text: WITH birth AS( birthTuples AS ( cohortT AS ( SELECT p, Min(t) as birthTime SELECT p, c as cohort, birthTime SELECT p, a, cohort, birthRole, gold FROM D role as birthRole TimeDiff(D.t, birthTime) as age WHERE a = "launch" FROM D, birth FROM D, birthTuples GROUP BY p WHERE D.p = birth.p AND WHERE D.p = birthTuples.p), ), D.t = birth.birthTime), (c) (a) (b) cohortSize AS ( SELECT cohort, size, age, Sum(gold) SELECT cohort, Count(distinct p) as size FROM cohortT, cohortSize FROM cohortT WHERE cohortT.cohort = cohortSize.cohort GROUP BY cohort birthRole = "dwarf" AND a = "shop" AND age > 0 ), GROUP BY cohort, age (d) (e) Figure 2: The SQL query Q of our example analysis task. s superior to its mostly optimized counterpart, demon- can be performed by an intelligent optimizer. How- strating the necessity of extending database systems to ever, our evaluation shows that few database systems cater to cohort analytics, rather than simply running can perform such an optimization. an SQL statement over a traditional DBMS. To speed up the processing of the analysis task, we can The rest of the paper is organized as follows: Section adopt a materialized view (MV) approach that stores some presents the SQL based and the materialized view based intermediate results. For example, we can materialize the approaches for processing cohort queries. Section presents intermediate table cohortT produced by the sub-query in the foundations of cohort analysis. Section presents our Qs (Figure 2(c)) as follows. proposed columnar database based scheme for cohort query processing. Section 17 reports the experimental results. We CREATE VIEW MATERIALIZED cohorts as cohortT present related work in Section 17 and conclude in Section 17. With cohorts, we can express the query QS in a simpler SQL statement consisting of a single sub-query (Figure 2(d)) 2. NON-INTRUSIVE APPROACHES TO CO- and an outer query (Figure 2(e)). The performance of the re- sulting SQL expression is also improved since it only involves HORT ANALYTICS a single join. However, the materialized view approach also suﬀers from a number of issues. A least intrusive approach to supporting cohort analytics is to use an existing relational DBMS and express the cohort • The cost of generating the MV is still high since it analysis task as a SQL query. We illustrate such an approach involves two joins (Figure 2(b) and 2(c)). using the following cohort analysis task: • The storage space for the MV is huge if the approach Example 1. Given the launch birth action and the activ- is used as a general cohort query processing strategy. ity table as shown in Table 1 (denoted as D), for players who Figure 2(c) only includes a single calculated birth at- play the dwarf role at their birth time, cohort those players tribute birthRole as it is the only attribute appearing based on their birth countries and report the total gold that in the birth selection condition (i.e., the condition of country launch cohorts spent since they were born. playing as the dwarf role at birth time) of the analy- sis task. However, if other calculated birth attributes Figure 2 shows the corresponding SQL query Q for this s are also involved in the birth selection condition, we task. To save space, we use p, a, t, c abbreviations re- need to include those attributes in the MV as well. In spectively to denote the player, action, time, and country the extreme case, every possible birth attribute shall attribute in Table 1. The Q employs four sub-queries (i.e., s be included in the MV, doubling the storage space as Figure 2(a) – Figure 2(d)) and one outer query (i.e., Fig- compared to the original activity table. ure 2(e)) to produce the results. Overall, this SQL approach • performs poorly for three reasons: The MV only answers cohort queries introduced by • launch birth action. If another birth action (e.g., The SQL statement Qs is verbose, and its complexity shop) is used, one more MV is required. Obviously, renders it prone to mistakes. this per birth action per MV approach does not scale • even for a small number of birth actions due to the The SQL statement Q requires many joins to perform s cost of MV construction and maintenance. the analysis task. As we shall see in our experimental • The query performance is still not optimal. By the def- study, such a query processing scheme can be up to 5 inition of the analysis task, if a player does not play orders of magnitude slower than our proposed solution. • as dwarf role when that player is born, we should ex- It requires manual tuning. For example, one may no- clude all activity tuples of that player in the result set. tice that we can push the selection condition (i.e., Ideally, this ﬁltering operation can be performed by birthRole = "dwarf") from the outer query (Figure 2(e)) simply checking the single birth activity tuple of that to the inner sub-query (Figure 2(c)) to reduce the size player. If the birth activity tuple indicates that the of intermediate tables. Ideally, such an optimization player is not qualiﬁed, we can safely skip all activity 3

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text: selection

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text: ALMA observations of molecular medium in galaxies 5 able integration time for local IR-luminous galaxies, and they will provide an ambiguous basis to calibrate other proposed diagnostic methods. the HCN enhancement: A signiﬁcant enhancement of HCN emission with respect to other molecules such as CO and HCO+ at the centers of local AGNs was ﬁrst indicated by Jackson et al. (1993) and Tacconi et al. (1994) based on the NMA and PdBI observations of HCN(1-0) in NGC 1068. Since then, mounting supporting evidence for such elevated HCN emission at the centers of local Seyfert galaxies has been accumulated (e.g., Kohno et al. 2001; Imanishi et al. 2006; Krips et al. 2008), giving an empirical AGN diagnostic. There are on-going debates on the origin of such elevated HCN, however. X-ray induced chemistry is one of the attractive scenarios because the CND of NGC 1068 is suggested to be a giant X-ray dominated region (XDR; Maloney et al. 1996) based on detections of HOC+ and other molecules that are preferentially enhanced in the XDR (Usero et al. 2004), although chemical models often do not reproduce such HCN enhancement in the XDR (e.g., Meijerinket al. 2007). More recently, spatially resolved ALMA observations ∼ of HCN(4-3) and HCO+(4-3) at the CND of NGC 1068 at a 50 pc resolution reveal that the HCN(4-3)/HCO+(4-3) line ratios are indeed highly elevated at the CND but not signiﬁcantly at the very position of the active nucleus (Garc´ıa-Burillo et al. 2014; Viti et al. 2014). This indicates that HCN enhancement is not simply caused by X-ray irradiation. In fact, recent HCN(4-3)/HCO+(4-3) measurements of the high-luminosity × − type-1 AGN NGC 7469 (Izumi et al. 2015a) with L = 2 1043 erg s 1 suggest that the X degree of HCN enhancement is not very diﬀerent from that in NGC 1097 (Izumi et al. × − 2013), a much lower luminosity type-1 AGN (L = 7 1040 erg s 1), also adding further X evidence against the XDR origin of enhanced HCN in AGNs (see also Mart´ın et al. 2015). Mechanical heating, which is caused by the dissipation of shock-driven energy (Loenen et al. 2008) and high-temperature-induced chemistry (Harada et al. 2010; Harada et al. 2013; Izumi et al. 2013) are other possible mechanisms that could explain such HCN enhancement. Elevated HCN/HCO+ ratios in massive (and presumably shocked) molec- ular outﬂows driven by the powerful AGN in Mrk 231 would be consistent with this view (Aalto et al. 2012). A possible HCN maser ampliﬁcation is also proposed (Matsushita et al. 2015) in the case of the HCN enhancement in M51 (Kohno et al. 1996). Figure 3. Column density ratios of HCO to H13CO+ as a function of the column density of H13CO+, including two new detections of HCO absorption (J1717-337 and NRAO530; Ando et al. 2015). Molecular absorption systems toward J1717-337, NRAO530, 3C111, and BL-LAC show elevated HCO/H13CO+ column density ratios similar to the value in the Horsehead Nebula (Gerin et al. 2009), indicating that the diﬀuse media in the Galaxy seen toward these radio sources are exposed in the intense UV ﬁeld. Figure modiﬁed from Ando et al. (2015). ##LTFigure## ##LTFigure## ##LTLine##

text: ALMA

label: paragraph

text: observations

text: of

text: molecular

text: medium

text: in

text: galaxies

text: able

text: integration

text: time

text: for

text: local

text: IR-luminous

text: galaxies,

text: and

text: they

text: will

text: provide

text: an

text: ambiguous

text: basis

text: to

text: calibrate

text: other

text: proposed

text: diagnostic

text: methods.

text: the

text: HCN

text: enhancement:

text: A

text: signiﬁcant

text: enhancement

text: of

text: HCN

text: emission

text: with

text: respect

text: to

text: other

text: molecules

text: such

text: as

text: CO

text: and

text: HCO+

text: at

text: the

text: centers

text: of

text: local

text: AGNs

text: was

text: ﬁrst

text: indicated

text: by

text: Jackson

text: et

text: al.

text: and

text: Tacconi

text: et

text: al.

text: based

text: on

text: the

text: NMA

text: and

text: PdBI

text: observations

text: of

text: in

text: NGC

text: Since

text: then,

text: mounting

text: supporting

text: evidence

text: for

text: such

text: elevated

text: HCN

text: emission

text: at

text: the

text: centers

text: of

text: local

text: Seyfert

text: galaxies

text: has

text: been

text: accumulated

text: (e.g.,

text: Kohno

text: et

text: al.

text: Imanishi

text: et

text: al.

text: Krips

text: et

text: al.

text: giving

text: an

text: empirical

text: AGN

text: diagnostic.

text: There

text: are

text: on-going

text: debates

text: on

text: the

text: origin

text: of

text: such

text: elevated

text: HCN,

text: however.

text: X-ray

text: induced

text: chemistry

text: is

text: one

text: of

text: the

text: attractive

text: scenarios

text: because

text: the

text: CND

text: of

text: NGC

text: is

text: suggested

text: to

text: be

text: a

text: giant

text: X-ray

text: dominated

text: region

text: (XDR;

text: Maloney

text: et

text: al.

text: based

text: on

text: detections

text: of

text: HOC+

text: and

text: other

text: molecules

text: that

text: are

text: preferentially

text: enhanced

text: in

text: the

text: XDR

text: (Usero

text: et

text: al.

text: although

text: chemical

text: models

text: often

text: do

text: not

text: reproduce

text: such

text: HCN

text: enhancement

text: in

text: the

text: XDR

text: (e.g.,

text: Meijerinket

text: al.

text: More

text: recently,

text: spatially

text: resolved

text: ALMA

text: observations

text: of

text: and

text: at

text: the

text: CND

text: of

text: NGC

text: at

text: a

text: pc

text: resolution

text: reveal

text: that

text: the

text: line

text: ratios

text: are

text: indeed

text: highly

text: elevated

text: at

text: the

text: CND

text: but

text: not

text: signiﬁcantly

text: at

text: the

text: very

text: position

text: of

text: the

text: active

text: nucleus

text: (Garc´ıa-Burillo

text: et

text: al.

text: Viti

text: et

text: al.

text: This

text: indicates

text: that

text: HCN

text: enhancement

text: is

text: not

text: simply

text: caused

text: by

text: X-ray

text: irradiation.

text: In

text: fact,

text: recent

text: measurements

text: of

text: the

text: high-luminosity

text: AGN

text: NGC

text: (Izumi

text: et

text: al.

text: with

text: L

text: erg

text: s

text: suggest

text: that

text: the

text: X

text: degree

text: of

text: HCN

text: enhancement

text: is

text: not

text: very

text: diﬀerent

text: from

text: that

text: in

text: NGC

text: (Izumi

text: et

text: al.

text: a

text: much

text: lower

text: luminosity

text: AGN

text: (L

text: erg

text: s

text: also

text: adding

text: further

text: X

text: evidence

text: against

text: the

text: XDR

text: origin

text: of

text: enhanced

text: HCN

text: in

text: AGNs

text: (see

text: also

text: Mart´ın

text: et

text: al.

text: Mechanical

text: heating,

text: which

text: is

text: caused

text: by

text: the

text: dissipation

text: of

text: shock-driven

text: energy

text: (Loenen

text: et

text: al.

text: and

text: high-temperature-induced

text: chemistry

text: (Harada

text: et

text: al.

text: Harada

text: et

text: al.

text: Izumi

text: et

text: al.

text: are

text: other

text: possible

text: mechanisms

text: that

text: could

text: explain

text: such

text: HCN

text: enhancement.

text: Elevated

text: HCN/HCO+

text: ratios

text: in

text: massive

text: (and

text: presumably

text: shocked)

text: molec-

text: ular

text: outﬂows

text: driven

text: by

text: the

text: powerful

text: AGN

text: in

text: Mrk

text: would

text: be

text: consistent

text: with

text: this

text: view

text: (Aalto

text: et

text: al.

text: A

text: possible

text: HCN

text: maser

text: ampliﬁcation

text: is

text: also

text: proposed

text: (Matsushita

text: et

text: al.

text: in

text: the

text: case

text: of

text: the

text: HCN

text: enhancement

text: in

text: (Kohno

text: et

text: al.

text: Figure

text: Column

text: density

text: ratios

text: of

text: HCO

text: to

text: as

text: a

text: function

text: of

text: the

text: column

text: density

text: of

text: including

text: two

text: new

text: detections

text: of

text: HCO

text: absorption

text: and

text: Ando

text: et

text: al.

text: Molecular

text: absorption

text: systems

text: toward

text: and

text: BL-LAC

text: show

text: elevated

text: column

text: density

text: ratios

text: similar

text: to

text: the

text: value

text: in

text: the

text: Horsehead

text: Nebula

text: (Gerin

text: et

text: al.

text: indicating

text: that

text: the

text: diﬀuse

text: media

text: in

text: the

text: Galaxy

text: seen

text: toward

text: these

text: radio

text: sources

text: are

text: exposed

text: in

text: the

text: intense

text: UV

text: ﬁeld.

text: Figure

text: modiﬁed

text: from

text: Ando

text: et

text: al.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: - + - 2 2 AZ(π ,l l )A*Z-1, plab=16 GeV/c, Ml+l-=4 GeV 1.0 0.9 CT α=0.904 0.8 Glauber 0.7 α=0.786 0.6 0.5 0.4 0.3 0.2 0 50 100 150 200 A FIG. 7. (color online) Target nucleus mass number dependence of the transparency for the − − ∗ A (π , l+l )A − reaction at p = 16 GeV/c and M2 − = 4 GeV2. The calculations are done Z Z 1 lab l+l for nuclei 12C, 27Al, 40Ca, 56Fe, 63Cu, 75As, 84Kr, 120Sn, 142Ce, 197Au, and 208Pb. The result obtained within the quantum diﬀusion model is shown by solid circles while the Glauber model result is shown by solid triangles. The ﬁts with a power law of Eq.(??) are shown by lines marked with the value of α. in the point-like conﬁguration across its entire trajectory within the nucleus, while for the 197Au nucleus the pion dynamics is still in the region of quantum diﬀusion. − − Fig. ?? shows the mass dependence of the transparency for the (π , l l+) reaction at − ﬁxed values of the beam momentum, p = 16 GeV/c, and l+l invariant mass, M2 − = 4 lab l+l GeV2. We performed the ﬁts of the calculated mass number dependence of the transparency by the power law − T (cid:39) Aα 1 . (17) The Glauber model results can be well reproduced with α = 0.786 ± 0.003. This is not so far from the surface dominated production mechanism (α = 2/3). The power law ﬁt of the quantum diﬀusion calculation is less good, although it is clear that the obtained value, 16 T ##LTFigure##

font: default

text: AZ(π

text: ,l

text: l

text: GeV/c,

text: GeV

text: CT

text: Glauber

text: A

text: FIG.

text: (color

label: caption

text: online)

text: Target

text: nucleus

text: mass

text: number

text: dependence

text: of

text: the

text: transparency

text: for

text: the

text: A

text: (π

text: l+l

text: )A

text: reaction

text: at

text: p

text: GeV/c

text: and

text: The

text: calculations

text: are

text: done

text: Z

text: Z

text: lab

text: l+l

text: for

text: nuclei

text: and

text: The

text: result

text: obtained

text: within

text: the

text: quantum

text: diﬀusion

text: model

text: is

text: shown

text: by

text: solid

text: circles

text: while

text: the

text: Glauber

text: model

text: result

text: is

text: shown

text: by

text: solid

text: triangles.

text: The

text: ﬁts

text: with

text: a

text: power

text: law

text: of

text: Eq.(??)

text: are

text: shown

text: by

text: lines

text: marked

text: with

text: the

text: value

text: of

text: α.

text: in

text: the

text: point-like

text: conﬁguration

text: across

text: its

text: entire

text: trajectory

text: within

text: the

text: nucleus,

text: while

text: for

text: the

text: nucleus

text: the

text: pion

text: dynamics

text: is

text: still

text: in

text: the

text: region

text: of

text: quantum

text: diﬀusion.

text: Fig.

text: shows

text: the

text: mass

text: dependence

text: of

text: the

text: transparency

text: for

text: the

text: (π

text: l

text: l+)

text: reaction

text: at

text: ﬁxed

text: values

text: of

text: the

text: beam

text: momentum,

text: p

text: GeV/c,

text: and

text: l+l

text: invariant

text: mass,

text: lab

text: l+l

text: We

text: performed

text: the

text: ﬁts

text: of

text: the

text: calculated

text: mass

text: number

text: dependence

text: of

text: the

text: transparency

text: by

text: the

text: power

text: law

text: T

text: Aα

text: The

text: Glauber

text: model

text: results

text: can

text: be

text: well

text: reproduced

text: with

text: α

text: This

text: is

text: not

text: so

text: far

text: from

text: the

text: surface

text: dominated

text: production

text: mechanism

text: (α

text: The

text: power

text: law

text: ﬁt

text: of

text: the

text: quantum

text: diﬀusion

text: calculation

text: is

text: less

text: good,

text: although

text: it

text: is

text: clear

text: that

text: the

text: obtained

text: value,

text: T

text: ##LTFigure##


text: for heavy targets than for light targets. By a similar mechanism, squeezing of the incoming pion is expected for the semiexclusive − A(π , J/ψ) reaction (as well as the production of χ-mesons) which is driven by the small − (charmonium mass) 1 scale. Thus, the color transparency signal should be observable in this case too. Taking into account the uncertainty of the degree of squeezing of the charmonium in the hard interaction point, the genuine J/ψN cross section can be determined with an accuracy better than ∼ 20% from the ratio of transparencies for the J/ψ and pure dilepton outgoing channels. − We have also given predictions for the nuclear transparency in the case of π -induced ρ0 production on nuclei using the Glauber model. The comparison with the ρ meson photo- production would allow to check whether similar conﬁgurations in the ρ meson dominate for the interaction in the vacuum and non-vacuum channels. The eﬀects discussed in this work can be experimentally studied with pionic secondary beam which will be available at the J-PARC and, possibly, at GSI/FAIR. ACKNOWLEDGMENTS AL and MB acknowledge ﬁnancial support of HIC for FAIR within the framework of the Hessian LOEWE program. MS’s work is supported by US Department of Energy grants under contract DE-FG02-93ER40771. [] S. J. Brodsky, in Proceedings of the 13th International Symposium on Multiparticle Dynamics, edited by W. Kittel, W. Metzger, and A. Stergiou (World Scientiﬁc, Singapore, 1982) p. 963. [] A. H. Mueller, in Proceedings of the 17th Rencontres de Moriond, Vol. I, edited by J. Tran Thanh Van (Editions Frontieres, Gif-sur-Yvette, France, 1982) p. 13. [] L. L. Frankfurt, G. A. Miller, and M. Strikman, Ann. Rev. Nucl. Part. Sci. 44, 501 (1994). [] P. Jain, B. Pire, and J. P. Ralston, Phys. Rept. 271, 67 (1996). [] D. Dutta, K. Haﬁdi, and M. Strikman, Prog.Part.Nucl.Phys. 69, 1 (2013). [] S. J. Brodsky, L. Frankfurt, J. F. Gunion, A. H. Mueller, and M. Strikman, Phys. Rev. D50, 3134 (1994). [] J. C. Collins, L. Frankfurt, and M. Strikman, Phys.Rev. D56, 2982 (1997). 24 ##LTLine## ##LTLine## ##LTLine##

text: for

text: heavy

text: targets

text: than

text: for

text: light

text: targets.

text: By

text: a

text: similar

text: mechanism,

text: squeezing

text: of

text: the

text: incoming

text: pion

text: is

text: expected

text: for

text: the

text: semiexclusive

text: A(π

text: J/ψ)

text: reaction

text: (as

text: well

text: as

text: the

text: production

text: of

text: χ-mesons)

text: which

text: is

text: driven

text: by

text: the

text: small

text: (charmonium

text: mass)

text: scale.

text: Thus,

text: the

text: color

text: transparency

text: signal

text: should

text: be

text: observable

text: in

text: this

text: case

text: too.

text: Taking

text: into

text: account

text: the

text: uncertainty

text: of

text: the

text: degree

text: of

text: squeezing

text: of

text: the

text: charmonium

text: in

text: the

text: hard

text: interaction

text: point,

text: the

text: genuine

text: J/ψN

text: cross

text: section

text: can

text: be

text: determined

text: with

text: an

text: accuracy

text: better

text: than

text: from

text: the

text: ratio

text: of

text: transparencies

text: for

text: the

text: J/ψ

text: and

text: pure

text: dilepton

text: outgoing

text: channels.

text: We

text: have

text: also

text: given

text: predictions

text: for

text: the

text: nuclear

text: transparency

text: in

text: the

text: case

text: of

text: π

text: -induced

text: ρ0

text: production

text: on

text: nuclei

text: using

text: the

text: Glauber

text: model.

text: The

text: comparison

text: with

text: the

text: ρ

text: meson

text: photo-

text: production

text: would

text: allow

text: to

text: check

text: whether

text: similar

text: conﬁgurations

text: in

text: the

text: ρ

text: meson

text: dominate

text: for

text: the

text: interaction

text: in

text: the

text: vacuum

text: and

text: non-vacuum

text: channels.

text: The

text: eﬀects

text: discussed

text: in

text: this

text: work

text: can

text: be

text: experimentally

text: studied

text: with

text: pionic

text: secondary

text: beam

text: which

text: will

text: be

text: available

text: at

text: the

text: J-PARC

text: and,

text: possibly,

text: at

text: GSI/FAIR.

text: ACKNOWLEDGMENTS

text: AL

text: and

text: MB

text: acknowledge

text: ﬁnancial

text: support

text: of

text: HIC

text: for

text: FAIR

text: within

text: the

text: framework

text: of

text: the

text: Hessian

text: LOEWE

text: program.

text: MS’s

text: work

text: is

text: supported

text: by

text: US

text: Department

text: of

text: Energy

text: grants

text: under

text: contract

text: S.

text: J.

text: Brodsky,

text: in

text: Proceedings

text: of

text: the

text: International

text: Symposium

text: on

text: Multiparticle

text: Dynamics,

text: edited

text: by

text: W.

text: Kittel,

text: W.

text: Metzger,

text: and

text: A.

text: Stergiou

text: (World

text: Scientiﬁc,

text: Singapore,

text: p.

text: A.

text: H.

text: Mueller,

text: in

text: Proceedings

text: of

text: the

text: Rencontres

text: de

text: Moriond,

text: Vol.

text: I,

text: edited

text: by

text: J.

text: Tran

text: Thanh

text: Van

text: (Editions

text: Frontieres,

text: Gif-sur-Yvette,

text: France,

text: p.

text: L.

text: L.

text: Frankfurt,

text: G.

text: A.

text: Miller,

text: and

text: M.

text: Strikman,

text: Ann.

text: Rev.

text: Nucl.

text: Part.

text: Sci.

text: P.

text: Jain,

text: B.

text: Pire,

text: and

text: J.

text: P.

text: Ralston,

text: Phys.

text: Rept.

text: D.

text: Dutta,

text: K.

text: Haﬁdi,

text: and

text: M.

text: Strikman,

text: Prog.Part.Nucl.Phys.

text: S.

text: J.

text: Brodsky,

text: L.

text: Frankfurt,

text: J.

text: F.

text: Gunion,

text: A.

text: H.

text: Mueller,

text: and

text: M.

text: Strikman,

text: Phys.

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text: L.

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text: M.

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text: Phys.Rev.

text: ##LTLine##

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text: [] E. Aitala et al. (E791 Collaboration), Phys.Rev.Lett. 86, 4773 (2001). [] L. Frankfurt, G. A. Miller, and M. Strikman, Phys. Lett. B304, 1 (1993). [] M. D. Sokoloﬀ et al., Phys. Rev. Lett. 57, 3003 (1986). [] S. Chekanov et al. (ZEUS Collaboration), Nucl.Phys. B695, 3 (2004). [] S. Chekanov et al. (ZEUS Collaboration), PMC Phys. A1, 6 (2007). [] B. Clasie et al., Phys. Rev. Lett. 99, 242502 (2007). [] L. El Fassi et al. (CLAS), Phys. Lett. B712, 326 (2012). [] G. A. Miller and M. Strikman, Phys. Rev. C82, 025205 (2010). [] G. T. Howell and G. A. Miller, Phys. Rev. C88, 035202 (2013). [] S. Kumano, in 21st International Symposium on Spin Physics (SPIN 2014) Beijing, China, October 20-24, 2014 (2015) arXiv:1504.05264 [hep-ph]. [] E. R. Berger, M. Diehl, and B. Pire, Phys. Lett. B523, 265 (2001). [] S. V. Goloskokov and P. Kroll, Phys. Lett. B748, 323 (2015). [] L. Mankiewicz, G. Piller, and A. Radyushkin, Eur. Phys. J. C10, 307 (1999). [] L. L. Frankfurt, P. V. Pobylitsa, M. V. Polyakov, and M. Strikman, Phys. Rev. D60, 014010 (1999). [] D. Mueller, B. Pire, L. Szymanowski, and J. Wagner, Phys. Rev. D86, 031502 (2012). [] J. W. Qiu, Talk presented at the KEK workshop on Hadron physics with high-momentum hadron beams at J-PARC, March 13-16, 2015, http://research.kek.jp/group/hadron10/j-parc-hm-2015/. [] L. L. Frankfurt, M. M. Sargsian, and M. I. Strikman, Phys. Rev. C 56, 1124 (1997). [] M. M. Sargsian, Int. J. Mod. Phys. E10, 405 (2001). [] A. B. Larionov, M. Strikman, and M. Bleicher, Phys.Rev. C89, 014621 (2014). [] G. Farrar, H. Liu, L. Frankfurt, and M. Strikman, Phys. Rev. Lett. 61, 686 (1988). [] J. Hu¨fner, B. Kopeliovich, and J. Nemchik, Phys. Lett. B 383, 362 (1996). [] L. Frankfurt, G. A. Miller, and M. Strikman, Phys. Rev. C78, 015208 (2008). [] K. Gallmeister, M. Kaskulov, and U. Mosel, Phys. Rev. C83, 015201 (2011). [] R. L. Anderson et al., Phys. Rev. Lett. 38, 263 (1977). [] G. R. Farrar, L. L. Frankfurt, M. I. Strikman, and H. Liu, Nucl. Phys. B 345, 125 (1990). [] C. Gerschel and J. Hu¨fner, Z. Phys. C 56, 171 (1992). [] D. Kharzeev, C. Lourenco, M. Nardi, and H. Satz, Z. Phys. C 74, 307 (1997). 25

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text: π− π− ρ0 π− J/ψ π+ p n p n AZ (a) A∗Z−1 AZ (b) A∗Z−1 − − ∗ FIG. 2. The Feynman graphs describing the amplitudes of the semiexclusive A (π , π+π )A − Z Z 1 − ∗ process mediated by ρ0 (a) and A (π , J/ψ)A − process (b). Thick short-dashed lines denote Z Z 1 − π p → V n transition amplitudes, V = ρ0, J/ψ. Other notations are similar to Fig. 1. for the processes with CT by replacing the pion-nucleon cross section by the eﬀective one in the pion survival probability (3) as   (cid:90)   z (cid:48) (cid:48) (cid:48) PCT (b, −∞, z) = exp − dz σeﬀ (p , z − z )ρ(b, z ) . (8) π,surv πN π −∞ − − ∗ − The process A (π , l+l )A − with large invariant mass of l+l pair represents a clean Z Z 1 way to measure CT for the incoming pion. It is also possible to study other ﬁnal states in − the π -nucleus reactions with charge exchange in similar kinematical conditions. Two of − − ∗ − ∗ such channels, A (π , π+π )A − with intermediate ρ0 and A (π , J/ψ)A − , are shown Z Z 1 Z Z 1 in Fig. 2. These two reactions could provide complementary information to the already existing studies of the ρ [? ? ? ? ] and J/ψ [? ? ? ? ? ? ? ? ? ] attenuation in the nuclear medium. The corresponding transparencies can be expressed as (cid:90) 1 T = d3rP (b, −∞, z)ρ (r)P (b, z, +∞) , (9) V π,surv p V,surv Z with V = ρ, J/ψ. The survival probabilities of the vector mesons are expressed as (cid:90)   ∞ + (cid:48) (cid:48) (cid:48) P (b, z, +∞) = exp − dz σeﬀ (p , z − z)ρ(b, z ) . (10) V,surv V N V z 8 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: π−

text: π−

text: ρ0

text: π−

text: J/ψ

text: π+

text: p

text: n

text: p

text: n

text: AZ

text: (a)

text: AZ

text: (b)

text: FIG.

text: The

text: Feynman

text: graphs

text: describing

text: the

text: amplitudes

text: of

text: the

text: semiexclusive

text: A

text: (π

text: π+π

text: )A

text: Z

text: Z

text: process

text: mediated

text: by

text: ρ0

text: (a)

text: and

text: A

text: (π

text: J/ψ)A

text: process

text: (b).

text: Thick

text: short-dashed

text: lines

text: denote

text: Z

text: Z

text: π

text: p

text: V

text: n

text: transition

text: amplitudes,

text: V

text: J/ψ.

text: Other

text: notations

text: are

text: similar

text: to

text: Fig.

text: for

text: the

text: processes

text: with

text: CT

text: by

text: replacing

text: the

text: pion-nucleon

text: cross

text: section

text: by

text: the

text: eﬀective

text: one

text: in

text: the

text: pion

text: survival

text: probability

text: as

text: z

text: PCT

text: (b,

text: z)

text: exp

text: dz

text: σeﬀ

text: (p

text: z

text: z

text: )ρ(b,

text: z

text: π,surv

text: πN

text: π

text: The

text: process

text: A

text: (π

text: l+l

text: )A

text: with

text: large

text: invariant

text: mass

text: of

text: l+l

text: pair

text: represents

text: a

text: clean

text: Z

text: Z

text: way

text: to

text: measure

text: CT

text: for

text: the

text: incoming

text: pion.

text: It

text: is

text: also

text: possible

text: to

text: study

text: other

text: ﬁnal

text: states

text: in

text: the

text: π

text: -nucleus

text: reactions

text: with

text: charge

text: exchange

text: in

text: similar

text: kinematical

text: conditions.

text: Two

text: of

text: such

text: channels,

text: A

text: (π

text: π+π

text: )A

text: with

text: intermediate

text: ρ0

text: and

text: A

text: (π

text: J/ψ)A

text: are

text: shown

text: Z

text: Z

text: Z

text: Z

text: in

text: Fig.

text: These

text: two

text: reactions

text: could

text: provide

text: complementary

text: information

text: to

text: the

text: already

text: existing

text: studies

text: of

text: the

text: ρ

text: and

text: J/ψ

text: attenuation

text: in

text: the

text: nuclear

text: medium.

text: The

text: corresponding

text: transparencies

text: can

text: be

text: expressed

text: as

text: T

text: (b,

text: z)ρ

text: (r)P

text: (b,

text: z,

text: V

text: π,surv

text: p

text: V,surv

text: Z

text: with

text: V

text: ρ,

text: J/ψ.

text: The

text: survival

text: probabilities

text: of

text: the

text: vector

text: mesons

text: are

text: expressed

text: as

text: P

text: (b,

text: z,

text: exp

text: dz

text: σeﬀ

text: (p

text: z

text: z)ρ(b,

text: z

text: V,surv

text: V

text: N

text: V

text: z

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: In the case of J/ψ production, since the size of the J/ψ is much smaller than for pion − the CT eﬀects are important. Moreover, the overlap integral in the π qq¯ vertex may select larger transverse momenta in the J/ψ wave function. Thus, we evaluate the eﬀective J/ψN cross section within the quantum diﬀusion model which gives  (cid:32) (cid:33)     z (cid:104)n2k2(cid:105) z σeﬀ (p , z) = σ + t 1 − Θ(l − z) + Θ(z − l ) , (11) J/ψN J/ψ J/ψN l M2 l J/ψ J/ψ J/ψ J/ψ J/ψ similar to Eq.(6) for the πN eﬀective cross section. Note that in the J/ψ case character- istic k are much larger than for the pion - on the scale of 0.8 − 1 GeV/c. This means that t the J/ψN cross section is reduced by approximately a factor of 2 at the hard interaction point. The interaction cross sections of the qq¯ conﬁgurations with the nucleon just before and after the hard interaction should be the same. To match this condition in calculations of J/ψ production, we have modiﬁed the (cid:104)k2(cid:105)1/2 value in Eq.(6) for the πN eﬀective cross t section. Since the J/ψN total cross section, σ , is not well known, we will apply the J/ψN value σ = 4 mb for the default calculations. This value is motivated by the analysis of J/ψN the J/ψ transparencies taking into account the contribution from the radiative decays of χ c states [? ]. The J/ψ coherence length can be evaluated as (cf. [? ? ]) 2p J/ψ l (cid:39) . (12) J/ψ m2 (cid:48) − m2 ψ J/ψ For the energies discussed here l remains small and hence expansion eﬀects are small. J/ψ In the case of ρ-meson production, the expansion eﬀects are negligible since we are dealing with a soft process. The survival probability of the ρ-meson should, however, be corrected for the ρ decay inside nucleus. This leads to the following expression for the ρN eﬀective cross section (cf. [? ]): (cid:32) (cid:33) (cid:34) (cid:32) (cid:33)(cid:35) Γ Γ σeﬀ (p , z) = σ exp − ρ z + 2σ (p /2) 1 − exp − ρ z , (13) ρN ρ ρN γ πN ρ γ (cid:113) where Γ = 0.149 GeV is the decay width of the ρ meson at rest and γ = m2 + p2/m ρ ρ ρ ρ 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: the

text: case

text: of

text: J/ψ

text: production,

text: since

text: the

text: size

text: of

text: the

text: J/ψ

text: is

text: much

text: smaller

text: than

text: for

text: pion

text: the

text: CT

text: eﬀects

text: are

text: important.

text: Moreover,

text: the

text: overlap

text: integral

text: in

text: the

text: π

text: qq¯

text: vertex

text: may

text: select

text: larger

text: transverse

text: momenta

text: in

text: the

text: J/ψ

text: wave

text: function.

text: Thus,

text: we

text: evaluate

text: the

text: eﬀective

text: J/ψN

text: cross

text: section

text: within

text: the

text: quantum

text: diﬀusion

text: model

text: which

text: gives

label: equation

text: z

text: z

text: σeﬀ

text: (p

text: z)

text: σ

text: t

text: Θ(l

text: z)

text: Θ(z

text: l

text: J/ψN

text: J/ψ

text: J/ψN

text: l

text: l

text: J/ψ

text: J/ψ

text: J/ψ

text: J/ψ

text: J/ψ

text: similar

text: to

text: for

text: the

text: πN

text: eﬀective

text: cross

text: section.

text: Note

text: that

text: in

text: the

text: J/ψ

text: case

text: character-

text: istic

text: k

text: are

text: much

text: larger

text: than

text: for

text: the

text: pion

text: on

text: the

text: scale

text: of

text: GeV/c.

text: This

text: means

text: that

text: t

text: the

text: J/ψN

text: cross

text: section

text: is

text: reduced

text: by

text: approximately

text: a

text: factor

text: of

text: at

text: the

text: hard

text: interaction

text: point.

text: The

text: interaction

text: cross

text: sections

text: of

text: the

text: qq¯

text: conﬁgurations

text: with

text: the

text: nucleon

text: just

text: before

text: and

text: after

text: the

text: hard

text: interaction

text: should

text: be

text: the

text: same.

text: To

text: match

text: this

text: condition

text: in

text: calculations

text: of

text: J/ψ

text: production,

text: we

text: have

text: modiﬁed

text: the

text: value

text: in

text: for

text: the

text: πN

text: eﬀective

text: cross

text: t

text: section.

text: Since

text: the

text: J/ψN

text: total

text: cross

text: section,

text: σ

text: is

text: not

text: well

text: known,

text: we

text: will

text: apply

text: the

text: J/ψN

text: value

text: σ

text: mb

text: for

text: the

text: default

text: calculations.

text: This

text: value

text: is

text: motivated

text: by

text: the

text: analysis

text: of

text: J/ψN

text: the

text: J/ψ

text: transparencies

text: taking

text: into

text: account

text: the

text: contribution

text: from

text: the

text: radiative

text: decays

text: of

text: χ

text: c

text: states

text: The

text: J/ψ

text: coherence

text: length

text: can

text: be

text: evaluated

text: as

text: (cf.

text: J/ψ

text: l

text: J/ψ

text: ψ

text: J/ψ

text: For

text: the

text: energies

text: discussed

text: here

text: l

text: remains

text: small

text: and

text: hence

text: expansion

text: eﬀects

text: are

text: small.

text: J/ψ

text: In

text: the

text: case

text: of

text: ρ-meson

text: production,

text: the

text: expansion

text: eﬀects

text: are

text: negligible

text: since

text: we

text: are

text: dealing

text: with

text: a

text: soft

text: process.

text: The

text: survival

text: probability

text: of

text: the

text: ρ-meson

text: should,

text: however,

text: be

text: corrected

text: for

text: the

text: ρ

text: decay

text: inside

text: nucleus.

text: This

text: leads

text: to

text: the

text: following

text: expression

text: for

text: the

text: ρN

text: eﬀective

text: cross

text: section

text: (cf.

text: Γ

text: Γ

text: σeﬀ

text: (p

text: z)

text: σ

text: exp

text: ρ

text: z

text: 2σ

text: (p

text: exp

text: ρ

text: z

text: ρN

text: ρ

text: ρN

text: γ

text: πN

text: ρ

text: γ

text: where

text: Γ

text: GeV

text: is

text: the

text: decay

text: width

text: of

text: the

text: ρ

text: meson

text: at

text: rest

text: and

text: γ

text: ρ

text: ρ

text: ρ

text: ρ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ADAPTIVE ANISOTROPIC PETROV-GALERKIN METHODS FOR FIRST ORDER TRANSPORT EQUATIONS13 (cid:0) (cid:1) H Lemma 4.2. Assume that j is even and (Γ) is a horizontal Γ-chain of length l none of whose (cid:0) (cid:1) j,m2 elements intersects the left or right vertical part of ∂D. Then, replacing all pairs of contiguous H triangles in R (Γ) whose union is a pa(cid:0)rallelogram(cid:1) by this parallelogram while keeping all j+1 Hj,m2 other cells in R (Γ) the same, yields a collection j+1 j,m2 (cid:0) (cid:1) H MERGE (Γ) j,m2 H with the following properties: all cells Q in MERGE (Γ) that substantially intersect Γ, i.e., | ∩ | j,m2 − Q Γ > 0, form a horizontal Γ-chain of length not larger than 2l 1. Proof. The validity of the above claim is easily conﬁrmed by noting that by (4.11) ι = ι (cid:48) for (cid:48) ∈ H P P any two P, P and using the maximality of a horizontal Γ-chain. j,m2 (cid:0) (cid:1) (cid:91) (cid:8) (cid:0) (cid:1) (cid:9) H For arbitrary, ﬁnite union of horizontal Γ-chains we denote H H H ∈ H MERGE := MERGE (Γ) : (Γ) . j,m2 j,m2 The statement in Lemma 4.2 regarding the fact that Γ is only intersected by parallelograms after merging, is not necessarily true when the ﬁrst or last parallelogram P in a horizontal Γ-chain (cid:54) intersects the left or right vertical boundary part of D, respectively. In fact, when ι = 0 the P left and rightmost triangles cannot be merged but may still be substantially intersected by Γ. To account for this fact, note ﬁrst that due to the constraints on Γ there exist at most a ﬁnite number of intersections of Γ with vertical boundary portions of ∂D. We may assume without loss of generality that J is large enough (depending only on the parameters κ, L, ω) such that a cell C 0 in contains at most one intersection of Γ with the vertical boundary 0 { ∈ { } ∈ } (∂D)v := (x¯, x ) : x¯ 0, 1 , x [0, 1] . 2 2 Let us denote C { ∈ C ∩ ∩ (cid:54) ∅} C(cid:48) C \ C (4.16) B := P : P Γ (∂D)v = , (Γ) := (Γ) B. 0 0 0 0 0 C As said before #( B) is ﬁnite depending only on κ, L, ω. 0 C(cid:48) C(cid:48) We are now prepared to describe the reﬁnement process ﬁrst only for the cells in (Γ), := C \ C C(cid:48) 0 0 B. As mentioned before (Γ) can be decomposed into horizontal Γ-chains so that the 0 0 0 collection (cid:0) (cid:1) C { ∈ ∈ C(cid:48) } C(cid:48) (4.17) an = MERGE( Q R (P) : P (Γ) ) = MERGE R ( (Γ)) 0,0 0 0 1 0 C is well deﬁned and has, by Lemma 4.2, the following property: all cells Q in an that substan- 0,0 tially intersect Γ are parallelograms with the same shear orientation for each vertical level. C Given an, we can initialize the collection 0,0 C { (cid:48) ∈ ∈ C ∪ C(cid:48) | ∩ | } (4.18) 0is,0o := Q (Riso)(cid:96)00(Q)(Q) : Q 0a,n0 0, Q Γ = 0 | ∩ | of those cells that result from an isotropic reﬁnement of the cells Q with Q Γ = 0, see Figure 3 and the deﬁnition (4.13) of the reﬁnement depth (cid:96)0(Q). 0 C C On account of Lemma 4.2, we can now recursively deﬁne ( an) and ( iso) , j > 0 by j j,0 j j,0 C { (cid:48) ∈ ∈ C | ∩ | (cid:54) } ≤ − (4.19) an := MERGE( Q R (Q) : Q a−n , Q Γ = 0 ), 0 < j 2J J , j,0 j j 1,0 0 and

text: ADAPTIVE

text: ANISOTROPIC

text: PETROV-GALERKIN

text: METHODS

text: FOR

text: FIRST

text: ORDER

text: TRANSPORT

text: H

text: Lemma

text: Assume

text: that

text: j

text: is

text: even

text: and

text: (Γ)

text: is

text: a

text: horizontal

text: Γ-chain

text: of

text: length

text: l

text: of

text: whose

text: elements

text: intersects

text: the

text: left

text: or

text: right

text: vertical

text: part

text: of

text: ∂D.

text: Then,

text: replacing

text: all

text: pairs

text: of

text: contiguous

text: H

text: triangles

text: in

text: R

text: (Γ)

text: whose

text: union

text: is

text: a

text: by

text: this

text: parallelogram

text: while

text: keeping

text: all

text: other

text: cells

text: in

text: R

text: (Γ)

text: the

text: same,

text: yields

text: a

text: collection

text: H

text: MERGE

text: (Γ)

text: H

text: with

text: the

text: following

text: properties:

text: all

text: cells

text: Q

text: in

text: MERGE

text: (Γ)

text: that

text: substantially

text: intersect

text: Γ,

text: i.e.,

text: Q

text: Γ

text: form

text: a

text: horizontal

text: Γ-chain

text: of

text: length

text: not

text: larger

text: than

text: Proof.

text: The

text: validity

text: of

text: the

text: above

text: claim

text: is

text: easily

text: conﬁrmed

text: by

text: noting

text: that

text: by

text: ι

text: ι

text: for

text: H

text: P

text: P

text: any

text: two

text: P,

text: P

text: and

text: using

text: the

text: maximality

text: of

text: a

text: horizontal

text: Γ-chain.

text: H

text: For

text: arbitrary,

text: ﬁnite

text: union

text: of

text: horizontal

text: Γ-chains

text: we

text: denote

text: H

text: H

text: H

text: H

text: MERGE

text: MERGE

text: (Γ)

text: (Γ)

text: The

text: statement

text: in

text: Lemma

text: regarding

text: the

text: fact

text: that

text: Γ

text: is

text: only

text: intersected

text: by

text: parallelograms

text: after

text: merging,

text: is

text: not

text: necessarily

text: true

text: when

text: the

text: ﬁrst

text: or

text: last

text: parallelogram

text: P

text: in

text: a

text: horizontal

text: Γ-chain

text: intersects

text: the

text: left

text: or

text: right

text: vertical

text: boundary

text: part

text: of

text: D,

text: respectively.

text: In

text: fact,

text: when

text: ι

text: the

text: P

text: left

text: and

text: rightmost

text: triangles

text: cannot

text: be

text: merged

text: but

text: may

text: still

text: be

text: substantially

text: intersected

text: by

text: Γ.

text: To

text: account

text: for

text: this

text: fact,

text: note

text: ﬁrst

text: that

text: due

text: to

text: the

text: constraints

text: on

text: Γ

text: there

text: exist

text: at

text: most

text: a

text: ﬁnite

text: number

text: of

text: intersections

text: of

text: Γ

text: with

text: vertical

text: boundary

text: portions

text: of

text: ∂D.

text: We

text: may

text: assume

text: without

text: loss

text: of

text: generality

text: that

text: J

text: is

text: large

text: enough

text: (depending

text: only

text: on

text: the

text: parameters

text: κ,

text: L,

text: ω)

text: such

text: that

text: a

text: cell

text: C

text: in

text: contains

text: at

text: most

text: one

text: intersection

text: of

text: Γ

text: with

text: the

text: vertical

text: boundary

text: (∂D)v

text: (x¯,

text: x

text: x¯

text: x

text: Let

text: us

text: denote

text: C

text: C

text: C

text: C

text: B

text: P

text: P

text: Γ

text: (∂D)v

text: (Γ)

text: (Γ)

text: B.

text: C

text: As

text: said

text: before

text: B)

text: is

text: ﬁnite

text: depending

text: only

text: on

text: κ,

text: L,

text: ω.

text: We

text: are

text: now

text: prepared

text: to

text: describe

text: the

text: reﬁnement

text: process

text: ﬁrst

text: only

text: for

text: the

text: cells

text: in

text: (Γ),

text: C

text: C

text: B.

text: As

text: mentioned

text: before

text: (Γ)

text: can

text: be

text: decomposed

text: into

text: horizontal

text: Γ-chains

text: so

text: that

text: the

text: collection

text: C

text: an

text: MERGE(

text: Q

text: R

text: (P)

text: P

text: (Γ)

text: MERGE

text: R

text: (Γ))

text: C

text: is

text: well

text: deﬁned

text: and

text: has,

text: by

text: Lemma

text: the

text: following

text: property:

text: all

text: cells

text: Q

text: in

text: an

text: that

text: substan-

text: tially

text: intersect

text: Γ

text: are

text: parallelograms

text: with

text: the

text: same

text: shear

text: orientation

text: for

text: each

text: vertical

text: level.

text: C

text: Given

text: an,

text: we

text: can

text: initialize

text: the

text: collection

text: C

text: C

text: Q

text: Q

text: Q

text: Γ

text: of

text: those

text: cells

text: that

text: result

text: from

text: an

text: isotropic

text: reﬁnement

text: of

text: the

text: cells

text: Q

text: with

text: Q

text: Γ

text: see

text: Figure

text: and

text: the

text: deﬁnition

text: of

text: the

text: reﬁnement

text: depth

text: C

text: C

text: On

text: account

text: of

text: Lemma

text: we

text: can

text: now

text: recursively

text: deﬁne

text: an)

text: and

text: iso)

text: j

text: by

text: j

text: j

text: C

text: C

text: an

text: MERGE(

text: Q

text: R

text: (Q)

text: Q

text: a−n

text: Q

text: Γ

text: j

text: J

text: j

text: j

text: and


text: ADAPTIVE ANISOTROPIC PETROV-GALERKIN METHODS FOR FIRST ORDER TRANSPORT EQUATIONS29 stable than the isotropic ones. The L approximation error and δ estimates can be found in Table 2 2. 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a) (b) 100 10−1 10−2 10−3 102 (c) (d) Figure 12. (a) Adaptive partition for the trial space X . (b) Adaptive partition 5 for the test space Z . (c) Approximate solution (345 basis elements). (d) Approx- 5 imation error in L (D) versus the number of degrees of freedom (blue) and the 2 theoretical rate (red). Figure 13. Solution and partition of the 6th isotropically adaptive cycle for the test problem (5.2). The trial space and test spaces consist of linear and bilinear ﬁnite elements, respectively. The partition of the test space is obtaind from the the corresponding partition of the trial space by reﬁning each cell once. In summary, the experiments conﬁrm that the employed anisotropic adaptive reﬁnements es- sentially recover the optimal rates relating the achieved accuracy to the used number of degrees ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: 30 W. DAHMEN, G. KUTYNIOK, W.-Q LIM, C. SCHWAB, AND G. WELPER Anisotropic Isotropic n Estimated δ L error Rate n Estimated δ L error Rate 2 2 · − · − 48 0.14 9.71 10 2 52 0.25 8.93 10 2 0.37 · − · − 108 0.19 6.87 10 2 0.43 100 0.26 7.54 10 2 0.26 · − · − 162 0.22 4.85 10 2 0.86 160 0.29 6.53 10 2 0.31 · − · − 201 0.18 3.44 10 2 1.59 256 0.33 5.49 10 2 0.37 · − · − 285 0.19 2.74 10 2 0.65 388 0.38 4.87 10 2 0.29 · − · − 345 0.18 2.31 10 2 0.89 556 0.44 4.4 10 2 0.28 Table 2. Numerical estimates for the stability constant δ, L approximation error 2 and convergence rates. of freedom. Overall this results in a better rate than the one achieved by isotropic reﬁnements. Perhaps more importantly, the anisotropy appears to greatly promote the stability of the scheme, i.e., the projections form the spaces Z (although not much larger in dimension than X ) come j j suﬃciently close to the optimal (computationally infeasible) test spaces to give rise to very well conditioned variational problems. Moreover, pointwise over- and undershoots of the approxima- tions analogous to Gibbs’ phenomena near jump discontinuities when approximating in L (D) in 2 these examples is much less pronounced than with isotropic reﬁnements. 6. The Parametric Case We now consider parametric transport problems of the type (2.9) and adhere to the notation and terminology introduced in Section . We recall (2.10), i.e. the in- and outﬂow boundaries { ∈ ± · } ∈ S ∂D±((cid:126)s) := x ∂D : (cid:126)b((cid:126)s) (cid:126)n(x) > 0 , (cid:126)s , ∈ S now depend on the transport direction (cid:126)s . Similar to [22, 23], we will now investigate the pos- sibility of sparse tensor product approximations of the parametric transport problem (2.9)–(2.10), using the directionally adaptive multi-level approximation rates in L (D) which were obtained in 2 Theorem 4.1. We outline in this section some circumstances under which the preceding anisotropic approx- imations lead in connection with sparse tensor product concepts to the eﬃcient computation of · simple quantities of interest of the parametric ﬁelds u( (cid:126)s). · ∈ ∗ We assume given a continuous, linear functional G( ) (L2(D)) , independent of the transport (cid:90) direction (cid:126)s. The goal of computation is now to approximate an integral quantity over all directions (cid:126)s, such as the total emission: · (6.1) I(G(u)) := G(u( ,(cid:126)s))d(cid:126)s . S In order to obtain superior rates of sparse tensor product approximations, we require that the C solution u((cid:126)s, x) of (2.9)–(2.10) belongs to the cartoon class uniformly over all directions (cid:126)s, with a certain amount of regularity with respect to (cid:126)s. We work under the following assumptions. Assumption 6.1. ∈ S (1) Condition (2.2) holds uniformly with respect to the direction (cid:126)s , (2) There exist parameters κ, L, M, ω (as in Theorem 4.1) such that problem (2.9) admits for ∈ S · ∈ C each (cid:126)s a unique solution u((cid:126)s, ) (κ, L, M, ω) and ◦ ∈ S ≤ (3) G u C0,α( ) for some 0 < α 1. We remark that this requirement necessitates ∈ S C regularity with respect to (cid:126)s and in the class of admissible cartoons. Note that for (uniformly in (cid:126)s) C1-convection ﬁelds, piecewise smooth boundary conditions and data f in a cartoon class with C1-pieces separated by a C2-curve, the solutions belong to a cartoon class with parameters depending on the data (right hand side, boundary data, convection ﬁeld) ##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: W.

text: DAHMEN,

text: G.

text: KUTYNIOK,

text: W.-Q

text: LIM,

text: C.

text: SCHWAB,

text: AND

text: G.

text: WELPER

text: Anisotropic

text: Isotropic

text: n

text: Estimated

text: δ

text: L

text: error

text: Rate

text: n

text: Estimated

text: δ

text: L

text: error

text: Rate

text: Table

text: Numerical

text: estimates

text: for

text: the

text: stability

text: constant

text: δ,

text: L

text: approximation

text: error

text: and

text: convergence

text: rates.

text: of

text: freedom.

text: Overall

text: this

text: results

text: in

text: a

text: better

text: rate

text: than

text: the

text: one

text: achieved

text: by

text: isotropic

text: reﬁnements.

text: Perhaps

text: more

text: importantly,

text: the

text: anisotropy

text: appears

text: to

text: greatly

text: promote

text: the

text: stability

text: of

text: the

text: scheme,

text: i.e.,

text: the

text: projections

text: form

text: the

text: spaces

text: Z

text: (although

text: not

text: much

text: larger

text: in

text: dimension

text: than

text: X

text: come

text: j

text: j

text: suﬃciently

text: close

text: to

text: the

text: optimal

text: (computationally

text: infeasible)

text: test

text: spaces

text: to

text: give

text: rise

text: to

text: very

text: well

text: conditioned

text: variational

text: problems.

text: Moreover,

text: pointwise

text: over-

text: and

text: undershoots

text: of

text: the

text: approxima-

text: tions

text: analogous

text: to

text: Gibbs’

text: phenomena

text: near

text: jump

text: discontinuities

text: when

text: approximating

text: in

text: L

text: (D)

text: in

text: these

text: examples

text: is

text: much

text: less

text: pronounced

text: than

text: with

text: isotropic

text: reﬁnements.

text: The

text: Parametric

text: Case

text: We

text: now

text: consider

text: parametric

text: transport

text: problems

text: of

text: the

text: type

text: and

text: adhere

text: to

text: the

text: notation

text: and

text: terminology

text: introduced

text: in

text: Section

text: We

text: recall

text: i.e.

text: the

text: in-

text: and

text: outﬂow

text: boundaries

text: S

text: x

text: ∂D

text: S

text: now

text: depend

text: on

text: the

text: transport

text: direction

text: Similar

text: to

text: we

text: will

text: now

text: investigate

text: the

text: pos-

text: sibility

text: of

text: sparse

text: tensor

text: product

text: approximations

text: of

text: the

text: parametric

text: transport

text: problem

text: using

text: the

text: directionally

text: adaptive

text: multi-level

text: approximation

text: rates

text: in

text: L

text: (D)

text: which

text: were

text: obtained

text: in

text: Theorem

text: We

text: outline

text: in

text: this

text: section

text: some

text: circumstances

text: under

text: which

text: the

text: preceding

text: anisotropic

text: approx-

text: imations

text: lead

text: in

text: connection

text: with

text: sparse

text: tensor

text: product

text: concepts

text: to

text: the

text: eﬃcient

text: computation

text: of

text: simple

text: quantities

text: of

text: interest

text: of

text: the

text: parametric

text: ﬁelds

text: u(

text: We

text: assume

text: given

text: a

text: continuous,

text: linear

text: functional

text: G(

text: independent

text: of

text: the

text: transport

text: direction

text: The

text: goal

text: of

text: computation

text: is

text: now

text: to

text: approximate

text: an

text: integral

text: quantity

text: over

text: all

text: directions

text: such

text: as

text: the

text: total

text: emission:

text: I(G(u))

text: G(u(

text: S

text: In

text: order

text: to

text: obtain

text: superior

text: rates

text: of

text: sparse

text: tensor

text: product

text: approximations,

text: we

text: require

text: that

text: the

text: C

text: solution

text: x)

text: of

text: belongs

text: to

text: the

text: cartoon

text: class

text: uniformly

text: over

text: all

text: directions

text: with

text: a

text: certain

text: amount

text: of

text: regularity

text: with

text: respect

text: to

text: We

text: work

text: under

text: the

text: following

text: assumptions.

text: Assumption

text: S

text: Condition

text: holds

text: uniformly

text: with

text: respect

text: to

text: the

text: direction

text: There

text: exist

text: parameters

text: κ,

text: L,

text: M,

text: ω

text: (as

text: in

text: Theorem

text: such

text: that

text: problem

text: admits

text: for

text: S

text: C

text: each

text: a

text: unique

text: solution

text: (κ,

text: L,

text: M,

text: ω)

text: and

text: S

text: G

text: u

text: for

text: some

text: α

text: We

text: remark

text: that

text: this

text: requirement

text: necessitates

text: S

text: C

text: regularity

text: with

text: respect

text: to

text: and

text: in

text: the

text: class

text: of

text: admissible

text: cartoons.

text: Note

text: that

text: for

text: (uniformly

text: in

text: ﬁelds,

text: piecewise

text: smooth

text: boundary

text: conditions

text: and

text: data

text: f

text: in

text: a

text: cartoon

text: class

text: with

text: separated

text: by

text: a

text: the

text: solutions

text: belong

text: to

text: a

text: cartoon

text: class

text: with

text: parameters

text: depending

text: on

text: the

text: data

text: (right

text: hand

text: side,

text: boundary

text: data,

text: convection

text: ﬁeld)

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text: 36 W. DAHMEN, G. KUTYNIOK, W.-Q LIM, C. SCHWAB, AND G. WELPER

text: W.

text: DAHMEN,

text: G.

text: KUTYNIOK,

text: W.-Q

text: LIM,

text: C.

text: SCHWAB,

text: AND

text: G.

text: WELPER