text: 21 (cid:2) (cid:3) For a general null network, equation (4.2) takes the form − max Tr(ST AS) Tr(ST (γP )S) , ∈S S where P is the adjacency matrix of the null network. That is, modularity maxi- mization ﬁnds block-diagonal structure in A (ﬁrst term) that is not in γP (second term). It is common to avoid using the U null network in applications because “it is not a good representation of most real-world networks” [47]. The extent to which one wants a null network to be a good representation of an observed network de- pends on the features in an application for which one wants to control. We argue that whether an NG null network is more appropriate than a U null network for a given situation depends at least in part on one’s interpretation of node strength for that application. As we discussed in Section , the strength of a node in correlation matrices is given by the covariance between its standardized time series and the mean time series. When using the NG null network, it thus follows that pairwise diﬀerences − B B (cid:48) (cid:48) in the modularity quality functio(cid:80)n depend on corr(zˆ , zˆ ), corr(zˆ (cid:48), zˆ (cid:48)), and ij i j ∈ { (cid:48) (cid:48)} i j i j corr(zˆ , zˆ ), where k i, j, i , j , the quantity zˆ is the standardized time series of k tot i asset i deﬁned in subsection , and zˆ = N zˆ . When using the U null network, tot i=1 i pairwise diﬀerences in the modularity quality function depend only on the observed edge weights corr(zˆ , zˆ ) and corr(zˆ (cid:48), zˆ (cid:48)). In contrast, the term corr(zˆ , zˆ ) intro- i j i j k tot duces a dependence between the communities that one ﬁnds with the NG null network and the extent to which nodes in those communities are representative [as measured by corr(zˆ , zˆ )] of the mean time series for the sample. In situations in which one k tot may wish to vary one’s node sample (e.g., by changing the size of asset classes), one needs to bear such dependencies in mind when interpreting the communities that one obtains. In Section , we set the inter-layer connection weights to 0 in the multilayer net- work. The solution to the multilayer modularity-maximization problem (3.4) then depends solely on the values in the modularity matrix of each time layer, and the multilayer modularity-maximization problem reduces to performing single-layer mod- ularity maximization on each layer independently. Recall the multilay(cid:88)er m(cid:88)odularity-maximization(cid:88)prob(cid:88)lem |T | |T |− N 1 N max ∈C B δ(c , c ) + 2ω δ(c , c ) . C ijs is js is is+1 s=1 i,j=1 s=1 i=1 |T | A solution to this problem is a partition of an N -node multilayer network. Its communities can contain nodes from the same layer and nodes from diﬀerent layers. (cid:54) Nodes from diﬀerent layers can be the same node at diﬀerent times ((i , i ) with s = r) (cid:54) (cid:54) s r or diﬀerent nodes at diﬀerent times ((i , j ) with i = j and s = r). We say that a s r node i remains in the same community (respectively, changes communities) between s consecutive layers s and s + 1 if δ(c , c ) = 1 (respectively, δ(c , c ) = 0). is is+1 is is+1 Positive ordinal, diagonal, and uniform inter-layer connections favor nodes re- maining in the same community between consecutive layers. Every time a node does not change communities between two consecutive layers (i.e., δ(c , c ) = 1), a posi- is is+1 tive contribution of 2ω is added to the multilayer quality function. One thereby favors communities that do not to change in time because community assignments are tran- sitive: if δ(c , c ) = 1 and δ(c , c ) = δ(c , c ) = 1, then δ(c , c ) = 1. is js is is+1 js js+1 is+1 js+1

text: For

text: a

text: general

text: network,

text: equation

text: takes

text: the

text: form

text: max

text: Tr(ST

text: AS)

text: Tr(ST

text: (γP

text: )S)

text: ∈S

text: S

text: where

text: P

text: is

text: the

text: adjacency

text: matrix

text: of

text: the

text: network.

text: That

text: is,

text: modularity

text: maxi-

text: mization

text: ﬁnds

text: block-diagonal

text: structure

text: in

text: A

text: (ﬁrst

text: term)

text: that

text: is

text: not

text: in

text: γP

text: (second

text: term).

text: It

text: is

text: common

text: to

text: avoid

text: using

text: the

text: U

text: network

text: in

text: applications

text: because

text: “it

text: is

text: not

text: a

text: good

text: representation

text: of

text: most

text: real-world

text: networks”

text: The

text: extent

text: to

text: which

text: one

text: wants

text: a

text: network

text: to

text: be

text: a

text: good

text: representation

text: of

text: an

text: observed

text: network

text: de-

text: pends

text: on

text: the

text: features

text: in

text: an

text: application

text: for

text: which

text: one

text: wants

text: to

text: control.

text: We

text: argue

text: that

text: whether

text: an

text: NG

text: network

text: is

text: more

text: appropriate

text: than

text: a

text: U

text: network

text: for

text: a

text: given

text: situation

text: depends

text: at

text: least

text: in

text: part

text: on

text: one’s

text: interpretation

text: of

text: node

text: strength

text: for

text: that

text: application.

text: As

text: we

text: discussed

text: in

text: Section

text: the

text: strength

text: of

text: a

text: node

text: in

text: correlation

text: matrices

text: is

text: given

text: by

text: the

text: covariance

text: between

text: its

text: standardized

text: time

text: series

text: and

text: the

text: mean

text: time

text: series.

text: When

text: using

text: the

text: NG

text: network,

text: it

text: thus

text: follows

text: that

text: pairwise

text: diﬀerences

text: B

text: B

text: in

text: the

text: modularity

text: quality

text: depend

text: on

text: corr(zˆ

text: zˆ

text: corr(zˆ

text: zˆ

text: and

text: ij

text: i

text: j

text: i

text: j

text: i

text: j

text: corr(zˆ

text: zˆ

text: where

text: k

text: i,

text: j,

text: i

text: j

text: the

text: quantity

text: zˆ

text: is

text: the

text: standardized

text: time

text: series

text: of

text: k

text: tot

text: i

text: asset

text: i

text: deﬁned

text: in

text: subsection

text: and

text: zˆ

text: N

text: zˆ

text: When

text: using

text: the

text: U

text: network,

text: tot

text: i

text: pairwise

text: diﬀerences

text: in

text: the

text: modularity

text: quality

text: function

text: depend

text: only

text: on

text: the

text: observed

text: edge

text: weights

text: corr(zˆ

text: zˆ

text: and

text: corr(zˆ

text: zˆ

text: In

text: contrast,

text: the

text: term

text: corr(zˆ

text: zˆ

text: intro-

text: i

text: j

text: i

text: j

text: k

text: tot

text: duces

text: a

text: dependence

text: between

text: the

text: communities

text: that

text: one

text: ﬁnds

text: with

text: the

text: NG

text: network

text: and

text: the

text: extent

text: to

text: which

text: nodes

text: in

text: those

text: communities

text: are

text: representative

text: [as

text: measured

text: by

text: corr(zˆ

text: zˆ

text: of

text: the

text: mean

text: time

text: series

text: for

text: the

text: sample.

text: In

text: situations

text: in

text: which

text: one

text: k

text: tot

text: may

text: wish

text: to

text: vary

text: one’s

text: node

text: sample

text: (e.g.,

text: by

text: changing

text: the

text: size

text: of

text: asset

text: classes),

text: one

text: needs

text: to

text: bear

text: such

text: dependencies

text: in

text: mind

text: when

text: interpreting

text: the

text: communities

text: that

text: one

text: obtains.

text: In

text: Section

text: we

text: set

text: the

text: inter-layer

text: connection

text: weights

text: to

text: in

text: the

text: multilayer

text: net-

text: work.

text: The

text: solution

text: to

text: the

text: multilayer

text: modularity-maximization

text: problem

text: then

text: depends

text: solely

text: on

text: the

text: values

text: in

text: the

text: modularity

text: matrix

text: of

text: each

text: time

text: layer,

text: and

text: the

text: multilayer

text: modularity-maximization

text: problem

text: reduces

text: to

text: performing

text: single-layer

text: mod-

text: ularity

text: maximization

text: on

text: each

text: layer

text: independently.

text: Recall

text: the

text: |T

text: |T

text: N

text: N

text: max

text: ∈C

text: B

text: δ(c

text: c

text: 2ω

text: δ(c

text: c

text: C

text: ijs

text: is

text: js

text: is

text: |T

text: A

text: solution

text: to

text: this

text: problem

text: is

text: a

text: partition

text: of

text: an

text: N

text: -node

text: multilayer

text: network.

text: Its

text: communities

text: can

text: contain

text: nodes

text: from

text: the

text: same

text: layer

text: and

text: nodes

text: from

text: diﬀerent

text: layers.

text: Nodes

text: from

text: diﬀerent

text: layers

text: can

text: be

text: the

text: same

text: node

text: at

text: diﬀerent

text: times

text: ((i

text: i

text: with

text: s

text: r)

text: s

text: r

text: or

text: diﬀerent

text: nodes

text: at

text: diﬀerent

text: times

text: ((i

text: j

text: with

text: i

text: j

text: and

text: s

text: r).

text: We

text: say

text: that

text: a

text: s

text: r

text: node

text: i

text: remains

text: in

text: the

text: same

text: community

text: (respectively,

text: changes

text: communities)

text: between

text: s

text: consecutive

text: layers

text: s

text: and

text: s

text: if

text: δ(c

text: c

text: (respectively,

text: δ(c

text: c

text: is

text: is

text: Positive

text: ordinal,

text: diagonal,

text: and

text: uniform

text: inter-layer

text: connections

text: favor

text: nodes

text: re-

text: maining

text: in

text: the

text: same

text: community

text: between

text: consecutive

text: layers.

text: Every

text: time

text: a

text: node

text: does

text: not

text: change

text: communities

text: between

text: two

text: consecutive

text: layers

text: (i.e.,

text: δ(c

text: c

text: a

text: posi-

text: is

text: tive

text: contribution

text: of

text: 2ω

text: is

text: added

text: to

text: the

text: multilayer

text: quality

text: function.

text: One

text: thereby

text: favors

text: communities

text: that

text: do

text: not

text: to

text: change

text: in

text: time

text: because

text: community

text: assignments

text: are

text: tran-

text: sitive:

text: if

text: δ(c

text: c

text: and

text: δ(c

text: c

text: δ(c

text: c

text: then

text: δ(c

text: c

text: is

text: js

text: is

text: js


text: 27 (cid:18) (cid:19) (cid:26) (cid:27) some s, into one set: (cid:91) (cid:91) |T | |T | (cid:48)(cid:48) (cid:48) \ { } ∪ C = C C C . is is s=1 s=1 Consequently, (cid:48)(cid:48)| ≥ | |T | − Q(C B , . . . , B|T |; ω) Q(C B , . . . , B|T |; ω) + 2ω( 1) , 1 1 |T | ≥ so C is not optimal. (Note that 2 for a network with more than a single layer, T − so 2ω( 1) is strictly positive for ω > 0.) Proposition 5.2. | ∅ ∈ { |T | − } | ∅ If C = for some r 1, . . . , 1 , then C = for all s > r , l r l s ∈ ∈ C where C Cω and Cω (ω). l max max max Proposition 5.2 ensures that if a community becomes empty in a given layer, then it remains empty in all subsequent layers. We omit the proof as this result follows B directly from the sparsity pattern of and our assumption that optimal solutions do not contain disconnected components. | | ⇔ | Proposition 5.3. Cω = Cω Pers(Cω ) = N. max s max s+1 max s Proposition 5.3 connects the notion of persistence between a pair of layers to the notion of change in community structure within layers. Various numerical experiments that have been performed with ordinal diagonal and uniform inter-layer coupling consist of varying the value of ω and using information about when nodes change communities between layers as an indication of change in community structure within these layers [6,7,44]. The equivalence relation in Proposition 5.3 motivates the use of | Pers(C) (or a variant thereof) as an indication of intra-layer change in community s structure. Proof. ⇐ : This follows straightforwardly by transitivity of community assignments: if δ(c , c ) = js js+1 δ(c , c ) = 1 for all i, j, then δ(c , c ) = 1 if and only if δ(c , c ) = 1 for all is is+1 is js is+1 js+1 i, j. (This direction holds for any multilayer partition; it need not be optimal.) ⇒ : Let C be a multilayer partition that does not contain disconnected components | | | ∈ { |T |} such that C = C and Pers(C) < N for some s 1, . . . , . We show that s s+1 s ∈ | (cid:54) ∅ C is not optimal.13 Consider a set C C such that C = . If δ(c , c ) = 1 l ∈ | l s is is+1 (respectively, δ(c , c ) = 0) for some i C , then δ(c , c ) = 1 (respectively, is is+1 ∈ | s l s js js+1 δ(c , c ) = 0) for all j C by transitivity of community assignments and be- js js|+1 | s l s | cause C = C by hypothesis. Because Pers(C) < N by hypothesis, there exists s s+1 | ∈ s ∈ | at least one set of nodes C , with C C, such that δ(c , c ) = 0 for all i C . | l s l is is+1 s ∈ l|s Let C denote the set of nodes in layer s + 1 that contains i for all i C . k s+1 ∪ | { s+1 } s l s Consider the set of nodes ≤ C in C that are in layers 1, . . . , s and the set of ∪ | r s l r {l |T |} nodes C in C that are in layers s + 1, . . . , . Because δ(c , c ) = 0 for r>s k r k is is+1 | | 13Imposing Pers(C) s = N is not suﬃcient because changing Pers(C) s can change partitions induced on individual layers. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: some

text: s,

text: into

text: one

text: set:

text: |T

text: |T

text: C

text: C

text: C

text: C

text: is

text: is

text: Consequently,

text: |T

text: Q(C

text: B

text: B|T

text: ω)

text: Q(C

text: B

text: B|T

text: ω)

text: |T

text: so

text: C

text: is

text: not

text: optimal.

text: (Note

text: that

text: for

text: a

text: network

text: with

text: more

text: than

text: a

text: single

text: layer,

text: T

text: so

text: is

text: strictly

text: positive

text: for

text: ω

text: Proposition

text: |T

text: If

text: C

text: for

text: some

text: r

text: then

text: C

text: for

text: all

text: s

text: r

text: l

text: r

text: l

text: s

text: C

text: where

text: C

text: Cω

text: and

text: Cω

text: (ω).

text: l

text: max

text: max

text: max

text: Proposition

text: ensures

text: that

text: if

text: a

text: community

text: becomes

text: empty

text: in

text: a

text: given

text: layer,

text: then

text: it

text: remains

text: empty

text: in

text: all

text: subsequent

text: layers.

text: We

text: omit

text: the

text: proof

text: as

text: this

text: result

text: follows

text: B

text: directly

text: from

text: the

text: sparsity

text: pattern

text: of

text: and

text: our

text: assumption

text: that

text: optimal

text: solutions

text: do

text: not

text: contain

text: disconnected

text: components.

text: Proposition

text: Cω

text: Cω

text: Pers(Cω

text: N.

text: max

text: s

text: max

text: max

text: s

text: Proposition

text: connects

text: the

text: notion

text: of

text: persistence

text: between

text: a

text: pair

text: of

text: layers

text: to

text: the

text: notion

text: of

text: change

text: in

text: community

text: structure

text: within

text: layers.

text: Various

text: numerical

text: experiments

text: that

text: have

text: been

text: performed

text: with

text: ordinal

text: diagonal

text: and

text: uniform

text: inter-layer

text: coupling

text: consist

text: of

text: varying

text: the

text: value

text: of

text: ω

text: and

text: using

text: information

text: about

text: when

text: nodes

text: change

text: communities

text: between

text: layers

text: as

text: an

text: indication

text: of

text: change

text: in

text: community

text: structure

text: within

text: these

text: layers

text: The

text: equivalence

text: relation

text: in

text: Proposition

text: motivates

text: the

text: use

text: of

text: Pers(C)

text: (or

text: a

text: variant

text: thereof)

text: as

text: an

text: indication

text: of

text: intra-layer

text: change

text: in

text: community

text: s

text: structure.

text: Proof.

text: This

text: follows

text: straightforwardly

text: by

text: transitivity

text: of

text: community

text: assignments:

text: if

text: δ(c

text: c

text: js

text: δ(c

text: c

text: for

text: all

text: i,

text: j,

text: then

text: δ(c

text: c

text: if

text: and

text: only

text: if

text: δ(c

text: c

text: for

text: all

text: is

text: is

text: js

text: i,

text: j.

text: (This

text: direction

text: holds

text: for

text: any

text: multilayer

text: partition;

text: it

text: need

text: not

text: be

text: optimal.)

text: Let

text: C

text: be

text: a

text: multilayer

text: partition

text: that

text: does

text: not

text: contain

text: disconnected

text: components

text: |T

text: such

text: that

text: C

text: C

text: and

text: Pers(C)

text: N

text: for

text: some

text: s

text: We

text: show

text: that

text: s

text: s

text: C

text: is

text: not

text: Consider

text: a

text: set

text: C

text: C

text: such

text: that

text: C

text: If

text: δ(c

text: c

text: l

text: l

text: s

text: is

text: (respectively,

text: δ(c

text: c

text: for

text: some

text: i

text: C

text: then

text: δ(c

text: c

text: (respectively,

text: is

text: s

text: l

text: s

text: js

text: δ(c

text: c

text: for

text: all

text: j

text: C

text: by

text: transitivity

text: of

text: community

text: assignments

text: and

text: be-

text: js

text: s

text: l

text: s

text: cause

text: C

text: C

text: by

text: hypothesis.

text: Because

text: Pers(C)

text: N

text: by

text: hypothesis,

text: there

text: exists

text: s

text: s

text: at

text: least

text: one

text: set

text: of

text: nodes

text: C

text: with

text: C

text: C,

text: such

text: that

text: δ(c

text: c

text: for

text: all

text: i

text: C

text: l

text: s

text: l

text: is

text: s

text: l|s

text: Let

text: C

text: denote

text: the

text: set

text: of

text: nodes

text: in

text: layer

text: s

text: that

text: contains

text: i

text: for

text: all

text: i

text: C

text: k

text: s

text: l

text: s

text: Consider

text: the

text: set

text: of

text: nodes

text: C

text: in

text: C

text: that

text: are

text: in

text: layers

text: s

text: and

text: the

text: set

text: of

text: r

text: s

text: l

text: r

text: {l

text: |T

text: nodes

text: C

text: in

text: C

text: that

text: are

text: in

text: layers

text: s

text: Because

text: δ(c

text: c

text: for

text: r>s

text: k

text: r

text: k

text: is

text: Pers(C)

text: s

text: N

text: is

text: not

text: suﬃcient

text: because

text: changing

text: Pers(C)

text: s

text: can

text: change

text: partitions

text: induced

text: on

text: individual

text: layers.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ecnetsisrep dezilamroN snoitubirtnoc reyal−artnI 35 1 0.095 0.98 0.09 0.96 0.085 0.94 0.92 0.08 0 10 20 30 40 50 0 10 20 30 40 50 ω ω (a) Persistence versus ω for (b) Intra-layer contributions γ = 1 versus ω for γ = 1 2001 Gov. 2002 10 Corp. 2003 20 30 2004 Equ. 40 2005 50 2006 Cur. 60 2007 Met. 70 Fue. 2008 80 Com. 2009 90 2010 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 10 20 30 40 50 60 70 80 90 (c) Output partition for (d) Persistence between all (e) Mean co-classiﬁcation (γ,ω) = (1,0.1) pairs of layers index between all pairs of nodes over all layers 0 0.5 1 Fig. 5.5. Numerical experiments with data set 1. We sample the set of inter-layer edge weights uniformly from the interval [0,50] with a discretization step of 0.1 (so there are 501 values of ω in (cid:80) (cid:80) (cid:104) (cid:105) (cid:80) (cid:80) total), and we use|Tth|e−uniform null network (i.e., Pijs = As ) with γ = 1. (a) The persistence normalized by N( 1) for each value of ω averaged over 20 runs of LouvainRand. (b) The |T | |T | intra-layer modularity contributions s=1 Ni,j=1 Bijsδ(cis,cjs) normalized by s=1 Ni,j=1 Aijs for each value of ω averaged over 20 runs of LouvainRand. (d) Sample output multilayer partition. Each point on the horizontal axis represents a single time window, and each position on the vertical axis is an asset. We order the assets by asset class, and the colors represent communities. (e) (cid:80) Association matrix of normalized persistence values between all pairs of layers averaged over all ∈ values of ω [0,50] in our sample and 20 runs for each value. The normalized persistence between { } a pair of layers s,r is Ni=1 δ(cis,cir)/N. (f) Association matrix indicating the co-classiﬁcation of nodes averaged over the set of partitions induced on each layer for each value of ω and 20 runs of LouvainRand. (cid:80) be larger for larger values of inter-layer coupling, and in Fig. 5.5(b) that intra-layer |T | modularity contributions (which we normalize by (1T A 1)) tend to be smaller s=1 s for larger values of inter-layer coupling. The increase of persistence and the decrease of intra-layer modularity contributions need not be monotonic, because we are a ﬁnding a set of local optima for each value of ω rather than the set of global optima. In Fig. 5.5(c), we show a sample output of the multilayer partition (which contains 35 communities). Some of the changes in community structure correspond to known events (e.g., Lehman Bankruptcy in September 2008 [marked by an increase in the size of the equity asset class]). Observe that the two largest communities are the ones that contain the government bond assets and the equity assets. In particular, the community that contains equities becomes noticeably larger in 2006 and in 2008. For larger values of the resolution parameter, this community instead becomes noticeably ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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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label: reference

text: S.

text: Gonz´alez-Bail´on,

text: J.

text: Borge-Holthoefer,

text: A.

text: Rivero,

text: and

text: Y.

text: Moreno.

text: The

text: dynamics

text: of

text: protest

text: recruitment

text: through

text: an

text: online

text: network.

text: Sci.

text: Rep.,

text: B.

text: H.

text: Good,

text: Y.-A.

text: de

text: Montjoye,

text: and

text: A.

text: Clauset.

text: Performance

text: of

text: modularity

text: maximization

text: in

text: practical

text: contexts.

text: Phys.

text: Rev.

text: E,

text: R.

text: Guimer`a,

text: M.

text: Sales-Pardo,

text: and

text: L.

text: A.

text: N.

text: Amaral.

text: Functional

text: cartography

text: of

text: complex

text: metabolic

text: networks.

text: Nature,

text: T.

text: Hoﬀmann,

text: M.

text: A.

text: Porter,

text: and

text: R.

text: Lambiotte.

text: Social

text: structure

text: of

text: Facebook

text: networks.

text: Phys.

text: Rev.

text: E,

text: P.

text: Holme

text: and

text: J.

text: Saram¨aki.

text: Temporal

text: networks.

text: Phys.

text: Rep.,

text: J.

text: Hopcroft,

text: O.

text: Khan,

text: B.

text: Kulli,

text: and

text: B.

text: Selman.

text: Tracking

text: evolving

text: communities

text: in

text: large

text: linked

text: networks.

text: Proc.

text: Natl.

text: Acad.

text: Sci.

text: U.S.A.,

text: L.

text: G.

text: S.

text: Jeub,

text: P.

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text: M.

text: A.

text: Porter,

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text: J.

text: Mucha,

text: and

text: M.

text: W.

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text: act

text: locally:

text: The

text: detection

text: of

text: small,

text: medium-sized,

text: and

text: large

text: communities

text: in

text: large

text: networks.

text: Phys.

text: Rev.

text: E,

text: in

text: press

text: I.

text: S.

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text: Mon. Not. R. Astron. Soc. 000, 1–13 (2014) Printed 27 May 2020 (MN LATEX style ﬁle v2.2) Abell 1033: birth of a radio phoenix F. de Gasperin1, G. A. Ogrean1,2, R. J. van Weeren2, W. A. Dawson3, M. Brüggen1, A. Bonafede1, A. Simionescu4 1 Universität Hamburg, Hamburger Sternwarte, Gojenbergsweg 112, D-21029, Hamburg, Germany 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 3 Lawrence Livermore National Lab, 7000 East Avenue, Livermore, CA 94550, USA 4 Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan ABSTRACT Extended steep-spectrum radio emission in a galaxy cluster is usually associated with a re- cent merger. However, given the complex scenario of galaxy cluster mergers, many of the discovered sources hardly ﬁt into the strict boundaries of a precise taxonomy. This is espe- cially true for radio phoenixes that do not have very well deﬁned observational criteria. Radio phoenixes are aged radio galaxy lobes whose emission is reactivated by compression or other mechanisms. Here, we present the detection of a radio phoenix close to the moment of its formation. The source is located in Abell 1033, a peculiar galaxy cluster which underwent a recent merger. To support our claim, we present unpublished Westerbork Synthesis Radio Telescope and Chandra observations together with archival data from the Very Large Array and the Sloan Digital Sky Survey. We discover the presence of two sub-clusters displaced along the N-S direction. The two sub-clusters probably underwent a recent merger which is the cause of a moderately perturbed X-ray brightness distribution. A steep-spectrum extended radio source very close to an AGN is proposed to be a newly born radio phoenix: the AGN lobes have been displaced/compressed by shocks formed during the merger event. This sce- nario explains the source location, morphology, spectral index, and brightness. Finally, we show evidence of a density discontinuity close to the radio phoenix and discuss the conse- quences of its presence. Key words: galaxies: clusters: individual: Abell 1033 – large-scale structure of Universe – radio continuum: general – X-rays: galaxies: clusters 1 INTRODUCTION through, for example, diﬀusive shock acceleration (Drury 1983; Blandford & Eichler 1987; Jones & Ellison 1991). Radio halos are Galaxy clusters form via a hierarchical sequence of major and diﬀuse, unpolarized, centrally located sources. They may be caused − minor mergers. A merger event dissipates up to 1063 1064 ergs into by large turbulent regions formed during cluster collisions and can heating the gas but also through large-scale intra-cluster medium reaccelerate populations of long-lived synchrotron-dark CR elec- (ICM) motions such as shocks and turbulence. These processes trons to radio bright energies (Donnert et al. 2013). These pop- are related to the formation of non-thermal components in the ulations are generated by AGN activity, shocks or supernovae in ICM: cosmic rays (CR) and magnetic ﬁelds (e.g. Brunetti & Jones galaxies (Brunetti & Jones 2014). 2014). Relativistic particles diﬀusing into magnetic ﬁelds emit syn- A third, less studied, type of emission in merging clusters are chrotron radiation. Therefore, non-thermal components in galaxy the so-called radio phoenixes (Slee et al. 2001; Ensslin & Bruggen clusters are most evident once their radio emission is traced. Cur- 2002; Kempner et al. 2004; van Weeren et al. 2009). When shocks rent observations strongly support this idea of a tight connection from a cluster merger pass through an old AGN lobe, they compress between cluster-scale synchrotron emission and mergers, ﬁnding the fossil radio plasma. Because the density in the fossil plasma indeed these radio sources always hosted in dynamically disturbed is much lower than the surrounding ICM, these shocks may be- systems (Cassano et al. 2010, 2013). come subsonic compression waves inside the plasma, as shown by Radio relics and radio halos are two types of radio sources Ensslin & Gopal-Krishna (2001). This compression is able to re- associated with non-thermal cluster components. Radio relics (or energizing the electrons in the fossil plasma so they shine again at gischt) are located in cluster outskirts. They are long (1–2 Mpc) radio wavelengths. ∼ and thin ( 100 kpc) sources. They are polarized and are likely Here, we present the detection of a radio phoenix located in signatures of electrons accelerated by large-scale accretion shocks Abell 1033, a peculiar galaxy cluster which underwent a recent (cid:13) c 2014 RAS

text: Mon.

text: Not.

text: R.

text: Astron.

text: Soc.

text: Printed

text: May

text: (MN

text: LATEX

text: style

text: ﬁle

text: Abell

text: birth

text: of

text: a

text: radio

text: phoenix

text: F.

label: author

text: de

text: G.

text: A.

text: R.

text: J.

text: van

text: W.

text: A.

text: M.

text: A.

text: A.

text: Universität

text: Hamburg,

text: Hamburger

text: Sternwarte,

text: Gojenbergsweg

text: Hamburg,

text: Germany

text: Harvard-Smithsonian

text: Center

text: for

text: Astrophysics,

text: Garden

text: Street,

text: Cambridge,

text: MA

text: USA

text: Lawrence

text: Livermore

text: National

text: Lab,

text: East

text: Avenue,

text: Livermore,

text: CA

text: USA

text: Japan

text: Aerospace

text: Exploration

text: Agency,

text: Yoshinodai,

text: Sagamihara,

text: Kanagawa

text: Japan

text: ABSTRACT

text: Extended

label: abstract

text: steep-spectrum

text: radio

text: emission

text: in

text: a

text: galaxy

text: cluster

text: is

text: usually

text: associated

text: with

text: a

text: re-

text: cent

text: merger.

text: However,

text: given

text: the

text: complex

text: scenario

text: of

text: galaxy

text: cluster

text: mergers,

text: many

text: of

text: the

text: discovered

text: sources

text: hardly

text: ﬁt

text: into

text: the

text: strict

text: boundaries

text: of

text: a

text: precise

text: taxonomy.

text: This

text: is

text: espe-

text: cially

text: true

text: for

text: radio

text: phoenixes

text: that

text: do

text: not

text: have

text: very

text: well

text: deﬁned

text: observational

text: criteria.

text: Radio

text: phoenixes

text: are

text: aged

text: radio

text: galaxy

text: lobes

text: whose

text: emission

text: is

text: reactivated

text: by

text: compression

text: or

text: other

text: mechanisms.

text: Here,

text: we

text: present

text: the

text: detection

text: of

text: a

text: radio

text: phoenix

text: close

text: to

text: the

text: moment

text: of

text: its

text: formation.

text: The

text: source

text: is

text: located

text: in

text: Abell

text: a

text: peculiar

text: galaxy

text: cluster

text: which

text: underwent

text: a

text: recent

text: merger.

text: To

text: support

text: our

text: claim,

text: we

text: present

text: unpublished

text: Westerbork

text: Synthesis

text: Radio

text: Telescope

text: and

text: Chandra

text: observations

text: together

text: with

text: archival

text: data

text: from

text: the

text: Very

text: Large

text: Array

text: and

text: the

text: Sloan

text: Digital

text: Sky

text: Survey.

text: We

text: discover

text: the

text: presence

text: of

text: two

text: sub-clusters

text: displaced

text: along

text: the

text: N-S

text: direction.

text: The

text: two

text: sub-clusters

text: probably

text: underwent

text: a

text: recent

text: merger

text: which

text: is

text: the

text: cause

text: of

text: a

text: moderately

text: perturbed

text: X-ray

text: brightness

text: distribution.

text: A

text: steep-spectrum

text: extended

text: radio

text: source

text: very

text: close

text: to

text: an

text: AGN

text: is

text: proposed

text: to

text: be

text: a

text: newly

text: born

text: radio

text: phoenix:

text: the

text: AGN

text: lobes

text: have

text: been

text: displaced/compressed

text: by

text: shocks

text: formed

text: during

text: the

text: merger

text: event.

text: This

text: sce-

text: nario

text: explains

text: the

text: source

text: location,

text: morphology,

text: spectral

text: index,

text: and

text: brightness.

text: Finally,

text: we

text: show

text: evidence

text: of

text: a

text: density

text: discontinuity

text: close

text: to

text: the

text: radio

text: phoenix

text: and

text: discuss

text: the

text: conse-

text: quences

text: of

text: its

text: presence.

text: Key

text: words:

text: galaxies:

text: clusters:

text: individual:

text: Abell

text: large-scale

text: structure

text: of

text: Universe

text: radio

text: continuum:

text: general

text: X-rays:

text: galaxies:

text: clusters

text: INTRODUCTION

text: through,

text: for

text: example,

text: diﬀusive

text: shock

text: acceleration

text: (Drury

text: Blandford

text: Eichler

text: Jones

text: Ellison

text: Radio

text: halos

text: are

text: Galaxy

text: clusters

text: form

text: via

text: a

text: hierarchical

text: sequence

text: of

text: major

text: and

text: diﬀuse,

text: unpolarized,

text: centrally

text: located

text: sources.

text: They

text: may

text: be

text: caused

text: minor

text: mergers.

text: A

text: merger

text: event

text: dissipates

text: up

text: to

text: ergs

text: into

text: by

text: large

text: turbulent

text: regions

text: formed

text: during

text: cluster

text: collisions

text: and

text: can

text: heating

text: the

text: gas

text: but

text: also

text: through

text: large-scale

text: intra-cluster

text: medium

text: reaccelerate

text: populations

text: of

text: long-lived

text: synchrotron-dark

text: CR

text: elec-

text: (ICM)

text: motions

text: such

text: as

text: shocks

text: and

text: turbulence.

text: These

text: processes

text: trons

text: to

text: radio

text: bright

text: energies

text: (Donnert

text: et

text: al.

text: These

text: pop-

text: are

text: related

text: to

text: the

text: formation

text: of

text: non-thermal

text: components

text: in

text: the

text: ulations

text: are

text: generated

text: by

text: AGN

text: activity,

text: shocks

text: or

text: supernovae

text: in

text: ICM:

text: cosmic

text: rays

text: (CR)

text: and

text: magnetic

text: ﬁelds

text: (e.g.

text: Brunetti

text: Jones

text: galaxies

text: (Brunetti

text: Jones

text: Relativistic

text: particles

text: diﬀusing

text: into

text: magnetic

text: ﬁelds

text: emit

text: syn-

text: A

text: third,

text: less

text: studied,

text: type

text: of

text: emission

text: in

text: merging

text: clusters

text: are

text: chrotron

text: radiation.

text: Therefore,

text: non-thermal

text: components

text: in

text: galaxy

text: the

text: so-called

text: radio

text: phoenixes

text: (Slee

text: et

text: al.

text: Ensslin

text: Bruggen

text: clusters

text: are

text: most

text: evident

text: once

text: their

text: radio

text: emission

text: is

text: traced.

text: Cur-

text: Kempner

text: et

text: al.

text: van

text: Weeren

text: et

text: al.

text: When

text: shocks

text: rent

text: observations

text: strongly

text: support

text: this

text: idea

text: of

text: a

text: tight

text: connection

text: from

text: a

text: cluster

text: merger

text: pass

text: through

text: an

text: old

text: AGN

text: lobe,

text: they

text: compress

text: between

text: cluster-scale

text: synchrotron

text: emission

text: and

text: mergers,

text: ﬁnding

text: the

text: fossil

text: radio

text: plasma.

text: Because

text: the

text: density

text: in

text: the

text: fossil

text: plasma

text: indeed

text: these

text: radio

text: sources

text: always

text: hosted

text: in

text: dynamically

text: disturbed

text: is

text: much

text: lower

text: than

text: the

text: surrounding

text: ICM,

text: these

text: shocks

text: may

text: be-

text: systems

text: (Cassano

text: et

text: al.

text: come

text: subsonic

text: compression

text: waves

text: inside

text: the

text: plasma,

text: as

text: shown

text: by

text: Radio

text: relics

text: and

text: radio

text: halos

text: are

text: two

text: types

text: of

text: radio

text: sources

text: Ensslin

text: Gopal-Krishna

text: This

text: compression

text: is

text: able

text: to

text: re-

text: associated

text: with

text: non-thermal

text: cluster

text: components.

text: Radio

text: relics

text: (or

text: energizing

text: the

text: electrons

text: in

text: the

text: fossil

text: plasma

text: so

text: they

text: shine

text: again

text: at

text: gischt)

text: are

text: located

text: in

text: cluster

text: outskirts.

text: They

text: are

text: long

text: Mpc)

text: radio

text: wavelengths.

text: and

text: thin

text: kpc)

text: sources.

text: They

text: are

text: polarized

text: and

text: are

text: likely

text: Here,

text: we

text: present

text: the

text: detection

text: of

text: a

text: radio

text: phoenix

text: located

text: in

text: signatures

text: of

text: electrons

text: accelerated

text: by

text: large-scale

text: accretion

text: shocks

text: Abell

text: a

text: peculiar

text: galaxy

text: cluster

text: which

text: underwent

text: a

text: recent

text: c

text: RAS


text: A PRECONDITIONER FOR A PRIMAL-DUAL NEWTON CONJUGATE GRADIENTS METHOD FOR COMPRESSED SENSING PROBLEMS December 30, 2014, revised in July 2, 2015 ∗ † ‡ IOANNIS DASSIOS AND KIMON FOUNTOULAKIS AND JACEK GONDZIO Abstract. In this paper we are concerned with the solution of Compressed Sensing (CS) problems where the signals to be recovered are sparse in coherent and redundant dictionaries. We extend the primal-dual Newton Conjugate Gradients (pdNCG) method in [11] for CS problems. We provide an inexpensive and provably eﬀective preconditioning technique for linear systems using pdNCG. Numerical results are presented on CS problems which demonstrate the performance of pdNCG with the proposed preconditioner compared to state-of-the-art existing solvers. Key words. compressed sensing, (cid:96)1-analysis, total-variation, second-order methods, Newton conjugate gradients ∈ R CS is concerned with recovering a signal x˜ n by observing a linear combination of the signal ˜b = Ax˜, where A ∈ Rm×n is an under-determined linear operator with m < n and ˜b ∈ Rm are the observed measurements. Although this system has inﬁnitely many solutions, reconstruction of x˜ is possible due to its assumed properties. In particular, x˜ is ∈ assumed to have a sparse image through a coherent and redundant dictionary W × R C ≤ ∗ En l, where E = or and n l. More precisely, W x˜, is sparse, i.e. it has only few non-zero components, where the star superscript denotes the conjugate transpose. If ∗ W x˜ is sparse, under certain conditions on matrices A and W (discussed in Subsection ) the optimal solution of the linear problem (cid:107) ∗ (cid:107) minimize W x , subject to: Ax = ˜b 1 (cid:107) · (cid:107) is x˜, where is the (cid:96) -norm. 1 1 Frequently measurements ˜b might be contaminated with noise, i.e. one measures b = ˜b + e instead, where e is a vector of noise, usually modelled as Gaussian with ∗ zero-mean and bounded Euclidean norm. In addition, in realistic applications, W x˜ might not be exactly sparse, but its mass might be concentrated only on few of its components, while the rest are rapidly decaying. In this case, (again under certain conditions on matrices A and W) the optimal solution of the following problem (cid:107) ∗ (cid:107) 1(cid:107) − (cid:107) (1.1) minimize f (x) := c W x + Ax b 2, c 1 2 2 ∗ I. DASSIOS IS WITH THE SCHOOL OF MATHEMATICS AND MAXWELL INSTITUTE, THE UNIVERSITY OF EDINBURGH, PETER GUTHRIE TAIT ROAD, EDINBURGH EH9 3FD, UNITED KINGDOM E-MAIL: IDASSIOS@ED.AC.UK. I. DASSIOS IS SUPPORTED BY EPSRC GRANT EP/I017127/1 † K. FOUNTOULAKIS IS WITH THE SCHOOL OF MATHEMATICS AND MAXWELL INSTI- TUTE, THE UNIVERSITY OF EDINBURGH, PETER GUTHRIE TAIT ROAD, EDINBURGH EH9 3FD, UNITED KINGDOM E-MAIL: K.FOUNTOULAKIS@SMS.ED.AC.UK. ‡ J. GONDZIO IS WITH THE SCHOOL OF MATHEMATICS AND MAXWELL INSTITUTE, THE UNIVERSITY OF EDINBURGH, PETER GUTHRIE TAIT ROAD, EDINBURGH EH9 3FD, UNITED KINGDOM E-MAIL: J.GONDZIO@ED.AC.UK. 1 ##LTLine## ##LTLine##

text: A

text: PRECONDITIONER

text: FOR

text: A

text: PRIMAL-DUAL

text: NEWTON

text: CONJUGATE

text: GRADIENTS

text: METHOD

text: FOR

text: COMPRESSED

text: SENSING

text: PROBLEMS

text: December

text: revised

text: in

text: July

text: IOANNIS

text: DASSIOS

text: AND

text: KIMON

text: FOUNTOULAKIS

text: AND

text: JACEK

text: GONDZIO

text: Abstract.

text: In

text: this

text: paper

text: we

text: are

text: concerned

text: with

text: the

text: solution

text: of

text: Compressed

text: Sensing

text: (CS)

text: problems

text: where

text: the

text: signals

text: to

text: be

text: recovered

text: are

text: sparse

text: in

text: coherent

text: and

text: redundant

text: dictionaries.

text: We

text: extend

text: the

text: primal-dual

text: Newton

text: Conjugate

text: Gradients

text: (pdNCG)

text: method

text: in

text: for

text: CS

text: problems.

text: We

text: provide

text: an

text: inexpensive

text: and

text: provably

text: eﬀective

text: preconditioning

text: technique

text: for

text: linear

text: systems

text: using

text: pdNCG.

text: Numerical

text: results

text: are

text: presented

text: on

text: CS

text: problems

text: which

text: demonstrate

text: the

text: performance

text: of

text: pdNCG

text: with

text: the

text: proposed

text: preconditioner

text: compared

text: to

text: state-of-the-art

text: existing

text: solvers.

text: Key

text: words.

text: compressed

text: sensing,

text: total-variation,

text: second-order

text: methods,

text: Newton

text: conjugate

text: gradients

text: R

text: CS

text: is

text: concerned

text: with

text: recovering

text: a

text: signal

text: x˜

text: n

text: by

text: observing

text: a

text: linear

text: combination

text: of

text: the

text: signal

text: ˜b

text: Ax˜,

text: where

text: A

text: Rm×n

text: is

text: an

text: under-determined

text: linear

text: operator

text: with

text: m

text: n

text: and

text: ˜b

text: Rm

text: are

text: the

text: observed

text: measurements.

text: Although

text: this

text: system

text: has

text: inﬁnitely

text: many

text: solutions,

text: reconstruction

text: of

text: x˜

text: is

text: possible

text: due

text: to

text: its

text: assumed

text: properties.

text: In

text: particular,

text: x˜

text: is

text: assumed

text: to

text: have

text: a

text: sparse

text: image

text: through

text: a

text: coherent

text: and

text: redundant

text: dictionary

text: W

text: R

text: C

text: En

text: l,

text: where

text: E

text: or

text: and

text: n

text: l.

text: More

text: precisely,

text: W

text: x˜,

text: is

text: sparse,

text: i.e.

text: it

text: has

text: only

text: few

text: non-zero

text: components,

text: where

text: the

text: star

text: superscript

text: denotes

text: the

text: conjugate

text: transpose.

text: If

text: W

text: x˜

text: is

text: sparse,

text: under

text: certain

text: conditions

text: on

text: matrices

text: A

text: and

text: W

text: (discussed

text: in

text: Subsection

text: the

text: optimal

text: solution

text: of

text: the

text: linear

text: problem

text: minimize

text: W

text: x

text: subject

text: to:

text: Ax

text: ˜b

text: is

text: x˜,

text: where

text: is

text: the

text: -norm.

text: Frequently

text: measurements

text: ˜b

text: might

text: be

text: contaminated

text: with

text: noise,

text: i.e.

text: one

text: measures

text: b

text: ˜b

text: e

text: instead,

text: where

text: e

text: is

text: a

text: vector

text: of

text: noise,

text: usually

text: modelled

text: as

text: Gaussian

text: with

text: zero-mean

text: and

text: bounded

text: Euclidean

text: norm.

text: In

text: addition,

text: in

text: realistic

text: applications,

text: W

text: x˜

text: might

text: not

text: be

text: exactly

text: sparse,

text: but

text: its

text: mass

text: might

text: be

text: concentrated

text: only

text: on

text: few

text: of

text: its

text: components,

text: while

text: the

text: rest

text: are

text: rapidly

text: decaying.

text: In

text: this

text: case,

text: (again

text: under

text: certain

text: conditions

text: on

text: matrices

text: A

text: and

text: W)

text: the

text: optimal

text: solution

text: of

text: the

text: following

text: problem

text: minimize

text: f

text: (x)

text: c

text: W

text: x

text: Ax

text: b

text: c

text: I.

text: DASSIOS

text: IS

text: WITH

text: THE

text: SCHOOL

text: OF

text: MATHEMATICS

text: AND

text: MAXWELL

text: INSTITUTE,

text: THE

text: UNIVERSITY

text: OF

text: EDINBURGH,

text: PETER

text: GUTHRIE

text: TAIT

text: ROAD,

text: EDINBURGH

text: UNITED

text: KINGDOM

text: E-MAIL:

text: IDASSIOS@ED.AC.UK.

text: I.

text: DASSIOS

text: IS

text: SUPPORTED

text: BY

text: EPSRC

text: GRANT

text: K.

text: FOUNTOULAKIS

text: IS

text: WITH

text: THE

text: SCHOOL

text: OF

text: MATHEMATICS

text: AND

text: MAXWELL

text: INSTI-

text: TUTE,

text: THE

text: UNIVERSITY

text: OF

text: EDINBURGH,

text: PETER

text: GUTHRIE

text: TAIT

text: ROAD,

text: EDINBURGH

text: UNITED

text: KINGDOM

text: E-MAIL:

text: K.FOUNTOULAKIS@SMS.ED.AC.UK.

text: J.

text: GONDZIO

text: IS

text: WITH

text: THE

text: SCHOOL

text: OF

text: MATHEMATICS

text: AND

text: MAXWELL

text: INSTITUTE,

text: THE

text: UNIVERSITY

text: OF

text: EDINBURGH,

text: PETER

text: GUTHRIE

text: TAIT

text: ROAD,

text: EDINBURGH

text: UNITED

text: KINGDOM

text: E-MAIL:

text: J.GONDZIO@ED.AC.UK.

text: ##LTLine##

text: ##LTLine##


text: matrix Bˆ have large variations and there are many small eigenvalues close to zero. Notice that the preconditioner was eﬀective not only at optimality as it was predicted by theory, but through all iterations of pdNCG. This is because starting from the ∗ ∀ zero solution the iterates W xk were maintained approximately sparse k. Additionally, in Figure 5.2 we show the number of CG/PCG iterations and time required for the unpreconditioned and the preconditioned cases of the same experi- ment. Observe in Figures 5.2a (µ = 1.0e-3) and 5.2b (µ = 1.0e-5) that the number of PCG iterations are much less than the number of CG iterations. Surprisingly, the number of CG iterations required for the experiment with µ = 1.0e-3 were more than the number of iterations for CG for the experiment with µ = 1.0e-5. Although, matrix Bˆ has worse condition number in the latter case, see the values of the vertical axis in Figures 5.1a and 5.1b. We believe that this is because of the slightly better clustering of the eigenvalues of matrix Bˆ for the experiments with µ = 1.0e-5; see Figures 5.1a and 5.1b. Finally, PCG was faster than CG in terms of required time for convergence, see Figures 5.2c and 5.2d. In this subsection we discuss how we can solve systems with the proposed pre- conditioner N˜. The simplest case is when W is an orthogonal matrix. In this case it is readily veriﬁed that solving systems with matrix N˜ costs two matrix vector products with (cid:124) matrices W, W and the inversion of a diagonal matrix. Therefore, this operation is inexpensive. Especially, when matrix W is a DCT or a Wavelets transform for which (cid:124) O O matrix-vector products with W and W can be calculated in (n log n) and (n) time, respectively. Let us now consider iTV problems. Let p be the vertical number of pixels of the v image to be reconstructed and p be the horizontal number of pixels. For simplicity h we will assume that the image is square, hence, p = p = p . Additionally, we assume v h × that the image is handled in a vectorized form, i.e., instead of an image of size p p × we have a vectorized image of size p2 1 where the columns of the image are stuck ∈ C × one after the other. In this case, for iTV the W n n matrix in problem (1.1) is − square with n = p2, rank-deﬁcient with rank(W) = n 1. Matrix W corresponds to a discretization of the nabla operator and it measures local diﬀerences of pixels when applied on a vectorized image. In particular, √ − W = W + 1W , v h ∈ R × ∈ R × where W n n and W n n. Matrix W measures vertical diﬀerences of pixels v h v when applied on a vectorized image and it has the following non-zero components: − [W ] − − = 1 and [W ] − − = 1 v p(j 1)+i,p(j 1)+i v p(j 1)+i,p(j 1)+i+1 ∀ · · · ∀ · · · − j = 1, 2, , p and i = 1, 2, , p 1. Matrix W measures horizontal diﬀerences of h pixels when applied on a vectorized image and it has the following non-zero pattern: − [W ] − − = 1 and [W ] − − = 1 h p(j 1)+i,p(j 1)+i h p(j 1)+i,p(j 1)+i+p ∀ · · · − ∀ · · · j = 1, 2, , p 1 and i = 1, 2, , p. Observe that matrix N˜ in this case is at most seven-diagonal and it has the following block tridiagonal form 14 ##LTLine##

text: matrix

text: Bˆ

text: have

text: large

text: variations

text: and

text: there

text: are

text: many

text: small

text: eigenvalues

text: close

text: to

text: zero.

text: Notice

text: that

text: the

text: preconditioner

text: was

text: eﬀective

text: not

text: only

text: at

text: optimality

text: as

text: it

text: was

text: predicted

text: by

text: theory,

text: but

text: through

text: all

text: iterations

text: of

text: pdNCG.

text: This

text: is

text: because

text: starting

text: from

text: the

text: zero

text: solution

text: the

text: iterates

text: W

text: xk

text: were

text: maintained

text: approximately

text: sparse

text: k.

text: Additionally,

text: in

text: Figure

text: we

text: show

text: the

text: number

text: of

text: CG/PCG

text: iterations

text: and

text: time

text: required

text: for

text: the

text: unpreconditioned

text: and

text: the

text: preconditioned

text: cases

text: of

text: the

text: same

text: experi-

text: ment.

text: Observe

text: in

text: Figures

text: (µ

text: and

text: (µ

text: that

text: the

text: number

text: of

text: PCG

text: iterations

text: are

text: much

text: less

text: than

text: the

text: number

text: of

text: CG

text: iterations.

text: Surprisingly,

text: the

text: number

text: of

text: CG

text: iterations

text: required

text: for

text: the

text: experiment

text: with

text: µ

text: were

text: more

text: than

text: the

text: number

text: of

text: iterations

text: for

text: CG

text: for

text: the

text: experiment

text: with

text: µ

text: Although,

text: matrix

text: Bˆ

text: has

text: worse

text: condition

text: number

text: in

text: the

text: latter

text: case,

text: see

text: the

text: values

text: of

text: the

text: vertical

text: axis

text: in

text: Figures

text: and

text: We

text: believe

text: that

text: this

text: is

text: because

text: of

text: the

text: slightly

text: better

text: clustering

text: of

text: the

text: eigenvalues

text: of

text: matrix

text: Bˆ

text: for

text: the

text: experiments

text: with

text: µ

text: see

text: Figures

text: and

text: Finally,

text: PCG

text: was

text: faster

text: than

text: CG

text: in

text: terms

text: of

text: required

text: time

text: for

text: convergence,

text: see

text: Figures

text: and

text: In

text: this

text: subsection

text: we

text: discuss

text: how

text: we

text: can

text: solve

text: systems

text: with

text: the

text: proposed

text: pre-

text: conditioner

text: N˜.

text: The

text: simplest

text: case

text: is

text: when

text: W

text: is

text: an

text: orthogonal

text: matrix.

text: In

text: this

text: case

text: it

text: is

text: readily

text: veriﬁed

text: that

text: solving

text: systems

text: with

text: matrix

text: N˜

text: costs

text: two

text: matrix

text: vector

text: products

text: with

text: matrices

text: W,

text: W

text: and

text: the

text: inversion

text: of

text: a

text: diagonal

text: matrix.

text: Therefore,

text: this

text: operation

text: is

text: inexpensive.

text: Especially,

text: when

text: matrix

text: W

text: is

text: a

text: DCT

text: or

text: a

text: Wavelets

text: transform

text: for

text: which

text: O

text: O

text: matrix-vector

text: products

text: with

text: W

text: and

text: W

text: can

text: be

text: calculated

text: in

text: (n

text: log

text: n)

text: and

text: (n)

text: time,

text: respectively.

text: Let

text: us

text: now

text: consider

text: iTV

text: problems.

text: Let

text: p

text: be

text: the

text: vertical

text: number

text: of

text: pixels

text: of

text: the

text: v

text: image

text: to

text: be

text: reconstructed

text: and

text: p

text: be

text: the

text: horizontal

text: number

text: of

text: pixels.

text: For

text: simplicity

text: h

text: we

text: will

text: assume

text: that

text: the

text: image

text: is

text: square,

text: hence,

text: p

text: p

text: p

text: Additionally,

text: we

text: assume

text: v

text: h

text: that

text: the

text: image

text: is

text: handled

text: in

text: a

text: vectorized

text: form,

text: i.e.,

text: instead

text: of

text: an

text: image

text: of

text: size

text: p

text: p

text: we

text: have

text: a

text: vectorized

text: image

text: of

text: size

text: where

text: the

text: columns

text: of

text: the

text: image

text: are

text: stuck

text: C

text: one

text: after

text: the

text: other.

text: In

text: this

text: case,

text: for

text: iTV

text: the

text: W

text: n

text: n

text: matrix

text: in

text: problem

text: is

text: square

text: with

text: n

text: rank-deﬁcient

text: with

text: rank(W)

text: n

text: Matrix

text: W

text: corresponds

text: to

text: a

text: discretization

text: of

text: the

text: nabla

text: operator

text: and

text: it

text: measures

text: local

text: diﬀerences

text: of

text: pixels

text: when

text: applied

text: on

text: a

text: vectorized

text: image.

text: In

text: particular,

text: W

text: W

text: v

text: h

text: R

text: R

text: where

text: W

text: n

text: n

text: and

text: W

text: n

text: n.

text: Matrix

text: W

text: measures

text: vertical

text: diﬀerences

text: of

text: pixels

text: v

text: h

text: v

text: when

text: applied

text: on

text: a

text: vectorized

text: image

text: and

text: it

text: has

text: the

text: following

text: non-zero

text: components:

text: [W

text: and

text: [W

text: v

text: p(j

text: v

text: p(j

text: j

text: p

text: and

text: i

text: p

text: Matrix

text: W

text: measures

text: horizontal

text: diﬀerences

text: of

text: h

text: pixels

text: when

text: applied

text: on

text: a

text: vectorized

text: image

text: and

text: it

text: has

text: the

text: following

text: non-zero

text: pattern:

text: [W

text: and

text: [W

text: h

text: p(j

text: h

text: p(j

text: j

text: p

text: and

text: i

text: p.

text: Observe

text: that

text: matrix

text: N˜

text: in

text: this

text: case

text: is

text: at

text: most

text: seven-diagonal

text: and

text: it

text: has

text: the

text: following

text: block

text: tridiagonal

text: form

text: ##LTLine##


text: (a) Unpreconditioned (b) Unpreconditioned (c) Preconditioned (d) Preconditioned − Fig. 5.1: Spectra of λ(Bˆ), λ(N˜ 1Bˆ) when pdNCG is applied with smoothing parame- ter µ = 1.0e-3 (left column of sub-ﬁgures) and µ = 1.0e-5 (right column of sub-ﬁgures). Matrix A in Bˆ is a 2D DCT, n = 210, m = n/4 and c = 2.29e-2. Seventeen systems are solved in total for each experiment.      (cid:124)  C K   1 1 (cid:124)  K C K   1 2 2 (cid:124)   K C K   2 3 3  (5.14) N˜ =  K ... ...  , 3 ... C − K(cid:124)− p 1 p 1 K − C p 1 p ∈ R × ∀ · · · ∈ R × where C p p are tridiagonal matrices i = 1, 2, , p and K p p are i ∀ − i upper bidiagonal matrices i = 1, 2, . . . , p 1. Solving systems with the symmetric O positive deﬁnite block tridiagonal matrix N˜ can be done in (p4) time by calculating its Cholesky decomposition without re-ordering. More precisely, the Cholesky factor (cid:124) L˜ of N˜ = L˜L˜ is of the form 15 ##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## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## 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text: (a)

text: Unpreconditioned

label: caption

text: (b)

text: Unpreconditioned

text: (c)

text: Preconditioned

text: (d)

text: Preconditioned

text: Fig.

text: Spectra

text: of

text: λ(Bˆ),

text: λ(N˜

text: when

text: pdNCG

text: is

text: applied

text: with

text: smoothing

text: parame-

text: ter

text: µ

text: (left

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