-to-evaluate

text: limit-state

text: function

text: by

text: a

text: fast-to-evaluate

text: surrogate

text: based

text: on

text: a

text: relatively

text: small

text: number

text: of

text: runs

text: of

text: the

text: original

text: one.

text: The

text: meta-model

text: is

text: then

text: used

text: in

text: the

text: structural

text: reliability

text: analysis.

text: In

text: particular,

text: meta-

text: models

text: with

text: adaptive

text: experimental

text: designs

text: have

text: been

text: proven

text: to

text: be

text: eﬃcient

text: in

text: the

text: estimation

text: of

text: failure

text: probabilities.

text: As

text: an

text: example,

text: (Bichon

text: et

text: al.,

text: Echard

text: et

text: al.,

text: Dubourg

text: et

text: al.,

text: combine

text: Kriging

text: (Gaussian

text: process

text: models)

text: (Santner

text: et

text: al.,

text: and

text: structural

text: reliability

text: analysis

text: in

text: the

text: context

text: of

text: probability

text: theory.

text: However,

text: only

text: a

text: few

text: authors

text: combine

text: meta-modelling

text: techniques

text: with

text: imprecise

text: probabilities

text: for

text: structural

text: reliability

text: analysis.

text: (Balesdent

text: et

text: al.,

text: Morio

text: and

text: Balesdent,

text: use

text: an

text: adaptive-

text: Kriging

text: importance

text: sampling

text: algorithm

text: for

text: distributions

text: with

text: interval-valued

text: parameters.

text: A

text: combi-

text: nation

text: of

text: Karhunen-Lo`eve

text: expansions

text: and

text: evidence

text: theory

text: is

text: presented

text: in

text: (Oberguggenberger,

text: (Zhang

text: et

text: al.,

text: use

text: radial

text: basis

text: expansions

text: in

text: the

text: presence

text: of

text: evidence

text: theory.

text: Based

text: on

text: these

text: developments,

text: a

text: set

text: of

text: algorithms

text: is

text: proposed

text: in

text: this

text: paper

text: to

text: further

text: increase

text: the

text: eﬃciency

text: of

text: structural

text: reliability

text: analysis

text: in

text: the

text: presence

text: of

text: epistemic

text: uncertainties.

text: In

text: particular,

text: free

text: and

text: parametric

text: p-boxes

text: are

text: presented

text: separately

text: and

text: compared

text: later

text: in

text: this

text: paper.

text: The

text: application


text: 6.6 Comparison to free p-boxes ISRA for parametric p-boxes consists of two interconnected Kriging models, the experimental de- signs of which are enriched quasi simultaneously, as shown in Figure 3. In the context of free p-boxes, two levels of meta-models are computed subsequently (see Figure 2). Additionally, an optimization op- eration is required for free p-boxes, which increases the complexity and potentially the computational eﬀort in this case. The increased complexity originates from the basic deﬁnition of the free p-box, which can be interpreted as a generalization of the parametric p-box, as discussed in Section . Assuming, however, that the computational costs are dominated by the evaluation of the limit-state function, the total computational costs for estimating the boundary values of the failure probabilities may be similar for free and parametric p-boxes. The total number of limit-state function evaluations may be similar due to the same method (i.e. AK-MCS) used to surrogate the limit-state surface G(x) = 0. 7 Applications 7.1 Two-dimensional toy function 7.1.1 Problem statement The ﬁrst example is a simple two-dimensional problem used to illustrate the machinery of the proposed approaches. The limit-state function is deﬁned as: g (x) = x − x2. (29) 1 1 2 P Failure is deﬁned as g (x) ≤ 0 and hence the failure probability is P = (g (x) ≤ 0). The two 1 f 1 variables X are modelled by p-boxes. In the following, two cases are distinguished: (i) X is modelled i i by free p-boxes and (ii) X is modelled by parametric p-boxes. In the former case, the bounds of the i free p-box are deﬁned as: F (x ) = FN (x |µ = 2.5, σ = 1) , F (x ) = FN (x |µ = 1.5, σ = 1) , i = 1, 2, (30) Xi i i Xi i i where FN (x|µ, σ) denotes the CDF of a Gaussian distribution with mean value µ and standard deviation σ. For the parametric p-box case, the distribution parameters θ are chosen so that the boundary curves are the same as in the case of free p-boxes: F (x ) = FN (x |µ, σ = 1) , µ ∈ [1.5, 2.5]. (31) Xi i i 7.1.2 Analysis The settings for the ISRA are the following. The meta-models and Monte Carlo simulations are computed using the Matlab-based uncertainty quantiﬁcation framework UQLab (Marelli and Sudret, 2014), including its implementation of AK-MCS (Marelli et al., 2015). All Kriging models are deﬁned by a Gaussian autocorrelation function and a constant trend (f(x) = 1, a.k.a. ordinary Kriging). The number of samples in the initial experimental design are N0 = 12 and N0 = 4 for AK-MCS and 1 2 EGO, respectively. The initial experimental designs are generated by the Latin-hypercube sampling 13 ##LTLine## ##LTLine##

text: Comparison

text: to

text: free

text: p-boxes

text: ISRA

text: for

text: parametric

text: p-boxes

text: consists

text: of

text: two

text: interconnected

text: Kriging

text: models,

text: the

text: experimental

text: de-

text: signs

text: of

text: which

text: are

text: enriched

text: quasi

text: simultaneously,

text: as

text: shown

text: in

text: Figure

text: In

text: the

text: context

text: of

text: free

text: p-boxes,

text: two

text: levels

text: of

text: meta-models

text: are

text: computed

text: subsequently

text: (see

text: Figure

text: Additionally,

text: an

text: optimization

text: op-

text: eration

text: is

text: required

text: for

text: free

text: p-boxes,

text: which

text: increases

text: the

text: complexity

text: and

text: potentially

text: the

text: computational

text: eﬀort

text: in

text: this

text: case.

text: The

text: increased

text: complexity

text: originates

text: from

text: the

text: basic

text: deﬁnition

text: of

text: the

text: free

text: p-box,

text: which

text: can

text: be

text: interpreted

text: as

text: a

text: generalization

text: of

text: the

text: parametric

text: p-box,

text: as

text: discussed

text: in

text: Section

text: Assuming,

text: however,

text: that

text: the

text: computational

text: costs

text: are

text: dominated

text: by

text: the

text: evaluation

text: of

text: the

text: limit-state

text: function,

text: the

text: total

text: computational

text: costs

text: for

text: estimating

text: the

text: boundary

text: values

text: of

text: the

text: failure

text: probabilities

text: may

text: be

text: similar

text: for

text: free

text: and

text: parametric

text: p-boxes.

text: The

text: total

text: number

text: of

text: limit-state

text: function

text: evaluations

text: may

text: be

text: similar

text: due

text: to

text: the

text: same

text: method

text: (i.e.

text: AK-MCS)

text: used

text: to

text: surrogate

text: the

text: limit-state

text: surface

text: G(x)

text: Applications

text: Two-dimensional

text: toy

text: function

text: Problem

text: statement

text: The

text: ﬁrst

text: example

text: is

text: a

text: simple

text: two-dimensional

text: problem

text: used

text: to

text: illustrate

text: the

text: machinery

text: of

text: the

text: proposed

text: approaches.

text: The

text: limit-state

text: function

text: is

text: deﬁned

text: as:

text: g

text: (x)

text: x

text: P

text: Failure

text: is

text: deﬁned

text: as

text: g

text: (x)

text: and

text: hence

text: the

text: failure

text: probability

text: is

text: P

text: (g

text: (x)

text: The

text: two

text: f

text: variables

text: X

text: are

text: modelled

text: by

text: p-boxes.

text: In

text: the

text: following,

text: two

text: cases

text: are

text: distinguished:

text: (i)

text: X

text: is

text: modelled

text: i

text: i

text: by

text: free

text: p-boxes

text: and

text: (ii)

text: X

text: is

text: modelled

text: by

text: parametric

text: p-boxes.

text: In

text: the

text: former

text: case,

text: the

text: bounds

text: of

text: the

text: i

text: free

text: p-box

text: are

text: deﬁned

text: as:

text: F

text: (x

text: FN

text: (x

text: |µ

text: σ

text: F

text: (x

text: FN

text: (x

text: |µ

text: σ

text: i

text: Xi

text: i

text: i

text: Xi

text: i

text: i

text: where

text: FN

text: (x|µ,

text: σ)

text: denotes

text: the

text: CDF

text: of

text: a

text: Gaussian

text: distribution

text: with

text: mean

text: value

text: µ

text: and

text: standard

text: deviation

text: σ.

text: For

text: the

text: parametric

text: p-box

text: case,

text: the

text: distribution

text: parameters

text: θ

text: are

text: chosen

text: so

text: that

text: the

text: boundary

text: curves

text: are

text: the

text: same

text: as

text: in

text: the

text: case

text: of

text: free

text: p-boxes:

text: F

text: (x

text: FN

text: (x

text: |µ,

text: σ

text: µ

text: Xi

text: i

text: i

text: Analysis

text: The

text: settings

text: for

text: the

text: ISRA

text: are

text: the

text: following.

text: The

text: meta-models

text: and

text: Monte

text: Carlo

text: simulations

text: are

text: computed

text: using

text: the

text: Matlab-based

text: uncertainty

text: quantiﬁcation

text: framework

text: UQLab

text: (Marelli

text: and

text: Sudret,

text: including

text: its

text: implementation

text: of

text: AK-MCS

text: (Marelli

text: et

text: al.,

text: All

text: Kriging

text: models

text: are

text: deﬁned

text: by

text: a

text: Gaussian

text: autocorrelation

text: function

text: and

text: a

text: constant

text: trend

text: (f(x)

text: a.k.a.

text: ordinary

text: Kriging).

text: The

text: number

text: of

text: samples

text: in

text: the

text: initial

text: experimental

text: design

text: are

text: and

text: for

text: AK-MCS

text: and

text: EGO,

text: respectively.

text: The

text: initial

text: experimental

text: designs

text: are

text: generated

text: by

text: the

text: Latin-hypercube

text: sampling

text: ##LTLine##

text: ##LTLine##


text: than the simultaneous optimization in this case. This behaviour has been also observed on other toy functions. Based on this observation, the remaining application examples are analysed using separate optimizations. Comparison The algorithms for free and parametric p-boxes originate from two diﬀerent paradigms: interval analysis and nested Monte Carlo simulations, respectively. However, the cost statistics shows that the two proposed two-level meta-modelling algorithms perform similarly in terms of model eval- uations. Free p-boxes required N = 14.7 evaluations of the limit-state function whereas parametric 1 p-boxes required N = 15.4 on average. This can be intuitively understood due to the identical 1 limit-state surface in both settings. Contrarily to N , the number of evaluations on the second-level 1 meta-models, i.e. N , are not comparable for the case of free and parametric p-boxes, due to the 2 diﬀerent algorithms used. 7.2 Single-degree-of-freedom oscillator 7.2.1 Problem setup In order to visualize the eﬀects of non-monotone functions on the failure probability, the following example is analysed. Consider the non-linear undamped single-degree-of-freedom (SDOF) oscillator sketched in Figure 6 (Echard et al., 2011, 2013; Schueremans and Van Gemert, 2005). The correspond- ing limit-state function reads: (cid:12) (cid:18) (cid:19)(cid:12) (cid:12) 2F ω T (cid:12) g (r, F , t , k , k , m) = 3r − (cid:12) 1 sin 0 1 (cid:12) , (32) SDOF 1 1 1 2 (cid:12)mω2 2 (cid:12) 0 where m is the mass, k , k are the spring constants of the primary and secondary springs, r is the 1 2 displacement at which(cid:113)the secondary spring yields, t1 is the duration of the loading, F1 is the amplitude of the force and ω0 = k1m+k2 is the natural frequency of the oscillator. Failure is deﬁned as gSDOF ≤ 0 P and the associated failure probability is P = (g ≤ 0). f SDOF F(t) k 1 F F(t) 1 k r m 2 t t 1 Figure 6: SDOF oscillator – geometry sketch and deﬁnition of the variables The input vector is modelled by a mix of probabilistic variables and p-boxes accounting for the diﬀerent levels of knowledge. The description of the input variables is provided in Table 3. It is assumed that the spring stiﬀnesses and the mass are known well. Hence {k , k , m} are modelled 1 2 by precise CDFs. On the other side, knowledge on {r, F , t } is scarce. Hence, these variables are 1 1 modelled by p-boxes. The two cases of free and parametric p-boxes are distinguished and compared in the following. As seen in Table 3, the parametric p-box is characterized by a distribution function with interval-valued mean value. For the case of free p-boxes, the same p-box boundary curves are used as 18 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: than

text: the

text: simultaneous

text: optimization

text: in

text: this

text: case.

text: This

text: behaviour

text: has

text: been

text: also

text: observed

text: on

text: other

text: toy

text: functions.

text: Based

text: on

text: this

text: observation,

text: the

text: remaining

text: application

text: examples

text: are

text: analysed

text: using

text: separate

text: optimizations.

text: Comparison

text: The

text: algorithms

text: for

text: free

text: and

text: parametric

text: p-boxes

text: originate

text: from

text: two

text: diﬀerent

text: paradigms:

text: interval

text: analysis

text: and

text: nested

text: Monte

text: Carlo

text: simulations,

text: respectively.

text: However,

text: the

text: cost

text: statistics

text: shows

text: that

text: the

text: two

text: proposed

text: two-level

text: meta-modelling

text: algorithms

text: perform

text: similarly

text: in

text: terms

text: of

text: model

text: eval-

text: uations.

text: Free

text: p-boxes

text: required

text: N

text: evaluations

text: of

text: the

text: limit-state

text: function

text: whereas

text: parametric

text: p-boxes

text: required

text: N

text: on

text: average.

text: This

text: can

text: be

text: intuitively

text: understood

text: due

text: to

text: the

text: identical

text: limit-state

text: surface

text: in

text: both

text: settings.

text: Contrarily

text: to

text: N

text: the

text: number

text: of

text: evaluations

text: on

text: the

text: second-level

text: meta-models,

text: i.e.

text: N

text: are

text: not

text: comparable

text: for

text: the

text: case

text: of

text: free

text: and

text: parametric

text: p-boxes,

text: due

text: to

text: the

text: diﬀerent

text: algorithms

text: used.

text: Single-degree-of-freedom

text: oscillator

text: Problem

text: setup

text: In

text: order

text: to

text: visualize

text: the

text: eﬀects

text: of

text: non-monotone

text: functions

text: on

text: the

text: failure

text: probability,

text: the

text: following

text: example

text: is

text: analysed.

text: Consider

text: the

text: non-linear

text: undamped

text: single-degree-of-freedom

text: (SDOF)

text: oscillator

text: sketched

text: in

text: Figure

text: (Echard

text: et

text: al.,

text: Schueremans

text: and

text: Van

text: Gemert,

text: The

text: correspond-

text: ing

text: limit-state

text: function

text: reads:

text: ω

text: T

text: g

text: (r,

text: F

text: t

text: k

text: k

text: m)

text: sin

text: SDOF

text: where

text: m

text: is

text: the

text: mass,

text: k

text: k

text: are

text: the

text: spring

text: constants

text: of

text: the

text: primary

text: and

text: secondary

text: springs,

text: r

text: is

text: the

text: displacement

text: at

text: secondary

text: spring

text: yields,

text: is

text: the

text: duration

text: of

text: the

text: loading,

text: is

text: the

text: amplitude

text: of

text: the

text: force

text: and

text: ω0

text: is

text: the

text: natural

text: frequency

text: of

text: the

text: oscillator.

text: Failure

text: is

text: deﬁned

text: as

text: gSDOF

text: P

text: and

text: the

text: associated

text: failure

text: probability

text: is

text: P

text: (g

text: f

text: SDOF

text: F(t)

text: k

text: F

text: F(t)

text: k

text: r

text: m

text: t

text: t

text: Figure

text: SDOF

text: oscillator

text: geometry

text: sketch

text: and

text: deﬁnition

text: of

text: the

text: variables

text: The

text: input

text: vector

text: is

text: modelled

text: by

text: a

text: mix

text: of

text: probabilistic

text: variables

text: and

text: p-boxes

text: accounting

text: for

text: the

text: diﬀerent

text: levels

text: of

text: knowledge.

text: The

text: description

text: of

text: the

text: input

text: variables

text: is

text: provided

text: in

text: Table

text: It

text: is

text: assumed

text: that

text: the

text: spring

text: stiﬀnesses

text: and

text: the

text: mass

text: are

text: known

text: well.

text: Hence

text: {k

text: k

text: m}

text: are

text: modelled

text: by

text: precise

text: CDFs.

text: On

text: the

text: other

text: side,

text: knowledge

text: on

text: {r,

text: F

text: t

text: is

text: scarce.

text: Hence,

text: these

text: variables

text: are

text: modelled

text: by

text: p-boxes.

text: The

text: two

text: cases

text: of

text: free

text: and

text: parametric

text: p-boxes

text: are

text: distinguished

text: and

text: compared

text: in

text: the

text: following.

text: As

text: seen

text: in

text: Table

text: the

text: parametric

text: p-box

text: is

text: characterized

text: by

text: a

text: distribution

text: function

text: with

text: interval-valued

text: mean

text: value.

text: For

text: the

text: case

text: of

text: free

text: p-boxes,

text: the

text: same

text: p-box

text: boundary

text: curves

text: are

text: used

text: as

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: 7 ππ 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 00 -0.8 --ππ --ππ 00 ππ q x FIG. 4. Plot of F (q), Eq. (8), normalized to its absolute value at q = (0, 0), versus (q , q ) over x y the ﬁrst Brillouin zone using the same φ (k) gap functions and cut-oﬀ around the Fermi surface as d in Fig. 3. Momentum transfers near (π, π) (red shaded region) lead to a positive contribution from the spin-ﬂuctuations to the coupling strength λ . Spin-ﬂuctuations with momentum transfers near d (0, 0) (blue shaded region) give a negative contribution. Regions near (π, π) which are accessible via polarized inelastic neutron scattering provide the dominant contribution to the strength of the d-wave pairing interaction. A closer inspection of the momentum regions accessible near the zone center by RIXS would also be useful in understanding the decrease of pairing strength. The DQMC results reported here and elsewhere are consistent with the weakening of spectral intensity and hardening of the spin excitations observed near the magnetic zone center (π, π) in those measurements [2, 13–16]. Thus we conclude that the evolution of the spin spectrum of excitations with doping in the Hubbard model is consistent with the existing data in the cuprates and can account for the reduction of d-wave pairing strength with doping. ACKNOWLEDGMENTS We acknowledge helpful discussions with Christian Mendl, Yvonne Kung, and Chunjing Jia. Computational work was performed on the Sherlock cluster at Stanford University. This work was supported by the U.S. Department of Energy, Oﬃce of Basic Energy Sciences, q y ##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##

text: ππ

text: --ππ

text: --ππ

text: ππ

text: q

text: x

text: FIG.

text: Plot

text: of

text: F

text: (q),

text: Eq.

text: normalized

text: to

text: its

text: absolute

text: value

text: at

text: q

text: versus

text: (q

text: q

text: over

text: x

text: y

text: the

text: ﬁrst

text: Brillouin

text: zone

text: using

text: the

text: same

text: φ

text: (k)

text: gap

text: functions

text: and

text: cut-oﬀ

text: around

text: the

text: Fermi

text: surface

text: as

text: d

text: in

text: Fig.

text: Momentum

text: transfers

text: near

text: (π,

text: π)

text: (red

text: shaded

text: region)

text: lead

text: to

text: a

text: positive

text: contribution

text: from

text: the

text: spin-ﬂuctuations

text: to

text: the

text: coupling

text: strength

text: λ

text: Spin-ﬂuctuations

text: with

text: momentum

text: transfers

text: near

text: d

text: (blue

text: shaded

text: region)

text: give

text: a

text: negative

text: contribution.

text: Regions

text: near

text: (π,

text: π)

text: which

text: are

text: accessible

text: via

text: polarized

text: inelastic

text: neutron

text: scattering

text: provide

text: the

text: dominant

text: contribution

text: to

text: the

text: strength

text: of

text: the

text: d-wave

text: pairing

text: interaction.

text: A

text: closer

text: inspection

text: of

text: the

text: momentum

text: regions

text: accessible

text: near

text: the

text: zone

text: center

text: by

text: RIXS

text: would

text: also

text: be

text: useful

text: in

text: understanding

text: the

text: decrease

text: of

text: pairing

text: strength.

text: The

text: DQMC

text: results

text: reported

text: here

text: and

text: elsewhere

text: are

text: consistent

text: with

text: the

text: weakening

text: of

text: spectral

text: intensity

text: and

text: hardening

text: of

text: the

text: spin

text: excitations

text: observed

text: near

text: the

text: magnetic

text: zone

text: center

text: (π,

text: π)

text: in

text: those

text: measurements

text: Thus

text: we

text: conclude

text: that

text: the

text: evolution

text: of

text: the

text: spin

text: spectrum

text: of

text: excitations

text: with

text: doping

text: in

text: the

text: Hubbard

text: model

text: is

text: consistent

text: with

text: the

text: existing

text: data

text: in

text: the

text: cuprates

text: and

text: can

text: account

text: for

text: the

text: reduction

text: of

text: d-wave

text: pairing

text: strength

text: with

text: doping.

text: ACKNOWLEDGMENTS

text: We

text: acknowledge

text: helpful

text: discussions

text: with

text: Christian

text: Mendl,

text: Yvonne

text: Kung,

text: and

text: Chunjing

text: Jia.

text: Computational

text: work

text: was

text: performed

text: on

text: the

text: Sherlock

text: cluster

text: at

text: Stanford

text: University.

text: This

text: work

text: was

text: supported

text: by

text: the

text: U.S.

text: Department

text: of

text: Energy,

text: Oﬃce

text: of

text: Basic

text: Energy

text: Sciences,

text: q

text: y

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##

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: ##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: ##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 (cid:0) (cid:1) (cid:107) (cid:107) considering the fact(cid:13)(cid:13)that for every(cid:0) realizatio(cid:1)n(cid:13)(cid:13)of the random activation matrix(cid:13)(cid:13)we have Φ(t(cid:13)(cid:13)) = 1, we obtain (cid:13)(cid:13)(cid:16) − −(cid:0) ∗ ≤ 2(1 − δ) + αM 1/2α(cid:1)L(cid:17)εΛ − (cid:0)− ∗ (cid:1)(cid:13)(cid:13) (cid:13)D(t 1)1/2 x(t) x x(t 1) x 2+ (cid:13) 2 − − − − − − − − ∗ I εΦ(t) + εΦ(t) D(t 1) 1/2BD(t 1) 1/2 2 D(t 1)1/2 x(t 1) x . F This inequality holds for any random activation of the agents. We now note that conditioned on − , matrix Φ(t) and x(t) − (cid:0) (cid:1) t 1 aarned rhaanvdeom varia(cid:104)b(cid:13)(cid:13)les and x(t (cid:0) 1) is dete(cid:1)r(cid:13)(cid:13)m(cid:12)(cid:12)(cid:12)inisti(cid:105)c, we can hence take expectatio(cid:13)(cid:13)n from both sid(cid:13)(cid:13)es of the above inequality E(cid:104)(cid:13)(cid:13)(cid:13)D(cid:16) (t − 1)1/2 x(t) −(cid:0)x∗ F − ≤ 2(1 − δ) + αM(cid:1) (cid:17)1/2αLεΛ x((cid:0)t − 1) − x∗ 2+(cid:1)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) (cid:105) t 1 2 E I − εΦ(t) + εΦ(t) D(t − 1)−1/2BD(t − 1)−1/(cid:0)2 2 D(t − 1)1(cid:1)/2 x(t − 1) − x∗ F − . t 1 We now consider the(cid:104)(cid:13)(cid:13)result of Lem(cid:0)ma IV.11 w(cid:1)(cid:13)(cid:13)it(cid:12)(cid:12)(cid:12)h y =(cid:105)D(t(cid:0)− 1)1/2 x(t − 1(cid:1)) − x∗ to(cid:13)(cid:13)obtain (cid:13)(cid:13) (cid:13) (cid:0) (cid:1)(cid:13) − 1/2 E (cid:13)D(t − 1)1/2 x(t) − x∗ F(cid:13)− ≤ 2(1 δ) + αM αLεΛ x(t − 1) − x∗ 2+ t 1 2 − − − ∗ Γ D(t(cid:13) 1)1/2 x(t (cid:13)1) x . (cid:13) (cid:0) (cid:1)(cid:13) 2 (cid:13) (cid:13) (cid:0) (cid:1)(cid:13) (cid:13) We also have − − ∗ ≤ 1 − − − ∗ x(t 1) x − D(t 2)1/2 x(t 1) x , (cid:13) (cid:13) µ D(t 2)1/2(cid:13) (cid:0) (cid:1)(cid:13) (cid:13) (cid:13) min (cid:13) (cid:13) hence, x(t − 1) − x∗ 2 ≤ − 1 D(t − 2)1/2 (cid:13)(cid:13)x(t − 1) − x∗(cid:0) 2 (cid:1)(cid:13)(cid:13) (cid:13) (cid:0) (cid:1)(cid:13) 2(1 ∆) + αm (cid:13) (cid:13) − − − ∗ In order to have a recursion we use the result of Lemma IV.12 to bound D(t 1)1/2 x(t 1) x in terms of D(t − 2)1/2(cid:104)(cid:13)(cid:13)x(t − 1) − x∗(cid:0) and obt(cid:1)a(cid:13)(cid:13)in(cid:12)(cid:12)(cid:12) (cid:105) (cid:0) (cid:0) (cid:1) (cid:1) (cid:13)(cid:13) (cid:0) (cid:1)(cid:13)(cid:13) − E (cid:16)D(t − 1(cid:13)(cid:13))1/2 x(t)(cid:13)(cid:13)− x(cid:17)∗(cid:13)(cid:13) F − ≤ (cid:0)2(1 δ) −+ αM(cid:1)(cid:13)(cid:13) 1/2αLεΛ D(t − 2)1/2 x(t − 1) − x∗ 2+ t 1 2 2(1 ∆) + αm (30) − − − − ∗ Γ 1 + C g(t 2) 1/2 D(t 2)1/2 x(t 1) x . 2 1 (cid:0) (cid:1) We now con(cid:104)s(cid:13)ider iteration(cid:0)t, taking e(cid:1)x(cid:13)p(cid:105)ectations from both sides of Eq. ((cid:104)3(cid:13)0) and using (cid:0)the law of tota(cid:1)l(cid:13)ex(cid:105)pectation yield (cid:13) (cid:13) (cid:0) (cid:1) (cid:13) (cid:13) − E D(cid:104)(cid:16)(t − 1)1(cid:13)(cid:13)/2 x(t) −(cid:13)(cid:13)x∗ (cid:17)(cid:13)(cid:13) ≤ 2(1 δ(cid:0)) −+ αM 1/2αL(cid:1)ε(cid:13)(cid:13)Λ(cid:105)E D(t − 2)1/2 x(t − 1) − x∗ 2 + 2 2(1 ∆) + αm (31) Γ E 1 + C g(t − 2) 1/2 D(t − 2)1/2 x(t − 1) − x∗ . 2 1 (cid:104)(cid:13) (cid:0) (cid:1)(cid:13)(cid:105) (cid:104)(cid:13) (cid:13) (cid:13) (cid:0) (cid:1)(cid:13)(cid:105) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) The second expected value in the right hand side of the previous inequality is equal to E D(t − 2)1/2 x(t − 1) − x∗ + C E g(t − 2) 1/2 D(t − 2)1/2 x(t − 1) − x∗ 1 We note that the Cauchy Schwarz inequality in t(cid:16)he contex(cid:17)t of the(cid:2) exp(cid:3)ec(cid:2)tatio(cid:3)n states that for any two random variables X and E E E Y such that [X], [Y ], and [XY ] exist, we have E 2 ≤ E E [XY ] X2 Y 2 . (cid:32) (cid:33) (cid:32) (cid:33) There(cid:104)fo(cid:13)re, (cid:13) (cid:13) (cid:0) (cid:1)(cid:13)(cid:105) (cid:104)(cid:13) (cid:13)(cid:105) (cid:104)(cid:13) (cid:0) (cid:1)(cid:13) (cid:105) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 1/2 1/2 E g(t − 2) 1/2 D(t − 2)1/2 x(t − 1) − x∗ ≤ E g(t − 2) E D(t − 2)1/2 x(t − 1) − x∗ 2 (cid:0) (cid:1) (cid:104)(cid:13) (cid:13)(cid:105) (cid:13) (cid:13) (cid:22) We next use the result of Lemma IV.6 on the properties of strongly convex functions together with the fact that H(t) 2(1 − δ) + αM I = 1 I, to ﬁnd an upp(cid:13)(cid:13)er bound(cid:13)(cid:13)for (cid:16)E (cid:16)g(t(cid:0)− 2) a(cid:1)s follow(cid:17)(cid:17)s λ − ≤ 2 − − ∗ 1/2 g(t 2) F x(t 2) F . λ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: considering

text: the

text: for

text: the

text: random

text: activation

text: have

text: we

text: obtain

text: δ)

text: αM

text: x(t)

text: x

text: x(t

text: x

text: I

text: εΦ(t)

text: εΦ(t)

text: D(t

text: D(t

text: x(t

text: x

text: F

text: This

text: inequality

text: holds

text: for

text: any

text: random

text: activation

text: of

text: the

text: agents.

text: We

text: now

text: note

text: that

text: conditioned

text: on

text: matrix

text: Φ(t)

text: and

text: x(t)

text: t

text: aarned

text: rhaanvdeom

text: and

text: x(t

text: is

text: we

text: can

text: hence

text: take

text: from

text: both

text: of

text: the

text: above

text: inequality

text: (t

text: x(t)

text: F

text: δ)

text: x∗

text: t

text: E

text: I

text: εΦ(t)

text: εΦ(t)

text: D(t

text: D(t

text: x(t

text: x∗

text: F

text: t

text: We

text: now

text: consider

text: of

text: y

text: x(t

text: x∗

text: E

text: x(t)

text: x∗

text: δ)

text: αM

text: αLεΛ

text: x(t

text: x∗

text: t

text: Γ

text: x(t

text: x

text: We

text: also

text: have

text: x(t

text: x

text: D(t

text: x(t

text: x

text: µ

text: D(t

text: min

text: hence,

text: x(t

text: x∗

text: D(t

text: αm

text: In

text: order

text: to

text: have

text: a

text: recursion

text: we

text: use

text: the

text: result

text: of

text: Lemma

text: to

text: bound

text: D(t

text: x(t

text: x

text: in

text: terms

text: of

text: D(t

text: and

text: E

text: F

text: δ)

text: D(t

text: x(t

text: x∗

text: t

text: αm

text: Γ

text: C

text: g(t

text: D(t

text: x(t

text: x

text: We

text: now

text: taking

text: from

text: both

text: sides

text: of

text: Eq.

text: and

text: using

text: law

text: of

text: yield

text: E

text: x(t)

text: αM

text: D(t

text: x(t

text: x∗

text: αm

text: Γ

text: E

text: C

text: g(t

text: D(t

text: x(t

text: x∗

text: The

text: second

text: expected

text: value

text: in

text: the

text: right

text: hand

text: side

text: of

text: the

text: previous

text: inequality

text: is

text: equal

text: to

text: E

text: D(t

text: x(t

text: x∗

text: C

text: E

text: g(t

text: D(t

text: x(t

text: x∗

text: We

text: note

text: that

text: the

text: Cauchy

text: Schwarz

text: inequality

text: in

text: of

text: states

text: that

text: for

text: any

text: two

text: random

text: variables

text: X

text: and

text: E

text: E

text: E

text: Y

text: such

text: that

text: [X],

text: [Y

text: and

text: [XY

text: exist,

text: we

text: have

text: E

text: E

text: E

text: [XY

text: Y

text: E

text: g(t

text: D(t

text: x(t

text: x∗

text: E

text: g(t

text: E

text: D(t

text: x(t

text: x∗

text: We

text: next

text: use

text: the

text: result

text: of

text: Lemma

text: on

text: the

text: properties

text: of

text: strongly

text: convex

text: functions

text: together

text: with

text: the

text: fact

text: that

text: H(t)

text: δ)

text: αM

text: I

text: I,

text: to

text: ﬁnd

text: an

text: λ

text: g(t

text: F

text: x(t

text: F

text: λ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##