text: SPECTRAL DIAGONAL ENSEMBLE KALMAN FILTERS IVAN KASANICKY´∗ AND JAN MANDEL∗† AND MARTIN VEJMELKA∗ Abstract. A new type of ensemble Kalman ﬁlter is developed, which is based on replacing the sample covariance in the analysis step by its diagonal in a spectral basis. It is proved that this technique improves the aproximation of the covariance when the covariance itself is diagonal in the spectral basis, as is the case, e.g., for a second-order stationary random ﬁeld and the Fourier basis. The method is extended by wavelets to the case when the state variables are random ﬁelds which are not spatially homogeneous. Eﬃcient implementations by the fast Fourier transform (FFT) and discrete wavelet transform (DWT) are presented for several types of observations, including high-dimensional data given on a part of the domain, such as radar and satellite images. Computational experiments conﬁrm that the method performs well on the Lorenz 96 problem and the shallow water equations with very small ensembles and over multiple analysis cycles.. Data assimilation consists of incorporating new data periodically into computations in progress, which is of interest in many ﬁelds, including weather forecasting (e.g., Kalnay, 2003; Lahoz et al., 2010). One data assimilation method is ﬁltering (e.g., Anderson and Moore, 1979), which is a sequential Bayesian estimation of the state at a given time given the data received up to that time. The probability distribution of the system state is advanced in time by a computational model, while the data is assimilated by modifying the probability distribution of the state by an application the Bayes theorem, called analysis. In the methods considered here, data is assimilated in discrete time steps, called analysis cycles, and the probability distributions are represented by their mean and covariance (thus making a tacit assumption that they are at least close to gaussian). When the state covariance is given externally, bayesian estimation becomes the classical optimal statistical interpolation (OSI). The Kalman ﬁlter (KF) uses the same computation as OSI in the analysis, but it evolves the covariance matrix of the state in time along with the model state. Since the covariance matrix can be large, the KF is not suitable for high-dimensional systems. The ensemble Kalman ﬁlter (EnKF) (Evensen, 2009) replaces the state covariance by the sample covariance computed from an ensemble of simulations, which represent the state probability distribution. It can be proved that the EnKF converges to the KF in the large ensemble limit (Kwiatkowski and Mandel, 2014; Le Gland et al., 2011; Mandel et al., 2011) in the gaussian case, but an acceptable approximation may require hundreds of ensemble members (Evensen, 2009), because of spurious long-distance correlations in the sample covariance due to its low rank. Localization techniques (e.g., Anderson, 2001; Furrer and Bengtsson, 2007; Hunt et al., 2007), essentially suppress long-distance covariance terms (Sakov and Bertino, 2010), which improves EnKF performance for small ensembles. FFT EnKF (Mandel et al., 2010a,b) was proposed as an alternative approach to localization, based on replacing the sample covariance in the EnKF by its diagonal in the Fourier space. This approach is motivated by the fact that a random ﬁeld in cartesian geometry is second order station- ary (that is, the covariance between the values at two points depends only on their distance vector) if and only if its covariance in the Fourier space is diagonal (e.g., Pannekoucke et al., 2007). On a sphere, an isotropic random ﬁeld has diagonal covariance in the basis of spherical harmonics (Boer, 1983), so similar algorithms can be developed there as well. However, the stationarity assumption does not allow the covariance to vary spatially. For this reason, the FFT EnKF was extended to wavelet EnKF (Beezley et al., 2011). The use of wavelets results in an automatic localization, which varies in space adaptively. For wavelets, the eﬀect of the diagonal spectral approximation is equivalent to a weighted spatial averaging of local covariance functions (Pannekoucke et al., 2007). 1Institute of Computer Science, Academy of Sciences of the Czech Republic 2Department of Mathematical and Statistical Sciences, University of Colorado Denver 1 ##LTLine##

text: SPECTRAL

text: DIAGONAL

text: ENSEMBLE

text: KALMAN

text: FILTERS

text: IVAN

text: KASANICKY´∗

text: AND

text: JAN

text: MANDEL∗†

text: AND

text: MARTIN

text: VEJMELKA∗

text: Abstract.

text: A

text: new

text: type

text: of

text: ensemble

text: Kalman

text: ﬁlter

text: is

text: developed,

text: which

text: is

text: based

text: on

text: replacing

text: the

text: sample

text: covariance

text: in

text: the

text: analysis

text: step

text: by

text: its

text: diagonal

text: in

text: a

text: spectral

text: basis.

text: It

text: is

text: proved

text: that

text: this

text: technique

text: improves

text: the

text: aproximation

text: of

text: the

text: covariance

text: when

text: the

text: covariance

text: itself

text: is

text: diagonal

text: in

text: the

text: spectral

text: basis,

text: as

text: is

text: the

text: case,

text: e.g.,

text: for

text: a

text: second-order

text: stationary

text: random

text: ﬁeld

text: and

text: the

text: Fourier

text: basis.

text: The

text: method

text: is

text: extended

text: by

text: wavelets

text: to

text: the

text: case

text: when

text: the

text: state

text: variables

text: are

text: random

text: ﬁelds

text: which

text: are

text: not

text: spatially

text: homogeneous.

text: Eﬃcient

text: implementations

text: by

text: the

text: fast

text: Fourier

text: transform

text: (FFT)

text: and

text: discrete

text: wavelet

text: transform

text: (DWT)

text: are

text: presented

text: for

text: several

text: types

text: of

text: observations,

text: including

text: high-dimensional

text: data

text: given

text: on

text: a

text: part

text: of

text: the

text: domain,

text: such

text: as

text: radar

text: and

text: satellite

text: images.

text: Computational

text: experiments

text: conﬁrm

text: that

text: the

text: method

text: performs

text: well

text: on

text: the

text: Lorenz

text: problem

text: and

text: the

text: shallow

text: water

text: equations

text: with

text: very

text: small

text: ensembles

text: and

text: over

text: multiple

text: analysis

text: cycles..

text: Data

text: assimilation

text: consists

text: of

text: incorporating

text: new

text: data

text: periodically

text: into

text: computations

text: in

text: progress,

text: which

text: is

text: of

text: interest

text: in

text: many

text: ﬁelds,

text: including

text: weather

text: forecasting

text: (e.g.,

text: Kalnay,

text: Lahoz

text: et

text: al.,

text: One

text: data

text: assimilation

text: method

text: is

text: ﬁltering

text: (e.g.,

text: Anderson

text: and

text: Moore,

text: which

text: is

text: a

text: sequential

text: Bayesian

text: estimation

text: of

text: the

text: state

text: at

text: a

text: given

text: time

text: given

text: the

text: data

text: received

text: up

text: to

text: that

text: time.

text: The

text: probability

text: distribution

text: of

text: the

text: system

text: state

text: is

text: advanced

text: in

text: time

text: by

text: a

text: computational

text: model,

text: while

text: the

text: data

text: is

text: assimilated

text: by

text: modifying

text: the

text: probability

text: distribution

text: of

text: the

text: state

text: by

text: an

text: application

text: the

text: Bayes

text: theorem,

text: called

text: analysis.

text: In

text: the

text: methods

text: considered

text: here,

text: data

text: is

text: assimilated

text: in

text: discrete

text: time

text: steps,

text: called

text: analysis

text: cycles,

text: and

text: the

text: probability

text: distributions

text: are

text: represented

text: by

text: their

text: mean

text: and

text: covariance

text: (thus

text: making

text: a

text: tacit

text: assumption

text: that

text: they

text: are

text: at

text: least

text: close

text: to

text: gaussian).

text: When

text: the

text: state

text: covariance

text: is

text: given

text: externally,

text: bayesian

text: estimation

text: becomes

text: the

text: classical

text: optimal

text: statistical

text: interpolation

text: (OSI).

text: The

text: Kalman

text: ﬁlter

text: (KF)

text: uses

text: the

text: same

text: computation

text: as

text: OSI

text: in

text: the

text: analysis,

text: but

text: it

text: evolves

text: the

text: covariance

text: matrix

text: of

text: the

text: state

text: in

text: time

text: along

text: with

text: the

text: model

text: state.

text: Since

text: the

text: covariance

text: matrix

text: can

text: be

text: large,

text: the

text: KF

text: is

text: not

text: suitable

text: for

text: high-dimensional

text: systems.

text: The

text: ensemble

text: Kalman

text: ﬁlter

text: (EnKF)

text: (Evensen,

text: replaces

text: the

text: state

text: covariance

text: by

text: the

text: sample

text: covariance

text: computed

text: from

text: an

text: ensemble

text: of

text: simulations,

text: which

text: represent

text: the

text: state

text: probability

text: distribution.

text: It

text: can

text: be

text: proved

text: that

text: the

text: EnKF

text: converges

text: to

text: the

text: KF

text: in

text: the

text: large

text: ensemble

text: limit

text: (Kwiatkowski

text: and

text: Mandel,

text: Le

text: Gland

text: et

text: al.,

text: Mandel

text: et

text: al.,

text: in

text: the

text: gaussian

text: case,

text: but

text: an

text: acceptable

text: approximation

text: may

text: require

text: hundreds

text: of

text: ensemble

text: members

text: (Evensen,

text: because

text: of

text: spurious

text: long-distance

text: correlations

text: in

text: the

text: sample

text: covariance

text: due

text: to

text: its

text: low

text: rank.

text: Localization

text: techniques

text: (e.g.,

text: Anderson,

text: Furrer

text: and

text: Bengtsson,

text: Hunt

text: et

text: al.,

text: essentially

text: suppress

text: long-distance

text: covariance

text: terms

text: (Sakov

text: and

text: Bertino,

text: which

text: improves

text: EnKF

text: performance

text: for

text: small

text: ensembles.

text: FFT

text: EnKF

text: (Mandel

text: et

text: al.,

text: was

text: proposed

text: as

text: an

text: alternative

text: approach

text: to

text: localization,

text: based

text: on

text: replacing

text: the

text: sample

text: covariance

text: in

text: the

text: EnKF

text: by

text: its

text: diagonal

text: in

text: the

text: Fourier

text: space.

text: This

text: approach

text: is

text: motivated

text: by

text: the

text: fact

text: that

text: a

text: random

text: ﬁeld

text: in

text: cartesian

text: geometry

text: is

text: second

text: order

text: station-

text: ary

text: (that

text: is,

text: the

text: covariance

text: between

text: the

text: values

text: at

text: two

text: points

text: depends

text: only

text: on

text: their

text: distance

text: vector)

text: if

text: and

text: only

text: if

text: its

text: covariance

text: in

text: the

text: Fourier

text: space

text: is

text: diagonal

text: (e.g.,

text: Pannekoucke

text: et

text: al.,

text: On

text: a

text: sphere,

text: an

text: isotropic

text: random

text: ﬁeld

text: has

text: diagonal

text: covariance

text: in

text: the

text: basis

text: of

text: spherical

text: harmonics

text: (Boer,

text: so

text: similar

text: algorithms

text: can

text: be

text: developed

text: there

text: as

text: well.

text: However,

text: the

text: stationarity

text: assumption

text: does

text: not

text: allow

text: the

text: covariance

text: to

text: vary

text: spatially.

text: For

text: this

text: reason,

text: the

text: FFT

text: EnKF

text: was

text: extended

text: to

text: wavelet

text: EnKF

text: (Beezley

text: et

text: al.,

text: The

text: use

text: of

text: wavelets

text: results

text: in

text: an

text: automatic

text: localization,

text: which

text: varies

text: in

text: space

text: adaptively.

text: For

text: wavelets,

text: the

text: eﬀect

text: of

text: the

text: diagonal

text: spectral

text: approximation

text: is

text: equivalent

text: to

text: a

text: weighted

text: spatial

text: averaging

text: of

text: local

text: covariance

text: functions

text: (Pannekoucke

text: et

text: al.,

text: of

text: Computer

text: Science,

text: Academy

text: of

text: Sciences

text: of

text: the

text: Czech

text: Republic

text: of

text: Mathematical

text: and

text: Statistical

text: Sciences,

text: University

text: of

text: Colorado

text: Denver

text: ##LTLine##


text: Appendix A. Properties of sample covariance matrix. ∼ R C Let Uk N (0, C) be independent random vectors in n or n. For each Uk, we have the Karhunen-Lo`eve decomposition (cid:88) n ∼ Uk = λ1/2θ u , θ N(0, 1) independent. (A.1) j j,k j j,k j=1 ≥ were λ 0 are the eigenvalues and u orthonormal eigenvectors of the covariance matrix C. j ∗ j Let F = [u , . . . , u ] . By a direct computation, we have in the basis of the eigenvectors: 1 n ∼ ∗ Lemma A.1. The random vector U = FU N (0, C ), where C = FCF is a diagonal F F F matrix with λ , . . . , λ on the diagonal. 1 n Next, we use (A.1) to compute an expansion of the sample covariance entries. Lemma A.2. Let CN be the sample covariance of U1 , . . . , UN, cf., (4.1). Then, F (cid:32) F F (cid:33) (cid:88) (cid:88) (cid:88) (λ λ )1/2 N − 1 N N (CN) = i −j θ θ θ θ . (A.2) F i,j N 1 i,k j,k N i,l j,m k=1 l=1 m=1 Proof. From the deﬁnition of the sample covariance, (cid:0) (cid:1) (cid:88) (cid:16) (cid:17) (cid:16) (cid:17) N ∗ CN = 1− (cid:32)Uk − U¯ Uk(cid:33)− (cid:32)U¯ (cid:33) F i.j N 1 (cid:88) F F(cid:88)i F F j (cid:88) k=1 ∗ N N N 1 − 1 − 1 = N − 1 (cid:32)(cid:88) U(cid:16) kF (cid:17)N(cid:16) (cid:17)UlF (cid:88)UkF(cid:16) N(cid:17) (cid:88)U(cid:16)mF (cid:17) (cid:33) k=1 l=1 i m=1 j N N N 1 (cid:32) ∗ − 1 (cid:33) = N − 1 (cid:88)UkF i UkF j (cid:88)N (cid:88)UkF l UlF m k=1 l=1 m=1 (λ λ )1/2 N − 1 N N = i −j θ θ θ θ . i,k j,k i,l j,m N 1 N k=1 l=1 m=1 Finally, we use the expansion (A.2) to derive the variance of the sample covariance entries. Lemma A.3. The variance of each entry o(cid:40)f CN is F 2λ2 −i if i = j, Var[(CNF )i,j] = Nλi−λj1 if i (cid:54)= j. N 1 Proof. The sample covariance is unbiased estimate of the true covariance, so from Lemma A.2, 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Appendix

label: section

text: A.

text: Properties

text: of

text: sample

text: covariance

text: matrix.

label: paragraph

text: R

text: C

text: Let

text: Uk

text: N

text: C)

text: be

text: independent

text: random

text: vectors

text: in

text: n

text: or

text: n.

text: For

text: each

text: Uk,

text: we

text: have

text: the

text: Karhunen-Lo`eve

text: decomposition

label: equation

text: n

text: Uk

text: u

text: θ

text: independent.

text: j

text: j,k

text: j

text: j,k

text: were

text: λ

text: are

text: the

text: eigenvalues

text: and

text: u

text: orthonormal

text: eigenvectors

text: of

text: the

text: covariance

text: matrix

text: C.

text: j

text: j

text: Let

text: F

text: [u

text: u

text: By

text: a

text: direct

text: computation,

text: we

text: have

text: in

text: the

text: basis

text: of

text: the

text: eigenvectors:

text: n

text: Lemma

text: The

text: random

text: vector

text: U

text: FU

text: N

text: C

text: where

text: C

text: FCF

text: is

text: a

text: diagonal

text: F

text: F

text: F

text: matrix

text: with

text: λ

text: λ

text: on

text: the

text: diagonal.

text: n

text: Next,

text: we

text: use

text: to

text: compute

text: an

text: expansion

text: of

text: the

text: sample

text: covariance

text: entries.

text: Lemma

text: Let

text: CN

text: be

text: the

text: sample

text: covariance

text: of

text: UN,

text: cf.,

text: Then,

text: F

text: F

text: F

text: (λ

text: λ

text: N

text: N

text: N

text: (CN)

text: i

text: −j

text: θ

text: θ

text: θ

text: θ

text: F

text: i,j

text: N

text: i,k

text: j,k

text: N

text: i,l

text: j,m

text: Proof.

text: From

text: the

text: deﬁnition

text: of

text: the

text: sample

text: covariance,

text: N

text: CN

text: 1−

text: U¯

text: F

text: i.j

text: N

text: F

text: F

text: F

text: j

text: N

text: N

text: N

text: N

text: kF

text: i

text: j

text: N

text: N

text: N

text: N

text: i

text: UkF

text: j

text: l

text: UlF

text: m

text: (λ

text: λ

text: N

text: N

text: N

text: i

text: −j

text: θ

text: θ

text: θ

text: θ

text: i,k

text: j,k

text: i,l

text: j,m

text: N

text: N

text: Finally,

text: we

text: use

text: the

text: expansion

text: to

text: derive

text: the

text: variance

text: of

text: the

text: sample

text: covariance

text: entries.

text: Lemma

text: The

text: variance

text: of

text: each

text: entry

text: CN

text: is

text: F

text: −i

text: if

text: i

text: j,

text: Var[(CNF

text: )i,j]

text: if

text: i

text: j.

text: N

text: Proof.

text: The

text: sample

text: covariance

text: is

text: unbiased

text: estimate

text: of

text: the

text: true

text: covariance,

text: so

text: from

text: Lemma

text: ##LTLine##

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text: (cid:2) (cid:3) (cid:2) (cid:2) (cid:3)(cid:3) (cid:2) (cid:3)     − − Var (CN) = E (CN) E (cid:88)(CN) 2 =(cid:88)E (CN) (C ) 2 F i,i  F i,i  F i,i F i,i  F i,i 2 (λ λ )1/2 N − 1 N − = E i −i (cid:34) θ2(cid:35) (θ θ ) λ  N 1 (cid:88) i,k N i,k i,l (cid:88) i k=1 k,l=1   2 λ2 N − 2λ2 N = (N −i 1)2 E θ(cid:88)i2,k N(N −i 1)2 E (cid:34)(cid:88)θi2,kθ(cid:35)i,lθi,m k=1  k,l,m=1 2 λ2 N − 2λ2 N + i− E θ θ −i E θ2 N2 (N 1)2 (cid:88) i,k i,l (N 1) i,k  k,l=1  k=1 2λ2 N + i− E θ θ + λ2. (A.3) N (N 1) i,k i,l i k,l=1  The random variables θi,k are i.i.d., so it follows that   3 if k = l = m = n, (cid:54)   1 if k = l, m = n, k = m,  (cid:54) E [θ θ θ θ ] = 1 if k = m, l = n, k = l, i,k i,l i,m i,n (cid:54) 1 if k = n, l = m, k = l, 0 otherwise, (cid:34) (cid:35) and we can com(cid:88)pute all the(cid:88)expe(cid:2)cted(cid:3)val(cid:88)ues in(cid:88)Eq. (A(cid:2) .3), (cid:3) 2 N N N N −  E θ2  = E θ4 + E θ2 θ2 = 3N + N(N 1) = N(N + 2), (cid:88) i,k (cid:88) (cid:2) i,k(cid:3) (cid:88) (cid:54) (cid:2) i,l (cid:3)i,k  k=1  k=1 k=1 l=1,l=k N N N − E  θ2 θ θ  = E θ4 + E θ2 θ2 = 3N + N(N 1) = N(N + 2), i,k i,l i,m i,k i,k i,l (cid:88) (cid:88) (cid:54) (cid:88) (cid:2) (cid:3) (cid:88) (cid:2) (cid:3) k,l,m=1  k=1 k,l=1,l=k 2 N N N N E (cid:34)θ θ (cid:35) = E [θ θ θ θ ] = E θ4 + 3 E θ2 θ2 = 3N2, (cid:88)i,k i,l (cid:88) (cid:2) (cid:3) i,k i,l i,m i,n i,k i,k i,l (cid:54) k,l=1 k,l,m,n=1 k=1 k,l=1,l=k N N   E θ2 = E θ2 = N, (cid:88) i,k (cid:88) (cid:2) i,k(cid:3)  k=1  k=1 N N E θ θ = E θ2 = N. i,k i,l i,k k,l=1 k=1 (cid:18) (cid:19) (cid:2) (cid:3) Together, we get N(N + 2) − 2(N + 2) 3 − 2N 2 2λ2 Var (CN) = λ2 − − + − − + − + 1 = −i . F i,i i (N 1)2 (N 1)2 (N 1)2 N 1 N 1 N 1 (cid:54) The variance of the oﬀ-diagonal entry (CN) , i = j, is F i,j 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Var

text: (CN)

text: E

text: (CN)

text: E

text: (CN)

text: (C

text: F

text: i,i

text: F

text: i,i

text: F

text: i,i

text: F

text: i,i

text: F

text: i,i

text: (λ

text: λ

text: N

text: N

text: E

text: i

text: −i

text: (θ

text: θ

text: λ

text: N

text: i,k

text: N

text: i,k

text: i,l

text: i

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text: N

text: N

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text: N(N

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text: variables

text: θi,k

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text: i.i.d.,

text: so

text: it

text: follows

text: that

text: if

text: k

text: l

text: m

text: n,

text: if

text: k

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text: E

text: [θ

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text: θ

text: θ

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text: k

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text: i,n

text: if

text: k

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text: l

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text: k

text: l,

text: otherwise,

text: and

text: we

text: can

text: all

text: N

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text: N

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text: E

text: θ2

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text: θ4

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text: θ2

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text: N(N

text: N(N

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text: θ

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text: [θ

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text: θ4

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text: θ2

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text: i,l

text: i,k

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text: N.

text: i,k

text: i,l

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text: Together,

text: we

text: get

text: N(N

text: Var

text: (CN)

text: λ2

text: −i

text: F

text: i,i

text: i

text: (N

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text: N

text: The

text: variance

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text: the

text: oﬀ-diagonal

text: entry

text: (CN)

text: i

text: j,

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text: F

text: i,j

text: ##LTLine##

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text: 8 THOMAS KRA¨MER ⊂ corresponding line L V is identiﬁed with the projective tangent space to the p Fano surface at that point: (cid:31) (cid:127) (cid:47)(cid:47) (cid:31) (cid:127) (cid:47)(cid:47) P ∗ L V Ω p P (cid:31) (cid:127) (cid:47)(cid:47) P T (S) T (A) p p ∈ P (cid:8) ⊂ P ∗ (cid:9) Identifying points H Ω with hyperplanes H Ω in the dual projective space, we obtain that P ∈ × P | ⊂ ∩ Λ = (p, H) S Ω L H V . S p ∈ P So the ﬁbre of the Gauss map γ over a general point H Ω is identiﬁed with S ∩ the 27 lines on the smooth cubic surface H V , and the monodromy group M is S the group of permutations of these 27 lines which is induced by variations of the ∈ P hyperplane H Ω. By [22, VI.20] this group is W(E ), so (2) follows. 6 ⊂ For part (3) we take a translate of the theta divisor Θ A which is stable under − −→ the involution ι = id : A A. The projectivized Gauss map factors over the quotient by this involution: P α (cid:47)(cid:47) (cid:104) (cid:105) β (cid:47)(cid:47) P Λ = Θ Θ/ ι Ω. Θ ∈ P From [10, sect. 1] we know that over a general point H Ω in the branch locus of − − the generically ﬁnite map β, the ﬁbre β 1(H) will consist of precisely r 1 distinct points where r = deg(β) = g!/2. Furthermore α is ´etale over the complement of the (cid:104) (cid:105) ◦ P → P ﬁnitely many singular points of Θ/ ι . Restricting β α to a general line 1 (cid:44) Ω we get by [14, th. 1.1] a branched cover X = Θ ×P P1 α(cid:48) (cid:47)(cid:47) X/(cid:104)ι(cid:105) β(cid:48) (cid:47)(cid:47) P1 Ω (cid:48) ◦ (cid:48) of irreducible curves such that the monodromy group of the cover α β coincides ◦ with the one of α β (the inclusion of a general line induces an epimorphism on the fundamental groups of the complements of the branch loci). For g > 2 we may (cid:48) (cid:48) assume that the double cover α is ´etale and that β is only simply ramiﬁed, and our claim follows from [5, th. 1] after passing to the normalizations of the respective (cid:3) curves. Note that the situation for g = 2 is diﬀerent but covered by (1). References 1. Andreev, E. M., Vinberg, E. B. and Elashvili, A. G., Orbits of greatest dimension in semi- simple linear Lie groups, Funct. Anal. Appl. 1 (1967), 257–261. 2. Arbarello, E., Cornalba, M., Griﬃths, P. A. and Harris, J., Geometry of algebraic curves, Grundlehren Math. Wiss., vol. 267, Springer Verlag, 1985. 3. Beauville, A., Les singularit´es du diviseur Θ de la jacobienne intermediaire de l’hypersurface P cubique dans 4, Algebraic threefolds (Varenna 1981), Lecture Notes in Math., vol. 947, Springer Verlag, 1982, pp. 190–208. 4. Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux Pervers, Ast´erisque 100 (1982). 5. Biggers, R. and Fried, M., Irreducibility of moduli spaces of cyclic unramiﬁed covers of genus g curves, Trans. Amer. Math. Soc. 295 (1986), 59–70. 6. Bourbaki, N., Groupes et alg`ebres de Lie — Chapitres 4 a` 6, Springer Verlag, 2009, Reprint of the ﬁrst edition from 1968. 7. Casalaina-Martin, S., Cubic threefolds and abelian varieties of dimension ﬁve, II, Math. Z. 260 (2008), 115–125. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THOMAS

text: KRA¨MER

text: corresponding

text: line

text: L

text: V

text: is

text: identiﬁed

text: with

text: the

text: projective

text: tangent

text: space

text: to

text: the

text: p

text: Fano

text: surface

text: at

text: that

text: point:

font: default

text: P

text: L

text: V

text: Ω

text: p

text: P

text: P

text: T

text: (S)

text: T

text: (A)

text: p

text: p

text: P

text: P

text: Identifying

text: points

text: H

text: Ω

text: with

text: hyperplanes

text: H

text: Ω

text: in

text: the

text: dual

text: projective

text: space,

text: we

text: obtain

text: that

text: P

text: P

text: Λ

text: (p,

text: H)

text: S

text: Ω

text: L

text: H

text: V

text: S

text: p

text: P

text: So

text: the

text: ﬁbre

text: of

text: the

text: Gauss

text: map

text: γ

text: over

text: a

text: general

text: point

text: H

text: Ω

text: is

text: identiﬁed

text: with

text: S

text: the

text: lines

text: on

text: the

text: smooth

text: cubic

text: surface

text: H

text: V

text: and

text: the

text: monodromy

text: group

text: M

text: is

text: S

text: the

text: group

text: of

text: permutations

text: of

text: these

text: lines

text: which

text: is

text: induced

text: by

text: variations

text: of

text: the

text: P

text: hyperplane

text: H

text: Ω.

text: By

text: this

text: group

text: is

text: W(E

text: so

text: follows.

text: For

text: part

text: we

text: take

text: a

text: translate

text: of

text: the

text: theta

text: divisor

text: Θ

text: A

text: which

text: is

text: stable

text: under

text: the

text: involution

text: ι

text: id

text: A

text: A.

text: The

text: projectivized

text: Gauss

text: map

text: factors

text: over

text: the

text: quotient

text: by

text: this

text: involution:

text: P

text: α

text: β

text: P

text: Λ

text: Θ

text: Θ/

text: ι

text: Ω.

text: Θ

text: P

text: From

text: sect.

text: we

text: know

text: that

text: over

text: a

text: general

text: point

text: H

text: Ω

text: in

text: the

text: branch

text: locus

text: of

text: the

text: generically

text: ﬁnite

text: map

text: β,

text: the

text: ﬁbre

text: β

text: will

text: consist

text: of

text: precisely

text: r

text: distinct

text: points

text: where

text: r

text: deg(β)

text: Furthermore

text: α

text: is

text: ´etale

text: over

text: the

text: complement

text: of

text: the

text: P

text: P

text: ﬁnitely

text: many

text: singular

text: points

text: of

text: Θ/

text: ι

text: Restricting

text: β

text: α

text: to

text: a

text: general

text: line

text: Ω

text: we

text: get

text: by

text: th.

text: a

text: branched

text: cover

text: X

text: Θ

text: ×P

text: Ω

text: of

text: irreducible

text: curves

text: such

text: that

text: the

text: monodromy

text: group

text: of

text: the

text: cover

text: α

text: β

text: coincides

text: with

text: the

text: one

text: of

text: α

text: β

text: (the

text: inclusion

text: of

text: a

text: general

text: line

text: induces

text: an

text: epimorphism

text: on

text: the

text: fundamental

text: groups

text: of

text: the

text: complements

text: of

text: the

text: branch

text: loci).

text: For

text: g

text: we

text: may

text: assume

text: that

text: the

text: double

text: cover

text: α

text: is

text: ´etale

text: and

text: that

text: β

text: is

text: only

text: simply

text: ramiﬁed,

text: and

text: our

text: claim

text: follows

text: from

text: th.

text: after

text: passing

text: to

text: the

text: normalizations

text: of

text: the

text: respective

text: curves.

text: Note

text: that

text: the

text: situation

text: for

text: g

text: is

text: diﬀerent

text: but

text: covered

text: by

text: References

text: Andreev,

text: E.

text: M.,

text: Vinberg,

text: E.

text: B.

text: and

text: Elashvili,

text: A.

text: G.,

text: Orbits

text: of

text: greatest

text: dimension

text: in

text: semi-

text: simple

text: linear

text: Lie

text: groups,

text: Funct.

text: Anal.

text: Appl.

text: Arbarello,

text: E.,

text: Cornalba,

text: M.,

text: Griﬃths,

text: P.

text: A.

text: and

text: Harris,

text: J.,

text: Geometry

text: of

text: algebraic

text: curves,

text: Grundlehren

text: Math.

text: Wiss.,

text: vol.

text: Springer

text: Verlag,

text: Beauville,

text: A.,

text: Les

text: singularit´es

text: du

text: diviseur

text: Θ

text: de

text: la

text: jacobienne

text: intermediaire

text: de

text: l’hypersurface

text: P

text: cubique

text: dans

text: Algebraic

text: threefolds

text: (Varenna

text: Lecture

text: Notes

text: in

text: Math.,

text: vol.

text: Springer

text: Verlag,

text: pp.

text: Beilinson,

text: A.,

text: Bernstein,

text: J.

text: and

text: Deligne,

text: P.,

text: Faisceaux

text: Pervers,

text: Ast´erisque

text: Biggers,

text: R.

text: and

text: Fried,

text: M.,

text: Irreducibility

text: of

text: moduli

text: spaces

text: of

text: cyclic

text: unramiﬁed

text: covers

text: of

text: genus

text: g

text: curves,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: Bourbaki,

text: N.,

text: Groupes

text: et

text: alg`ebres

text: de

text: Lie

text: Chapitres

text: a`

text: Springer

text: Verlag,

text: Reprint

text: of

text: the

text: ﬁrst

text: edition

text: from

text: Casalaina-Martin,

text: S.,

text: Cubic

text: threefolds

text: and

text: abelian

text: varieties

text: of

text: dimension

text: ﬁve,

text: II,

text: Math.

text: Z.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 Q. Zhao, C. Abeygunawardana, P. Talalay, and A. S. M. Ceriotti and D. E. Manolopoulos, Eﬃcient First- Mildvan, Nmr evidence for the participation of a low- Principles Calculation of the Quantum Kinetic Energy barrier hydrogen bond in the mechanism of delta 5-3- and Momentum Distribution of Nuclei, noop Phys. Rev. ketosteroid isomerase, Proc. Natl. Acad. Sci. U.S.A. Lett. 109, 100604 (2012). 93, 8220–8224 (1996). I. S. Uﬁmtsev and T. J. Martinez, Quantum chemistry M. Wolfsberg, W. Van Hook, P. Paneth, and L. P. N. on graphical processing units. 3. analytical energy gradi- Rebelo, Isotope Eﬀects: in the Chemical, Geological, and ents, geometry optimization, and ﬁrst principles molecu- Bio Sciences (Springer, Dordrecht, 2009). lar dynamics, Journal of Chemical Theory and Compu- tation, J. Chem. Theory Comput. 5, 2619–2628 (2009). M. Ceriotti and T. E. Markland, Eﬃcient methods and C. M. Isborn, A. W. Gotz, M. A. Clark, R. C. Walker, practical guidelines for simulating isotope eﬀects. noop and T. J. Martinez, Electronic absorption spectra from J. Chem. Phys. 138, 014112 (2013). mm and ab initio qm/mm molecular dynamics: Environ- O. Marsalek, P.-Y. Chen, R. Dupuis, M. Benoit, M. Meheut, mental eﬀects on the absorption spectrum of photoactive Z. Bacic, and M. E. Tuckerman, Eﬃcient calculation of yellow protein, Journal of Chemical Theory and Compu- free energy diﬀerences associated with isotopic substitu- tation, J. Chem. Theory Comput. 8, 5092–5106 (2012). tion using path-integral molecular dynamics, Journal of L. Wang, M. Ceriotti, and T. E. Markland, Quantum Chemical Theory and Computation, J. Chem. Theory ﬂuctuations and isotope eﬀects in ab initio descriptions Comput. 10, 1440–1453 (2014). of water, J. Chem. Phys. 141, 104502 (2014). A. Warshel and M. Levitt, Theoretical studies of enzymic M. Parrinello and A. 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text: Q.

text: Zhao,

text: C.

text: Abeygunawardana,

text: P.

text: Talalay,

text: and

text: A.

text: S.

text: M.

text: Ceriotti

text: and

text: D.

text: E.

text: Manolopoulos,

text: Eﬃcient

text: First-

text: Mildvan,

text: Nmr

text: evidence

text: for

text: the

text: participation

text: of

text: a

text: low-

text: Principles

text: Calculation

text: of

text: the

text: Quantum

text: Kinetic

text: Energy

text: barrier

text: hydrogen

text: bond

text: in

text: the

text: mechanism

text: of

text: delta

text: and

text: Momentum

text: Distribution

text: of

text: Nuclei,

text: noop

text: Phys.

text: Rev.

text: ketosteroid

text: isomerase,

text: Proc.

text: Natl.

text: Acad.

text: Sci.

text: U.S.A.

text: Lett.

text: I.

text: S.

text: Uﬁmtsev

text: and

text: T.

text: J.

text: Martinez,

text: Quantum

text: chemistry

text: M.

text: Wolfsberg,

text: W.

text: Van

text: Hook,

text: P.

text: Paneth,

text: and

text: L.

text: P.

text: N.

text: on

text: graphical

text: processing

text: units.

text: analytical

text: energy

text: gradi-

text: Rebelo,

text: Isotope

text: Eﬀects:

text: in

text: the

text: Chemical,

text: Geological,

text: and

text: ents,

text: geometry

text: optimization,

text: and

text: ﬁrst

text: principles

text: molecu-

text: Bio

text: Sciences

text: (Springer,

text: Dordrecht,

text: lar

text: dynamics,

text: Journal

text: of

text: Chemical

text: Theory

text: and

text: Compu-

text: tation,

text: J.

text: Chem.

text: Theory

text: Comput.

text: M.

text: Ceriotti

text: and

text: T.

text: E.

text: Markland,

text: Eﬃcient

text: methods

text: and

text: C.

text: M.

text: Isborn,

text: A.

text: W.

text: Gotz,

text: M.

text: A.

text: Clark,

text: R.

text: C.

text: Walker,

text: practical

text: guidelines

text: for

text: simulating

text: isotope

text: eﬀects.

text: noop

text: and

text: T.

text: J.

text: Martinez,

text: Electronic

text: absorption

text: spectra

text: from

text: J.

text: Chem.

text: Phys.

text: mm

text: and

text: ab

text: initio

text: qm/mm

text: molecular

text: dynamics:

text: Environ-

text: O.

text: Marsalek,

text: P.-Y.

text: Chen,

text: R.

text: Dupuis,

text: M.

text: Benoit,

text: M.

text: Meheut,

text: mental

text: eﬀects

text: on

text: the

text: absorption

text: spectrum

text: of

text: photoactive

text: Z.

text: Bacic,

text: and

text: M.

text: E.

text: Tuckerman,

text: Eﬃcient

text: calculation

text: of

text: yellow

text: protein,

text: Journal

text: of

text: Chemical

text: Theory

text: and

text: Compu-

text: free

text: energy

text: diﬀerences

text: associated

text: with

text: isotopic

text: substitu-

text: tation,

text: J.

text: Chem.

text: Theory

text: Comput.

text: tion

text: using

text: path-integral

text: molecular

text: dynamics,

text: Journal

text: of

text: L.

text: Wang,

text: M.

text: Ceriotti,

text: and

text: T.

text: E.

text: Markland,

text: Quantum

text: Chemical

text: Theory

text: and

text: Computation,

text: J.

text: Chem.

text: Theory

text: ﬂuctuations

text: and

text: isotope

text: eﬀects

text: in

text: ab

text: initio

text: descriptions

text: Comput.

text: of

text: water,

text: J.

text: Chem.

text: Phys.

text: A.

text: Warshel

text: and

text: M.

text: Levitt,

text: Theoretical

text: studies

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text: APS/123-QED Kondo lattice and antiferromagnetic behavior in quaternary CeTAl Si (T = Rh, Ir) single crystals 4 2 ∗ Arvind Maurya, R. Kulkarni, A. Thamizhavel and S. K. Dhar Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India. D. Paudyal The Ames Laboratory, U.S. Department of Energy, Iowa State University, Ames, Iowa 50011-3020, USA. (Dated: May 27, 2020) We report the synthesis and the magnetic properties of single crystalline CeRhAl Si and 4 2 CeIrAl Si and their non magnetic La-analogs. The single crystals of these quaternary compounds 4 2 were grown using Al-Si binary eutectic as ﬂux. The anisotropic magnetic properties of the cerium compounds were explored in detail by means of magnetic susceptibility, isothermal magnetization, electrical resistivity at ambient and applied pressures up to 12.6 kbar, magnetoresistivity and heat capacity measurements. Both CeRhAl Si and CeIrAl Si undergo two antiferromagnetic transi- 4 2 4 2 tions, ﬁrst from the paramagnetic to an antiferromagnetic state at T = 12.6 K and 15.5 K, followed N1 by a second transition at lower temperatures T = 9.4 K and 13.8 K (inferred from the peaks in N2 the heat capacity), respectively, in conformity with an earlier report in the literature. The para- magnetic susceptibility is highly anisotropic and its temperature dependence in the magnetically ordered state suggests the c-axis to be the relatively easy axis of magnetization. Concomitantly, isothermal magnetization at 2 K along the c-axis shows a sharp spin-ﬂop transition accompanied by a sizeable hysteresis, while it varies nearly linearly with ﬁeld along the [100] direction up to the highest ﬁeld 14 T of our measurement. The electrical resistivity provides evidence of the Kondo − interaction in both compounds, inferred from its lnT behavior in the paramagnetic region and the decrease of magnetic transition temperature with pressure. The heat capacity data conﬁrm the bulk nature of the two magnetic transitions in each compound, and further support the presence of Kondo interaction by a reduced value of the entropy associated with the magnetic ordering. From the heat capacity data below 1 K, the coeﬃcient of the linear term in the electronic heat capacity, γ, is inferred to be 195.6 and 49.4 mJ/mol K2 in CeRhAl Si and CeIrAl Si , respectively classifying 4 2 4 2 these materials as moderate heavy fermion compounds. The main features of the magnetoresis- tivity measured at a particular temperature correlate nicely with the isothermal magnetization at the same temperature in these two isostructural compounds. We have also carried out an analysis of the magnetization based on the point charge crystal electric ﬁeld model and derived the crystal electric ﬁeld energy levels which reproduce fairly well the peak seen in the Schottky heat capacity in the paramagnetic region. Further, we have also performed electronic structure calculations using (LSDA +U) approach, which provide physical insights on the observed magnetic behaviour of these two compounds. PACS numbers: 81.10.Fq, 75.50.Ee, 75.30.Kz, 75.10.Dg, 71.70.Ch, 75.50.Ee I. INTRODUCTION 1:1:4:2 compounds, with a local fourfold axial (4/mmm) symmetry at the Eu site. These are a new addition to several quaternary rare earth-based compounds al- Recently, the synthesis and magnetic properties of ready known in the literature with 1:1:4:2 stoichiome- quaternary EuTAl Si (T = Rh and Ir) single crystals, 4 2 try, which have been grown using aluminum as ﬂux; for using the Al-Si binary eutectic as ﬂux have been re- example, RNiAl (Ni Si − ), EuNiAl Si , RNiAl Ge , ported1,2. The two Eu compounds initially order into 4 x 2 x 4 2 4 2 RAuAl Ge and RAuAl (Au Ge − ) where R is a rare 4 2 4 x 1 x 2 an incommensurate amplitude modulated antiferromag- earth metal3,4. While most of the these compounds netic state at T = 11.7 and 14.7 K respectively, fol- N1 adopt the rhombohedral YNiAl Ge -type structure5, 4 2 lowed by a second transition to an equal moment state the phases RAuAl (Au Ge − ) and EuAu Al Ge 4 x 1 x 2 1.95 4 1.05 at lower temperature T . Though these two com- N2 crystallize in the KCu S -type structure5. Both struc- 4 3 pounds prima-facie are antiferromagnetic, the isother- ture types are characterized by the slabs of “AuAl X 4 2 mal magnetization curves at low temperatures (below (X = Si or Ge)” or “AuAl (Au Ge − ) ” stacked along 4 x 1 x 2 T ) show a hysteresis right near the origin with a rem- N2 the c-axis with layers of R atoms in between. The Ce nance; unlike any other antiferromagnetic material. The atoms in both CeAuAl Ge and CeAuAl (Au Ge − ) 4 2 4 x 1 x 2 EuTAl Si compounds adopt an ordered derivative of the 4 2 (x = 0.4) have been reported to be in a valence ﬂuc- ternary KCu S -type tetragonal, tP8, P4/mmm struc- 4 3 tuating state3. This suggests the possibility of strong ture, which leads to quaternary and truly stoichiometric

text: Kondo

label: title

text: lattice

text: and

text: antiferromagnetic

text: behavior

text: in

text: quaternary

text: CeTAl

text: Si

text: (T

text: Rh,

text: Ir)

text: single

text: crystals

text: Arvind

label: author

text: Maurya,

text: R.

text: Kulkarni,

text: A.

text: Thamizhavel

text: and

text: S.

text: K.

text: Dhar

text: Department

text: of

text: Condensed

text: Matter

text: Physics

text: and

text: Materials

text: Science,

text: Tata

text: Institute

text: of

text: Fundamental

text: Research,

text: Homi

text: Bhabha

text: Road,

text: Colaba,

text: Mumbai

text: India.

text: D.

text: Paudyal

text: The

text: Ames

text: Laboratory,

text: U.S.

text: Department

text: of

text: Energy,

text: Iowa

text: State

text: University,

text: Ames,

text: Iowa

text: USA.

text: (Dated:

text: May

label: date

text: We

label: abstract

text: report

text: the

text: synthesis

text: and

text: the

text: magnetic

text: properties

text: of

text: single

text: crystalline

text: CeRhAl

text: Si

text: and

text: CeIrAl

text: Si

text: and

text: their

text: non

text: magnetic

text: La-analogs.

text: The

text: single

text: crystals

text: of

text: these

text: quaternary

text: compounds

text: were

text: grown

text: using

text: Al-Si

text: binary

text: eutectic

text: as

text: ﬂux.

text: The

text: anisotropic

text: magnetic

text: properties

text: of

text: the

text: cerium

text: compounds

text: were

text: explored

text: in

text: detail

text: by

text: means

text: of

text: magnetic

text: susceptibility,

text: isothermal

text: magnetization,

text: electrical

text: resistivity

text: at

text: ambient

text: and

text: applied

text: pressures

text: up

text: to

text: kbar,

text: magnetoresistivity

text: and

text: heat

text: capacity

text: measurements.

text: Both

text: CeRhAl

text: Si

text: and

text: CeIrAl

text: Si

text: undergo

text: two

text: antiferromagnetic

text: transi-

text: tions,

text: ﬁrst

text: from

text: the

text: paramagnetic

text: to

text: an

text: antiferromagnetic

text: state

text: at

text: T

text: K

text: and

text: K,

text: followed

text: by

text: a

text: second

text: transition

text: at

text: lower

text: temperatures

text: T

text: K

text: and

text: K

text: (inferred

text: from

text: the

text: peaks

text: in

text: the

text: heat

text: capacity),

text: respectively,

text: in

text: conformity

text: with

text: an

text: earlier

text: report

text: in

text: the

text: literature.

text: The

text: para-

text: magnetic

text: susceptibility

text: is

text: highly

text: anisotropic

text: and

text: its

text: temperature

text: dependence

text: in

text: the

text: magnetically

text: ordered

text: state

text: suggests

text: the

text: c-axis

text: to

text: be

text: the

text: relatively

text: easy

text: axis

text: of

text: magnetization.

text: Concomitantly,

text: isothermal

text: magnetization

text: at

text: K

text: along

text: the

text: c-axis

text: shows

text: a

text: sharp

text: spin-ﬂop

text: transition

text: accompanied

text: by

text: a

text: sizeable

text: hysteresis,

text: while

text: it

text: varies

text: nearly

text: linearly

text: with

text: ﬁeld

text: along

text: the

text: direction

text: up

text: to

text: the

text: highest

text: ﬁeld

text: T

text: of

text: our

text: measurement.

text: The

text: electrical

text: resistivity

text: provides

text: evidence

text: of

text: the

text: Kondo

text: interaction

text: in

text: both

text: compounds,

text: inferred

text: from

text: its

text: lnT

text: behavior

text: in

text: the

text: paramagnetic

text: region

text: and

text: the

text: decrease

text: of

text: magnetic

text: transition

text: temperature

text: with

text: pressure.

text: The

text: heat

text: capacity

text: data

text: conﬁrm

text: the

text: bulk

text: nature

text: of

text: the

text: two

text: magnetic

text: transitions

text: in

text: each

text: compound,

text: and

text: further

text: support

text: the

text: presence

text: of

text: Kondo

text: interaction

text: by

text: a

text: reduced

text: value

text: of

text: the

text: entropy

text: associated

text: with

text: the

text: magnetic

text: ordering.

text: From

text: the

text: heat

text: capacity

text: data

text: below

text: K,

text: the

text: coeﬃcient

text: of

text: the

text: linear

text: term

text: in

text: the

text: electronic

text: heat

text: capacity,

text: γ,

text: is

text: inferred

text: to

text: be

text: and

text: mJ/mol

text: in

text: CeRhAl

text: Si

text: and

text: CeIrAl

text: Si

text: respectively

text: classifying

text: these

text: materials

text: as

text: moderate

text: heavy

text: fermion

text: compounds.

text: The

text: main

text: features

text: of

text: the

text: magnetoresis-

text: tivity

text: measured

text: at

text: a

text: particular

text: temperature

text: correlate

text: nicely

text: with

text: the

text: isothermal

text: magnetization

text: at

text: the

text: same

text: temperature

text: in

text: these

text: two

text: isostructural

text: compounds.

text: We

text: have

text: also

text: carried

text: out

text: an

text: analysis

text: of

text: the

text: magnetization

text: based

text: on

text: the

text: point

text: charge

text: crystal

text: electric

text: ﬁeld

text: model

text: and

text: derived

text: the

text: crystal

text: electric

text: ﬁeld

text: energy

text: levels

text: which

text: reproduce

text: fairly

text: well

text: the

text: peak

text: seen

text: in

text: the

text: Schottky

text: heat

text: capacity

text: in

text: the

text: paramagnetic

text: region.

text: Further,

text: we

text: have

text: also

text: performed

text: electronic

text: structure

text: calculations

text: using

text: (LSDA

text: +U)

text: approach,

text: which

text: provide

text: physical

text: insights

text: on

text: the

text: observed

text: magnetic

text: behaviour

text: of

text: these

text: two

text: compounds.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: compounds,

text: with

text: a

text: local

text: fourfold

text: axial

text: symmetry

text: at

text: the

text: Eu

text: site.

text: These

text: are

text: a

text: new

text: addition

text: to

text: several

text: quaternary

text: rare

text: earth-based

text: compounds

text: al-

text: Recently,

text: the

text: synthesis

text: and

text: magnetic

text: properties

text: of

text: ready

text: known

text: in

text: the

text: literature

text: with

text: stoichiome-

text: quaternary

text: EuTAl

text: Si

text: (T

text: Rh

text: and

text: Ir)

text: single

text: crystals,

text: try,

text: which

text: have

text: been

text: grown

text: using

text: aluminum

text: as

text: ﬂux;

text: for

text: using

text: the

text: Al-Si

text: binary

text: eutectic

text: as

text: ﬂux

text: have

text: been

text: re-

text: example,

text: RNiAl

text: (Ni

text: Si

text: EuNiAl

text: Si

text: RNiAl

text: Ge

text: The

text: two

text: Eu

text: compounds

text: initially

text: order

text: into

text: x

text: x

text: RAuAl

text: Ge

text: and

text: RAuAl

text: (Au

text: Ge

text: where

text: R

text: is

text: a

text: rare

text: x

text: x

text: an

text: incommensurate

text: amplitude

text: modulated

text: antiferromag-

text: earth

text: While

text: most

text: of

text: the

text: these

text: compounds

text: netic

text: state

text: at

text: T

text: and

text: K

text: respectively,

text: fol-

text: adopt

text: the

text: rhombohedral

text: YNiAl

text: Ge

text: -type

text: lowed

text: by

text: a

text: second

text: transition

text: to

text: an

text: equal

text: moment

text: state

text: the

text: phases

text: RAuAl

text: (Au

text: Ge

text: and

text: EuAu

text: Al

text: Ge

text: x

text: x

text: at

text: lower

text: temperature

text: T

text: Though

text: these

text: two

text: com-

text: crystallize

text: in

text: the

text: KCu

text: S

text: -type

text: Both

text: struc-

text: pounds

text: prima-facie

text: are

text: antiferromagnetic,

text: the

text: isother-

text: ture

text: types

text: are

text: characterized

text: by

text: the

text: slabs

text: of

text: “AuAl

text: X

text: mal

text: magnetization

text: curves

text: at

text: low

text: temperatures

text: (below

text: (X

text: Si

text: or

text: Ge)”

text: or

text: “AuAl

text: (Au

text: Ge

text: stacked

text: along

text: x

text: x

text: T

text: show

text: a

text: hysteresis

text: right

text: near

text: the

text: origin

text: with

text: a

text: rem-

text: the

text: c-axis

text: with

text: layers

text: of

text: R

text: atoms

text: in

text: between.

text: The

text: Ce

text: nance;

text: unlike

text: any

text: other

text: antiferromagnetic

text: material.

text: The

text: atoms

text: in

text: both

text: CeAuAl

text: Ge

text: and

text: CeAuAl

text: (Au

text: Ge

text: x

text: x

text: EuTAl

text: Si

text: compounds

text: adopt

text: an

text: ordered

text: derivative

text: of

text: the

text: (x

text: have

text: been

text: reported

text: to

text: be

text: in

text: a

text: valence

text: ﬂuc-

text: ternary

text: KCu

text: S

text: -type

text: tetragonal,

text: struc-

text: tuating

text: This

text: suggests

text: the

text: possibility

text: of

text: strong

text: ture,

text: which

text: leads

text: to

text: quaternary

text: and

text: truly

text: stoichiometric


text: 16 heinz Schwarz, Techn. Universita¨t Wien, Austria), ISBN Common Metals, 93, 219 (1983). 3-9501031-1-2 (2001). 35 T. Jarlborg, A. J. Freeman, and D. D. Koelling, J. Magn. 33 R.E. Baumbach, A. Gallagher, T. Besara, J. Sun, T. Magn. Mater 60, 291 (1986). Siegrist, D.J. Singh, J.D. Thompson, F. Ronning and E.D. 36 T. Jarlborg, A. J. Freeman, and D. D. Koelling, J. App. Bauer, Phys. Rev. B 91, 035102 (2015). Phys. 53, 2140 (1982). 34 W. E. Pickett and B. M. Klein, Journal of the Less-

text: heinz

label: reference

text: Schwarz,

text: Techn.

text: Universita¨t

text: Wien,

text: Austria),

text: ISBN

text: Common

text: Metals,

text: T.

text: Jarlborg,

text: A.

text: J.

text: Freeman,

text: and

text: D.

text: D.

text: Koelling,

text: J.

text: Magn.

text: R.E.

text: Baumbach,

text: A.

text: Gallagher,

text: T.

text: Besara,

text: J.

text: Sun,

text: T.

text: Magn.

text: Mater

text: Siegrist,

text: D.J.

text: Singh,

text: J.D.

text: Thompson,

text: F.

text: Ronning

text: and

text: E.D.

text: T.

text: Jarlborg,

text: A.

text: J.

text: Freeman,

text: and

text: D.

text: D.

text: Koelling,

text: J.

text: App.

text: Bauer,

text: Phys.

text: Rev.

text: B

text: Phys.

text: W.

text: E.

text: Pickett

text: and

text: B.

text: M.

text: Klein,

text: Journal

text: of

text: the

text: Less-


text: 10 relatively weaker in the Ir-analogue, its contribution to the heat capacity, furnishes the following values, γ = 8.7 MR is not signiﬁcant enough to cause a similar behaviour and 8.0 mJ/mol K2, β = 0.22 and 0.21 mJ/mol K4 for in CeIrAl Si . Possible diﬀerences in the positive MR the Rh and Ir compounds, respectively. Our values of γ 4 2 due to the cyclotron motion of the conduction electrons and β are comparable with those reported in ref. 9. The between the two compounds decrease with the increase 4f-derived entropy S was calculated by the following 4f (cid:90) of temperature. In the paramagnetic region the applied relation: ﬁeld will tend to suppress residual spin ﬂuctuations lead- ing to a negative MR as observed at 15 and 20 K in C CeRhAl4Si2. Clearly several competing mechanisms are S4f = T4f dT (1) contributing to MR which have slightly diﬀerent com- parative energy scales in the two compounds. where C was obtained by subtracting the heat ca- The MR in CeRhAl Si at selected temperatures, for 4f 4 2 pacity of the La-analog from the corresponding Ce- ﬁelds applied in the ab-plane, i.e. along [100] and [010] is compound and making the usual assumption of lattice qualitatively similar for current density parallel to [001] heat capacity being identical for the isotypic La and Ce and [100], respectively (see, Figs. 7c and 7e). A rela- compounds. Only 64% (72%) of entropy S for a dou- tively sharp upturn in MR is seen at 5 and 8 K around 4f ∼ ∼ blet ground state with eﬀective spin 1/2 (i.e. Rln2) is re- 12 and 6 T, respectively, which is most likely arising leased up to T for CeRh(Ir)Al Si indicating the pres- from changes in the spin reorientation. In CeIrAl Si ,for N1 4 2 (cid:107) (cid:107) 4 2 ence of Kondo interaction in these materials, assuming J [001] and H [100], we observe two steps with hys- insigniﬁcant short range order above T . The entropy teresis at 8 K which correlate well with the data shown N1 corresponding to full doublet ground state is recovered in the inset of (Fig. 2(d). A single step with hystere- ∼ at temperature 32 K (31 K). sis is seen in the 10 K data; however no corresponding The Kondo behavior of resistivity together with the re- anomaly is clearly seen in the magnetization. The MR (cid:107) (cid:107) duced value of entropy at the magnetic transition tem- in CeIrAl Si for J [100] and H [010] also shows 4 2 perature T imply a partial quenching of the 4f-derived some peculiar features. A ﬁeld induced hysteresis ap- m Ce magnetic moment by the Kondo interaction. In such pears at 5 K around 13 T, which splits into two at 8 and cases the degeneracy of the ground state doublet is par- 10 K at lower ﬁelds but vanishes again at 12 K. It may tially removed by the Kondo eﬀect, and it has been shown be noted that the ratios of resistivities of the Rh and Ir that S (T ) = S (T /T ), where S is the entropy as- compounds at 2, 10 and 20 K for J parallel to [001] are 4f m K m K 4f sociated with the magnetic ordering and S is the en- nearly 6, 2.4 and 1.2, respectively. As mentioned above K tropy at T due to the Kondo eﬀect with the Kondo the positive MR due to the cyclotron motion is expected m temperature of T 18. The speciﬁc heat and the entropy to be much larger in the Ir analogue, particularly at low K as a function of T/T for a spin 1/2 Kondo impurity is temperatures. K known19, and the ratio T /T can be determined using m K E. Heat Capacity the value of S . Using this procedure we get a single-ion 4f Kondo temperature T of 14 and 10 K in CeRhAl Si K 4 2 The heat capacity of CeRh(Ir)Al4Si2 was measured be- and CeIrAl4Si2, respectively. On the other hand if Tm is tween 100 mK and 150 K to gain more information about taken as the temperature at which the upturn in the heat the magnetically ordered Kondo lattice state such as the capacity begins than T of 12.3 and 7.7 K are obtained. K entropy associated with the magnetic ordering, values of We would like to emphasize that this procedures assumes the coeﬃcient of the linear term in the electronic heat negligible short range order above the magnetic transi- capacity γ, which is proportional to enhanced electron tion and may therefore overestimate the Kondo temper- eﬀective mass, and Kondo temperature. The heat ca- ature. The Kondo temperature of a lattice is believed to pacity of nonmagnetic analogs LaRh(Ir)Al4Si2 was also be lower than its value for a single impurity. The simple measured between 2 and 150 K. The data are plotted in analysis presented here supports the picture of two mag- Fig. 8. The heat capacity clearly exhibits two peaks in netically ordered compounds with a residual weak Kondo both cerium compounds (Figs. 8 (a) and 8(b)) conﬁrming interaction with a Kondo temperature lower or compara- the occurrence of two bulk phase transitions. The peak ble (in the case of Rh-analog) to the magnetic ordering temperatures are overall in good agreement with the cor- temperature. It may also be noticed that the T of the K responding peaks in the susceptibility. For CeIrAl4Si2 Ir compound is lower than that of the Rh analog. This the heat capacity was also measured in 5 and 8 T. The suggests larger electron correlation eﬀect in CeRhAl Si , 4 2 two peaks shift to lower temperatures in 5 T and they which is amply conﬁrmed by the low temperature heat virtually disappear in 8 T above the spin-ﬂop ﬁeld where capacity data discussed below. the ﬁeld induces a saturated paramagnetic state. The heat capacity below 1 K down to 100 mK, plot- The heat capacity data of La analog is typical of a non- ted as C/T versus T2 is shown in the insets of Figs. 8(a) magnetic reference compound. The plots of C/T vs. T2 and (b). An extrapolation of the data to T = 0 K gives below 5 K for the two La-compounds (not shown) are lin- γ = 195.6 and 49.4 mJ/mol K2 in the Rh and its Ir- ear and a ﬁt of the standard expression C/T = γ +βT2, sibling. The values of γ are an order of magnitude larger where γ and β are the electronic and phononic part of than the corresponding values in the La-analog and imply ##LTLine##

text: relatively

text: weaker

text: in

text: the

text: Ir-analogue,

text: its

text: contribution

text: to

text: the

text: heat

text: capacity,

text: furnishes

text: the

text: following

text: values,

text: γ

text: MR

text: is

text: not

text: signiﬁcant

text: enough

text: to

text: cause

text: a

text: similar

text: behaviour

text: and

text: mJ/mol

text: β

text: and

text: mJ/mol

text: for

text: in

text: CeIrAl

text: Si

text: Possible

text: diﬀerences

text: in

text: the

text: positive

text: MR

text: the

text: Rh

text: and

text: Ir

text: compounds,

text: respectively.

text: Our

text: values

text: of

text: γ

text: due

text: to

text: the

text: cyclotron

text: motion

text: of

text: the

text: conduction

text: electrons

text: and

text: β

text: are

text: comparable

text: with

text: those

text: reported

text: in

text: ref.

text: The

text: between

text: the

text: two

text: compounds

text: decrease

text: with

text: the

text: increase

text: entropy

text: S

text: was

text: calculated

text: by

text: the

text: following

text: of

text: temperature.

text: In

text: the

text: paramagnetic

text: region

text: the

text: applied

text: relation:

text: ﬁeld

text: will

text: tend

text: to

text: suppress

text: residual

text: spin

text: ﬂuctuations

text: lead-

text: ing

text: to

text: a

text: negative

text: MR

text: as

text: observed

text: at

text: and

text: K

text: in

text: C

text: Clearly

text: several

text: competing

text: mechanisms

text: are

text: dT

text: contributing

text: to

text: MR

text: which

text: have

text: slightly

text: diﬀerent

text: com-

text: parative

text: energy

text: scales

text: in

text: the

text: two

text: compounds.

text: where

text: C

text: was

text: obtained

text: by

text: subtracting

text: the

text: heat

text: ca-

text: The

text: MR

text: in

text: CeRhAl

text: Si

text: at

text: selected

text: temperatures,

text: for

text: pacity

text: of

text: the

text: La-analog

text: from

text: the

text: corresponding

text: Ce-

text: ﬁelds

text: applied

text: in

text: the

text: ab-plane,

text: i.e.

text: along

text: and

text: is

text: compound

text: and

text: making

text: the

text: usual

text: assumption

text: of

text: lattice

text: qualitatively

text: similar

text: for

text: current

text: density

text: parallel

text: to

text: heat

text: capacity

text: being

text: identical

text: for

text: the

text: isotypic

text: La

text: and

text: Ce

text: and

text: respectively

text: (see,

text: Figs.

text: and

text: A

text: rela-

text: compounds.

text: Only

text: of

text: entropy

text: S

text: for

text: a

text: dou-

text: tively

text: sharp

text: upturn

text: in

text: MR

text: is

text: seen

text: at

text: and

text: K

text: around

text: blet

text: ground

text: state

text: with

text: eﬀective

text: spin

text: (i.e.

text: is

text: re-

text: and

text: T,

text: respectively,

text: which

text: is

text: most

text: likely

text: arising

text: leased

text: up

text: to

text: T

text: for

text: CeRh(Ir)Al

text: Si

text: indicating

text: the

text: pres-

text: from

text: changes

text: in

text: the

text: spin

text: reorientation.

text: In

text: CeIrAl

text: Si

text: ,for

text: ence

text: of

text: Kondo

text: interaction

text: in

text: these

text: materials,

text: assuming

text: J

text: and

text: H

text: we

text: observe

text: two

text: steps

text: with

text: hys-

text: insigniﬁcant

text: short

text: range

text: order

text: above

text: T

text: The

text: entropy

text: teresis

text: at

text: K

text: which

text: correlate

text: well

text: with

text: the

text: data

text: shown

text: corresponding

text: to

text: full

text: doublet

text: ground

text: state

text: is

text: recovered

text: in

text: the

text: inset

text: of

text: (Fig.

text: A

text: single

text: step

text: with

text: hystere-

text: at

text: temperature

text: K

text: K).

text: sis

text: is

text: seen

text: in

text: the

text: K

text: data;

text: however

text: no

text: corresponding

text: The

text: Kondo

text: behavior

text: of

text: resistivity

text: together

text: with

text: the

text: re-

text: anomaly

text: is

text: clearly

text: seen

text: in

text: the

text: magnetization.

text: The

text: MR

text: duced

text: value

text: of

text: entropy

text: at

text: the

text: magnetic

text: transition

text: tem-

text: in

text: CeIrAl

text: Si

text: for

text: J

text: and

text: H

text: also

text: shows

text: perature

text: T

text: imply

text: a

text: partial

text: quenching

text: of

text: the

text: some

text: peculiar

text: features.

text: A

text: ﬁeld

text: induced

text: hysteresis

text: ap-

text: m

text: Ce

text: magnetic

text: moment

text: by

text: the

text: Kondo

text: interaction.

text: In

text: such

text: pears

text: at

text: K

text: around

text: T,

text: which

text: splits

text: into

text: two

text: at

text: and

text: cases

text: the

text: degeneracy

text: of

text: the

text: ground

text: state

text: doublet

text: is

text: par-

text: K

text: at

text: lower

text: ﬁelds

text: but

text: vanishes

text: again

text: at

text: K.

text: It

text: may

text: tially

text: removed

text: by

text: the

text: Kondo

text: eﬀect,

text: and

text: it

text: has

text: been

text: shown

text: be

text: noted

text: that

text: the

text: ratios

text: of

text: resistivities

text: of

text: the

text: Rh

text: and

text: Ir

text: that

text: S

text: (T

text: S

text: (T

text: /T

text: where

text: S

text: is

text: the

text: entropy

text: as-

text: compounds

text: at

text: and

text: K

text: for

text: J

text: parallel

text: to

text: are

text: m

text: K

text: m

text: K

text: sociated

text: with

text: the

text: magnetic

text: ordering

text: and

text: S

text: is

text: the

text: en-

text: nearly

text: and

text: respectively.

text: As

text: mentioned

text: above

text: K

text: tropy

text: at

text: T

text: due

text: to

text: the

text: Kondo

text: eﬀect

text: with

text: the

text: Kondo

text: the

text: positive

text: MR

text: due

text: to

text: the

text: cyclotron

text: motion

text: is

text: expected

text: m

text: temperature

text: of

text: T

text: The

text: speciﬁc

text: heat

text: and

text: the

text: entropy

text: to

text: be

text: much

text: larger

text: in

text: the

text: Ir

text: analogue,

text: particularly

text: at

text: low

text: K

text: as

text: a

text: function

text: of

text: T/T

text: for

text: a

text: spin

text: Kondo

text: impurity

text: is

text: temperatures.

text: K

text: and

text: the

text: ratio

text: T

text: /T

text: can

text: be

text: determined

text: using

text: m

text: K

text: E.

text: Heat

text: Capacity

text: the

text: value

text: of

text: S

text: Using

text: this

text: procedure

text: we

text: get

text: a

text: single-ion

text: Kondo

text: temperature

text: T

text: of

text: and

text: K

text: in

text: CeRhAl

text: Si

text: K

text: The

text: heat

text: capacity

text: of

text: was

text: measured

text: be-

text: and

text: respectively.

text: On

text: the

text: other

text: hand

text: if

text: Tm

text: is

text: tween

text: mK

text: and

text: K

text: to

text: gain

text: more

text: information

text: about

text: taken

text: as

text: the

text: temperature

text: at

text: which

text: the

text: upturn

text: in

text: the

text: heat

text: the

text: magnetically

text: ordered

text: Kondo

text: lattice

text: state

text: such

text: as

text: the

text: capacity

text: begins

text: than

text: T

text: of

text: and

text: K

text: are

text: obtained.

text: K

text: entropy

text: associated

text: with

text: the

text: magnetic

text: ordering,

text: values

text: of

text: We

text: would

text: like

text: to

text: emphasize

text: that

text: this

text: procedures

text: assumes

text: the

text: coeﬃcient

text: of

text: the

text: linear

text: term

text: in

text: the

text: electronic

text: heat

text: negligible

text: short

text: range

text: order

text: above

text: the

text: magnetic

text: transi-

text: capacity

text: γ,

text: which

text: is

text: proportional

text: to

text: enhanced

text: electron

text: tion

text: and

text: may

text: therefore

text: overestimate

text: the

text: Kondo

text: temper-

text: eﬀective

text: mass,

text: and

text: Kondo

text: temperature.

text: The

text: heat

text: ca-

text: ature.

text: The

text: Kondo

text: temperature

text: of

text: a

text: lattice

text: is

text: believed

text: to

text: pacity

text: of

text: nonmagnetic

text: analogs

text: was

text: also

text: be

text: lower

text: than

text: its

text: value

text: for

text: a

text: single

text: impurity.

text: The

text: simple

text: measured

text: between

text: and

text: K.

text: The

text: data

text: are

text: plotted

text: in

text: analysis

text: presented

text: here

text: supports

text: the

text: picture

text: of

text: two

text: mag-

text: Fig.

text: The

text: heat

text: capacity

text: clearly

text: exhibits

text: two

text: peaks

text: in

text: netically

text: ordered

text: compounds

text: with

text: a

text: residual

text: weak

text: Kondo

text: both

text: cerium

text: compounds

text: (Figs.

text: (a)

text: and

text: conﬁrming

text: interaction

text: with

text: a

text: Kondo

text: temperature

text: lower

text: or

text: compara-

text: the

text: occurrence

text: of

text: two

text: bulk

text: phase

text: transitions.

text: The

text: peak

text: ble

text: (in

text: the

text: case

text: of

text: Rh-analog)

text: to

text: the

text: magnetic

text: ordering

text: temperatures

text: are

text: overall

text: in

text: good

text: agreement

text: with

text: the

text: cor-

text: temperature.

text: It

text: may

text: also

text: be

text: noticed

text: that

text: the

text: T

text: of

text: the

text: K

text: responding

text: peaks

text: in

text: the

text: susceptibility.

text: For

text: Ir

text: compound

text: is

text: lower

text: than

text: that

text: of

text: the

text: Rh

text: analog.

text: This

text: the

text: heat

text: capacity

text: was

text: also

text: measured

text: in

text: and

text: T.

text: The

text: suggests

text: larger

text: electron

text: correlation

text: eﬀect

text: in

text: CeRhAl

text: Si

text: two

text: peaks

text: shift

text: to

text: lower

text: temperatures

text: in

text: T

text: and

text: they

text: which

text: is

text: amply

text: conﬁrmed

text: by

text: the

text: low

text: temperature

text: heat

text: virtually

text: disappear

text: in

text: T

text: above

text: the

text: spin-ﬂop

text: ﬁeld

text: where

text: capacity

text: data

text: discussed

text: below.

text: the

text: ﬁeld

text: induces

text: a

text: saturated

text: paramagnetic

text: state.

text: The

text: heat

text: capacity

text: below

text: K

text: down

text: to

text: mK,

text: plot-

text: The

text: heat

text: capacity

text: data

text: of

text: La

text: analog

text: is

text: typical

text: of

text: a

text: non-

text: ted

text: as

text: C/T

text: versus

text: is

text: shown

text: in

text: the

text: insets

text: of

text: Figs.

text: magnetic

text: reference

text: compound.

text: The

text: plots

text: of

text: C/T

text: vs.

text: and

text: (b).

text: An

text: extrapolation

text: of

text: the

text: data

text: to

text: T

text: K

text: gives

text: below

text: K

text: for

text: the

text: two

text: La-compounds

text: (not

text: shown)

text: are

text: lin-

text: γ

text: and

text: mJ/mol

text: in

text: the

text: Rh

text: and

text: its

text: Ir-

text: ear

text: and

text: a

text: ﬁt

text: of

text: the

text: standard

text: expression

text: C/T

text: γ

text: sibling.

text: The

text: values

text: of

text: γ

text: are

text: an

text: order

text: of

text: magnitude

text: larger

text: where

text: γ

text: and

text: β

text: are

text: the

text: electronic

text: and

text: phononic

text: part

text: of

text: than

text: the

text: corresponding

text: values

text: in

text: the

text: La-analog

text: and

text: imply

text: ##LTLine##


text: 2 BANGTI JIN, RAYTCHO LAZAROV, AND ZHI ZHOU time of the particle motion follows a heavy tailed distribution, as opposed to a Gauss- ian process, which is characteristic of the normal diﬀusion equation. The macroscopic counterpart is a diﬀusion equation with a Caputo fractional derivative in time, i.e., (1.1). The derivation and study of accurate numerical methods for the model (1.1) with provable (possibly optimal-order) error bounds have attracted considerable interest in recent years, and a number of eﬃcient numerical schemes, notably the ﬁnite diﬀerence method, have been developed. There are two predominant approximations: the L1 type approximation and Gru¨nwald-Letnikov type approximation. The former is essentially of ﬁnite diﬀerence nature, whereas the latter is often based on convolution quadrature. We refer interested readers to [11, Section 1 and Table 1] for an updated overview of these numerical methods and relevant convergence rate results. In this paper, for reasons to be explained below, we revisit the L1 scheme, which was independently developed and analyzed in [22] and [16]. To this end, we divide the interval ∈ N [0, T] into a uniform grid with a time step size τ = T/N, N , so that 0 = t < t < 0 1 . . . < t = T, and t = nτ, n = 0, . . . , N. The L1 approximation of the Caputo fractional N n derivative ∂αu(x, t ) is given by [1(cid:90)6, Section 3] t n (cid:88) − 1 n 1 tj+1 ∂u(x, s) − − ∂αu(x, t ) = − (t (cid:90)s) α ds t n Γ(1 α) (cid:88) ∂s n j=0 tj − − ≈ 1− n 1 u(x, tj+1) u(x, tj) tj+1(t − s)−αds (cid:88)Γ(1 α) τ n j=0 tj (1.3) − − n 1 u(x, t − ) u(x, t − − ) = b n j n j 1 j τα (cid:88) j=0 − − − n 1 − = τ α[b u(x, t ) b − u(x, t ) + (b b − )u(x, t − )] =: Ln(u). 0 n n 1 0 j j 1 n j 1 j=1 where the weights b are given by j − − − − − b = ((j + 1)1 α j1 α)/Γ(2 α), j = 0, 1, . . . , N 1. j It was shown in [16, equation (3.3)] (see also [22, Lemma 4.1]) that the local truncation − error of the L1 approximation is bounded by cτ2 α for some constant c depending only on u, provided that the solution u is twice continuously diﬀerentiable. Further, upon dis- cretizing the spatial derivative(s) using the ﬁnite diﬀerence method, ﬁnite element method or spectral method, we arrive at a fully discrete scheme. Since its ﬁrst appearance, the L1 scheme has been extensively used in practice, and currently it is one of the most popular and successful numerical methods for solving the time fractional diﬀusion equation (1.1), including the case of nonsmooth data arising in inverse problems (see, e.g., [12]). Note that the fractional diﬀusion operator has only very limited smoothing property, especially for t close to zero. For example, for the homogeneous equation with an initial ∈ data v L2(Ω), we have the following stability estimate [20, Theorem 2.1] (cid:107) (cid:107) ≤ − (cid:107) (cid:107) ∂αu ct α v . t L2(Ω) L2(Ω) That is, the α-th order Caputo derivative is already unbounded, not to mention the high- order derivatives. In view of the limited regularity of the solution of (1.1), the high regularity required in the convergence analysis in these useful works is restrictive. Hence the C2-regularity assumption generally does not hold for problem (1.1), and the case of nonsmooth data is not covered by the existing error analysis. Our numerical − experiments indicate that the O(τ2 α) convergence rate actually does not hold even for smooth initial data v. This is clearly seen from the numerical results in Table 1, where rate denotes the empirical convergence rate, and the numbers in the bracket denote the ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BANGTI

text: JIN,

text: RAYTCHO

text: LAZAROV,

text: AND

text: ZHI

text: ZHOU

text: time

text: of

text: the

text: particle

text: motion

text: follows

text: a

text: heavy

text: tailed

text: distribution,

text: as

text: opposed

text: to

text: a

text: Gauss-

text: ian

text: process,

text: which

text: is

text: characteristic

text: of

text: the

text: normal

text: diﬀusion

text: equation.

text: The

text: macroscopic

text: counterpart

text: is

text: a

text: diﬀusion

text: equation

text: with

text: a

text: Caputo

text: fractional

text: derivative

text: in

text: time,

text: i.e.,

text: The

text: derivation

text: and

text: study

text: of

text: accurate

text: numerical

text: methods

text: for

text: the

text: model

text: with

text: provable

text: (possibly

text: optimal-order)

text: error

text: bounds

text: have

text: attracted

text: considerable

text: interest

text: in

text: recent

text: years,

text: and

text: a

text: number

text: of

text: eﬃcient

text: numerical

text: schemes,

text: notably

text: the

text: ﬁnite

text: diﬀerence

text: method,

text: have

text: been

text: developed.

text: There

text: are

text: two

text: predominant

text: approximations:

text: the

text: type

text: approximation

text: and

text: Gru¨nwald-Letnikov

text: type

text: approximation.

text: The

text: former

text: is

text: essentially

text: of

text: ﬁnite

text: diﬀerence

text: nature,

text: whereas

text: the

text: latter

text: is

text: often

text: based

text: on

text: convolution

text: quadrature.

text: We

text: refer

text: interested

text: readers

text: to

text: Section

text: and

text: Table

text: for

text: an

text: updated

text: overview

text: of

text: these

text: numerical

text: methods

text: and

text: relevant

text: convergence

text: rate

text: results.

text: In

text: this

text: paper,

text: for

text: reasons

text: to

text: be

text: explained

text: below,

text: we

text: revisit

text: the

text: scheme,

text: which

text: was

text: independently

text: developed

text: and

text: analyzed

text: in

text: and

text: To

text: this

text: end,

text: we

text: divide

text: the

text: interval

text: N

text: T]

text: into

text: a

text: uniform

text: grid

text: with

text: a

text: time

text: step

text: size

text: τ

text: T/N,

text: N

text: so

text: that

text: t

text: t

text: t

text: T,

text: and

text: t

text: nτ,

text: n

text: N.

text: The

text: approximation

text: of

text: the

text: Caputo

text: fractional

text: N

text: n

text: derivative

text: ∂αu(x,

text: t

text: is

text: given

text: by

text: Section

text: t

text: n

text: n

text: ∂u(x,

text: s)

text: ∂αu(x,

text: t

text: (t

text: α

text: ds

text: t

text: n

text: α)

text: ∂s

text: n

text: tj

text: 1−

text: n

text: u(x,

text: u(x,

text: tj)

text: s)−αds

text: α)

text: τ

text: n

text: tj

text: n

text: u(x,

text: t

text: u(x,

text: t

text: b

text: n

text: j

text: n

text: j

text: j

text: τα

text: n

text: τ

text: α[b

text: u(x,

text: t

text: b

text: u(x,

text: t

text: (b

text: b

text: )u(x,

text: t

text: Ln(u).

text: n

text: n

text: j

text: j

text: n

text: j

text: where

text: the

text: weights

text: b

text: are

text: given

text: by

text: j

text: b

text: ((j

text: α

text: α),

text: j

text: N

text: j

text: It

text: was

text: shown

text: in

text: equation

text: (see

text: also

text: Lemma

text: that

text: the

text: local

text: truncation

text: error

text: of

text: the

text: approximation

text: is

text: bounded

text: by

text: α

text: for

text: some

text: constant

text: c

text: depending

text: only

text: on

text: u,

text: provided

text: that

text: the

text: solution

text: u

text: is

text: twice

text: continuously

text: diﬀerentiable.

text: Further,

text: upon

text: dis-

text: cretizing

text: the

text: spatial

text: derivative(s)

text: using

text: the

text: ﬁnite

text: diﬀerence

text: method,

text: ﬁnite

text: element

text: method

text: or

text: spectral

text: method,

text: we

text: arrive

text: at

text: a

text: fully

text: discrete

text: scheme.

text: Since

text: its

text: ﬁrst

text: appearance,

text: the

text: scheme

text: has

text: been

text: extensively

text: used

text: in

text: practice,

text: and

text: currently

text: it

text: is

text: one

text: of

text: the

text: most

text: popular

text: and

text: successful

text: numerical

text: methods

text: for

text: solving

text: the

text: time

text: fractional

text: diﬀusion

text: equation

text: including

text: the

text: case

text: of

text: nonsmooth

text: data

text: arising

text: in

text: inverse

text: problems

text: (see,

text: e.g.,

text: Note

text: that

text: the

text: fractional

text: diﬀusion

text: operator

text: has

text: only

text: very

text: limited

text: smoothing

text: property,

text: especially

text: for

text: t

text: close

text: to

text: zero.

text: For

text: example,

text: for

text: the

text: homogeneous

text: equation

text: with

text: an

text: initial

text: data

text: v

text: we

text: have

text: the

text: following

text: stability

text: estimate

text: Theorem

text: ∂αu

text: ct

text: α

text: v

text: t

text: That

text: is,

text: the

text: α-th

text: order

text: Caputo

text: derivative

text: is

text: already

text: unbounded,

text: not

text: to

text: mention

text: the

text: high-

text: order

text: derivatives.

text: In

text: view

text: of

text: the

text: limited

text: regularity

text: of

text: the

text: solution

text: of

text: the

text: high

text: regularity

text: required

text: in

text: the

text: convergence

text: analysis

text: in

text: these

text: useful

text: works

text: is

text: restrictive.

text: Hence

text: the

text: assumption

text: generally

text: does

text: not

text: hold

text: for

text: problem

text: and

text: the

text: case

text: of

text: nonsmooth

text: data

text: is

text: not

text: covered

text: by

text: the

text: existing

text: error

text: analysis.

text: Our

text: numerical

text: experiments

text: indicate

text: that

text: the

text: α)

text: convergence

text: rate

text: actually

text: does

text: not

text: hold

text: even

text: for

text: smooth

text: initial

text: data

text: v.

text: This

text: is

text: clearly

text: seen

text: from

text: the

text: numerical

text: results

text: in

text: Table

text: where

text: rate

text: denotes

text: the

text: empirical

text: convergence

text: rate,

text: and

text: the

text: numbers

text: in

text: the

text: bracket

text: denote

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 BANGTI JIN, RAYTCHO LAZAROV, AND ZHI ZHOU ∈ ∩ Next we turn to smooth initial data, i.e., v D(A) = H2(Ω) H1(Ω). To this end, we 0 ﬁrst state an alternative estimate on the solution kernels. − − − Lemma 3.8. Let θ be close to π/2, and δ < π/2τ. Further, let Ks(z) = z 1(zα+A ) 1 − − − ∈ 1 h and Ks(z) = χ(z) 1(χ (z) + A ) 1. Then for any z Σ , there hold 2 1 h δ,θ (cid:107) − (cid:107) ≤ | |− Ks(z) Ks(z) c z ατ. 1 2 − − − − Proof. Like before, we set Bs(z) = (zα + A ) 1 and Bs(z) = (χ (z) + A ) 1. Then 1 h 2 1 h by Lemmas 3.4 and 3.5, we have (cid:107) − (cid:107) ≤ | − |(cid:107) − (cid:107)(cid:107) − (cid:107) Bs(z) Bs(z) χ (z) zα (zα + A ) 1 (χ (z) + A ) 1 1 2 1 h 1 h ≤ | | − | |− | |− ≤ | | − c z 2τ2 α z α χ (z) 1 c z 1 ατ. 1 (cid:3) The rest follows analogously to the derivation (3.3). ∈ Now we can state an error estimate for Av L2(Ω). ∈ Theorem 3.2. Let u and Un be the solutions of problems (2.5) and (2.8) with v H˙ 2(Ω), h ≡ h U0 = v = R v and f 0, respectively. Then there holds h h h (cid:107) − (cid:107) ≤ − (cid:107) (cid:107) u (t ) Un cτtα 1 Av . h n h L2(Ω) n L2(Ω) − − − − − − Proof. Let Ks(z) = z 1(zα + A ) 1 and Ks(z) = χ(z) 1(χ (z) + A ) 1. Then we 1 (cid:90) h 2 1 h can rewrite the error as − 1 (cid:90) w (t ) Wn = eztnKs(z)A v dz h n h 2πi \ 1 h h (3.11) Γθ,δ Γτ 1 − − + eztn(Ks(z) e zτKs(z))A v dz = I + II. 2πi 1 2 h h Γτ ∈ By Lemma 3.8 we have for z Γ τ (cid:107) − − (cid:107) ≤ | |− Ks(z) e zτKs(z) c z ατ. 1 2 ∈ (cid:32) (cid:33) By setting δ = 1/t and for all z Γ (cid:90), we derive the following bou(cid:90)nd for the term II n δ,θ (cid:107) (cid:107) ≤ (cid:107) (cid:107) π/(τ sinθ) − θ − II cτ A v ertn cosθr αdr + ecosψtα 1dψ (3.12) L2(Ω) h h L2(Ω) 1/tn −θ n ≤ − (cid:107) (cid:107) ctα 1τ A v . n h h L2(Ω) ∈ (cid:90) Now (2.3) implies that for all z Γ δ,θ ∞ (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:90) − − I c A v ertn cosθr α 1 dr L2(Ω) h h L2(Ω) (3.13) π/(∞τ sinθ) ≤ (cid:107) (cid:107) − ≤ − (cid:107) (cid:107) cτ A v ertn cosθr α dr cτtα 1 A v . h h L2(Ω) n h h L2(Ω) 0 − Then the desired result follows directly from (3.12), (3.13) and the identities Un − h (cid:3) u (t ) = Wn w (t ) and A R = P A. h n h n h h h ##LTLine## ##LTLine##

text: BANGTI

text: JIN,

text: RAYTCHO

text: LAZAROV,

text: AND

text: ZHI

text: ZHOU

text: Next

text: we

text: turn

text: to

text: smooth

text: initial

text: data,

text: i.e.,

text: v

text: D(A)

text: To

text: this

text: end,

text: we

text: ﬁrst

text: state

text: an

text: alternative

text: estimate

text: on

text: the

text: solution

text: kernels.

text: Lemma

text: Let

text: θ

text: be

text: close

text: to

text: and

text: δ

text: Further,

text: let

text: Ks(z)

text: z

text: h

text: and

text: Ks(z)

text: χ(z)

text: (z)

text: A

text: Then

text: for

text: any

text: z

text: Σ

text: there

text: hold

text: h

text: δ,θ

text: Ks(z)

text: Ks(z)

text: c

text: z

text: ατ.

text: Proof.

text: Like

text: before,

text: we

text: set

text: Bs(z)

text: (zα

text: A

text: and

text: Bs(z)

text: (χ

text: (z)

text: A

text: Then

text: h

text: h

text: by

text: Lemmas

text: and

text: we

text: have

text: Bs(z)

text: Bs(z)

text: χ

text: (z)

text: zα

text: (zα

text: A

text: (χ

text: (z)

text: A

text: h

text: h

text: c

text: z

text: α

text: z

text: α

text: χ

text: (z)

text: c

text: z

text: ατ.

text: The

text: rest

text: follows

text: analogously

text: to

text: the

text: derivation

text: Now

text: we

text: can

text: state

text: an

text: error

text: estimate

text: for

text: Av

text: Theorem

text: Let

text: u

text: and

text: Un

text: be

text: the

text: solutions

text: of

text: problems

text: and

text: with

text: v

text: H˙

text: h

text: h

text: v

text: R

text: v

text: and

text: f

text: respectively.

text: Then

text: there

text: holds

text: h

text: h

text: h

text: u

text: (t

text: Un

text: cτtα

text: Av

text: h

text: n

text: h

text: n

text: Proof.

text: Let

text: Ks(z)

text: z

text: A

text: and

text: Ks(z)

text: χ(z)

text: (z)

text: A

text: Then

text: we

text: h

text: h

text: can

text: rewrite

text: the

text: error

text: as

text: w

text: (t

text: Wn

text: eztnKs(z)A

text: v

text: dz

text: h

text: n

text: h

text: h

text: h

text: Γθ,δ

text: Γτ

text: eztn(Ks(z)

text: e

text: zτKs(z))A

text: v

text: dz

text: I

text: II.

text: h

text: h

text: Γτ

text: By

text: Lemma

text: we

text: have

text: for

text: z

text: Γ

text: τ

text: Ks(z)

text: e

text: zτKs(z)

text: c

text: z

text: ατ.

text: By

text: setting

text: δ

text: and

text: for

text: all

text: z

text: Γ

text: we

text: derive

text: the

text: following

text: for

text: the

text: term

text: II

text: n

text: δ,θ

text: π/(τ

text: sinθ)

text: θ

text: II

text: cτ

text: A

text: v

text: ertn

text: cosθr

text: αdr

text: ecosψtα

text: h

text: h

text: −θ

text: n

text: ctα

text: 1τ

text: A

text: v

text: n

text: h

text: h

text: Now

text: implies

text: that

text: for

text: all

text: z

text: Γ

text: δ,θ

text: I

text: c

text: A

text: v

text: ertn

text: cosθr

text: α

text: dr

text: h

text: h

text: π/(∞τ

text: sinθ)

text: cτ

text: A

text: v

text: ertn

text: cosθr

text: α

text: dr

text: cτtα

text: A

text: v

text: h

text: h

text: n

text: h

text: h

text: Then

text: the

text: desired

text: result

text: follows

text: directly

text: from

text: and

text: the

text: identities

text: Un

text: h

text: u

text: (t

text: Wn

text: w

text: (t

text: and

text: A

text: R

text: P

text: A.

text: h

text: n

text: h

text: n

text: h

text: h

text: h

text: ##LTLine##

text: ##LTLine##