Marc Briane
email: mbriane@insa-rennes.fr

Homogenization of linear transport equations. A new approach. Homogénéisation d'équations de transport linéaires. Une nouvelle approche

Keywords: homogenization, transport equation, dynamic flow, rectification Mots clés : homogénéisation, équation de transport, flot dynamique, redressement Mathematics Subject Classification: 35B27, 35F05, 37C10 1

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Résumé

Cet article propose une nouvelle approche de l'homogénéisation des équations de transport linéaires induites par une suite uniformément bornée de champs de vecteurs bε(x) et dont les solutions uε(t, x) coïncident en t = 0 avec une suite bornée de L p loc (R N ) pour un certain p ∈ (1, ∞). En supposant que la suite bε • ∇w 1 ε est compacte dans L q loc (R N ) (q exposant conjugué de p) pour un champ de gradients ∇w 1 ε borné dans L N loc (R N ) N et qu'il existe une suite uniformément bornée σε > 0 telle que σε bε est à divergence nulle si N = 2 ou est un produit vectoriel de (N-1) gradients bornés dans L N loc (R N ) N si N ≥ 3, on montre que la suite σε uε converge faiblement vers une solution d'une équation de transport. Il s'avère que la compacité de bε • ∇w 1 ε remplace la condition d'ergodicité du cas périodique bidimensionnel classique et permet de traiter des champs de vecteurs non périodiques en toute dimension. Le résultat d'homogénéisation est illustré par différents exemples généraux.

Introduction

In this paper we study the homogenization of the sequence of linear transport equations indexed by ε > 0,

   ∂u ε ∂t -b ε (x) • ∇ x u ε = 0 in (0, T ) × R N , u ε (0, •) = u 0 ε in R N .
(1.1)

where N ≥ 2, T > 0 and p ∈ [1, ∞] with conjugate exponent q. Equation (1.1) is associated with the flow Y ε (t, x) defined by

   ∂Y ε (t, x) ∂t = b ε (Y ε (t, x)), t ∈ R Y ε (0, x) = x ∈ R N .
(1.2)

Using the DiPerna-Lions transport theory [8, Corollary II.1], if for instance b ε is a vector field in L ∞ (R N ) N ∩ W 1,q loc (R N ) N with bounded divergence and the initial condition u 0 ε is in L p (R N ), then there exists a unique solution u ε (t, x) to equation (1.1) in L ∞ (0, T ; L p (R N )).

Tartar [START_REF] Tartar | Nonlocal effects induced by homogenization[END_REF] showed that the homogenization of first-order hyperbolic equations may lead to nonlocal effective equations with memory effects. In this framework Amirat, Hamdache and Ziani obtained memory effects for the homogenization of the transport equation (1.1), first in dimension two when b ε • ∇ x u ε = b ε (t, x 2 ) ∂ x1 u ε [START_REF] Amirat | Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux" (French) [Homogenization of a system of first-order hyperbolic equations and application to miscible flows in a porous medium[END_REF], and in a more general setting [START_REF] Amirat | Homogénéisation par décomposition en fréquences d'une équation de transport dans R n " (French) [Homogenization by decomposition into frequencies of a transport equation in R n ][END_REF][START_REF] Amirat | Homogenisation of parametrised families of hyperbolic problems[END_REF] they derived an integral representation of the memory term of the homogenized equation using Young's measures and a Radon transform.

In the present paper, rather than determining the "closure" set of the homogenized equations of (1.1) as in [START_REF] Amirat | Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux" (French) [Homogenization of a system of first-order hyperbolic equations and application to miscible flows in a porous medium[END_REF][START_REF] Amirat | Homogénéisation par décomposition en fréquences d'une équation de transport dans R n " (French) [Homogenization by decomposition into frequencies of a transport equation in R n ][END_REF][START_REF] Amirat | Homogenisation of parametrised families of hyperbolic problems[END_REF], we provide some sufficient conditions under which the homogenized equation of (1.1) remains a transport equation, namely the nature of equation (1.1) is preserved through homogenization. Assuming that b ε (x) = b(x/ε) is a divergence free periodically oscillating vector field, Brenier [START_REF] Brenier | Remarks on some linear hyperbolic equations with oscillatory coefficients[END_REF] showed thanks to the ergodic theorem that the solution u ε of (1.1) converges. This result was improved in dimension two by Hou and Xin [START_REF] Hou | Homogenization of linear transport equations with oscillatory vector fields[END_REF] who proved thanks to a two-scale convergence approach that the homogenization of (1.1) leads to the averaged transport equation with ´T2 b(y) dy (T N denotes the N -dimensional torus). Golse [START_REF] Golse | Moyennisation des champs de vecteurs et EDP[END_REF]Theorem 8] extended these results to the locally periodic case b ε (x) = b(x, x/ε) with div y b(x, •) = 0, assuming some ergodic property involving the flow associated with -b(x, •). In [START_REF] Golse | Moyennisation des champs de vecteurs et EDP[END_REF][START_REF] Golse | On perturbations of dynamical systems and the velocity averaging method for PDEs[END_REF] and in the generalizations [START_REF] Dumas | Multiphase averaging for generalized flows on manifolds[END_REF][START_REF] Dumas | The averaging method for perturbations of mixing flows[END_REF] with an application to Physics [START_REF] Dumas | A mathematical theory of planar particle channeling in crystals[END_REF], Golse et al. also studied the perturbed differential system satisfied by the pair (

I ε , J ε ) in R N × T N :        dI ε dt = εf (I ε , J ε ), t ∈ R dJ ε dt = ω(I ε ) + εg(I ε , J ε ), (1.3)
with f (I, •), g(I, •) periodic, which after an ε-rescaling is associated with a Liouville partial differential equation of type (1.1) but more complicated. Assuming among others that ω satisfies the Kolmogorov non-degeneracy type ergodic condition

meas I ∈ R N , |I| ≤ R : |ω(I) • ξ| < α -→ α→0 0 uniformly in ξ ∈ S N -1 , (1.4)
or some variant, Golse et al. obtained an error estimate between the solution to system (1.3) and the averaged system with ´TN f (I, y) dy, as well as the velocity averaging (homogenization) of the associated Liouville partial differential equation. They also extended this result to a non-periodic framework.

Returning to the periodic case b ε (x) = b(x/ε), Tassa [START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF] replaced in dimension two the divergence free of b by the existence of a periodic positive regular function σ such that div (σb) = 0 in R 2 , (

i.e. σ is an invariant measure for b by the Liouville theorem. The main assumption of the periodic framework b ε (x) = b(x/ε) of [START_REF] Brenier | Remarks on some linear hyperbolic equations with oscillatory coefficients[END_REF][START_REF] Hou | Homogenization of linear transport equations with oscillatory vector fields[END_REF][START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF]

v, b • ∇v = 0 in R 2 ⇒ ∇v = 0 in R 2 , (1.6)
together with b = 0 in R which is equivalent to the irrationality of the rotation number. In the locally periodic case b ε (x) = b(x, x/ε) of [START_REF] Golse | Moyennisation des champs de vecteurs et EDP[END_REF]Section 8] the ergodicity assumption states that for a.e.

x ∈ R N , the fluctuation of the vector field b(x, •) around its average value is ergodic with respect to the flow associated with -b(x, •).

In the present paper, besides the closure results of [START_REF] Amirat | Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux" (French) [Homogenization of a system of first-order hyperbolic equations and application to miscible flows in a porous medium[END_REF][START_REF] Amirat | Homogénéisation par décomposition en fréquences d'une équation de transport dans R n " (French) [Homogenization by decomposition into frequencies of a transport equation in R n ][END_REF][START_REF] Amirat | Homogenisation of parametrised families of hyperbolic problems[END_REF] and the ergodic approaches of [START_REF] Brenier | Remarks on some linear hyperbolic equations with oscillatory coefficients[END_REF][START_REF] Hou | Homogenization of linear transport equations with oscillatory vector fields[END_REF][START_REF] Golse | Moyennisation des champs de vecteurs et EDP[END_REF][START_REF] Golse | On perturbations of dynamical systems and the velocity averaging method for PDEs[END_REF][START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF], we propose a new approach which holds both in a non-periodic framework and in any dimension, assuming that the vector field b ε satisfies a non-ergodic condition which preserves the nature of equation (1.1) through homogenization. More precisely, the ergodic assumption (1.6) or (1.7) of the periodic framework is now replaced by the existence of a sequence

w 1 ε in C 1 (R N ) and q ∈ (1, ∞) such that 0 < b ε • ∇w 1 ε → θ 0 > 0 strongly in L q loc (R N ), (1.8)
which is equivalent in the periodic case to the existence of a periodic gradient ∇w

satisfying b • ∇w = 1 in R N . (1.9)
Moreover, the invariant measure σ of the periodic case is replaced by a sequence σ ε satisfying 0 < c -1 < σ ε < c for some constant c > 1, and (see Remark 2.1 for an equivalent expression)

div (σ ε b ε ) = 0 if N = 2 and σ ε b ε = ∇w 2 ε × • • • × ∇w N ε if N ≥ 3. (1.10)
The case where σ ε b ε is only divergence free in dimension N ≥ 3 remains open. In this way the vector field b ε is naturally associated with the vector field W ε := (w 1 ε , . . . , w N ε ) which induces a global rectification of the field b ε in the direction e 1 (see Remark 2.1). Then, assuming in addition to (1.8), (1.10) that W ε is uniformly proper (see condition (2.1) below) and converges both in C 0 loc (R N ) N and weakly in W 1,N loc (R N ) N , we prove (see Theorem 2.2) that up to a subsequence σ ε u ε converges weakly in L ∞ (0, T ; L p (R N )) to a solution v to the transport equation

     ∂v ∂t -ξ 0 • ∇ x v σ 0 = 0 in (0, T ) × R N v(0, •) = v 0 in R N , (1.11)
where σ 0 is the weak- * limit of σ ε in L ∞ (R N ), ξ 0 is the weak limit of σ ε b ε in

L N N -1 loc (R N ) N and v 0 the weak limit of σ ε u 0 ε in L p (R N ). Moreover, if σ ε converges strongly to σ 0 in L 1 loc (R N ) (see Remark 2.4
) or u 0 ε converges strongly to u 0 in L p loc (R N ), then up to a subsequence u ε converges weakly in L ∞ (0, T ; L p (R N )) to a solution u to the transport equation

     ∂u ∂t - ξ 0 σ 0 • ∇ x u = 0 in (0, T ) × R N u(0, •) = u 0 in R N .
(1.12)

The convergence of u ε also turns out to be strong in L ∞ (0, T ; L 2 loc (R N )) if u 0 ε converges strongly to u 0 in L p loc (R N ) with p > 2 (see the second part of Theorem 2.2). The compactness condition (1.8) is the main assumption of Theorem 2.2. It is equivalent to the compactness of the product σ ε det(DW ε ) which is connected to the vector field b ε by (1.10). The examples of Section 3 show that this condition may be satisfied in quite general situations.

Section 2 is devoted to the statement of the main result and to its proof. Section 3 deals by three applications of Theorem 2.2. In Section 3.1 we study the case of a diffeomorphism W ε on R 2 such that det(DW ε ) is compact in L p loc (R 2 ) for some q ∈ (1, ∞). In Section 3.2 we extend the periodic case of [START_REF] Brenier | Remarks on some linear hyperbolic equations with oscillatory coefficients[END_REF][START_REF] Hou | Homogenization of linear transport equations with oscillatory vector fields[END_REF][START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF] with b ε (x) = b(x/ε) and the periodic case of [START_REF] Briane | Isotropic realizability of fields and reconstruction of invariant measures under positivity properties. Asymptotics of the flow by a nonergodic approach[END_REF]Sec. 4] on the asymptotic of the flow associated with b, in the light of Theorem 2.2 with a periodically oscillating function σ ε (x) = σ(x/ε) (see Proposition 3.1). In Section 3.3 we consider the case of a diffeomorphism W ε which agrees at a fixed time t to a flow X ε (t, •) associated with a suitable vector field a ε (see Proposition 3.2). In this general setting assumption (1.8) holds simply when div a ε is compact in L q loc (R N ) for some q ∈ (1, ∞).

Notations

• (e 1 , . . . , e N ) denotes the canonical basis of R N .

• • denotes the scalar product in R N and | • | the associated norm.

• I N is the unit matrix of R N ×N , and R ⊥ is the clockwise 90 • rotation matrix in R 2×2 . • For M ∈ R N ×N , M T denotes the transpose of M . • Y N := [0, 1)
N , and f denotes the average-value of a function f ∈ L 1 (Y N ).

• For any open set Ω of R N and k ∈ N ∪ {∞}, C k c (Ω), respectively C k b (Ω)
, denotes the space of the C k functions with compact support in Ω, respectively bounded in Ω.

• For k ∈ N ∪ {∞} and p ∈ [1, ∞], C k (Y N ) denotes the space of the Y N -periodic functions in C k (R N ), and L p (Y N ) denotes the space of the Y N -periodic functions in L p loc (R N ) (i.e. in L p (K) for any compact set K of R N ). • For u ∈ L 1 loc (R N ) and U = (U j ) 1≤j≤d ∈ L 1 loc (R N ) N . ∇ x u := (∂ x1 u, . . . , ∂ x N u) and DU := ∂ xi U j 1≤i,j≤d .
The matrix-valued function DU stands for the transposition of the Jacobian matrix of the vector field U .

• For ξ 1 , . . . , ξ N in R N , the cross product ξ 2 × • • • × ξ N is defined by ξ 1 • (ξ 2 × • • • × ξ N ) = det (ξ 1 , ξ 2 , . . . , ξ N ) for ξ 1 ∈ R N , (1.13)
where det is the determinant with respect to the canonical basis (e 1 , . . . , e N ).

• o ε denotes a term which tends to zero as ε → 0.

• C denotes a constant which may vary from line to line.

The main result

Let W ε = (w 1 ε , . . . , w N ε ), ε > 0, be a sequence of vector fields in C 1 (R N ) N which is uniformly proper, i.e. for any compact set K of R N there exists a compact set K of R N satisfying W -1 ε (K) ⊂ K for any small enough ε > 0, ( 2.1)
and let W ∈ C 1 (R N ) N be such that

W ε → W in C 0 loc (R N ) N and W ε W in W 1,N loc (R N ) N . (2.2) Let b ε be a vector field in C 0 b (R N ) N ∩ W 1,q loc (R N ) N with bounded divergence and let σ ε be a positive function in C 0 (R N ) ∩ W 1,q loc (R N ) satisfying for some constant c > 1, c -1 ≤ σ ε ≤ c and σ ε b ε = R ⊥ ∇w 2 ε if N = 2 ∇w 2 ε × • • • × ∇w N ε if N ≥ 3, in R N . (2.3)
Also assume that for p ∈ (1, ∞) with conjugate exponent q, there exists a positive function θ 0 in C 0 (R N ) such that

θ ε := b ε • ∇w 1 ε > 0 in R N and θ ε → θ 0 > 0 strongly in L q loc (R N ). (2.4)
Finally, assume:

• either that there exists a constant B > 0 such that

|div b ε | ≤ B a.e. in R N , (2.5)
• or the regularity condition

b ε ∈ C 1 b (R N ) N , σ ε ∈ C 1 (R N ) and u 0 ε ∈ C 1 (R N ). (2.6) Remark 2.1. The definition (2.
3) of b ε can be also written for any dimension N ≥ 2 as the existence of (N -1) gradients ∇w 2 ε , . . . , ∇w N ε satisfying

∀ ξ ∈ R N , σ ε b ε • ξ = det ξ, ∇w 2 ε , . . . , ∇w N ε . (2.7)
In dimension N ≥ 3 this is exactly the definition of the cross product

∇w 2 ε ו • •×∇w N ε (see (1.13)). In dimension N = 2 this means exactly that σ ε b ε = R ⊥ ∇w 2 ε , which is equivalent to div (σ ε b ε ) = 0 in R 2 . ( 2
diffeomorphism W ε DW T ε b ε = θ ε e 1 in R N , (2.9)
in the direction e 1 with the compact range θ ε .

Then, we have the following homogenization result.

Theorem 2.2. Let T > 0, let p ∈ (1, ∞) and let u 0 ε be a bounded sequence in L p (R N ), N ≥ 2. Assume that conditions (2.1) to (2.4) together with (2.5) or (2.6) hold true. Let u ε be the solution to the transport equation (1.1) and set

v ε := σ ε u ε . Then, up to a subsequence v ε converges weakly in L ∞ (0, T ; L p (R N )) to a solution v to the transport equation      ∂v ∂t -ξ 0 • ∇ x v σ 0 = 0 in (0, T ) × R N v(0, •) = v 0 in R N , (2.10)
where (Cof denotes the cofactors matrix)

ξ 0 = Cof (DW ) e 1 ∈ C 0 (R N ) N , (2.11)
σ ε b ε ξ 0 in L N N -1 loc (R N ) N , σ ε σ 0 in L ∞ (R N ) * , σ ε u 0 ε v 0 in L p (R N ). (2.12) Moreover, if in addition b ε ∈ W 1,p/(p-2) loc (R N ) N with p > 2 and the sequence u 0 ε converges strongly to u 0 ∈ L p loc (R N ) with σ 0 ∈ W 1,∞ (R) and ξ 0 ∈ L ∞ (R N ) N ∩ W 1,p/(p-2) loc (R N ) N , then u ε converges strongly in L ∞ (0, T ; L 2 loc (R N )) to the solution u to the transport equation      ∂u ∂t - ξ 0 σ 0 • ∇ x u = 0 in (0, T ) × R N u(0, •) = u 0 in R N .
(2.13) Remark 2.4. In addition to the conditions (2.1) to (2.4) assume that σ ε converges strongly in L 1 loc (R N ) to σ 0 ∈ W 1,q loc (R N ). Then, we have v = σ 0 u and v 0 = σ 0 u 0 where u 0 is the weak limit of u 0 ε in L p (R N ), which implies that equation (2.10) is equivalent to equation (2.13). Therefore, u ε converges weakly in L ∞ (0, T ; L p (R N )) to a solution u to the transport equation (2.13).

Remark 2.3. If in Theorem 2.2 we assume in addition that σ 0 is in W 1,∞ (R N ) and ξ 0 belongs to L ∞ (R N ) N ∩W 1,q loc (R N ) N ,
To prove Theorem 2.2 we need the following L p -estimate.

Lemma 2.5. Let b ε ∈ L ∞ (R N ) N ∩ W 1,q
loc (R N ) N with bounded divergence be such that • either estimate (2.5) holds true,

• or both conditions (2.3) and (2.6) hold true.

Then, there exists a constant C > 0 such that for any u 0 ε ∈ L p (R N ) with p ∈ [1, ∞), the solution u ε to equation (1.1) satisfies the estimate

u ε (t, •) L p (R N ) ≤ C u 0 ε L p (R N ) for a.e. t ∈ (0, T ), (2.14)
Proof of Theorem 2.2. First of all, note that by (2.3) and (2.4) we have

det(DW ε ) = σ ε θ ε > 0 in R N . (2.15)
This combined with property (2.1) and Hadamard-Caccioppoli's theorem [START_REF] Caccioppoli | Sugli elementi uniti delle trasformazioni funzionali: un teorema di esistenza e unicit ed alcune sue applicazioni[END_REF] (or Hadamard-Lévy's theorem) implies that W ε is a C 1 -diffeomorphism on R N . Moreover, since by (2.15) det(DW ε ) is positive and by (2.2) W ε converges weakly in W 1,N loc (R N ) N , by virtue of Müller's theorem [START_REF] Müller | A surprising higher integrability property of mappings with positive determinant[END_REF] det(DW ε ) weakly converges to det(DW ) in L 1 loc (R N ). Hence, passing to the limit in (2.15) together with the strong convergence (2.4) of θ ε , the weak convergence (2.12) of σ ε and the boundedness (2.3) of σ ε we get that

det(DW ) = σ 0 θ 0 ≥ c -1 θ 0 > 0 a.e. in R N , (2.16)
which taking into account the continuity of DW and θ 0 implies that det(DW ) > 0 in R N . Moreover, again by the uniform character of (2.1) W is a proper mapping. Therefore, W is also a C 1 -diffeomorphism on R N . The weak formulation of equation (1.1) is that for any function

φ ∈ C 1 c ([0, T ) × R N ), ˆT 0 ˆRN u ε ∂φ ∂t dx dt + ˆRN u 0 ε (x) φ(0, x) dx = ˆT 0 ˆRN u ε div (φ b ε ) dx dt.
(2.17)

Using a density argument with σ ε ∈ W 1,q loc (R N ), we can replace the test function φ by σ ε ϕ for any ϕ ∈ C 1 c ([0, T ) × R N ). This combined with the divergence free of σ ε b ε leads us to the new formulation

ˆT 0 ˆRN σ ε u ε ∂ϕ ∂t dx dt + ˆRN σ ε (x) u 0 ε (x) ϕ(0, x) dx = ˆT 0 ˆRN u ε σ ε b ε • ∇ x ϕ dx dt.
(2.18) We pass easily to the limit in the left hand-side of (2.18). The delicate point comes from the right-hand side of (2.18).

By the L p -estimate (2.14) of Lemma 2.5 combined with the uniform boundedness of σ ε in (2.3) there exists a subsequence, still denoted by ε, such that

v ε = σ ε u ε converges weakly to some function v in L ∞ (0, T ; L p (R N )). Let ψ ∈ C 1 c ([0, T ) × R N ) the support of which is contained in some compact set [t 0 , t 1 ] × K of [0, T ) × R N , and define ϕ ε (t, x) := ψ(t, W ε (x)) for (t, x) ∈ (0, T ) × R N , (2.19)
so that ∇ x ϕ ε (t, x) = DW ε (x)∇ y ψ(t, y). Hence, making the change of variables y = W ε (x) and using (2.9) we deduce that

ˆT 0 ˆRN v ε (t, x) b ε (x) • ∇ x ϕ ε (t, x) dx dt = ˆT 0 ˆW -1 ε (K) v ε (t, x) b ε (x) • ∇ x ϕ ε (t, x) dx dt = ˆT 0 ˆK v ε (t, W -1 ε (y)) θ ε (W -1 ε (y)) e 1 • ∇ y ψ(t, y) det(DW -1 ε )(y) dy dt.
(2.20) First, using successively the Hölder inequality combined with the L p -estimate (2.14), the inclusion (2.1) and the L q -strong convergence (2.4) of θ ε , we have

ˆT 0 ˆK v ε (t, W -1 ε (y)) (θ ε -θ 0 )(W -1 ε (y)) e 1 • ∇ y ψ(t, y) det(DW -1 ε )(y) dy dt ≤ C ψ ˆT 0 ˆK v ε (t, W -1 ε (y)) p det(DW -1 ε )(y) dy 1 p × ˆK (θ ε -θ 0 )(W -1 ε (y)) q det(DW -1 ε )(y) dy 1 q dt ≤ C ψ ˆT 0 v ε (t, •) L p (K ) θ ε -θ 0 L q (K ) dt = o ε , which implies that ˆT 0 ˆK v ε (t, W -1 ε (y)) θ ε (W -1 ε (y)) e 1 • ∇ y ψ(t, y) det(DW -1 ε )(y) dy dt = ˆT 0 ˆK v ε (t, W -1 ε (y)) θ 0 (W -1 ε (y)) e 1 • ∇ y ψ(t, y) det(DW -1 ε )(y) dy dt + o ε .
Next, by the uniform convergence (2.2)

∇ y ψ(t, W ε (x)) → ∇ y ψ(t, W (x)) in C 0 loc ([0, T ] × R N ).
Then, making the inverse change of variables x = W -1 ε (y) together with (2.1) and using the weak convergence of v ε to v in L ∞ (0, T ; L p (R N )), we have

ˆT 0 ˆK v ε (t, W -1 ε (y)) θ 0 (W -1 ε (y)) e 1 • ∇ y ψ(t, y) det(DW -1 ε )(y) dy dt = ˆT 0 ˆK v ε (t, x) θ 0 (x) e 1 • ∇ y ψ(t, W ε (x)) dx dt = ˆT 0 ˆK v(t, x) θ 0 (x) e 1 • ∇ y ψ(t, W (x)) dx dt + o ε . Let ϕ ∈ C 1 c ([0, T ) × R N ) and define similarly to (2.19) ϕ(t, x) := ψ(t, W (x)) for (t, x) ∈ [0, T ) × R N ,
so that ∇ x ϕ(t, x) = DW (x)∇ y ψ(t, y). Therefore, passing to the limit in (2.20) we obtain that

ˆT 0 ˆRN v ε (t, x) b ε (x) • ∇ x ϕ ε (t, x) dx dt = ˆT 0 ˆRN v(t, x) θ 0 (x) DW (x) T -1 e 1 • ∇ x ϕ(t, x) dx dt + o ε .
(2.21)

On the other hand, using (2.9), (2.3) and the Murat-Tartar div-curl lemma in L N N -1 -L N (see, e.g., [17, Theorem 2]) with convergences (2.2), (2.4), (2.12) we get that

DW T ε (σ ε b ε ) = σ ε θ ε e 1 DW T ξ 0 = σ 0 θ 0 e 1 weakly in L 1 loc (R N ). ( 2
ˆT 0 ˆRN v ε b ε • ∇ x ϕ ε dx dt -→ ε→0 ˆT 0 ˆRN v σ 0 ξ 0 • ∇ x ϕ dx dt.
Finally, passing to the limit in formula (2.18) with ϕ ε , it follows that for any ϕ ∈

C 1 c ([0, T ) × R N ), ˆT 0 ˆRN v ∂ϕ ∂t dx dt + ˆRN v 0 (x) ϕ(0, x) dx = ˆT 0 ˆRN v σ 0 ξ 0 • ∇ x ϕ dx dt,
which taking into account that ξ 0 is divergence free yields the weak formulation of the desired limit equation (2.10). This concludes the proof of the first part of Theorem 2.2. Now, assume in addition that b ε ∈ W

φ ∈ C 1 c ([0, T ) × R N ), ˆT 0 ˆRN u 2 ε ∂φ ∂t dx dt + ˆRN (u 0 ε ) 2 (x) φ(0, x) dx = ˆT 0 ˆRN u 2 ε div (φ b ε ) dx dt,
Replacing u ε by u 2 ε in the first part of Theorem 2.2 and using the strong convergence of u 0 ε we get that the sequence σ ε u 2 ε converges weakly in L ∞ (0, T ; L p/2 (R N )) to the solution w to the transport equation

     ∂w ∂t -ξ 0 • ∇ x w σ 0 = ∂w ∂t - ξ 0 σ 0 • ∇ x w + ξ 0 • ∇σ 0 σ 2 0 w = 0 in (0, T ) × R N w(0, •) = σ 0 (u 0 ) 2 in R N .
(2.23) Note that by virtue of [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]Corollary II.1] the solution w to equation (2.23) is unique due to the regularities

σ 0 ∈ W 1,∞ (R N ), ξ 0 ∈ L ∞ (R N ) N ∩ W
     ∂(v 2 ) ∂t - ξ 0 σ 0 • ∇ x (v 2 ) + 2 ξ 0 • ∇σ 0 σ 0 v 2 = 0 in (0, T ) × R N v 2 (0, •) = (σ 0 u 0 ) 2 in R N , or equivalently, for any φ ∈ C 1 c ([0, T ) × R N ), ˆT 0 ˆRN v 2 ∂φ ∂t dx dt + ˆRN (σ 0 u 0 ) 2 (x) φ(0, x) dx = ˆT 0 ˆRN v 2 div φ ξ 0 σ 0 dx dt + ˆT 0 ˆRN 2 v 2 ξ 0 • ∇σ 0 σ 2 0 φ dx dt.
Replacing the test function φ by ϕ/σ 0 by a density argument, it follows that for any

function ϕ ∈ C 1 c ([0, T ) × R N ), ˆT 0 ˆRN v 2 σ 0 ∂ϕ ∂t dx dt + ˆRN σ 0 (x) (u 0 ) 2 (x) ϕ(0, x) dx = ˆT 0 ˆRN v 2 div ϕ ξ 0 σ 2 0 dx dt + ˆT 0 ˆRN 2 v 2 ξ 0 • ∇σ 0 σ 3 0 ϕ dx dt = ˆT 0 ˆRN v 2 σ 0 div ϕ ξ 0 σ 0 dx dt + ˆT 0 ˆRN v 2 σ 0 ξ 0 • ∇σ 0 σ 2 0 ϕ dx dt,
which shows that v 2 /σ 0 is also a solution to equation (2.23). By uniqueness we thus get that w = v 2 /σ 0 . Similarly, the solution u to equation (2.13) agrees with v/σ 0 . Finally, using these two equalities we have for any compact set

K of R N , ˆT 0 ˆK σ ε (u ε -u) 2 dx dt = ˆT 0 ˆK(σ ε u 2 ε -2 σ ε u ε u + σ ε u 2 ) dx dt -→ ε→0 ˆT 0 ˆK(w -2 v u + σ 0 u 2 ) dx dt = 0,
which concludes the proof of Theorem 2.2.

Proof of Lemma 2.5. If the uniform boundedness (2.5) of div b ε is satisfied, then using the estimate [START_REF] Murat | Compacité par compensation[END_REF] of [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]Proposition II.1] for the solution to the regularized equation of (1.1) and the lower semi-continuity of the L p -norm (p < ∞) we get estimate (2.14). Otherwise, assume that conditions (2.3) and (2.6) hold true. Using the regularity of the data the proof is based on an explicit expression of the solution to equation (1.1) from the flow Y ε associated with the vector field b ε by

   ∂Y ε (t, x) ∂t = b ε (Y ε (t, x)), t ∈ R Y ε (0, x) = x ∈ R N . (2.24) Let u 0 ε be a function in C 1 (R N ) ∩ L p (R N ).
It is classical that the regular solution u ε to the transport equation (1.1) is given by

u ε (t, x) = u 0 ε (Y ε (t, x)) for (t, x) ∈ [0, T ] × R N . ( 2
= Y ε (t, x) ⇔ x = Y ε (-t, y), we get that ˆRN u 0 ε (Y ε (t, x)) p dx = ˆRN u 0 ε (y) p det(D y Y ε (-t, y)) dy. (2.26)
Moreover, by (2.24) and the Liouville formula we have for any (τ,

y) ∈ R × R N , det(D y Y ε (τ, y)) = exp ˆτ 0 (div b ε )(Y ε (s, y)) ds .
However, since by (2.3)

σ ε b ε is divergence free, we have ˆτ 0 (div b ε )(Y ε (s, y)) ds = - ˆτ 0 ∇σ ε • b ε σ ε (Y ε (s, y)) ds = - ˆτ 0 ∂ ∂s ln σ ε (Y ε (s, y)) ds = ln σ ε (y) σ ε (Y ε (τ, y))
.

This combined with the boundedness of σ ε in condition (2.3) implies that

∀ (τ, y) ∈ R × R N , 0 < det(D y Y ε (τ, y)) = σ ε (y) σ ε (Y ε (τ, y)) ≤ c 2 .
Hence, we deduce from (2.26) that

ˆRN |u ε (x)| p dx = ˆRN u 0 ε (Y ε (t, x)) p dx ≤ c 2 ˆRN u 0 ε (y) p dy,
which yields the desired estimate (2.14). This concludes the proof of Lemma 2.5.

Examples

The purpose of this section is to illustrate the homogenization of the transport equation (1.1) by various oscillating fields b ε which satisfy the assumptions of Theorem 2.2. It means giving examples of diffeomorphism W ε on R N satisfying the rectification (2.9) of the vector field b ε where the sequence θ ε > 0 is compact in L q loc (R N ) for some q ∈ (1, ∞).

First example

Let α ε , α ∈ C 1 (R) be such that for some constant c > 0,

α ε → α in C 0 loc (R), α ε ≥ c in R, α ε → α in L 2 loc (R), (3.1)
and let β ε , β ∈ C 1 (R) be such that for some constant C > 0,

β ε → β in C 0 loc (R), |β ε | ≤ C in R, β ε is bounded in L ∞ loc (R), (3.2)
Consider the vector field

W ε ∈ C 1 (R N ) N defined by W ε (x) := α ε (x 1 ) exp β ε (α ε (x 1 )α ε (x 2 )) , α ε (x 2 ) exp -β ε (α ε (x 1 )α ε (x 2 )) , x ∈ R 2 , (3.3)
which is based on the characterization of the holomorphic mappings on C 2 with constant Jacobian [START_REF] Nishimura | French) [Injective holomorphic mappings of two variables with constant Jacobian[END_REF]. The gradient of W ε is given by

             ∇w 1 ε (x) = exp β ε (α ε (x 1 )α ε (x 2 )) α ε (x 1 ) 1 + α ε (x 1 )α ε (x 2 )β ε (α ε (x 1 )α ε (x 2 )) α ε (x 2 )α 2 ε (x 1 )β ε (α ε (x 1 )α ε (x 2 ) ∇w 2 ε (x) = exp -β ε (α ε (x 1 )α ε (x 2 )) -α ε (x 1 )α 2 ε (x 2 )β ε (α ε (x 1 )α ε (x 2 )) α ε (x 2 ) 1 -α ε (x 1 ) α ε (x 2 )β ε (α ε (x 1 )α ε (x 2 )) . Also define b ε := R ⊥ ∇w 2
ε and σ ε := 1, so that conditions (2.3) and (2.5) are fulfilled. By (3.1) and (3.2) we have

W ε (x) C 0 loc (R 2 ) -→ W (x) := α(x 1 ) exp β(α(x 1 )α(x 2 )) , α(x 2 ) exp -β(α(x 1 )α(x 2 )) , W ε W in H 1 loc (R 2 ),
so that conditions (2.2) are satisfied, and

b ε • ∇w 1 ε (x) = det(DW ε )(x) = α ε (x 1 ) α ε (x 2 ) → α (x 1 ) α (x 2 ) in L 2 loc (R 2 ), (3.4)
so that condition (2.4) is satisfied with p = 2. Moreover, since by (3.1)

∀ t ∈ R, |α ε (t) -α ε (0)| ≥ c |t|,
the sequence α ε (0) converges, and β ε is uniformly bounded in R, condition (2.1) holds for W ε . Note that the oscillations of the drift b ε in equation (1.1) are only due to the oscillations of the sequence β ε which does not appear in the convergence (3.4) of the Jacobian.

The periodic case

This section extends the periodic framework of [START_REF] Brenier | Remarks on some linear hyperbolic equations with oscillatory coefficients[END_REF][START_REF] Hou | Homogenization of linear transport equations with oscillatory vector fields[END_REF][START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF], [START_REF] Golse | Moyennisation des champs de vecteurs et EDP[END_REF]Theorem 8], and [START_REF] Briane | Isotropic realizability of fields and reconstruction of invariant measures under positivity properties. Asymptotics of the flow by a nonergodic approach[END_REF]Corollary 4.4].

Let W = (w 1 , . . . , w N ) be a vector field in C 2 (R N ) N , and let M be a matrix in

R N ×N such that x → W (x) -M x ∈ C 1 (Y N ) N and σ := det(DW ) > 0 in R N .
(3.5)

Consider the periodic vector field b ∈ C 1 (Y N ) N defined by

σ b := R ⊥ ∇w 2 if N = 2 ∇w 2 × • • • × ∇w N if N ≥ 3. (3.6)
We have the following result.

Proposition 3.1. Let u 0 ε ∈ C 1 (R N ) be a bounded sequence in L p (R N ) with p ∈ (1, ∞). Assume that conditions (3.5) and (3.6) hold true. Then, the vector fields W ε and b ε defined by

W ε (x) := ε W x ε and b ε (x) := b x ε for x ∈ R N , (3.7)
satisfy the assumptions of Theorem 2.2. Moreover, for any sequence u 0 ε in L p (R N ) such that σ(x/ε) u 0 ε converges weakly to v 0 in L p (R N ), the solution u ε to equation (1.1) is such that σ(x/ε) u ε converges weakly in L ∞ (0, T ; L p (R N )) to the solution v to the equation (2.10) with σ 0 = σ and ξ 0 = σ b .

Proof of Proposition 3.1. By the quasi-affinity of the determinant (see, e.g., [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF]Sec. 4.3.2]) and by (3.5) we have det(M ) = det DW = det(DW ) > 0, and by (3.7) there exists a constant C > 0 such that

∀ x ∈ R N , |W ε (x) -M x| ≤ Cε, (3.8)
which implies condition (2.1). Moreover, estimate (3.8) and the uniform bounded of DW ε imply easily the convergences (2.2) with the limit W (x) := M x.

On the other hand, the definitions (3.5) of W , σ and the definition (3.6) of b show clearly that condition (2.3) and the regularity (2.6) hold true. Moreover, we have

θ := b • ∇w 1 = det(DW ) σ = 1 in R N , which implies (2.4) since θ ε (x) = θ(x/ε) = 1.
Moreover, let u 0 ε be a sequence in L p (R N ) such that σ(x/ε) u ε converges weakly to v 0 in L p (R N ). By virtue of Theorem 2.2 combined with Remark 2.3 and using the weak limit of a periodically oscillating sequence, the sequence σ(x/ε) u ε converges weakly in L p (R N ) to the solution v to the equation (2.10) with σ 0 = σ and ξ 0 = σ b . The proof of Proposition 3.1 is now complete.

The dynamic flow case

In this section we construct a sequence W ε from a dynamic flow associated with a suitable but quite general sequence of vector fields a ε .

Let a ε , a be vector fields in

C 1 (R N ) N such that a ε → a in C 0 loc (R N ) N , a ε a in W 1,∞ loc (R N ) N * , (3.9)
and for some constant A > 0,

|a ε | + |div a ε | ≤ A in R N . (3.10) Also assume that there exists q ∈ (1, ∞) such that div a ε → div a in L q loc (R N ). ( 3

.11)

Consider the dynamic flow X ε associated with the vector field a ε defined by

   ∂X ε (t, x) ∂t = a ε (X ε (t, x)), t ∈ R X ε (0, x) = x ∈ R d , (3.12
) and let X be the limit flow associated with the limit vector field a.

Then, from any sequence of flows X ε we may derive a general sequence of vector fields b ε inducing the homogenization of the transport equation (1.1). Proposition 3.2. Let u 0 ε be a bounded sequence in L p (R N ) with p ∈ (1, ∞). Assume that conditions (3.9), (3.10), (3.11) hold true. For a fixed t > 0, define the vector field W ε := X ε (t, •) from R N into R N , and the vector field b ε by (2.3) with σ ε = 1. Then, the sequences W ε and b ε satisfy the assumptions of Theorem 2.2. Moreover, for any sequence u 0 ε converging weakly to u 0 in L p (R N ), the solution u ε to equation (1.1) converges weakly in L ∞ (0, T ; L p (R N )) to a solution u to the equation (2.13) where σ 0 = 1 and ξ 0 = Cof (D x X(t, x)) e 1 .

Remark 3.3.

There is a strong correspondance between the conditions (3.9)-(3.10) and (3.11) satisfied by the vector field a ε , and respectively the conditions (2.2) and (2.4) satisfied by the vector fields W ε and b ε .

Proof of Proposition 3.2. First of all, conditions (2.3) and (2.5) are straightforward, since σ ε = 1 and b ε is divergence free. Fix T > 0. By (3.10) we have

∀ t ∈ [0, T ], ∀ x ∈ R N , |X ε (t, x) -x| ≤ A T, (3.13)
so that the uniform property (2.1) is satisfied by W ε .

Let K be a compact set of R N . Again by (3.13) there exists a compact set

K of R N such that X ε (t, x), (t, x) ∈ [0, T ] × K ⊂ K . (3.14)
Let δ > 0. Since a ε converges uniformly to a in K and a ∈ C 1 (R N ) is k-Lipschitz in K for some k > 0, we have for any small enough ε > 0 and for any t ∈ [0, T ], for any x, y ∈ K, X ε (t, x) -X ε (t, y) ≤ |x -y| + ˆt 0 a ε (X ε (s, x)) -a ε (X ε (s, y)) ds ≤ δ + |x -y| + k ˆt 0 X ε (s, x) -X ε (s, y) ds.

Hence, by Gronwall's inequality (see, e.g., [14, Sec. 17.3]) we get that for any small enough ε > 0, ∀ t ∈ [0, T ], ∀ x, y ∈ K, |X ε (t, x) -X ε (t, y) ≤ (δ + |x -y|) e kt , which by (3.10) implies that for any small enough ε > 0, ∀ s, t ∈ [0, T ], ∀ x, y ∈ K, |X ε (s, x) -X ε (t, y) ≤ A |s -t| + (δ + |x -y|) e kt , namely X ε is uniformly equicontinuous in the compact set [0, T ] × K. Therefore, by virtue of Ascoli's theorem this combined with (3.14) and (3.9) implies that up to a subsequence X ε converges uniformly in [0, T ] × K to a solution X to ∀ t ∈ [0, T ], ∀ x ∈ K, X(t, x) = x + ˆt 0 a(X(s, x)) ds, i.e. X is the flow associated with the vector field a. Since a belongs to C 1 b (R N ), the flow X is uniquely determined (see, e.g., [START_REF] Hirsch | Differential equations, Dynamical Systems, and an Introduction to Chaos[END_REF]Sec. 17.4]). Therefore, the whole sequence X ε converges uniformly to X in [0, T ]×K. Moreover, by the differentiability of the flow (see, e.g., [START_REF] Hirsch | Differential equations, Dynamical Systems, and an Introduction to Chaos[END_REF]Sec. 17 which using (3.9), (3.14) and Gronwall's inequality implies that there exists a constant c > 0 such that

∀ t ∈ [0, T ], ∀ x ∈ K, |D x X ε (t, x)| ≤ |I N | e ct .
Therefore, convergences (2.2) hold true.

On the other hand, by the Liouville formula associated with equation (3.15) and estimate (3.10) we get that there exists a constant c > 1 such that ∀ t ∈ [0, T ], ∀ x ∈ K, c -1 ≤ det (D x X ε (t, x)) = exp ˆt 0 (div a ε )(X ε (s, x)) ds ≤ c, (3.16) which implies the existence of a constant C > 0 such that for any t ∈ [0, T ] and x ∈ K, det (D x X ε (t, x)) -det (D x X(t, x))

≤ C
ˆT 0 |div a ε -div a|(X ε (s, x)) ds + C ˆT 0 (div a)(X ε (s, x)) -(div a)(X(s, x)) ds.

Hence, using successively Jensen's inequality with respect to the integral in s, Fubini's theorem and the change of variables y = X ε (s, x) together with (3.14) and (3.16), it follows that there exists a constant C > 0 such that for any t ∈ [0, T ],

det (D x X ε (t, •)) -det (D x X(t, •)) L q (K)

≤ C div a ε -div a L q (K ) + C sup This combined with convergence (3.11) and the uniform convergence of X ε to X in the compact set [0, T ] × K implies the convergence (2.4) of θ ε = det(D x X ε (t, •)).

Finally, let u 0 ε be a sequence in L p (R N ) converging weakly to u 0 in L p (R N ). By virtue of Theorem 2.2 combined with Remark 2.4 and recalling that σ ε = 1, the sequence u ε converges weakly in L p (R N ) to a solution u to the equation (2.13) where σ 0 = 1 and by (2.11) ξ 0 = Cof (D x X(t, •)) e 1 in R N .

Proposition 3.2 is thus proved.

. 8 )

 8 However, in dimension N ≥ 3 condition (2.3) is stronger than σ ε b ε divergence free. The definition (2.3) of b ε and the definition (2.4) of θ ε are equivalent to the global rectification of the field b ε by the

  then by virtue of [8, Corollary II.1] there exists a unique solution v to the transport equation (2.10).

1

  ,p/(p-2) loc (R N ) N with p > 2 and u 0 ε converges strongly tou 0 in L p (R N ) with σ 0 ∈ W 1,∞ (R N ) and ξ 0 ∈ L ∞ (R N ) N ∩W1,p/(p-2) loc (R N ) N . By [8, Theorem II.3 and Corollary II.1] u 2 ε is the unique solution to the equation (1.1) with initial condition (u 0 ε ) 2 , or equivalently, for any

1

  ,p/(p-2) loc (R N ) N with divergence free. Moreover, again by [8, Theorem II.3 and Corollary II.1] v 2 is the unique solution to the equation induced by(2.10)

  .25) Let t ∈ [0, T ]. Making the change of variables combined with the semi-group property of the flow y

  .6]) we have∀ t ∈ [0, T ], ∀ x ∈ K, D x X ε (t, x) = I N + ˆt 0 D x X ε (s, x) D x a ε (X ε (s, x)) ds,(3.15)

[ 0 ,

 0 T ]×K (div a)(X ε ) -(div a)(X) .

  is the ergodicity of the flow associated with b (see, e.g., [20, Lect. 1], or [19, Chap. II, § 5]), namely any periodic invariant function by the flow is constant, or equivalently, for any periodic regular function

Acknowledgments. The author is very grateful to the unknown referees for several comments which have improved the presentation of the paper and enlarged the bibliography.


Sylvie Nonin

Feng Jiang

Dinshaw Patel

Imino proton exchange and base-pair kinetics in the AMP-RNA aptamer complex


Tonmoy Dey

Kento Sato

Bogdan Nicolae

Jian Guo

Jens Domke

Weikuan Yu

Franck Cappello

Kathryn Mohror

Optimizing Asynchronous Multi-Level Checkpoint/Restart Configurations with Machine Learning

Keywords: Machine Learning, Neural Network, Multi-Level Checkpointing (MLC)

With the emergence of versatile storage systems, multi-level checkpointing (MLC) has become a common approach to gain efficiency. However, multi-level checkpoint/restart can cause enormous I/O traffic on HPC systems. To use multilevel checkpointing efficiently, it is important to optimize checkpoint/restart configurations. Current approaches, namely modeling and simulation, are either inaccurate or slow in determining the optimal configuration for a large scale system. In this paper, we show that machine learning models can be used in combination with accurate simulation to determine the optimal checkpoint configurations. We also demonstrate that more advanced techniques such as neural networks can further improve the performance in optimizing checkpoint configurations.

I. INTRODUCTION

Current petascale systems in High-performance computing (HPC) deal with enormous workloads. Such workloads require a massive number of components simultaneously. Though the HPC systems are built using highly reliable components, with the sheer number of components, there is an increasing frequency of component failures, resulting in degradation of system and application reliability. To reliably run applications on such large-scale systems, a common technique is checkpoint/restart (CR) [START_REF] Vaidya | On checkpoint latency[END_REF], in which the system writes a snapshot of the application's state at fixed intervals to persistent storage, which is called a checkpoint. Application states can later be restored to the last saved checkpoint in case of a failure. Though CR is useful for large scale systems, its overhead can be an enormous challenge on large-scale ssystems.

One of approaches to reduce the overhead of CR is to determine the optimal checkpoint interval and checkpoint count. Poorly determined checkpoint interval makes system resilience worse. There are two approaches for obtaining the optimal checkpoint interval and checkpoint count values for any given configuration, namely the modeling approach [START_REF] Moody | Detailed modeling, design, and evaluation of a scalable multi-level checkpointing system[END_REF] and the simulation approach [START_REF] Sato | Design and modeling of a non-blocking checkpointing system[END_REF]. The modeling approach mainly formulates an analytical solution to obtain optimal values, whereas the simulation approach runs the application across multiple failures to check different scenarios for obtaining the optimal values.

In simple checkpoint models [START_REF] Vaidya | On checkpoint latency[END_REF] with synchronous checkpointing, where the checkpointing process mainly comprises of compute, checkpoint, and recovery state in a serial manner without other concurrent operations in the background. such modeling is beneficial in formulating an analytical solution for the optimal interval. However, with fast local storage becoming commercially available, asynchronous multi-level checkpointing (Async-MLC) [START_REF] Sato | Design and modeling of a non-blocking checkpointing system[END_REF] has become a common approach for efficient checkpointing. As shown in Figure 1, in Async-MLC, compute, checkpoint and recovery occur asynchronously from the application computation. This allows most of the checkpoint operations to happen in the background without delaying the critical compute operations in the applicaiton, thus minimzing the checkpoint overhead. This is particularly attractive when a parallel file system (PFS) is equipped with multi-level storage devices, which allows the checkpointing operations to happen on lower-level storage devices while the application I/O continues with upper-level storage devices.

Although introduction of multi-level checkpointing has greatly improved its performance over the simple model, however without the optimal interval and checkpoint count configuration, the performance of the system will still be degraded due to the checkpoint overhead. To obtain the optimal configuration of checkpoint interval and checkpoint count in MLC, the modeling approach is ineffective as it faces significant difficulty to formulate an analytical solution unless we simplify the model and/or make strong assumptions. Another approach is simulation. It is very effective in determining the optimal checkpoint configuration, however, it is very slow as it runs thousands of scenarios before obtaining the optimal values and this makes the simulation approach impractical for real world usage.

In this paper, we try to obtain the optimal checkpoint configuration for a given HPC system using the effectiveness and accuracy of the simulation approach while reducing the time taken by simulation to obtain the optimal result. We achieve this by combining machine learning methods with the simulation approach where we apply our machine learning models on existing simulated data to learn from the existing data and obtain the optimal checkpoint configuration with minimal error. The major contributions of this paper are:

Multi-Level Checkpointing

• Development of multi-level checkpoint simulator to replicate the behavior of real world large scale systems. chaining With our approach and design optimizations, we show that our models can predict the optimized parameter values when trained with the simulation approach. We also show that using more advanced deep neural network techniques can improve the performance of neural network over the machine learning models by up to 50%. which can further be used in future research works to optimize the checkpoint restart configuration of large scale systems.

II. BACKGROUND AND MOTIVATION

Checkpoint and restart is a commonly used technique for large scale systems running for a long time. Checkpoint is a snapshot of an application's state taken periodically and stored on to the available storage devices and it is used to restart the application when a failure occurs. Though CR is very important for large scale systems, however it is also one of the major contributors to the slowdown because of input output (I/O) workloads in HPC systems. To improve the efficiency of such systems, it is very crucial to use the optimal CR configuration [START_REF] Di | Optimization of multi-level checkpoint model for large scale hpc applications[END_REF]. Checkpoint interval is one of the most important checkpoint parameter that can configured to optimize the performance of large scale systems. Finding the optimal interval is very important as shorter interval will lead to frequent checkpoints where the system will spend more I/O time for saving checkpoints whereas longer interval can lead a less resilient system where a huge number of computation may be lost during a failure. Even with state of the art CR For simple checkpoint models where execution states can be categorized into compute, checkpoint and recovery state without any overlapping, determining the optimal checkpoint interval can be relatively simple. However, this approach to checkpointing is inefficient due to its synchronous manner of execution and with the emergence of fast local storage such as non volatile memory (NVM), multi-level checkpointing (MLC) has become a standard approach for efficient checkpointing. In MLC, the system stores checkpoints on the local node storage in addition to the parity node storage and the PFS. Unlike the simple checkpoint model, the execution takes place in an asynchronous manner (Async MLC) such that all operations can continue while saving the checkpoint in the background. For Async MLC, in addition to the checkpoint interval it is also very important to determine the optimal checkpoint count for each checkpoint levels.

There are mainly two approaches for optimizing the configuration of CR, namely modeling and simulation approach. However, with Async MLC, CR becomes more complicated due to the additional levels of checkpointing and the complicated scenarios that could arise from such design. It becomes very challenging for the existing approaches to determine the optimal configuration as the design of the system becomes more complex. The modeling approach is not accurate for such complicated CR scenarios and cannot provide accurate results, and though the simulation approach is accurate, it is very slow due to the large number of scenarios it has to simulate to obtain the optimal configuration. In this paper, we focus on maintaining the accuracy of the simulation without spending the performance of CR. We achieve this by predicting the optimized checkpoint count and interval for large scale systems using machine learning on the data generated by the simulator as shown in Figure 2.

III. NON-BLOCKING CHECKPOINT/RESTART SIMULATOR

Our multi-level checkpointing simulator is designed to replicate the behavior of real-world large scale systems for managing checkpoints across different levels of the storage hierarchy. The simulator supports three levels of checkpointing schemes, LOCAL, PARITY, and PFS. The first level of checkpointing is LOCAL in which the system saves the checkpoints on the storage in the local node and no redundant data stored across other nodes. In the PARITY scheme, the systems write checkpoints to the local storage along with the storage of other parity nodes. The redundant data are stored on the parity nodes so that the system can recover a copy of the file, provided a node, and its parity nodes do not fail simultaneously. Reliability is further enhanced, some systems encode the checkpoint data across the parity nodes so that the resiliency of the system can be enhanced. One of the standard encoding schemes is XOR encoding, where the redundant data are collectively written to the parity nodes to withstand single node failures in a group. However, it can only restart successfully if two or more nodes from the same parity group do not fail at the same time. Another one is Reed-Solomon encoding, which allows the system to withstand failures as long as half the nodes in the parity group do not fail simultaneously. Unlike the other two schemes, in the PFS parity scheme the system saves and restores the checkpoints from the PFS. The PFS scheme is only required when there is a catastrophic failure where a large number of nodes fail simultaneously requiring the entire system to be restored from the PFS.

Our simulator stores the checkpoint data at different storage level, each of which may have a different cost and level of resilience. Each of the checkpointing storage is a level, for which lowest level or level 1 checkpoints are the least resilient with minimum overhead, while storing the checkpoints in PFS is the most resilient with maximum overhead. In our simulator, a level L failure is a failure that requires a checkpoint at level L or higher for restarting while a level L recovery refers to the process of restarting an application using a checkpoint saved at level L. Since lower level failures such as process failures occur more frequently as compared to the higher level failure such as PFS failure, the system records one or more lower level checkpoints for every higher level checkpoint.

The performance of the checkpointing system is dependent on several characteristics of the system such as the overhead of saving the checkpoint to different levels of storage and also the cost of recovering from these storage levels. The overhead of saving and recovering the checkpoint from different levels of storage are not the only factors to take into account, the performance is also dependent on how frequently does each of these storage levels fail as failure of a checkpoint level causes the system to lose compute time and delays the progress. Considering all these characteristics of a system, our simulator was developed. It simulates multiple failures using different configurations to determine the optimal interval and checkpoint count for the different levels. This simulator can provide an overview of multi-level checkpoint systems for current and future systems and motivate systems to optimize checkpoint configuration that provide adequate overall reliability and efficiency. The simulator can be used to optimize performance of a given multi-level checkpointing implementation on a specific HPC system. For a given configuration of a large scale system, we simulate the performance of the system across

IV. MACHINE LEARNING FOR CHECKPOINT/RESTART

The count and interval are two of the most important checkpoint parameters for optimizing the CR configuration. For determining the optimized checkpoint interval, the simulation approach is accurate but it can be slow as it explores different CR parameters before determining the optimized checkpoint interval and because of this, it is not a practical approach for real world scenarios such as while submitting a job. Our objective was to determine the optimized interval faster than the simulation approach without losing much of its accuracy. For our approach, we generated a limited amount of data using the simulator which provides us with optimized interval and count for the provided CR configurations and we use that data to determine the optimized count and interval for other CR configurations using the AI techniques described in the following sections.

A. Machine Learning

After running simple machine learning models such as linear and polynomial regression on the simulated data, we observed that these models were not able to learn from the data and provided inaccurate predictions for checkpoint count and interval. However, running more complex machine learning models for predicting checkpoint count and checkpoint interval showed that the optimized count can be predicted very accurately without much modification to the algorithms. Using different machine learning algorithms on the simulated data, we got the best results for predicting the checkpoint count using Random forest, Support Vector Clustering and Gaussian Naive Bayes. While random forest algorithm performed best among the different machine learning models for predicting the checkpoint interval. Since the random forest model could predict the optimized checkpoint count for the two level checkpoint model with an accuracy greater than 99%, we did not design a neural network for predicting checkpoint count. We designed a neural network only for predicting the optimized checkpoint interval, to obtain better performance than the machine learning models. For designing our neural network, we started with a simple neural network consisting of two hidden layers with 10 nodes on each hidden layer and ReLU as the activation function. The neural network showed an increase in error by a factor of 2.5 in comparison to the machine learning models for predicting the optimized interval. To improve the performance without significantly increasing training time, we added an extra hidden layer and increased the node on each hidden layer to 25. Though adding a layer and more nodes improved the performance, however, this baseline model still predicted the optimized checkpoint interval with an error that was greater than two times the error shown by the machine learning models. To improve the performance of this baseline model, we optimized its design using the techniques described in the following sections.

1) Daisy Chaining: Our initial design, consist of two independent models, the random forest model for predicting the checkpoint count and the neural network for predicting checkpoint interval. However, our analysis shows a strong correlation between the checkpoint count and interval with interval being inversely proportional to the checkpoint count. Since our random forest model was able to predict the checkpoint count very accurately, in order to improve the performance we fed the output from checkpoint count prediction as an input to the neural network as shown in Figure 3. The optimization reduced the error by 40% over the baseline model.

2) Parameter Reduction: After further analysis we observed some of the input parameters were inter dependent on each other and were impacting the performance of the neural network. On removing these input parameters, we observed a further reduction in error by 40% over the daisy chained model.

V. IMPLEMENTATION

The simulator has been developed to replicate the behavior of real-world scenarios when using a multi-level checkpoint for large scale systems. The simulator is provided with three critical parameters for each level, checkpoint overhead, checkpoint restart time, and failure rates. The checkpoint overhead of each level corresponds to the delay incurred when a checkpoint is stored at the desired level. Though the checkpoint storing mechanism is asynchronous and the majority of it is performed in the background, there still remains a fragment of time during which other useful operations need to be halted to perform the checkpointing. The checkpoint restart time refers to the time taken to retrieve the checkpoint data from the last stored checkpoint at the desired level, and failure rates provide how frequently the checkpoint levels fail.

The checkpoint overhead, restart time, and failure rates are used by the simulator as shown in Figure 4 to provide the user with elapsed time and the efficiency of the system. The efficiency corresponds to the amount of time that it would for the system to complete the execution without checkpoint to the increase in execution time due to checkpoint [START_REF] Vaidya | On checkpoint latency[END_REF]. The elapsed time and efficiency are obtained by running the simulator for 3000 different failure scenarios. These failure scenarios are created based on the failure rate [START_REF] Moody | Detailed modeling, design, and evaluation of a scalable multi-level checkpointing system[END_REF] provided by the user and generates a time interval until the next failure of one of the checkpoint levels. These time intervals are not fixed and vary across different simulations. Once the time interval to the subsequent failure is complete, the simulator uses the checkpoint restart time of the failed checkpoint level to determine the delay that will be incurred and add the delay to the total elapsed time. However, when no such failure occurs as there is still time until the next crash, the checkpoint uses the overhead parameter value of the checkpoint level where the current application status will be stored to determine the delay for storing the data at that desired level and add the delay to the elapsed time. At the end of all the failure scenarios, the simulator determines the total elapsed time and the efficiency of the system for that particular configuration, which is determined by comparing the elapsed time of the checkpoint system with delays to the elapsed time without any checkpoint delays.

The above simulation run provides the elapsed time and efficiency of a system with a specific configuration. The simulator we have developed not only provides the elapsed time and efficiency for a system but also performs simulation across different settings of the checkpoint system to determine the optimal checkpoint count for each level of the multi-level checkpoint system. The optimized configurations are obtained by modulating the checkpoint configuration values starting with checkpoint interval. As shown in Figure 4, the simulator initially performs optimization of the checkpoint interval by comparing the current efficiency with the efficiency of the previous configuration until the maxima are obtained. The same procedure is also followed for both checkpoint count of level 1 and level 2. After the peaks for all the configuration parameters are captured by the simulator, the user is provided with the optimized configuration for the given system based on the overhead, restart time, and failure rates.

Once a significant amount of data is collected from the simulator, the information is passed on to different machine learning and neural network models to predict the optimized checkpoint configuration for other systems with different overhead, restart time, and failure rates. The models that were used from the scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine learning in python[END_REF] module, a python module integrating a wide range of state-of-the-art machine learning algorithms for training and predicting the optimized checkpoint configuration will be explained in the following section.

A. Random Forest

Random forest is a machine learning method used for classification and regression that operates by building multiple decision trees in randomly selected subspace of the feature space at training time and predicting the output class based on the combined decision of all the trees for classification or mean prediction (regression) of the individual trees. Random decision forests correct for decision trees' habit of overfitting to their training set [START_REF] Tin | Random decision forests[END_REF]

B. Gaussian Naive Bayes

The naive Baiyes classifier is a learning method that assumes that features are independent given class. It is a simple probabilistic classifiers that learns based on Bayes' theorem with an assumption of strong independence between the features. To allow a systematic study of classification accuracy for several classes of randomly generated problems it uses Monte Carlo simulations. It works well for completely independent features and functionally dependent features. [START_REF] Rish | An empirical study of the naive bayes classifier[END_REF] C. Support Vector Clustering It is a clustering method which uses the approach of support vector machines. It uses Gaussian kernel to map the data points in a high dimensional feature space, where we search for the minimal enclosing sphere. When mapped back from the enclosing sphere to data space, can separate into several components, each enclosing a separate cluster of points. The width of the Gaussian kernel controls the scale at which

D. Neural Network

Neural networks are a wide class of flexible nonlinear regression and discriminant models, data reduction models, and nonlinear dynamical systems. They consist of an often large number of neurons, i.e., simple linear or nonlinear computing elements, interconnected in often complex ways and often organized into layers [START_REF] Warren S Sarle | Neural networks and statistical models[END_REF]. In our implementation the neural network used ReLU [START_REF] Fred | Deep learning using rectified linear units (relu)[END_REF] as the activation function, Adam optimizer as the solver with a batch size of 200 and learning rate of 0.001.

VI. EVALUATION

For our evaluation, initially, we started with a two-level checkpoint model and then further evaluated on a threelevel checkpoint model. The following sections describe the evaluation method and observations from both the multi-level checkpoint models.

A. Two-Level Checkpoint Model

Two independent data sets were generated for evaluation using the two-level checkpoint simulator. The training data was simulated by providing realistic overhead, latency, and restart time obtained using characteristics of some of the Top500 systems. We used the base failure rate of 2 . 10 -6 and 4 . 10 -7 failures/sec for Level-1 (L1) and Level-2 (L2) respectively [START_REF] Moody | Detailed modeling, design, and evaluation of a scalable multi-level checkpointing system[END_REF] and explored the impact of these failure rates by increasing them up to 50X the base value. To validate our models for unique CR configurations, the test data set was simulated using data mutually exclusive to the training data set but within the parameter space of the training data set Our random forest model could predict optimized checkpoint count with 99.365% accuracy when compared to the simulated result. For predicting the checkpoint interval with its values ranging up to 9960 seconds, both machine learning models could determine the optimized interval with a mean absolute error of 49.5 seconds. As shown in Figure 5, our baseline neural network predicted optimized interval with a mean absolute error of 116.63 seconds. However, with our optimizations described in Section 2.2, we see that daisy chaining shows a 40% performance improvement over the baseline design followed by a further 40% improvement with parameter reduction over the daisy chained model. The optimized neural network model performs 18% better than the machine learning models with a mean absolute error 40.7 seconds as shown in Figure 6. With majority of our dataset in the region 1000 -8000 seconds, this converts to 0.5% -4% mean error from the simulated values of the optimized interval for two-level checkpoint model.

B. Three-Level Checkpoint Model

For evaluating the three-level checkpoint model, we generated a dataset consisting of 112,000 different system configurations with varying checkpoint overhead, restart time, and failure rates for each level. We used the base checkpoint overhead and restart time of 0.5, 4.5, and 1052 seconds for Level-1 (L1), Level-2 (L2), and Level-3 (L3)checkpoints and a base failure rate of 2 . 10 -7 , 1.8 . 10 -6 and 4 . 10 -7 failures/sec, respectively [START_REF] Moody | Detailed modeling, design, and evaluation of a scalable multi-level checkpointing system[END_REF] and similar to the two-level checkpoint model, we evaluated the performance of different models by modulating these parameters between their base value and upper limit of 50X the base value. However, unlike the two-level checkpoint model, only one large dataset was generated using the parameter values ranging between their base value and upper limit. This dataset was further randomly broken into the training set and testing set for evaluation of the performance.

The training set was used for training the machine learning, and neural network model and validation for predicting the checkpoint count accurately for each level was performed on the test set. As shown in Figure 7 , the machine learning Fig. 8: Neural Network Models Performance for three-level checkpoint model models show very similar accuracy for both level-1 and level-2 checkpoint count prediction with Support Vector Clustering performing the best with an accuracy of 23.89% and 25.50% for level-1 and level-2 count prediction. Random forest gives an accuracy of 22.93% and 23.31% respectively, and Gaussian Naive Bayes performs the worst among the tested machine learning models with an accuracy of 19.63% and 22.83% for level-1 and level-2 prediction.

Following the machine learning model evaluation, the neural network was trained and tested using the same set of data. The baseline neural network with 3 hidden layers and 25 hidden layer nodes gave an accuracy of 29.35% for predicting level-1 checkpoint count, while level-2 checkpoint count also gave a similar result with an accuracy of 29.99%. To evaluate checkpoint count prediction with a much more complex neural network, the hidden layers were increased from 3 to 13 layers and then further increased to 23 layers, and this was accompanied by an increase in the number of nodes in the hidden layers. The increase in the number of hidden layers and the hidden layer nodes does not improve the accuracy. As shown in Figure 8, the accuracy remains within 1% of the baseline neural network accuracy.

On comparing the prediction accuracy of the neural network with the three machine learning models, we see a noticeable improvement in checkpoint count prediction accuracy in the neural network. As shown in Figure 9, the neural network outperforms the machine learning models by a margin of 24% -51% for Level-1 count prediction. For Level-2 count

VII. RELATED WORK

Multi-level checkpointing [5] [6] [2] has become a promising technique for dealing with fault-tolerance in large scale or even exascale systems. With the increase in demand for such large scale systems [START_REF] Wang | Modeling coordinated checkpointing for large-scale supercomputers[END_REF] [10] [START_REF] Zheng | A scalable double inmemory checkpoint and restart scheme towards exascale[END_REF], it has been studied thoroughly . The multi-level checkpoint-restart model providing an effective way to control the checkpoint overhead. In this paper, we focus on multi-level CR model and optimize the checkpoint count for multiple levels.

SCR [START_REF] Moody | Detailed modeling, design, and evaluation of a scalable multi-level checkpointing system[END_REF] is a library which uses Markov model to fit the multi-level CR mechanism. There are some assumptions made in this paper such as the failure rates at different checkpoint levels are completely independent and also assumes that the failure rates at different levels are known beforehand. The other assumptions include that the checkpoint overhead remains constant throughout the job as well as infinite pool of spare nodes available at all time. Using the Markov model, Sato Kento et al. [START_REF] Sato | Design and modeling of a non-blocking checkpointing system[END_REF] combines the benefits of non-blocking and multi-level checkpointing. The work presents the design of the system and models its performance to show that the system can improve efficiency by 1.1 to 2.0 on future machines. Additionally, applications using that checkpointing system can achieve high efficiency when using a PFS with lower bandwidth.

FTI [START_REF] Bautista-Gomez | Fti: high performance fault tolerance interface for hybrid systems[END_REF] is another multi-level checkpointing proposed by Bautista-Gomez et al. which uses local SSDs and a PFS. Since PFS usage is costly when compared to local storage, the model reduces PFS usage by encoding the data using Reed-Solomon (RS) encoding [START_REF] Chen | A scalable checkpoint encoding algorithm for diskless checkpointing[END_REF] for highly resilient cached checkpoints. However, increasing failure rates require checkpoints to a PFS more frequently. However, increasing failure rates require checkpoints to a PFS more frequently. Thus, checkpointing to a PFS is crucial for future multi-checkpointing systems. Sheng et al. [START_REF] Di | Optimization of multi-level checkpoint model for large scale hpc applications[END_REF] builds a mathematical model to fit the multilevel CR mechanism with large scale applications for various types of failures and theoretically optimize the entire execution performance for each parallel application by selecting the best checkpoint level combination and corresponding checkpoint intervals. They evaluated their optimal solutions using both simulation with millions of cores and real environment with real-world MPI programs running on hundreds of cores and showed that with optimal checkpoint intervals at each level, the system outperforms other state-of-the-art solutions by up to 50 percent.

VIII. CONCLUSION

In this paper, we present an idea to combine the simulation approach with machine learning models to reduce the time taken to determine the optimized parameter values of checkpoint interval and checkpoint count for different configurations of CR. With our approach and design optimizations, we show that our models can predict the optimized parameter values when trained with the simulation approach. We have also demonstrated that using techniques such as neural networks can improve the performance over the machine learning models with neural network sometime exceeding the performance of a machine learning model by 50%. We plan to validate the results on real systems in the future. From our study and exploration, we can say that using much more advanced techniques can further improve this performance and predict optimized checkpoint configuration accurately.

Fig. 1 :

 1 Fig. 1: Multi Level Checkpointing

Fig. 2 :

 2 Fig. 2: Combining Simulation Approach with Machine Learning

Fig. 3 :

 3 Fig. 3: Daisy Chaining output from checkpoint count prediction as an input to the neural network for checkpoint interval prediction

Fig. 4 :

 4 Fig. 4: Workflow for Optimizing Checkpoint Configuration

Fig. 5 :

 5 Fig. 5: Performance improvement with NN optimization

Fig. 6 :

 6 Fig. 6: Performance comparison between Machine Learning and Neural Network model for two-level checkpoint model

22.93% 19.63% 23.89% 23.31% 22.83% 25.50% 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% RA NDOM FOREST GAUSSIAN NAIV E BAYES SUPPORT V ECTOR CL USTERING MACHINE LEARNING MODELS

				Count_L1
				Count_L2
	Fig. 7: Machine Learning Models Performance for three-level
	checkpoint model
	30.40%			30.39%
		30.22%
	30.20%	30.12%	30.16%
				30.05%
	30.00%	29.99%
	29.80%
		29.64%		29.65%
	29.60%	29.57%
			29.44%	29.44%
	29.40%	29.35%
	29.20%
	29.00%
	28.80%

Perofrmance Improvement Neural Network Performance Improvement vs Machine Learning Models

	60.0000%
				51.0520%
	50.0000%
	40.0000%
				33.1202%
	30.0000%	29.3348%	30.3900%		Count_L1
					24.1105%	Count_L2
	20.0000%				19.1765%
	10.0000%
	0.0000%
		Random Forest	Gaussian Naïve Bayes	Support Vector Clustering
	Fig. 9: Neural Network Models Performance Improvement vs
	Machine Learning Models for three-level checkpoint model
	prediction, it shows a performance improvement ranging from
	from 19% -33%.

ACKNOWLEDGMENT

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This material was based upon work supported by the U.S. Department of Energy, Office of Science, under contract DE-AC02-06CH11357. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. This work is also supported in part by the National Science Foundation awards 561041, 1564647, 1763547, and 1822737. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.