text: grmE b grmj 8 → −0.06 mass case, ν 0.25. In particular, for ν = 0.25 we see that the EOB prediction for Emrg is larger than the × − b −0.08 NR one by 4.9 10 3 and that the EOB prediction for × − jmrg is larger than the NR one by 3.1 10 2. [These −0.1 diﬀerences are larger than those apparent in the corre- NR: SXS sponding endpoints of Fig. 5 because we are now com- −0.12 NR: Llama puting EEOB(jEOB) − ENR(jNR ) while the vertical dis- EOB b mrg b mrg SXS NR fit placement of the endpoints of Fig. 5 corresponded to −0.14 − 0 0.05 0.1 0.15 0.2 0.25 EEOB(jEOB) ENR(jEOB)]. In the same ﬁgure we also 3.8 dibsplay mLrlgama mebrgermdrgata, denoted by asterisks. We ﬁnd that Llama and SXS data are fully consistent within 3.6 − the 10 4 level. Finally, the ν-dependence of the SXS 3.4 NR Emrg can be ﬁtted by means of a second order poly- b nomial, while jmrg(ν) is well ﬁtted by a straight line, 3.2 NR: SXS namely 3 NR: Llama EOB SXS NR fit 2.80 0.05 0.1 0.15 0.2 0.25 Emrg = −0.062951 − 0.176949ν − 0.216264ν2, (9) ν b,SXS − jmrg = 3.406583 2.421052ν. (10) SXS FIG. 6. Binding energy and angular momentum at merger | | (i.e. at the maximum of h ) for nonspinning binaries versus 22 the symmetric mass ratio ν. The plot shows: (i) the highly V. NR CALIBRATION OF THE EOB accurate numerical agreement between SXS and Llama ﬁnal SPIN-ORBIT COUPLING conﬁgurations for q = 1 and q = 2; (ii) the good numerical consistency between EOB predictions and actual NR states. Let us now turn to discussing the numerical comple- tion of the EOB spinning model. As mentioned above, the spinning sector of the model we use here is exactly merger. The diﬀerence at the NR merger point is of order the same as described in Paper I. In the latter work it × − 4 10 4, which is barely visible in the top panel of Fig. 4 was shown that it was possible to obtain a good EOBNR (and it is approximately twice the estimated uncertainty phasing agreement by tuning a single eﬀective functional there). parameter c (χ) that was entering the spin-orbit coupling 3 For the other nonspinning conﬁgurations in Table I, at next-to-next-to-next leading order5. The best values one obtains curves quite analogous to the q = 8 case. The of the parameter c , for various spin values, are listed 3 relevant information is displayed in Fig. 5 that collects in Table I of Paper I. Since the orbital A function has ≡ − orb the ∆EEOBNR(j) EEOB(j) ENR(j) diﬀerences for all changed, in the present work, because of (i) the use of the b b b nonspinning data but one (the q = 1.5 one). Note that complete analytical 4PN information and (ii) the conse- the end points indicated by square markers in Fig. 5, quently modiﬁed functional dependence of a6(ν) given c record the values of the function ∆EEOBNR(j) at the by Eq. (5), we now need to look for a new determination b corresponding EOB merger values of j, jEOB. On the of c (χ). In doing so, we also consider here more NR mrg 3 other hand, the stars on the x-axis of the ﬁgure mark the simulations than in Paper I, notably taking into account corresponding NR merger values of j, jNR . all available nonprecessing data in the SXS catalog. The mrg For q = 1, we use here the SXS:BBH:0066 dataset with spinning SXS conﬁgurations we use are listed in Table I: ∼ ≤ ≤ 28 orbits. The vectorial shifts (∆j0, ∆E0) we use are the mass ratio varies in the range 1 q 8 and there are b b listed in Table III in the Appendix. For all conﬁgura- several conﬁgurations where only one of the two black- tions, one ﬁnds that the EOB/NR agreement is of the holes is spinning. A priori, one expects the NNNLO ef- − order 10 5 at the beginnin−g of the inspiral and grows fective parameter c3 to be a function of both the mass only up to a few parts in 10 4 around merger. These dif- ratio and the spins of the binary. The determination of c3 ferences are comparable to (though slightly larger than) is done in two steps. In a ﬁrst step, we separately consid- the corresponding error bars computed, as above, taking ered each binary conﬁguration and determine a prelimi- the diﬀerence between the two highest resolutions avail- nary best value of c3 by minimizing the EOB-NR phase able. diﬀerence, after alignment in the inspiral phase, so to be In Fig. 6 we complement the above results by focus- compatible with (and typically smaller than) the NR un- ing on the EOB/NR comparison for merger characteris- certainty. This procedure is rather straightforward, as it tics, (j , E ) as a function of ν. The ﬁgure shows mrg mrg that the EOB/NR disagreement increases with ν: from × − × − a diﬀerence of 3.07 10 4 for Emrg and 1.7 10 4 for b jmrg for q = 9.989 (ν = 0.0827) the EOB model ends 5 The analysis of Paper I was limited to the equal-mass, equal-spin up slightly overestimating the NR values in the equal- case. ##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## 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text: grmE

text: b

text: grmj

text: mass

text: case,

text: ν

text: In

text: particular,

text: for

text: ν

text: we

text: see

text: that

text: the

text: EOB

text: prediction

text: for

text: Emrg

text: is

text: larger

text: than

text: the

text: b

text: NR

text: one

text: by

text: and

text: that

text: the

text: EOB

text: prediction

text: for

text: jmrg

text: is

text: larger

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text: diﬀerences

text: are

text: larger

text: than

text: those

text: apparent

text: in

text: the

text: corre-

text: NR:

text: SXS

text: sponding

text: endpoints

text: of

text: Fig.

text: because

text: we

text: are

text: now

text: com-

text: NR:

text: Llama

text: puting

text: EEOB(jEOB)

text: ENR(jNR

text: while

text: the

text: vertical

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text: EOB

text: b

text: mrg

text: b

text: mrg

text: SXS

text: NR

text: fit

text: placement

text: of

text: the

text: endpoints

text: of

text: Fig.

text: corresponded

text: to

text: EEOB(jEOB)

text: ENR(jEOB)].

text: In

text: the

text: same

text: ﬁgure

text: we

text: also

text: dibsplay

text: mLrlgama

text: mebrgermdrgata,

text: denoted

text: by

text: asterisks.

text: We

text: ﬁnd

text: that

text: Llama

text: and

text: SXS

text: data

text: are

text: fully

text: consistent

text: within

text: the

text: level.

text: Finally,

text: the

text: ν-dependence

text: of

text: the

text: SXS

text: NR

text: Emrg

text: can

text: be

text: ﬁtted

text: by

text: means

text: of

text: a

text: second

text: order

text: poly-

text: b

text: nomial,

text: while

text: jmrg(ν)

text: is

text: well

text: ﬁtted

text: by

text: a

text: straight

text: line,

text: NR:

text: SXS

text: namely

text: NR:

text: Llama

text: EOB

text: SXS

text: NR

text: fit

text: Emrg

text: ν

text: b,SXS

text: jmrg

text: SXS

text: FIG.

text: Binding

text: energy

text: and

text: angular

text: momentum

text: at

text: merger

text: (i.e.

text: at

text: the

text: maximum

text: of

text: h

text: for

text: nonspinning

text: binaries

text: versus

text: the

text: symmetric

text: mass

text: ratio

text: ν.

text: The

text: plot

text: shows:

text: (i)

text: the

text: highly

text: V.

text: NR

text: CALIBRATION

text: OF

text: THE

text: EOB

text: accurate

text: numerical

text: agreement

text: between

text: SXS

text: and

text: Llama

text: ﬁnal

text: SPIN-ORBIT

text: COUPLING

text: conﬁgurations

text: for

text: q

text: and

text: q

text: (ii)

text: the

text: good

text: numerical

text: consistency

text: between

text: EOB

text: predictions

text: and

text: actual

text: NR

text: states.

text: Let

text: us

text: now

text: turn

text: to

text: discussing

text: the

text: numerical

text: comple-

text: tion

text: of

text: the

text: EOB

text: spinning

text: model.

text: As

text: mentioned

text: above,

text: the

text: spinning

text: sector

text: of

text: the

text: model

text: we

text: use

text: here

text: is

text: exactly

text: merger.

text: The

text: diﬀerence

text: at

text: the

text: NR

text: merger

text: point

text: is

text: of

text: order

text: the

text: same

text: as

text: described

text: in

text: Paper

text: I.

text: In

text: the

text: latter

text: work

text: it

text: which

text: is

text: barely

text: visible

text: in

text: the

text: top

text: panel

text: of

text: Fig.

text: was

text: shown

text: that

text: it

text: was

text: possible

text: to

text: obtain

text: a

text: good

text: EOBNR

text: (and

text: it

text: is

text: approximately

text: twice

text: the

text: estimated

text: uncertainty

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text: agreement

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text: tuning

text: a

text: single

text: eﬀective

text: functional

text: there).

text: parameter

text: c

text: (χ)

text: that

text: was

text: entering

text: the

text: spin-orbit

text: coupling

text: For

text: the

text: other

text: nonspinning

text: conﬁgurations

text: in

text: Table

text: I,

text: at

text: next-to-next-to-next

text: leading

text: The

text: best

text: values

text: one

text: obtains

text: curves

text: quite

text: analogous

text: to

text: the

text: q

text: case.

text: The

text: of

text: the

text: parameter

text: c

text: for

text: various

text: spin

text: values,

text: are

text: listed

text: relevant

text: information

text: is

text: displayed

text: in

text: Fig.

text: that

text: collects

text: in

text: Table

text: I

text: of

text: Paper

text: I.

text: Since

text: the

text: orbital

text: A

text: function

text: has

text: orb

text: the

text: ∆EEOBNR(j)

text: EEOB(j)

text: ENR(j)

text: diﬀerences

text: for

text: all

text: changed,

text: in

text: the

text: present

text: work,

text: because

text: of

text: (i)

text: the

text: use

text: of

text: the

text: b

text: b

text: b

text: nonspinning

text: data

text: but

text: one

text: (the

text: q

text: one).

text: Note

text: that

text: complete

text: analytical

text: information

text: and

text: (ii)

text: the

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text: the

text: end

text: points

text: indicated

text: by

text: square

text: markers

text: in

text: Fig.

text: quently

text: modiﬁed

text: functional

text: dependence

text: of

text: given

text: c

text: record

text: the

text: values

text: of

text: the

text: function

text: ∆EEOBNR(j)

text: at

text: the

text: by

text: Eq.

text: we

text: now

text: need

text: to

text: look

text: for

text: a

text: new

text: determination

text: b

text: corresponding

text: EOB

text: merger

text: values

text: of

text: j,

text: jEOB.

text: On

text: the

text: of

text: c

text: (χ).

text: In

text: doing

text: so,

text: we

text: also

text: consider

text: here

text: more

text: NR

text: mrg

text: other

text: hand,

text: the

text: stars

text: on

text: the

text: x-axis

text: of

text: the

text: ﬁgure

text: mark

text: the

text: simulations

text: than

text: in

text: Paper

text: I,

text: notably

text: taking

text: into

text: account

text: corresponding

text: NR

text: merger

text: values

text: of

text: j,

text: jNR

text: all

text: available

text: nonprecessing

text: data

text: in

text: the

text: SXS

text: catalog.

text: The

text: mrg

text: For

text: q

text: we

text: use

text: here

text: the

text: dataset

text: with

text: spinning

text: SXS

text: conﬁgurations

text: we

text: use

text: are

text: listed

text: in

text: Table

text: I:

text: orbits.

text: The

text: vectorial

text: shifts

text: we

text: use

text: are

text: the

text: mass

text: ratio

text: varies

text: in

text: the

text: range

text: q

text: and

text: there

text: are

text: b

text: b

text: listed

text: in

text: Table

text: III

text: in

text: the

text: Appendix.

text: For

text: all

text: conﬁgura-

text: several

text: conﬁgurations

text: where

text: only

text: one

text: of

text: the

text: two

text: black-

text: tions,

text: one

text: ﬁnds

text: that

text: the

text: EOB/NR

text: agreement

text: is

text: of

text: the

text: holes

text: is

text: spinning.

text: A

text: priori,

text: one

text: expects

text: the

text: NNNLO

text: ef-

text: order

text: at

text: the

text: beginnin−g

text: of

text: the

text: inspiral

text: and

text: grows

text: fective

text: parameter

text: to

text: be

text: a

text: function

text: of

text: both

text: the

text: mass

text: only

text: up

text: to

text: a

text: few

text: parts

text: in

text: around

text: merger.

text: These

text: dif-

text: ratio

text: and

text: the

text: spins

text: of

text: the

text: binary.

text: The

text: determination

text: of

text: ferences

text: are

text: comparable

text: to

text: (though

text: slightly

text: larger

text: than)

text: is

text: done

text: in

text: two

text: steps.

text: In

text: a

text: ﬁrst

text: step,

text: we

text: separately

text: consid-

text: the

text: corresponding

text: error

text: bars

text: computed,

text: as

text: above,

text: taking

text: ered

text: each

text: binary

text: conﬁguration

text: and

text: determine

text: a

text: prelimi-

text: the

text: diﬀerence

text: between

text: the

text: two

text: highest

text: resolutions

text: avail-

text: nary

text: best

text: value

text: of

text: by

text: minimizing

text: the

text: EOB-NR

text: phase

text: able.

text: diﬀerence,

text: after

text: alignment

text: in

text: the

text: inspiral

text: phase,

text: so

text: to

text: be

text: In

text: Fig.

text: we

text: complement

text: the

text: above

text: results

text: by

text: focus-

text: compatible

text: with

text: (and

text: typically

text: smaller

text: than)

text: the

text: NR

text: un-

text: ing

text: on

text: the

text: EOB/NR

text: comparison

text: for

text: merger

text: characteris-

text: certainty.

text: This

text: procedure

text: is

text: rather

text: straightforward,

text: as

text: it

text: tics,

text: (j

text: E

text: as

text: a

text: function

text: of

text: ν.

text: The

text: ﬁgure

text: shows

text: mrg

text: mrg

text: that

text: the

text: EOB/NR

text: disagreement

text: increases

text: with

text: ν:

text: from

text: a

text: diﬀerence

text: of

text: for

text: Emrg

text: and

text: for

text: b

text: jmrg

text: for

text: q

text: (ν

text: the

text: EOB

text: model

text: ends

text: The

text: analysis

text: of

text: Paper

text: I

text: was

text: limited

text: to

text: the

text: equal-mass,

text: equal-spin

text: up

text: slightly

text: overestimating

text: the

text: NR

text: values

text: in

text: the

text: equal-

text: case.

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text: On the Approximation Theory of Linear Variational Subspace Design ∗ Jianbo Ye Zhixin Yan College of Information Sciences and Technology Department of Computer Science Pennsylvania State University, USA Worcester Polytechnic Institute, USA Abstract is as much as, if not more than, that spent on on-site modeling. In industrial deployments, companies have to hire many artists with Solving large-scale optimization on-the-ﬂy is often a difﬁcult task expertise skills for rigging a large set of models before those mod- for real-time computer graphics applications. To tackle this chal- els are used in productions. This poses the necessity for a fully lenge, model reduction is a well-adopted technique. Despite its use- automatic subspace generation method. This problems have re- fulness, model reduction often requires a handcrafted subspace that ceived attentions in the past. For example, data-driven methods spans a domain that hypothetically embodies desirable solutions. have been developed for deformable meshes, where a learning al- For many applications, obtaining such subspaces case-by-case ei- gorithm tries to capture the characteristics of deformable mesh se- ther is impossible or requires extensive human labors, hence does quences and applies to a different task [Sumner et al. 2005; Der not readily have a scalable solution for growing number of tasks. et al. 2006]. However, they still struggle to face two challenges: We propose linear variational subspace design for large-scale con- 1) Scalability: Like approaches relying on human inputs, obtaining strained quadratic programming, which can be computed automat- a deformable sequence of scanned meshes can also be expensive. ically without any human interventions. We provide meaningful No words to say if we want to build a deformable mesh database approximation error bound that substantiates the quality of calcu- containing large number of models with heterogeneous shapes. 2) lated subspace, and demonstrate its empirical success in interactive Applicability: Many models of complex geometries or topologies deformable modeling for triangular and tetrahedral meshes. are relatively difﬁcult to rig, and there are no easy ways to build a set of controllers with skinning weights to produce desirable defor- CR Categories: I.3.5 [Computer Graphics]: Computational Ge- mations. Though we see there have been several workarounds for ometry and Object Modeling I.3.6 [Computer Graphics]: Method- a domain speciﬁc mesh sets, such as faces and clothes, an automat- ology and Techniques—Interaction techniques ically computed subspace for arbitrary meshes, which is cheaply 1 Introduction obtained, still can be beneﬁcial, if not all, for fast prototyping or exploratory purposes: the set of constraints chosen on-site is ex- In computer graphics realm, solving optimization with a substan- ported for computing a deformation with full quality in the off-line tially large amount of variables is often an expensive task. In order stage. to speed up the computations, model reduction has been introduced as a useful technique, particularly for interactive and real-time ap- In this paper, we introduce an automatic and principled way to cre- plications. In solving a large-scale optimization problem, it typi- ate reduced models, which might be applied to other computation- cally assumes that a desired solution approximately lies in a man- ally intensive optimization scenarios other than mesh deformation. ifold of much lower dimension that is independent of the variable Our main idea is very simple: in solving a constrained quadratic size. Therefore, it is possible to cut down calculations to a compu- programming, we observe that Karush-Kuhn-Tucker (KKT) con- tationally practical level by only exploring variability (i.e., differ- dition implicitly deﬁnes an effective subspace that can be directly ent solutions subject to different constraints) in a suitably chosen reused for on-site subsequent optimization. We name this linear low-order space, meanwhile, attempting to produce visually con- variational subspace (for short, variational subspace). Our contri- vincing results just-in-time. In this paper, we re-examine model re- bution is to theoretically study the approximation error bound of duction techniques for quadratic optimization with uncertain linear variational subspace and to empirically validate its success in in- constraints, which has been widely used in interactively modeling teractive mesh modeling. The deformation framework is similar deformable surfaces and solids. to one used in [Jacobson et al. 2012].12 We further examine the deformation property of our proposed method, and compare with Modeling deformable meshes has been an established topic in com- physically based deformation [Ciarlet 2000; Grinspun et al. 2003; puter graphics for years [Sorkine et al. 2004; Yu et al. 2004]. Botsch et al. 2006; Sorkine and Alexa 2007; Chao et al. 2010] and Mesh deformation of high quality is accessible via off-line solv- conformal deformation[Paries et al. 2007; Crane et al. 2011]. ing a large-scale optimization whose variables are in complexity of mesh nodes. A studio work-ﬂow in mesh deformable model- 2 Mathematical Background ing often involves trial-and-error loops: an artist tries different sets of constraints and explores for desirable poses. In such processes, Consider minimizing a quadratic function f(X; q) = − an interactive technique helps to save the computation time where (1/2)XTHX qTX subject to linear constraints ATX = b, approximate solutions are ﬁrstly displayed for the purpose of guid- where X ∈ Rn is an overly high dimensional solution, H is ance before a ﬁnal solution is calculated and exported. Neverthe- a semi-deﬁnite positive matrix of size n × n, q ∈ Rn, A is a less, interactive techniques related to real-time mesh modeling has well-conditioned matrix of size n × m, and b ∈ Rm. Typically, been less successful than their off-line siblings till today. Exist- ing work based on model reduction often requires a high quality 1In independent work reported in a recent preprint [Wang et al. 2015], subspace as the input, which typically demands human interven- Wang et al. also propose a mesh deformation framework based on linear tions in constructing them. Exemplars include cage-based defor- variational subspace similar to ours with the difference that we in addition mations [Huang et al. 2006; Ben-Chen et al. 2009], LBS with addi- use linear variational subspace to model rotation errors in reduced-ARAP tional weights [Jacobson et al. 2012], LBS with skeletons [Shi et al. framework. Our deformation can be similar to theirs [Wang et al. 2015], if 2007], and LBS with point/region handles [Au et al. 2007; Sumner regularized coefﬁcient α is set to a large value. Therefore, the contribution et al. 2007]. The time spent on constructing such reduced models of our paper excluding the empirical efforts is the approximation theory for linear variational subspace. ∗ Email: jxy198@ist.psu.edu 2Implementations and demos: https://github.com/bobye ##LTLine## ##LTLine##

text: On

label: title

font: LIVDFZ+NimbusSanL-Bold

text: the

text: Approximation

text: Theory

text: of

text: Linear

text: Variational

text: Subspace

text: Design

label: paragraph

text: Jianbo

text: Ye

text: Zhixin

text: Yan

text: College

text: of

text: Information

text: Sciences

text: and

text: Technology

text: Department

text: of

text: Computer

text: Science

text: Pennsylvania

text: State

text: University,

text: USA

text: Worcester

text: Polytechnic

text: Institute,

text: USA

text: Abstract

text: is

text: as

text: much

text: as,

text: if

text: not

text: more

text: than,

text: that

text: spent

text: on

text: on-site

text: modeling.

text: In

text: industrial

text: deployments,

text: companies

text: have

text: to

text: hire

text: many

text: artists

text: with

text: Solving

label: abstract

text: large-scale

text: optimization

text: on-the-ﬂy

text: is

text: often

text: a

text: difﬁcult

text: task

text: expertise

text: skills

text: for

text: rigging

text: a

text: large

text: set

text: of

text: models

text: before

text: those

text: mod-

text: for

text: real-time

text: computer

text: graphics

text: applications.

text: To

text: tackle

text: this

text: chal-

text: els

text: are

text: used

text: in

text: productions.

text: This

text: poses

text: the

text: necessity

text: for

text: a

text: fully

text: lenge,

text: model

text: reduction

text: is

text: a

text: well-adopted

text: technique.

text: Despite

text: its

text: use-

text: automatic

text: subspace

text: generation

text: method.

text: This

text: problems

text: have

text: re-

text: fulness,

text: model

text: reduction

text: often

text: requires

text: a

text: handcrafted

text: subspace

text: that

text: ceived

text: attentions

text: in

text: the

text: past.

text: For

text: example,

text: data-driven

text: methods

text: spans

text: a

text: domain

text: that

text: hypothetically

text: embodies

text: desirable

text: solutions.

text: have

text: been

text: developed

text: for

text: deformable

text: meshes,

text: where

text: a

text: learning

text: al-

text: For

text: many

text: applications,

text: obtaining

text: such

text: subspaces

text: case-by-case

text: ei-

text: gorithm

text: tries

text: to

text: capture

text: the

text: characteristics

text: of

text: deformable

text: mesh

text: se-

text: ther

text: is

text: impossible

text: or

text: requires

text: extensive

text: human

text: labors,

text: hence

text: does

text: quences

text: and

text: applies

text: to

text: a

text: different

text: task

text: [Sumner

text: et

text: al.

text: Der

text: not

text: readily

text: have

text: a

text: scalable

text: solution

text: for

text: growing

text: number

text: of

text: tasks.

text: et

text: al.

text: However,

text: they

text: still

text: struggle

text: to

text: face

text: two

text: challenges:

text: We

text: propose

text: linear

text: variational

text: subspace

text: design

text: for

text: large-scale

text: con-

text: Scalability:

text: Like

text: approaches

text: relying

text: on

text: human

text: inputs,

text: obtaining

text: strained

text: quadratic

text: programming,

text: which

text: can

text: be

text: computed

text: automat-

text: a

text: deformable

text: sequence

text: of

text: scanned

text: meshes

text: can

text: also

text: be

text: expensive.

text: ically

text: without

text: any

text: human

text: interventions.

text: We

text: provide

text: meaningful

text: No

text: words

text: to

text: say

text: if

text: we

text: want

text: to

text: build

text: a

text: deformable

text: mesh

text: database

text: approximation

text: error

text: bound

text: that

text: substantiates

text: the

text: quality

text: of

text: calcu-

text: containing

text: large

text: number

text: of

text: models

text: with

text: heterogeneous

text: shapes.

text: lated

text: subspace,

text: and

text: demonstrate

text: its

text: empirical

text: success

text: in

text: interactive

text: Applicability:

text: Many

text: models

text: of

text: complex

text: geometries

text: or

text: topologies

text: deformable

text: modeling

text: for

text: triangular

text: and

text: tetrahedral

text: meshes.

text: are

text: relatively

text: difﬁcult

text: to

text: rig,

text: and

text: there

text: are

text: no

text: easy

text: ways

text: to

text: build

text: a

text: set

text: of

text: controllers

text: with

text: skinning

text: weights

text: to

text: produce

text: desirable

text: defor-

text: CR

text: Categories:

text: [Computer

text: Graphics]:

text: Computational

text: Ge-

text: mations.

text: Though

text: we

text: see

text: there

text: have

text: been

text: several

text: workarounds

text: for

text: ometry

text: and

text: Object

text: Modeling

text: [Computer

text: Graphics]:

text: Method-

text: a

text: domain

text: speciﬁc

text: mesh

text: sets,

text: such

text: as

text: faces

text: and

text: clothes,

text: an

text: automat-

text: ology

text: and

text: Techniques—Interaction

text: techniques

text: ically

text: computed

text: subspace

text: for

text: arbitrary

text: meshes,

text: which

text: is

text: cheaply

text: Introduction

text: obtained,

text: still

text: can

text: be

text: beneﬁcial,

text: if

text: not

text: all,

text: for

text: fast

text: prototyping

text: or

text: exploratory

text: purposes:

text: the

text: set

text: of

text: constraints

text: chosen

text: on-site

text: is

text: ex-

text: In

text: computer

text: graphics

text: realm,

text: solving

text: optimization

text: with

text: a

text: substan-

text: ported

text: for

text: computing

text: a

text: deformation

text: with

text: full

text: quality

text: in

text: the

text: off-line

text: tially

text: large

text: amount

text: of

text: variables

text: is

text: often

text: an

text: expensive

text: task.

text: In

text: order

text: stage.

text: to

text: speed

text: up

text: the

text: computations,

text: model

text: reduction

text: has

text: been

text: introduced

text: as

text: a

text: useful

text: technique,

text: particularly

text: for

text: interactive

text: and

text: real-time

text: ap-

text: In

text: this

text: paper,

text: we

text: introduce

text: an

text: automatic

text: and

text: principled

text: way

text: to

text: cre-

text: plications.

text: In

text: solving

text: a

text: large-scale

text: optimization

text: problem,

text: it

text: typi-

text: ate

text: reduced

text: models,

text: which

text: might

text: be

text: applied

text: to

text: other

text: computation-

text: cally

text: assumes

text: that

text: a

text: desired

text: solution

text: approximately

text: lies

text: in

text: a

text: man-

text: ally

text: intensive

text: optimization

text: scenarios

text: other

text: than

text: mesh

text: deformation.

text: ifold

text: of

text: much

text: lower

text: dimension

text: that

text: is

text: independent

text: of

text: the

text: variable

text: Our

text: main

text: idea

text: is

text: very

text: simple:

text: in

text: solving

text: a

text: constrained

text: quadratic

text: size.

text: Therefore,

text: it

text: is

text: possible

text: to

text: cut

text: down

text: calculations

text: to

text: a

text: compu-

text: programming,

text: we

text: observe

text: that

text: Karush-Kuhn-Tucker

text: (KKT)

text: con-

text: tationally

text: practical

text: level

text: by

text: only

text: exploring

text: variability

text: (i.e.,

text: differ-

text: dition

text: implicitly

text: deﬁnes

text: an

text: effective

text: subspace

text: that

text: can

text: be

text: directly

text: ent

text: solutions

text: subject

text: to

text: different

text: constraints)

text: in

text: a

text: suitably

text: chosen

text: reused

text: for

text: on-site

text: subsequent

text: optimization.

text: We

text: name

text: this

text: linear

text: low-order

text: space,

text: meanwhile,

text: attempting

text: to

text: produce

text: visually

text: con-

text: variational

text: subspace

text: (for

text: short,

text: variational

text: subspace).

text: Our

text: contri-

text: vincing

text: results

text: just-in-time.

text: In

text: this

text: paper,

text: we

text: re-examine

text: model

text: re-

text: bution

text: is

text: to

text: theoretically

text: study

text: the

text: approximation

text: error

text: bound

text: of

text: duction

text: techniques

text: for

text: quadratic

text: optimization

text: with

text: uncertain

text: linear

text: variational

text: subspace

text: and

text: to

text: empirically

text: validate

text: its

text: success

text: in

text: in-

text: constraints,

text: which

text: has

text: been

text: widely

text: used

text: in

text: interactively

text: modeling

text: teractive

text: mesh

text: modeling.

text: The

text: deformation

text: framework

text: is

text: similar

text: deformable

text: surfaces

text: and

text: solids.

text: to

text: one

text: used

text: in

text: [Jacobson

text: et

text: al.

text: We

text: further

text: examine

text: the

text: deformation

text: property

text: of

text: our

text: proposed

text: method,

text: and

text: compare

text: with

text: Modeling

text: deformable

text: meshes

text: has

text: been

text: an

text: established

text: topic

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text: physically

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text: and

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text: solv-

text: conformal

text: deformation[Paries

text: et

text: al.

text: Crane

text: et

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text: ing

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text: large-scale

text: optimization

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text: variables

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text: complexity

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text: mesh

text: nodes.

text: A

text: studio

text: work-ﬂow

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text: Mathematical

text: Background

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text: such

text: processes,

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text: minimizing

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text: quadratic

text: function

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text: q)

text: an

text: interactive

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text: computation

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text: where

text: qTX

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text: well-conditioned

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text: Typically,

text: been

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text: their

text: off-line

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text: today.

text: Exist-

text: ing

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text: model

text: reduction

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text: work

text: reported

text: in

text: a

text: recent

text: preprint

text: [Wang

text: et

text: al.

text: subspace

text: as

text: the

text: input,

text: which

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text: demands

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text: al.

text: also

text: propose

text: a

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text: deformation

text: framework

text: based

text: on

text: linear

text: tions

text: in

text: constructing

text: them.

text: Exemplars

text: include

text: cage-based

text: defor-

text: variational

text: subspace

text: similar

text: to

text: ours

text: with

text: the

text: difference

text: that

text: we

text: in

text: addition

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text: [Huang

text: et

text: al.

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text: et

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text: use

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text: large

text: value.

text: Therefore,

text: the

text: contribution

text: et

text: al.

text: The

text: time

text: spent

text: on

text: constructing

text: such

text: reduced

text: models

text: of

text: our

text: paper

text: excluding

text: the

text: empirical

text: efforts

text: is

text: the

text: approximation

text: theory

text: for

text: linear

text: variational

text: subspace.

text: Email:

text: and

text: demos:

text: ##LTLine##

text: ##LTLine##


text: Similarly, we have (cid:107) − (cid:107) (cid:107) − − − (cid:107) (cid:107) − (cid:107) Aˆ+ Aˆ+ = [(AˆTAˆ) 1 (AˆTAˆ ) 1]AˆTIˆ + Aˆ+ Aˆ+Iˆ p p p (cid:107) (cid:107)(cid:107) − (cid:107) ≤ Aˆ+ I Aˆ+IˆAˆ (cid:107) (cid:107) (cid:107) − (cid:107) − (cid:107) − (cid:107) + Aˆ+ 2 IˆAˆ Aˆ . 1 I Aˆ+IˆAˆ (cid:107) − (cid:107) (cid:107) − (cid:107) ≤ Corollary. Since I Aˆ+IˆAˆ = Aˆ+Aˆ Aˆ+IˆAˆ (cid:107)(cid:13)(cid:13)Aˆ+(cid:104)(cid:107)(cid:107)IˆAˆ − Aˆ(cid:107), we have (cid:105) (cid:13)(cid:13) (cid:13) (cid:13) (cid:107) (cid:107) Aˆ (AˆTAˆ)−1 − (AˆTp Aˆp)−1 AˆT ≤ 1c−on(cid:107)dI(−Aˆ)AˆA+ˆI+ˆAˆ(cid:107)(cid:107)IˆAˆ−Aˆ(cid:107), FinigAuRrAeP9:suLrefaftc:eOmroigdienlainlgm;oRdieglh;tM: Oidudrlem: eOthuot-dodf-esfhoarpmesdriesstot-rptoiosne part via shrinking. and (cid:107) (cid:107) − (cid:107) − (cid:107) (cid:107) − (cid:107) ≤ Aˆ+ 2(2 I Aˆ+IˆAˆ )(cid:107) − (cid:107) Aˆ+ Aˆ+p 1 − (cid:107)I − Aˆ+IˆAˆ(cid:107) IˆAˆ Aˆ . Here we prove the main result Proof. Based on Prop 4.6 and Prof 4.9. We can decompose the error term into two parts: (cid:107) ∗ − (cid:107) ≤ (cid:107) − (cid:107) (cid:107) − − − (cid:107) Figure 10: Difference between deformations with conformal fac- Xˆ Xˆ (Aˆ+)Tˆb (Aˆ+)ˆb + (Iˆ Aˆ Aˆ+)qˆ (I AˆAˆ+)qˆ . min min p p p tors (right) and without (middle). Seven constraint points are (44) marked as red balls. On one hand, based on Corollary , there exists a constant β = − 1 2−ρ > 0, such that by piece-wise linear form (see equation (8)). It is ﬁtted again by 1 ρ a single SVD in each frame. Then we update the reduced model (Phase 1 and 2) under updated frame coordinates, i.e., multiply the (cid:107) − (cid:107) ≤ (cid:107) − (cid:107)(cid:107) (cid:107) (Aˆ+)Tˆb (Aˆ+p )ˆb ≤ Aˆ(cid:107)+(cid:107)(cid:107) Aˆ+p(cid:107) (cid:107)ˆb − (cid:107) (45) isntrvuecrtsieonro(tbaatisoendroT0n EtoqP. e(q15a)n)d, wSe. sMhoeualndwahlsiloemduurlitnipglymreosthatrioecnorn- β1 ˆb Aˆ+ 2 IˆAˆ Aˆ to vertices V (cid:48), so as to display them in the original frame. 0 Afﬁne Patches On the other hand, from Corollary , we also have In deformable modeling, the user usually would like to constrain certain patches on the mesh to be rigid or ﬁxed, or more generally (cid:107) − − − (cid:107) afﬁne. Our framework can be accompanied by those requirements (Iˆ Aˆ Aˆ+)qˆ (I AˆAˆ+)qˆ (cid:48) p p in pre-computation. Vertex positions V on the deformed mesh can ≤ (cid:107) − − − (cid:107) (cid:107) − (cid:107)(cid:107) − (cid:107) (cid:48) (Iˆ Aˆ Aˆ+)qˆ (I AˆAˆ+)Iˆqˆ + I AˆAˆ+ Iˆqˆ qˆ be linearly expressed in terms of deformable vertices V and patch- p p 0 ≤ (cid:107) − − − (cid:107) (cid:107) (cid:107)(cid:107) − (cid:107) wise transformation t , d , i.e., (under a permutation) (cid:32)(Iˆ AˆpAˆ+p )qˆ(cid:33) Iˆ(I AˆAˆ+)Iˆqˆ + Aˆ+Iˆqˆ IˆAˆ Aˆ i i (cid:107) − (cid:107)(cid:107) − (cid:107) + I AˆAˆ+ Iˆqˆ qˆ (cid:48) (cid:48) V = [V ; V t + d ; . . . ; V t + d ] , (47) 0 1 1 1 s s s ≤ cond(Aˆ) (cid:107) (cid:107)(cid:107) (cid:107)(cid:107) − (cid:107) (cid:107) − (cid:107) 1 + − qˆ Aˆ+ IˆAˆ Aˆ + Iˆqˆ qˆ , 1 ρ (cid:48) (46) where V are deformable vertices, V , . . . , V are s afﬁne patches 0 1 s on the original mesh with prescribed transformation matrices t = where β = 1 + −ω > 0. Combining Eq. (45) and Eq. (46), [rte1se,n.t.a.t,iotns,],thaendﬁrdsitssptlaagceempreonbtlsemd i=s r[edfo1r,m. .u.la,tdesd]a.cUconrddeirntghliyssruecph- 2 1 ρ that the variational subspace is solved for variables of the de facto Eq. (39) is held. (cid:48) (cid:48) control layer [V , t, d, Q], instead of for [V , Q] (see equations 0 (14) and (15)). For simplicity, each afﬁne patch accompanies a sin- Implementation Details gle rotational proxy and a single linear proxy. In companion to our proposed algorithm, other algorithm details To improve the numerical stability in case one would like to con- less relevant to variational subspace is provided in this section, strain more than one vertex on a single afﬁne patch (e.g., constrain which we follow the notations used in section . four in rigid motion), we in addition append corresponding linear Global rotation adaption proxies for each variable of t. Therefore, the total degree of linear proxies is 3m + 9s. In our implementation, we also introduce global rotation r to di- 0 minish the approximation error of local rotation matrix rk incurred Conformal-like Deformations ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Similarly,

text: we

text: have

label: equation

text: Aˆ+

text: Aˆ+

text: [(AˆTAˆ)

text: (AˆTAˆ

text: Aˆ+

text: Aˆ+Iˆ

text: p

text: p

text: p

text: Aˆ+

text: I

text: Aˆ+IˆAˆ

text: Aˆ+

text: IˆAˆ

text: Aˆ

text: I

text: Aˆ+IˆAˆ

text: Corollary.

text: Since

text: I

text: Aˆ+IˆAˆ

text: Aˆ+Aˆ

text: Aˆ+IˆAˆ

text: we

text: have

text: Aˆ

text: (AˆTp

text: AˆT

label: caption

text: Oidudrlem:

text: part

text: via

text: shrinking.

text: and

text: Aˆ+

text: I

text: Aˆ+IˆAˆ

text: Aˆ+

text: Aˆ+p

text: IˆAˆ

text: Aˆ

text: Here

text: we

text: prove

text: the

text: main

text: result

text: Proof.

text: Based

text: on

text: Prop

text: and

text: Prof

text: We

text: can

text: decompose

text: the

text: error

text: term

text: into

text: two

text: parts:

text: Figure

text: Difference

text: between

text: deformations

text: with

text: conformal

text: fac-

text: Xˆ

text: Xˆ

text: (Aˆ+)Tˆb

text: (Aˆ+)ˆb

text: (Iˆ

text: Aˆ

text: Aˆ+)qˆ

text: (I

text: AˆAˆ+)qˆ

text: min

text: min

text: p

text: p

text: p

text: tors

text: (right)

text: and

text: without

text: (middle).

text: Seven

text: constraint

text: points

text: are

text: marked

text: as

text: red

text: balls.

text: On

text: one

text: hand,

text: based

text: on

text: Corollary

text: there

text: exists

text: a

text: constant

text: β

text: 2−ρ

text: such

text: that

text: by

text: piece-wise

text: linear

text: form

text: (see

text: equation

text: It

text: is

text: ﬁtted

text: again

text: by

text: ρ

text: a

text: single

text: SVD

text: in

text: each

text: frame.

text: Then

text: we

text: update

text: the

text: reduced

text: model

text: (Phase

text: and

text: under

text: updated

text: frame

text: coordinates,

text: i.e.,

text: multiply

text: the

text: (Aˆ+)Tˆb

text: (Aˆ+p

text: )ˆb

text: EtoqP.

text: wSe.

text: β1

text: ˆb

text: Aˆ+

text: IˆAˆ

text: Aˆ

text: to

text: vertices

text: V

text: so

text: as

text: to

text: display

text: them

text: in

text: the

text: original

text: frame.

text: Afﬁne

text: Patches

text: On

text: the

text: other

text: hand,

text: from

text: Corollary

text: we

text: also

text: have

text: In

text: deformable

text: modeling,

text: the

text: user

text: usually

text: would

text: like

text: to

text: constrain

text: certain

text: patches

text: on

text: the

text: mesh

text: to

text: be

text: rigid

text: or

text: ﬁxed,

text: or

text: more

text: generally

text: afﬁne.

text: Our

text: framework

text: can

text: be

text: accompanied

text: by

text: those

text: requirements

text: (Iˆ

text: Aˆ

text: Aˆ+)qˆ

text: (I

text: AˆAˆ+)qˆ

text: p

text: p

text: in

text: pre-computation.

text: Vertex

text: positions

text: V

text: on

text: the

text: deformed

text: mesh

text: can

text: (Iˆ

text: Aˆ

text: Aˆ+)qˆ

text: (I

text: AˆAˆ+)Iˆqˆ

text: I

text: AˆAˆ+

text: Iˆqˆ

text: qˆ

text: be

text: linearly

text: expressed

text: in

text: terms

text: of

text: deformable

text: vertices

text: V

text: and

text: patch-

text: p

text: p

text: wise

text: transformation

text: t

text: d

text: i.e.,

text: (under

text: a

text: permutation)

text: AˆpAˆ+p

text: Iˆ(I

text: AˆAˆ+)Iˆqˆ

text: Aˆ+Iˆqˆ

text: IˆAˆ

text: Aˆ

text: i

text: i

text: I

text: AˆAˆ+

text: Iˆqˆ

text: qˆ

text: V

text: [V

text: V

text: t

text: d

text: V

text: t

text: d

text: s

text: s

text: s

text: cond(Aˆ)

text: qˆ

text: Aˆ+

text: IˆAˆ

text: Aˆ

text: Iˆqˆ

text: qˆ

text: ρ

text: where

text: V

text: are

text: deformable

text: vertices,

text: V

text: V

text: are

text: s

text: afﬁne

text: patches

text: s

text: on

text: the

text: original

text: mesh

text: with

text: prescribed

text: transformation

text: matrices

text: t

text: where

text: β

text: −ω

text: Combining

text: Eq.

text: and

text: Eq.

text: i=s

text: ρ

text: that

text: the

text: variational

text: subspace

text: is

text: solved

text: for

text: variables

text: of

text: the

text: de

text: facto

text: Eq.

text: is

text: held.

text: control

text: layer

text: [V

text: t,

text: d,

text: Q],

text: instead

text: of

text: for

text: [V

text: Q]

text: (see

text: equations

text: and

text: For

text: simplicity,

text: each

text: afﬁne

text: patch

text: accompanies

text: a

text: sin-

text: Implementation

text: Details

text: gle

text: rotational

text: proxy

text: and

text: a

text: single

text: linear

text: proxy.

text: In

text: companion

text: to

text: our

text: proposed

text: algorithm,

text: other

text: algorithm

text: details

text: To

text: improve

text: the

text: numerical

text: stability

text: in

text: case

text: one

text: would

text: like

text: to

text: con-

text: less

text: relevant

text: to

text: variational

text: subspace

text: is

text: provided

text: in

text: this

text: section,

text: strain

text: more

text: than

text: one

text: vertex

text: on

text: a

text: single

text: afﬁne

text: patch

text: (e.g.,

text: constrain

text: which

text: we

text: follow

text: the

text: notations

text: used

text: in

text: section

text: four

text: in

text: rigid

text: motion),

text: we

text: in

text: addition

text: append

text: corresponding

text: linear

text: Global

text: rotation

text: adaption

text: proxies

text: for

text: each

text: variable

text: of

text: t.

text: Therefore,

text: the

text: total

text: degree

text: of

text: linear

text: proxies

text: is

text: In

text: our

text: implementation,

text: we

text: also

text: introduce

text: global

text: rotation

text: r

text: to

text: di-

text: minish

text: the

text: approximation

text: error

text: of

text: local

text: rotation

text: matrix

text: rk

text: incurred

text: Conformal-like

text: Deformations

text: ##LTFigure##

text: ##LTFigure##

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: We extend our model to conformal-like deformations in this sec- tion, by introducing scaling factors s for each rotational proxy. In- ∈ i ∈ s(cid:107)tsea(cid:107)d o∈f r[e1s/trψic,tψin]g, fsoir soSmOe c(3o)n,stwanetpψerm>it1.(cid:107)ssii(cid:107)2 SO(3), where i 2 ∈ ∈ Thus we can write s = ψ t , where ψ [1/ψ, ψ] and t i i i i i SO(3). The updating routine also contains two phases in corre- spondence, of which the ﬁrst phase is identical to former. For the second phase, we reformulate as follows. Similar to the previous Phase 2, we are again to ﬁt the consistent local frame s by optimizing the simpliﬁed energy i (cid:80) (cid:80) E (cid:48) − (cid:48) ◦ ⊗ (V (i), S) = constant [V (i); 0]TMT (T (Ψ 1 × )) 9 1 r (cid:107) − (cid:107) +1 c v v 2 ψ2 , 2 ∈G ijk i j ik k=1 (i,j) k (48) where T × is the vectorization of (t ), Ψ = [ψ , . . . , ψ ] and 9◦d 1 ⊗ i 1 d S = T (Ψ 1 × ). To solve for S, we use two steps: ﬁrst, 9 1 we ﬁx Ψ and optimize T, which is exactly the same as discussed. It is required to compute (M X(i) + M S(i)) and perform sin- N U gular value decomposition. Second, we ﬁx T and compute partial gradient w.r.t. Ψ E ∂ − ⊗ ◦ (cid:48) ◦ = (I × 1 × )T M[V (i); 0] + C Ψ ∂Ψ − d d ⊗ 1 9 ◦ ◦ = (I × 1 × )T (M X(i) + M S(i)) + C Ψ , d d 1 9 N U (49) where Cd×1 is pre-computed and (MNX(i) +E MUS(i)) is com- ∂ puted in the former step. Hence by setting = 0, Ψ is com- ∂Ψ puted. Fig. 10 illustrates the difference between deformations with conformal factors and without. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: extend

text: our

text: model

text: to

text: conformal-like

text: deformations

text: in

text: this

text: sec-

text: tion,

text: by

text: introducing

text: scaling

text: factors

text: s

text: for

text: each

text: rotational

text: proxy.

text: In-

text: i

text: o∈f

text: fsoir

text: soSmOe

text: where

text: i

text: Thus

text: we

text: can

text: write

text: s

text: ψ

text: t

text: where

text: ψ

text: ψ]

text: and

text: t

text: i

text: i

text: i

text: i

text: i

text: The

text: updating

text: routine

text: also

text: contains

text: two

text: phases

text: in

text: corre-

text: spondence,

text: of

text: which

text: the

text: ﬁrst

text: phase

text: is

text: identical

text: to

text: former.

text: For

text: the

text: second

text: phase,

text: we

text: reformulate

text: as

text: follows.

text: Similar

text: to

text: the

text: previous

text: Phase

text: we

text: are

text: again

text: to

text: ﬁt

text: the

text: consistent

text: local

text: frame

text: s

text: by

text: optimizing

text: the

text: simpliﬁed

text: energy

text: i

text: E

text: (V

text: (i),

text: S)

text: constant

text: [V

text: (i);

text: (T

text: (Ψ

text: r

text: c

text: v

text: v

text: ψ2

text: ∈G

text: ijk

text: i

text: j

text: ik

text: (i,j)

text: k

text: where

text: T

text: is

text: the

text: vectorization

text: of

text: (t

text: Ψ

text: [ψ

text: ψ

text: and

text: i

text: d

text: S

text: T

text: (Ψ

text: To

text: solve

text: for

text: S,

text: we

text: use

text: two

text: steps:

text: ﬁrst,

text: we

text: ﬁx

text: Ψ

text: and

text: optimize

text: T,

text: which

text: is

text: exactly

text: the

text: same

text: as

text: discussed.

text: It

text: is

text: required

text: to

text: compute

text: (M

text: X(i)

text: M

text: S(i))

text: and

text: perform

text: sin-

text: N

text: U

text: gular

text: value

text: decomposition.

text: Second,

text: we

text: ﬁx

text: T

text: and

text: compute

text: partial

text: gradient

text: w.r.t.

text: Ψ

text: E

text: (I

text: )T

text: M[V

text: (i);

text: C

text: Ψ

text: ∂Ψ

text: d

text: d

text: (I

text: )T

text: (M

text: X(i)

text: M

text: S(i))

text: C

text: Ψ

text: d

text: d

text: N

text: U

text: where

text: is

text: pre-computed

text: and

text: (MNX(i)

text: +E

text: MUS(i))

text: is

text: com-

text: puted

text: in

text: the

text: former

text: step.

text: Hence

text: by

text: setting

text: Ψ

text: is

text: com-

text: ∂Ψ

text: puted.

text: Fig.

text: illustrates

text: the

text: difference

text: between

text: deformations

text: with

text: conformal

text: factors

text: and

text: without.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 8: Approximation quality of our method in all images is demonstrated on the test suite of models introduced in [Botsch and Sorkine 2008]; From left to right: Bumpy Plane, Cylinder, Cactus, Bar. Multiple types of deformations, including bending, shifting and twisting, are tested. Deﬁnition 4.2. Two solutions subject to the form Eq. (1) (param- eterized by A, q, and b) are called quotient equivalent, if they It is observed that only X˜ appears in second-order term in the share the same companion problem deﬁned by Eq. (25), i.e. their objective function of Eq (23). Suppose the optimal solution to − Eq. (1) is X with a two-part decomposition (given by Eq. (20)) ˆb(A, b, q) = b ATX¯min are the same. This forms group equiva- min lence in the space of solutions. X = X˜ + X¯ , we then consider the following companion min min min optimization problem For example, let H be the Laplacian operator ∆, the solution would minimize the the L2 norm of ﬁrst-order gradient. In such case, two problems are of quotient equivalence if their optimal solutions pre- − − min f˜(Xˆ) = (1/2)X˜TLTLX˜ qTX˜ qTX¯ serve to an additive constant. We use the distance between two s.X˜t. ATX˜ = b − ATX¯ min (24) solution groups under the quotient equivalence to measure the ap- (I − L+L)X˜ = 0.min proximation error. It is the Mahalanobis distance provided by H, i.e. − − − − Remark −qTX¯ which appears in objective function of Eq. (24) dH(x, y) = (x y)TH(x y) = (Lx Ly)T(Lx Ly). min is a constant. We see Eq. (24) can be equivalently solved in two − steps: In the ﬁrst step, we solve the following problem Let LX = Xˆ, qˆ = L+q, Aˆ = L+A, and ˆb = b ATX¯ , we min rewrite Eq. (25) (but not equivalent) as − minX˜ f˜(X˜) + qT−X¯min = (1/2)X˜TLTLX˜ qTX˜ (25) min (1/2)XˆTXˆ − qˆTXˆ s.t. ATX˜ = b ATX¯min. s.Xˆt. AˆTXˆ = ˆb (26) − (I LL+)Xˆ = 0. And in the second stage, we project the solution to L+LX˜ , min where X˜ is the solution to Eq. (25). Theoremmi4n.3. Suppose X is the unique solution to Eq. (23), Similar to the treatment of Eq. (24), we can in ﬁrst solve ◦ min X˜ is the unique solution to Eq. (24), and X˜ is the unique so- min ◦ min lution to Eq. (25), then we have X˜min = L+LXmin = L+LX˜min. min (1/2)XˆTXˆ − qˆTXˆ Xˆ (27) s.t. AˆTXˆ = ˆb. Proof. We observe that L+LX satisﬁes the constraint of min Eq. (24), and objective functions of Eq. (23) and Eq. (24) coin- ◦ cide, i.e., f(Xmin) = f˜(L+LXmin). Therefore, as X˜min is the and then project the solution to LL+Xˆmin, where Xˆmin is the so- minimizer to Eq. (24), we have lution to Eq. (27). ≥ ◦ f(X ) f˜(X˜ ). Since we are always interested in distance measure d for different min min H solutions, the projection step in solving Eq. (24) is not necessary to ◦ ◦ On the other hand, f˜(X˜ ) = f(X˜ + X¯ ). Given X is compute d . The distance between two solution groups X and min min min min H 1 the minimizer to Eq. (23), we also have X of the quotient equivalence is therefore the Euclidean distance 2 f˜(X˜◦ ) ≥ f(X ). between Xˆ1 = LX1 and Xˆ2 = LX2. min min ◦ Hence, the equality holds for f˜(X˜ ) = f(X ). Given that the Similar to the treatment of Eq. (1) and Eq. (27), we can derive the min min optimum exists and is unique, we have two-stage problem from Eq. (2) as ◦ ⇒ ◦ X˜min + X¯min = Xmin X˜min = L+LXmin. − min (1/2)XˆTXˆ YˆTDˆTXˆ Xˆ The proof of X˜◦ = L+LX˜ is similar: observe that s.t. CˆTXˆ = Zˆ (28) f˜(X˜m◦in) = f˜(L+mLinX˜min) and L+mLinX˜min satisﬁes the constraint ATCZˆ = ˆb, of Eq. (24). ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: Approximation

text: quality

text: of

text: our

text: method

text: in

text: all

text: images

text: is

text: demonstrated

text: on

text: the

text: test

text: suite

text: of

text: models

text: introduced

text: in

text: [Botsch

text: and

text: Sorkine

text: From

text: left

text: to

text: right:

text: Bumpy

text: Plane,

text: Cylinder,

text: Cactus,

text: Bar.

text: Multiple

text: types

text: of

text: deformations,

text: including

text: bending,

text: shifting

text: and

text: twisting,

text: are

text: tested.

text: Deﬁnition

text: Two

text: solutions

text: subject

text: to

text: the

text: form

text: Eq.

text: (param-

text: eterized

text: by

text: A,

text: q,

text: and

text: b)

text: are

text: called

text: quotient

text: equivalent,

text: if

text: they

text: It

text: is

text: observed

text: that

text: only

text: X˜

text: appears

text: in

text: second-order

text: term

text: in

text: the

text: share

text: the

text: same

text: companion

text: problem

text: deﬁned

text: by

text: Eq.

text: i.e.

text: their

text: objective

text: function

text: of

text: Eq

text: Suppose

text: the

text: optimal

text: solution

text: to

text: Eq.

text: is

text: X

text: with

text: a

text: two-part

text: decomposition

text: (given

text: by

text: Eq.

text: ˆb(A,

text: b,

text: q)

text: b

text: ATX¯min

text: are

text: the

text: same.

text: This

text: forms

text: group

text: equiva-

text: min

text: lence

text: in

text: the

text: space

text: of

text: solutions.

text: X

text: X˜

text: X¯

text: we

text: then

text: consider

text: the

text: following

text: companion

text: min

text: min

text: min

text: optimization

text: problem

text: For

text: example,

text: let

text: H

text: be

text: the

text: Laplacian

text: operator

text: the

text: solution

text: would

text: minimize

text: the

text: the

text: norm

text: of

text: ﬁrst-order

text: gradient.

text: In

text: such

text: case,

text: two

text: problems

text: are

text: of

text: quotient

text: equivalence

text: if

text: their

text: optimal

text: solutions

text: pre-

text: min

text: f˜(Xˆ)

text: qTX˜

text: qTX¯

text: serve

text: to

text: an

text: additive

text: constant.

text: We

text: use

text: the

text: distance

text: between

text: two

text: s.X˜t.

text: ATX˜

text: b

text: ATX¯

text: min

text: solution

text: groups

text: under

text: the

text: quotient

text: equivalence

text: to

text: measure

text: the

text: ap-

text: (I

text: L+L)X˜

text: proximation

text: error.

text: It

text: is

text: the

text: Mahalanobis

text: distance

text: provided

text: by

text: H,

text: i.e.

text: Remark

text: −qTX¯

text: which

text: appears

text: in

text: objective

text: function

text: of

text: Eq.

text: dH(x,

text: y)

text: (x

text: y)TH(x

text: y)

text: (Lx

text: Ly)T(Lx

text: Ly).

text: min

text: is

text: a

text: constant.

text: We

text: see

text: Eq.

text: can

text: be

text: equivalently

text: solved

text: in

text: two

text: steps:

text: In

text: the

text: ﬁrst

text: step,

text: we

text: solve

text: the

text: following

text: problem

text: Let

text: LX

text: Xˆ,

text: qˆ

text: L+q,

text: Aˆ

text: L+A,

text: and

text: ˆb

text: b

text: ATX¯

text: we

text: min

text: rewrite

text: Eq.

text: (but

text: not

text: equivalent)

text: as

text: minX˜

text: f˜(X˜)

text: qT−X¯min

text: qTX˜

text: min

text: qˆTXˆ

text: s.t.

text: ATX˜

text: b

text: ATX¯min.

text: s.Xˆt.

text: AˆTXˆ

text: ˆb

text: (I

text: LL+)Xˆ

text: And

text: in

text: the

text: second

text: stage,

text: we

text: project

text: the

text: solution

text: to

text: L+LX˜

text: min

text: where

text: X˜

text: is

text: the

text: solution

text: to

text: Eq.

text: Suppose

text: X

text: is

text: the

text: unique

text: solution

text: to

text: Eq.

text: Similar

text: to

text: the

text: treatment

text: of

text: Eq.

text: we

text: can

text: in

text: ﬁrst

text: solve

text: min

text: X˜

text: is

text: the

text: unique

text: solution

text: to

text: Eq.

text: and

text: X˜

text: is

text: the

text: unique

text: so-

text: min

text: min

text: lution

text: to

text: Eq.

text: then

text: we

text: have

text: X˜min

text: L+LXmin

text: L+LX˜min.

text: min

text: qˆTXˆ

text: Xˆ

text: s.t.

text: AˆTXˆ

text: ˆb.

text: Proof.

text: We

text: observe

text: that

text: L+LX

text: satisﬁes

text: the

text: constraint

text: of

text: min

text: Eq.

text: and

text: objective

text: functions

text: of

text: Eq.

text: and

text: Eq.

text: coin-

text: cide,

text: i.e.,

text: f(Xmin)

text: f˜(L+LXmin).

text: Therefore,

text: as

text: X˜min

text: is

text: the

text: and

text: then

text: project

text: the

text: solution

text: to

text: LL+Xˆmin,

text: where

text: Xˆmin

text: is

text: the

text: so-

text: minimizer

text: to

text: Eq.

text: we

text: have

text: lution

text: to

text: Eq.

text: f(X

text: f˜(X˜

text: Since

text: we

text: are

text: always

text: interested

text: in

text: distance

text: measure

text: d

text: for

text: different

text: min

text: min

text: H

text: solutions,

text: the

text: projection

text: step

text: in

text: solving

text: Eq.

text: is

text: not

text: necessary

text: to

text: On

text: the

text: other

text: hand,

text: f˜(X˜

text: f(X˜

text: X¯

text: Given

text: X

text: is

text: compute

text: d

text: The

text: distance

text: between

text: two

text: solution

text: groups

text: X

text: and

text: min

text: min

text: min

text: min

text: H

text: the

text: minimizer

text: to

text: Eq.

text: we

text: also

text: have

text: X

text: of

text: the

text: quotient

text: equivalence

text: is

text: therefore

text: the

text: Euclidean

text: distance

text: f˜(X˜◦

text: f(X

text: between

text: and

text: min

text: min

text: Hence,

text: the

text: equality

text: holds

text: for

text: f˜(X˜

text: f(X

text: Given

text: that

text: the

text: Similar

text: to

text: the

text: treatment

text: of

text: Eq.

text: and

text: Eq.

text: we

text: can

text: derive

text: the

text: min

text: min

text: optimum

text: exists

text: and

text: is

text: unique,

text: we

text: have

text: two-stage

text: problem

text: from

text: Eq.

text: as

text: X˜min

text: X¯min

text: Xmin

text: X˜min

text: L+LXmin.

text: min

text: YˆTDˆTXˆ

text: Xˆ

text: The

text: proof

text: of

text: X˜◦

text: L+LX˜

text: is

text: similar:

text: observe

text: that

text: s.t.

text: CˆTXˆ

text: Zˆ

text: f˜(X˜m◦in)

text: f˜(L+mLinX˜min)

text: and

text: L+mLinX˜min

text: satisﬁes

text: the

text: constraint

text: ATCZˆ

text: ˆb,

text: of

text: Eq.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 The phase Φ corresponds to the same ﬂux and hence L to a vector potential A = 1 Φ . The vector potential cou- L L Φ 2m ples to the current j , which according to the continuity n = ν = ν . (C13) ρ π ξ equation is given by (cid:48) − √ √ In the thermodynamic limit, L = L Nξ, and there- − H − (cid:48) − − jρ = i[ , π2φρ] = 2vρKρΠρ. (C7) fore, LL = νν(cid:48) = 1(cid:48) nξ = 1 ν2m. The density of the new particles is n = N(cid:48) = −1− , and the new ﬁlling fac- L n 1 ξ ∼ tor is thus From the form of the coupling term j A, one can ρ immediately see that the second derivative of the energy (cid:48) with respect to the phase Φ is related to the coeﬃcient (cid:48) n π 1 L ν = (cid:48) = − − . (C14) of j2 term in the Luttinger liquid Hamiltonian Φ ν 1 2m ρ Since ξ = 0 in the new system, the Luttinger parameter ∂2E 2v K (cid:48) L GS = ρ ρ . (C8) is K = 1 and the scaling dimension x may be expressed ∂Φ2L π in eiρther(cid:18)2the new system(cid:19) or th(cid:18)e original one, (cid:19) We hence obtain the Lut(cid:118)tinger parameter (cid:117) (cid:117)(cid:116) x = 1 1 + (ν(cid:48))−2 · 1 = 1 1 + ν−2 · 1(1 − nξ)2 . 2 2 2 2 π L∂∂2EΦG2 S (C15) K = L . (C9) ρ 2 L1 ∂2∂EnG2S As the density should not exceed 1 , the integer m is ξ+1 a constrained by m < 1 ξ . One interesting case where Therefore, using Eq. (C3) and (C5), the low-energy 2ν ξ+1 the constraint is satisﬁed is physics of the model, Eq. (12) and (24), is described by a Luttinger liquid with ν = 1 , ξ = 2m + 1, Φ = 2πm , n = 2m , 2m+1 2m+1 (2m+1)2 (cid:48) (cid:48) (cid:48) (cid:48) 1 − ν = 1, ξ = 0, Φ = 2πm , n = 2m . K = (1 nξ)2. (C10) 2m+1 2m+1 ρ 2 (C16) 2. Mapping to a ν = 1 phase of zero-range For example, m = 1 corresponds to ν = 1, ξ = 1, (cid:48) 3 interacting fermions Φ = 2π, n = 2, n = 2. This example demonstrates that (cid:48)3 9 3 (cid:48) the ν = 1 gap for the ξ = 0 model is equivalent to ν = 1 3 L(cid:48)W=eLm−ay(Nfo−llo1w)ξ trhuengssaimnethme alipmpiitnUg f→rom∞ Lat ﬁrunnitgest⊥to, oshfotwheinogrtighiantalthpearretischleosu.ldItbeisgaalpwaoypsenrienlegvaanttsmasalxl t=⊥.43, and map the two-leg ladder to a reduced two-leg ladder. We restrict our attention to fermions, with the treatment Starting from bosons, one can reach similar mapping (cid:80) of bosons general(cid:48)ized below. The part H0 + Hint takes by including a Jordan-Wigner transformation30. This the shape of a ξ = 0 Hamiltonian. Since site j in the leads to Eq. (C16) with half integer m. For exam- − new lattice correspond to location j + ξ( jj(cid:48)=11 nj(cid:48)) in the pl(cid:48)e, m = 12 corresponds to ν = 12, ξ = 2, Φ = π2 , n = 14, original lattice,(cid:88)the(cid:16)inter-chain coupling becomes (cid:17) n = 1. Thus, it constitutes a mapping of the bosonic (cid:80) 2 (cid:48) ν = 1 state to the fermionic ν = 1 state. 2 H⊥(cid:48) = −t⊥ c†j,1cj,2eiΦ[j+ξ( jj−(cid:48)=11 nj(cid:48))] + h.c. , The choice of parameters discussed here illustrates an j important point. For weak interactions, the coeﬃcient of (C11) the cosine operator driving the FQH instability is pertur- batively small, behaving as a high power of the interac- which is nonlocal. However, the nonlocality disappears tions13. However, Eq. (C11) implies that for very strong for a special value of the ﬂux interactions, the coeﬃcient of the cosine operator be- comes simply t⊥, as in the non-interacting fermionic case. 2πm Indeed, this is observed by substituting the bosonized ex- Φ = , (C12) ξ pression for the density, (cid:88) √ where m is a nonnegative integer. Thus, the new par- j−1 (cid:48) · − 2 ticles are subject to the same value of ﬂux Φ = Φ. For a n (cid:48) = n x φ (x), (C17) j π ρ given ﬁlling ν, the density of the original particles is (cid:48) j =1 ##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: The

text: phase

text: Φ

text: corresponds

text: to

text: the

text: same

text: ﬂux

text: and

text: hence

text: L

text: to

text: a

text: vector

text: potential

text: A

text: Φ

text: The

text: vector

text: potential

text: cou-

text: L

text: L

text: Φ

text: ples

text: to

text: the

text: current

text: j

text: which

text: according

text: to

text: the

text: continuity

text: n

text: ν

text: ν

text: ρ

text: π

text: ξ

text: equation

text: is

text: given

text: by

text: In

text: the

text: thermodynamic

text: limit,

text: L

text: L

text: Nξ,

text: and

text: there-

text: H

text: jρ

text: i[

text: π2φρ]

text: 2vρKρΠρ.

text: fore,

text: LL

text: nξ

text: The

text: density

text: of

text: the

text: new

text: particles

text: is

text: n

text: −1−

text: and

text: the

text: new

text: ﬁlling

text: fac-

text: L

text: n

text: ξ

text: tor

text: is

text: thus

text: From

text: the

text: form

text: of

text: the

text: coupling

text: term

text: j

text: A,

text: one

text: can

text: ρ

text: immediately

text: see

text: that

text: the

text: second

text: derivative

text: of

text: the

text: energy

text: with

text: respect

text: to

text: the

text: phase

text: Φ

text: is

text: related

text: to

text: the

text: coeﬃcient

text: n

text: π

text: L

text: ν

text: of

text: term

text: in

text: the

text: Luttinger

text: liquid

text: Hamiltonian

text: Φ

text: ν

text: ρ

text: Since

text: ξ

text: in

text: the

text: new

text: system,

text: the

text: Luttinger

text: parameter

text: K

text: L

text: GS

text: ρ

text: ρ

text: is

text: K

text: and

text: the

text: scaling

text: dimension

text: x

text: may

text: be

text: expressed

text: ∂Φ2L

text: π

text: in

text: new

text: or

text: original

text: one,

text: We

text: hence

text: obtain

text: the

text: parameter

text: x

text: ν−2

text: π

text: S

text: K

text: L

text: ρ

text: As

text: the

text: density

text: should

text: not

text: exceed

text: the

text: integer

text: m

text: is

text: a

text: constrained

text: by

text: m

text: ξ

text: One

text: interesting

text: case

text: where

text: Therefore,

text: using

text: Eq.

text: and

text: the

text: low-energy

text: 2ν

text: the

text: constraint

text: is

text: satisﬁed

text: is

text: physics

text: of

text: the

text: model,

text: Eq.

text: and

text: is

text: described

text: by

text: a

text: Luttinger

text: liquid

text: with

text: ν

text: ξ

text: Φ

text: n

text: ν

text: ξ

text: Φ

text: n

text: K

text: ρ

text: Mapping

text: to

text: a

text: ν

text: phase

text: of

text: zero-range

text: For

text: example,

text: m

text: corresponds

text: to

text: ν

text: ξ

text: interacting

text: fermions

text: Φ

text: n

text: n

text: This

text: example

text: demonstrates

text: that

text: the

text: ν

text: gap

text: for

text: the

text: ξ

text: model

text: is

text: equivalent

text: to

text: ν

text: trhuengssaimnethme

text: alipmpiitnUg

text: f→rom∞

text: Lat

text: ﬁrunnitgest⊥to,

text: and

text: map

text: the

text: two-leg

text: ladder

text: to

text: a

text: reduced

text: two-leg

text: ladder.

text: We

text: restrict

text: our

text: attention

text: to

text: fermions,

text: with

text: the

text: treatment

text: Starting

text: from

text: bosons,

text: one

text: can

text: reach

text: similar

text: mapping

text: of

text: bosons

text: below.

text: The

text: part

text: Hint

text: takes

text: by

text: including

text: a

text: Jordan-Wigner

text: This

text: the

text: shape

text: of

text: a

text: ξ

text: Hamiltonian.

text: Since

text: site

text: j

text: in

text: the

text: leads

text: to

text: Eq.

text: with

text: half

text: integer

text: m.

text: For

text: exam-

text: new

text: lattice

text: correspond

text: to

text: location

text: j

text: ξ(

text: in

text: the

text: m

text: corresponds

text: to

text: ν

text: ξ

text: Φ

text: π2

text: n

text: original

text: coupling

text: becomes

text: n

text: Thus,

text: it

text: constitutes

text: a

text: mapping

text: of

text: the

text: bosonic

text: ν

text: state

text: to

text: the

text: fermionic

text: ν

text: state.

text: −t⊥

text: h.c.

text: The

text: choice

text: of

text: parameters

text: discussed

text: here

text: illustrates

text: an

text: j

text: important

text: point.

text: For

text: weak

text: interactions,

text: the

text: coeﬃcient

text: of

text: the

text: cosine

text: operator

text: driving

text: the

text: FQH

text: instability

text: is

text: pertur-

text: batively

text: small,

text: behaving

text: as

text: a

text: high

text: power

text: of

text: the

text: interac-

text: which

text: is

text: nonlocal.

text: However,

text: the

text: nonlocality

text: disappears

text: However,

text: Eq.

text: implies

text: that

text: for

text: very

text: strong

text: for

text: a

text: special

text: value

text: of

text: the

text: ﬂux

text: interactions,

text: the

text: coeﬃcient

text: of

text: the

text: cosine

text: operator

text: be-

text: comes

text: simply

text: t⊥,

text: as

text: in

text: the

text: non-interacting

text: fermionic

text: case.

text: Indeed,

text: this

text: is

text: observed

text: by

text: substituting

text: the

text: bosonized

text: ex-

text: Φ

text: ξ

text: pression

text: for

text: the

text: density,

text: where

text: m

text: is

text: a

text: nonnegative

text: integer.

text: Thus,

text: the

text: new

text: par-

text: ticles

text: are

text: subject

text: to

text: the

text: same

text: value

text: of

text: ﬂux

text: Φ

text: Φ.

text: For

text: a

text: n

text: n

text: x

text: φ

text: (x),

text: j

text: π

text: ρ

text: given

text: ﬁlling

text: ν,

text: the

text: density

text: of

text: the

text: original

text: particles

text: is

text: j

text: ##LTLine##

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text: ##LTLine##


text: (cid:54) 3.2.4 η = 0 Case, Third Rotation (cid:54) Let us now consider the η = 0 case. If we perform the same (1, 2) rotation V on (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) H = H + aˆηM as we did on H , the M part is transformed to η 0 η 0 η (cid:124) (cid:123)(cid:122) (cid:125) † (cid:48) (cid:48) V M (θ , θ , θ , δ)V = M (θ + ϕ, θ , θ , δ) η 12 13 23 η 12 13 23 (cid:48) = θ (cid:48) (cid:48) 12 = M (θ , θ , θ , δ) . (3.33) η 12 13 23 (cid:48) → (cid:48) → (cid:48) → | | − Using θ π, aˆs aˆ, and aˆc O((cid:15)2) δm2 as β is increased beyond 2, we 12 2 12 12 31 can approximate (cid:0) (cid:1) (cid:16) (cid:17) (cid:48) (cid:48) aˆηMη θ12, θ13, θ23, δ  ≈ (cid:48) π aˆηMη 2 , θ13, θ23, δ −  cos(2θ ) eiδs sin(2θ ) eiδc sin(2θ ) − 23 − 13 23 13 23 = aˆη e iδs sin(2θ ) s2 cos(2θ ) s c cos(2θ ) . (3.34) − − 13 23 13 23 −13 13 23 e iδc sin(2θ ) s c cos(2θ ) c2 cos(2θ ) 13 23 13 13 23 13 23 Performing the (2,(cid:16)3) rotation W(cid:17) next, we ﬁnd: (cid:16) (cid:17) † (cid:48) π W Mη 2 ,(cid:124)θ13(cid:123),(cid:122)θ2(cid:125)3, δ W (cid:48) π = M (cid:16) , θ + φ, θ(cid:17), δ η 2 13 23 (cid:48)  = θ  13 (cid:48) π (cid:48) = M η 2 , θ13, θ23, δ (cid:48) − (cid:48)  cos(2θ ) eiδs sin(2θ ) eiδc sin(2θ ) − (cid:48) 23 − (cid:48)13 23 (cid:48) (cid:48) 13 23 = e iδs sin(2θ ) s 2 cos(2θ ) s c cos(2θ ) , (3.35) − − 1(cid:48)3 23 (cid:48) 1(cid:48)3 23 −13 (cid:48)13 23 e iδc sin(2θ ) s c cos(2θ ) c 2 cos(2θ ) 13 23 13 13 23 13 23 (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) where s = sin θ and c = cos θ . The angle θ = θ + φ can be calculated 13 13 13 13 13 13 directly without the need to calculate φ using − (cid:48) (δm2 δm2 s2 ) sin 2θ tan 2θ = 3−1 21 12 1−3 , (3.36) 13 (δm2 δm2 s2 ) cos 2θ aˆ 31 21 12 13 (cid:48) and its β-dependence is shown in Fig. 3(b). As can be seen, θ increases rapidly to 13 π/2 when δm2 > 0, while damping quickly to zero when δm2 < 0, once β is increased 31 (cid:48) 31 (cid:48) above zero. Consequently, aˆ cos θ for the δm2 > 0 case, and aˆ sin θ for the δm2 < 0 − | 13 | |31 | 13 31 case plateau to c s (1 (cid:15)2s2 ) δm2 = O((cid:15)) δm2 as β is increased as shown in Fig. 5. 13 13 12 31 (cid:48) 31 | | Note that in the δm2 > 0 case, aˆ cos θ increases to O(1) δm2 in the vicinity of β = 0 31 13 31 – 18 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

label: section

text: η

text: Case,

text: Third

text: Rotation

text: Let

text: us

text: now

text: consider

text: the

text: η

text: case.

text: If

text: we

text: perform

text: the

text: same

text: rotation

text: V

text: on

text: H

text: H

text: aˆηM

text: as

text: we

text: did

text: on

text: H

text: the

text: M

text: part

text: is

text: transformed

text: to

text: η

text: η

text: η

text: V

text: M

text: (θ

text: θ

text: θ

text: δ)V

text: M

text: (θ

text: ϕ,

text: θ

text: θ

text: δ)

text: η

text: η

text: θ

text: M

text: (θ

text: θ

text: θ

text: δ)

text: η

text: Using

text: θ

text: π,

text: aˆs

text: aˆ,

text: and

text: aˆc

text: as

text: β

text: is

text: increased

text: beyond

text: we

text: can

text: approximate

text: aˆηMη

text: δ

text: π

text: aˆηMη

text: δ

text: eiδs

text: eiδc

text: aˆη

text: e

text: iδs

text: s

text: c

text: e

text: iδc

text: s

text: c

text: Performing

text: the

text: rotation

text: next,

text: we

text: ﬁnd:

text: π

text: W

text: Mη

text: δ

text: W

text: π

text: M

text: θ

text: φ,

text: δ

text: η

text: θ

text: π

text: M

text: η

text: δ

text: eiδs

text: eiδc

text: e

text: iδs

text: s

text: s

text: c

text: e

text: iδc

text: s

text: c

text: c

text: where

text: s

text: sin

text: θ

text: and

text: c

text: cos

text: θ

text: The

text: angle

text: θ

text: θ

text: φ

text: can

text: be

text: calculated

text: directly

text: without

text: the

text: need

text: to

text: calculate

text: φ

text: using

text: sin

text: 2θ

text: tan

text: 2θ

text: cos

text: 2θ

text: aˆ

text: and

text: its

text: β-dependence

text: is

text: shown

text: in

text: Fig.

text: As

text: can

text: be

text: seen,

text: θ

text: increases

text: rapidly

text: to

text: when

text: while

text: damping

text: quickly

text: to

text: zero

text: when

text: once

text: β

text: is

text: increased

text: above

text: zero.

text: Consequently,

text: aˆ

text: cos

text: θ

text: for

text: the

text: case,

text: and

text: aˆ

text: sin

text: θ

text: for

text: the

text: case

text: plateau

text: to

text: c

text: s

text: as

text: β

text: is

text: increased

text: as

text: shown

text: in

text: Fig.

text: Note

text: that

text: in

text: the

text: case,

text: aˆ

text: cos

text: θ

text: increases

text: to

text: in

text: the

text: vicinity

text: of

text: β

text: ##LTLine##

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text: ##LTLine##

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text: (cid:18) (cid:19) (cid:0) (cid:1) − (cid:61) ∗ ∗ ∆ − ∆ 4 U U U U sin2 21 sin ∆ sin2 31 sin ∆ . (3.87) α3 β3 α2 β2 2 31 2 21 In the above equations, we deﬁne (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) ≡ δm2 δm2 GeV L ≡ − ∆ ij L = 2.534 ij , δm2 m2 m2 . (3.88) ij 2E eV2 E km ij i j The transition probabilities in Eq. (3.86) and Eq. (3.87) contain several elements of the PMNS matrix U which are expressed in terms of three mixing angles θ , θ , θ , 12 23 13 and a CP-violating phase δ as shown in Eq. (3.52). The oscillation probabilities for the anti-neutrinos are obtained by replacing U with its complex conjugate. αi The neutrino oscillation probabilities in the presence of matter are obtained by replac- ing the vacuum expressions of the elements of the mixing matrix U and the mass-square diﬀerences ∆ with their eﬀective ‘running’ values in matter [21–23] : ij − ∼ ∼ → → (cid:48) → (cid:48) → (cid:48) → λ λ i j U U (θ θ , θ θ , θ θ ) , ∆ ∆ = L , (3.89) αi αi 12 12 13 13 23 23 ij ij 2E and for the anti-neutrinos − → (cid:118) → (cid:48) → (cid:48) → (cid:48) → (cid:118) λ λ i j U U (θ θ , θ θ , θ θ ) , ∆ ∆ = L . (3.90) αi αi 12 12 13 13 23 23 ij ij 2E To demonstrate the accuracy (or lack thereof in special cases) of our approximate an- alytical results, we compare the oscillation probabilities calculated with our approximate eﬀective running mixing angles and mass-squared diﬀerences with those calculated numer- ically for the same baseline and line-averaged constant matter density along it. For the mixing angles and mass-squared diﬀerences in vacuum, we use the benchmark values from Ref. [194] as listed in Table 1. In some plots, we take diﬀerent values of sin2 θ and δ 23 which we mention explicitly in the ﬁgure legends and captions. In this paper, all the plots (except in appendix ) are generated considering the line-averaged constant matter den- sity for a given baseline which has been estimated using the Preliminary Reference Earth Model (PREM) [195]. In appendix , we compare the exact numerical probabilities with line-averaged constant Earth density and varying Earth density proﬁle for 8770 km and 10000 km baselines. → In Fig. 9, we present our approximate ν ν oscillation probabilities (blue curves) as µ e a function of the neutrino energy against the exact numerical results (red curves) consider- − ing6 η = 0.1, ζ = 0 (left panels) and η = 0.1, ζ = 0 (right panels). The upper panels are 6We take ζ = 0 in our plots since we expect it to be hidden in the uncertainties in the matter density and neutrino energy. – 32 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: U

text: U

text: U

text: U

text: sin

text: sin

text: α3

text: β3

text: α2

text: β2

text: In

text: the

text: above

text: equations,

text: we

text: deﬁne

text: GeV

text: L

text: ij

text: L

text: ij

text: ij

text: E

text: km

text: ij

text: i

text: j

text: The

text: transition

text: probabilities

text: in

text: Eq.

text: and

text: Eq.

text: contain

text: several

text: elements

text: of

text: the

text: PMNS

text: matrix

text: U

text: which

text: are

text: expressed

text: in

text: terms

text: of

text: three

text: mixing

text: angles

text: θ

text: θ

text: θ

text: and

text: a

text: CP-violating

text: phase

text: δ

text: as

text: shown

text: in

text: Eq.

text: The

text: oscillation

text: probabilities

text: for

text: the

text: anti-neutrinos

text: are

text: obtained

text: by

text: replacing

text: U

text: with

text: its

text: complex

text: conjugate.

text: αi

text: The

text: neutrino

text: oscillation

text: probabilities

text: in

text: the

text: presence

text: of

text: matter

text: are

text: obtained

text: by

text: replac-

text: ing

text: the

text: vacuum

text: expressions

text: of

text: the

text: elements

text: of

text: the

text: mixing

text: matrix

text: U

text: and

text: the

text: mass-square

text: diﬀerences

text: with

text: their

text: eﬀective

text: ‘running’

text: values

text: in

text: matter

text: ij

text: λ

text: λ

text: i

text: j

text: U

text: U

text: (θ

text: θ

text: θ

text: θ

text: θ

text: θ

text: L

text: αi

text: αi

text: ij

text: ij

text: and

text: for

text: the

text: anti-neutrinos

text: λ

text: λ

text: i

text: j

text: U

text: U

text: (θ

text: θ

text: θ

text: θ

text: θ

text: θ

text: L

text: αi

text: αi

text: ij

text: ij

text: To

text: demonstrate

text: the

text: accuracy

text: (or

text: lack

text: thereof

text: in

text: special

text: cases)

text: of

text: our

text: approximate

text: an-

text: alytical

text: results,

text: we

text: compare

text: the

text: oscillation

text: probabilities

text: calculated

text: with

text: our

text: approximate

text: eﬀective

text: running

text: mixing

text: angles

text: and

text: mass-squared

text: diﬀerences

text: with

text: those

text: calculated

text: numer-

text: ically

text: for

text: the

text: same

text: baseline

text: and

text: line-averaged

text: constant

text: matter

text: density

text: along

text: it.

text: For

text: the

text: mixing

text: angles

text: and

text: mass-squared

text: diﬀerences

text: in

text: vacuum,

text: we

text: use

text: the

text: benchmark

text: values

text: from

text: Ref.

text: as

text: listed

text: in

text: Table

text: In

text: some

text: plots,

text: we

text: take

text: diﬀerent

text: values

text: of

text: θ

text: and

text: δ

text: which

text: we

text: mention

text: explicitly

text: in

text: the

text: ﬁgure

text: legends

text: and

text: captions.

text: In

text: this

text: paper,

text: all

text: the

text: plots

text: (except

text: in

text: appendix

text: are

text: generated

text: considering

text: the

text: line-averaged

text: constant

text: matter

text: den-

text: sity

text: for

text: a

text: given

text: baseline

text: which

text: has

text: been

text: estimated

text: using

text: the

text: Preliminary

text: Reference

text: Earth

text: Model

text: (PREM)

text: In

text: appendix

text: we

text: compare

text: the

text: exact

text: numerical

text: probabilities

text: with

text: line-averaged

text: constant

text: Earth

text: density

text: and

text: varying

text: Earth

text: density

text: proﬁle

text: for

text: km

text: and

text: km

text: baselines.

text: In

text: Fig.

text: we

text: present

text: our

text: approximate

text: ν

text: ν

text: oscillation

text: probabilities

text: (blue

text: curves)

text: as

text: µ

text: e

text: a

text: function

text: of

text: the

text: neutrino

text: energy

text: against

text: the

text: exact

text: numerical

text: results

text: (red

text: curves)

text: consider-

text: η

text: ζ

text: (left

text: panels)

text: and

text: η

text: ζ

text: (right

text: panels).

text: The

text: upper

text: panels

text: are

text: take

text: ζ

text: in

text: our

text: plots

text: since

text: we

text: expect

text: it

text: to

text: be

text: hidden

text: in

text: the

text: uncertainties

text: in

text: the

text: matter

text: density

text: and

text: neutrino

text: energy.

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##

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