Javier Moreno
email: moreno@math.univ-lyon1.fr

Iterative differential Galois theory: a model theoretic approach

Keywords:

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem holds. In addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations.

Introduction

Differential Galois theory is the study of extensions of differential fields with well-behaved automorphism groups. In contrast to classic algebraic Galois theory, the automorphism groups of differential Galois extensions turn out to be algebraic groups or, in more general settings, differential algebraic groups. The origins of differential Galois theory trace back to the works of Picard and Vessiot in the 1890s studying linear differential equations, but it was Kolchin (following the work of Ritt) who gave the first systematic and modern account of the subject in his seminal book [START_REF] Kolchin | Differential Algebra and Algebraic Groups[END_REF]. There he introduced, among other things, the notion of a strongly normal extension of differential fields as a non-linear generalization of the classic extensions discovered by Picard and Vessiot. Modern treatments of Kolchin's theory have been offered in recent years by Magid [9], Kovacic [START_REF] Kovacic | The differential Galois theory of strongly normal extensions[END_REF] and Umemura [START_REF] Umemura | Galois theory and Painlevé equations[END_REF], among others. The current general reference on the subject is the book by Singer and van der Put [START_REF] Van Der Put | Galois Theory of Linear Differential Equations[END_REF].

For basic technical reasons regarding the nature of constants in positive characteristic, most of the work on differential Galois theory has been restricted to the context of characteristic zero differential fields, but several attempts have been made to work around this obstacle [START_REF] André | Différentielles non-commutatives et théorie de Galois différentielle ou aux différences[END_REF] [START_REF] Van Der Put | Galois Theory of Linear Differential Equations[END_REF]. One of the most popular strategies is replacing the notion of a derivative for that of an iterative (Hasse-Schmidt) derivation. This approach was extensively studied (but with partial success) by Okugawa [START_REF] Okugawa | Basic properties of differential fields of arbitrary characteristic and the Picard-Vessiot theory[END_REF] [START_REF] Okugawa | Differential Algebra of Nonzero Characteristic[END_REF] and Shikishima-Tsuji [START_REF] Shikishima-Tsuji | Galois theory of differential fields of positive characteristic[END_REF],

and more recently by Matzat and van der Put [START_REF] Matzat | Differential Galois theory in positive characteristic[END_REF], who developed a full Picard-Vessiot theory for positive characteristic iterative differential fields with the use of the theory of torsors.

Differential Galois theory and model theory have had a long history. It all started with Poizat's paper Une théorie de Galois imaginaire [START_REF] Poizat | Une théorie de Galois imaginaire[END_REF] suggesting that Kolchin's main results could be obtained as a consequence of the ωstability of the theory of differentially closed fields and some work by Zilber and Hrushovski on the definability of automorphism groups [START_REF] Hrushovski | Unidimensional theories are superstable[END_REF] [START_REF] Zilber | Totally categorical theories: structural properties and non-finite axiomatizability in Model theory of algebra and arithmetic[END_REF]. After this, several people including Marker, Pillay and Sokolović [START_REF] Pillay | Differential Galois theory III: Some inverse problems[END_REF][15] [START_REF] Pillay | Differential Galois theory I[END_REF][18] [START_REF] Pillay | Sokolović, Superstable differential fields[END_REF] have contributed to the development and even generalization of Kolchin's results making use of these abstract tools from model theory.

Positive characteristic differential Galois theory, however, remained out of reach of model theory until Messmer, Wood and Ziegler proved (strongly based on Delon's work on separably closed fields of positive characteristic [START_REF] Delon | Separably Closed Fields[END_REF]) that the theory of fields equipped with stacks of commuting iterative derivations has a stable model companion with quantifier elimination and elimination of imaginaries [START_REF] Messmer | Separably closed fields with higher derivations[END_REF] [START_REF] Ziegler | Separably closed fields with Hasse derivations[END_REF]. After this, adapting some of his previous results in characteristic zero, and working along the lines of what Hrushovski suggested in [START_REF] Hrushovski | Computing the Galois group of a linear differential equation[END_REF], Pillay found alternative proofs of existence and uniqueness of iterative Picard-Vessiot extensions (results already proved by Matzat and van der Put) using now model-theoretic techniques [START_REF] Pillay | Two remarks on differential fields[END_REF].

Following Pillay, this article introduces a theory of strongly normal extensions for iterative differential fields of positive characteristic. These extensions generalize the Picard-Vessiot extensions developed by Matzat and Van Der Put as well as the strongly normal extensions proposed by Okugawa. Our results depend on the model theory of iterative differential fields.

After introducing in section 2 the basic definitions and model theory of iterative differential fields of positive characteristic, in section 3 we define what we mean by an iterative strongly normal extension (Definition 3.1). Then we prove that the Galois group of these extensions is isomorphic to the constant-rational points of an algebraic group defined over the constants of the base field (Theorem 3.5), we have good Galois correspondence (Theorem 3.15), and also what Kolchin called a G-primitive element theorem (Theorem 3.17).

In section 4 we start all over again, this time from the perspective of (logarithmic) differential equations. This requires us to make an overview of the notion of the arc bundle of an algebraic variety (Definition 4.4) and introduce what should be our logarithmic derivation (Definition 4.8). Once there we define what we mean by an iterative differential Galois extension for a given logarithmic differential equation (Definition 4.11) and prove that these extensions exist and are unique modulo isomorphism (Theorem 4.12).

Finally, in section 5, we show that, under certain hypothesis on the base field, iterative strongly normal extensions and iterative differential Galois extensions are just two faces of the same notion (Theorems 5.1 and 5.2).

We assume that the reader has working knowledge of the fundamentals of geometric model theory, and a fair understanding of the terminology of varieties from algebraic geometry and basic differential algebra.

Iterative differential algebra: Model theory and practice

The aim of this section is offering a brief introduction to the basic model theory of iterative derivations.

D ∂ : R → R[[ǫ]] : a → ∞ i=0 ∂ i (a)ǫ i is a ring homomorphism.
If, additionally, for any i, j ∈ ω we have that

∂ i • ∂ j = i+j i ∂ i+j ,
we say that ∂ is an iterative Hasse-Shmidt derivation or simply an iterative derivation.

A ring (field) R equipped with an iterative derivation ∂ is what we call an iterative differential ring (field) or ID-ring (field) and its ring (field) of constants, C R , is defined as the set where all the ∂ i vanish.

We say that an ID-field

F is non-trivial if ∂ 1 | F ≡ 0.
Let IDF p be the first-order theory of fields of characteristic p > 0 equipped with an iterative derivation ∂. The language we will consider is that of fields expanded with a sequence (∂ i ) i<ω of unary function symbols. This theory has a model companion, SCH p , the theory of separably closed ID-fields, K, of characteristic p, degree of imperfection 1 (i.e. [K :

K p ] = p) and K p = {x ∈ K : ∂ 1 (x) = 0}.
Given a model K of SCH p , we can see that C F = K p ∞ , an algebraically closed field. This theory is, in some sense, just another version of SCF p,1 , the theory of separably closed fields of characteristic p and degree of imperfection 1: Fact 2.2. Once a p-basis is fixed, every model of SCF p,1 can be expanded to a model of SCH p and, additionally, any highly enough saturated model of SCH p can be canonically equipped with a p-basis and its corresponding λ-functions are quantifier-free definable.

Proof. See [START_REF] Ziegler | Separably closed fields with Hasse derivations[END_REF].

And, as a consequence of this, Fact 2.3. SCH p is stable (non-superstable) and has quantifier elimination and elimination of imaginaries.

Proof. See [START_REF] Ziegler | Separably closed fields with Hasse derivations[END_REF].

Since SCH p is stable, we may let (U, ∂) be a saturated model of SCH p of large cardinality and C its field of constants.

As in the case of characteristic zero differentially closed fields, the field of constants of U is a pure algebraically closed field, that is, any definable subset is definable in the language of rings. Note that in this case, though, C is not definable but type-definable:

Fact 2.4. If Z ⊂ U m is definable in U m over A, then Z ∩ C m is definable in (C, +, • ) over dcl(A) ∩ C.
Proof. By stability and quantifier elimination.

We will now define three different closure operators of algebraic nature. Definition 2.5. Let A ⊂ U.

• The iterative differential closure of A, denoted A , will be the iterative differential subfield of U generated by the elements of A. If F is an ID-field and A is a set, by F A we mean F A . • The strict closure of A, denoted A s , will be the set obtained after closing A under pth-roots. • Finally, the relative algebraic closure of A, denoted A a , will be the field theoretic algebraic closure of A inside U.

Benoist [START_REF] Benoist | Théorie des modèles des corps munis d'une dérivation de Hasse[END_REF] gave useful algebraic characterizations of the model theoretic definable and algebraic closures in U in terms of these closure operators:

Fact 2.6. Let A ⊂ U. • dcl(A) = A s • acl(A) = A a
Proof. See Proposition II.2 and Proposition II.3 of [START_REF] Benoist | Théorie des modèles des corps munis d'une dérivation de Hasse[END_REF].

Iterative strongly normal extensions

From now on, let us fix a prime number p > 0. As in the previous chapter, let U be a saturated model of SCH p of large cardinality where any ID-field mentioned is embedded and let C the field of constants of U.

Let us also assume that (F, ∂) is an iterative differential field with ∂ 1 | F ≡ 0.

3.1. Definition and basic properties. Definition 3.1. An extension (F, ∂) < (K, ∂) of non-trivial definably closed ID-fields is said to be strongly normal if the following conditions hold:

(1) C F = C K , and C F is algebraically closed;

(2) K = F a s (= dcl(F a)) for some a = (a 1 , • • • , a m );

(3) Whenever σ : K ֒→ U is an embedding of K into U over F , then σ(K) ⊆ K C ; and finally, (4)

F a ∩ K = F d s for some d = (d 1 , • • • , d m ).
Following Kolchin, the Galois group of the strongly normal extension K/F , denoted Gal(K/F ) will be Aut ∂ (K C /F C ). The traditional Aut ∂ (K/F ) will be denoted instead gal(K/F ).

Our definition of a strongly normal theory does not differ much from the original one due to Kolchin in the characteristic zero case. We require, though, one property that in Kolchin's case is automatic: tp(a/F ) should have finite multiplicity. This is going to be crucial to assure the definability of our Galois group. The extra-condition (4) will do this for us: Fact 3.2. If K = F a s is an iterative strongly normal extension of F , then tp(a/F ) has finite multiplicity. This is a corollary of the following general lemma: Lemma 3.3. Let T be a stable theory and U a highly saturated model of T . Then, for any a ∈ U and F ⊆ U, we have that tp(a/F ) has finite multiplicity if and only if there exists a finite tuple c ∈ U eq such that acl eq (F ) ∩ dcl eq (aF ) = dcl eq (F c).

Proof. ⇒) Let p = tp(a/F ) and let p 1 , . . . , p m be the complete extensions of p to acl(F ), and p 1 = tp(a/acl(F )). The finite equivalence relation theorem provides us with a single F -definable finite equivalence relation E distinguishing the extensions of p to acl(A). Let c be the E-class of a. By definition c ∈ dcl eq (F a), and, since E is an equivalence relation over F with finitely many classes, c ∈ acl eq (F ).

On the other hand, let b ∈ dcl eq (aF ) ∩ acl eq (F ). Thus we have b = f (a) for some F -definable function f . Let σ be an automorphism of U fixing F c. Suppose σ(p 1 ) = p 1 . Then σ(p 1 ) = p i , for some i = 1 and so σ(c) = c, contradicting the choice of σ. Thus, the formula

σ(b) = f (x) is still in p 1 . But then, σ(b) = f (a) = b. This implies that b ∈ dcl eq (F c). ⇐) Let d = Cb(
tp(a/ acl eq (F ))). We know that d ⊆ acl eq (F ) and it is also clear that d ⊆ dcl(F a). Thus d ⊆ dcl eq (F c) for some c ∈ acl eq (F ). But this implies that d has finitely many conjugates over A, and so tp(a/F ) has finite multiplicity.

One consequence of condition (3) in the definition of iterative strongly normal extensions is the fact that tp(a/F ) is internal to

C: If tp(b/F ) = tp(a/F ) then b ∈ K C = dcl(F, a, C).
The fact that the type has finite multiplicity makes this definability uniform, as we show next. Proof. When p = tp(a/F ) is stationary, this is a standard fact (see, for instance, Theorem 2.19, p. 37, of [START_REF] Poizat | Stable Groups[END_REF]). For the general case, find a function for each complete extension of p to acl(F ) and then glue them together.

3.2.

The Galois group is an algebraic group. This subsection is devoted to prove the following key result: Theorem 3.5. There is an isomorphism of groups

µ : Gal(K/F ) → G(C), where G is algebraic group in U defined over C F . Furthermore, the action of Gal(K/F ) on X = tp(a/F ) U is (F ∪ {a})-definable.
The following lemma and its corollary, both crucial in the proof of this theorem, clarify the nature of Gal(K/F ) and its relation with the associated strongly normal extension. In particular, the lemma tells us that gal(K/F ) < Gal(K/F ). Lemma 3.6. Any embedding of K into U over F can be uniquely extended to an automorphism of K C fixing C pointwise.

Proof. It is enough to show that for any a ′ ∈ U such that tp(a/F ) = tp(a ′ /F ), we have tp(a/F C ) = tp(a ′ /F C ). To see this, take σ : K → U an embedding of ID-fields fixing F . Since tp(a/F ) = tp(σ(a)/F ), our assumption tells us that tp(a/F C ) = tp(σ(a)/F C ) and this, by homogeneity of U and stability of the theory, provides us with σ, a U-automorphism fixing F C and taking a to a ′ . Note that σ|

K C : K C → σ(K) C . Note also that σ(K) C = K C .
In order to prove the claim, consider the infinite tuples aF and a ′ F , where tp(a ′ /F ) = tp(a/F ). Since C is algebraically closed and type-definable over the empty set, tp(aF/C) and tp(a ′ F/C) are, respectively, the unique nonforking extensions of tp(aF/C F ) and tp(a ′ F/C F ). However, tp(aF/C F ) and tp(a ′ F/C F ) are equal, and, in consequence, the same is true about tp(aF/C) and tp(a ′ F/C).

Corollary 3.7. X , the set of realisations of tp(a/F ) in U, is a principal homogeneous space for Gal(K/F ).

Proof. As X ⊂ K C , then Gal(K/F ) acts on X . The fact that the action on X is transitive and free is a direct consequence of the previous lemma. Consider the function µ : Gal(K/F ) → Y : σ → h(a, σ(a)). Corollary 3.7 then tells us that µ is a bijection. Endow Y with the group operation induced by µ. This is, let us define d

• d ′ = µ(µ -1 (d) • µ -1 (d ′ )
). Note that this group operation is definable. Moreover, the induced action of Y on X turns out to be F ∪ a-definable.

Finally, the fact that C is totally transcendental plus the Weil-Van den Dries-Hrushovski theorem (See Theorem 4.13, p. 84 of [START_REF] Poizat | Stable Groups[END_REF], or [START_REF] Van Den Dries | Weil's group chunk theorem: A topological setting[END_REF]) tells us that (Y, •) is definably isomorphic to the set of C-rational points of an algebraic group G defined over C F . Identify G(C) and Y .

In addition, µ takes gal(K/F ) to the C F -rational points of G.

Fact 3.8. µ(gal(K/F )) = G(C F ) Proof. First, observe that for any σ ∈ Gal(K/F ), we have that σ(a) ∈ F a, µ(σ) s and µ(σ) ∈ F a, σ(a) s ∩ C. Now, if σ(a) ∈ K, then µ(σ) ∈ F a s ∩ C = C K = C F .
On the other hand, if µ(σ) ∈ C F , then, σ(a) ∈ F a s = K.

3.3.

Scaffolding. In characteristic zero, the model theoretic approach to differential Galois theory [START_REF] Pillay | Differential Galois theory I[END_REF] heavily depends on the existence of prime models, a basic consequence of the fact that DCF 0 is totally transcendental. Since SCH p is only stable and not even superstable, we need to rely in other tools to deal with the lack of decent differential closure. Just like in the linear case [START_REF] Pillay | Two remarks on differential fields[END_REF], the use of a suitable auxiliary structure as a scaffolding to handle the group inside U will do the trick.

Definition 3.9. Let K/F be an iterative strongly normal extension, with K = F a s . Now define M as the two-sorted structure (X , C), with relations induced by F -definable relations in U.

Fact 3.10. Let N be the structure whose universe is C, with relations induced by the F a s -definable sets in U. Then (M, a) is bi-interpretable with N .

Proof. On one hand, any intersection of C and a F a s -definable set in U is, inside M, an a-definable set. The other direction depends on the definability of the Galois group from the previous section.

As in the proof of theorem 3.5, let us define Y as the quotient of Z by E, where Z is the type-definable set {c ∈ C : u(a, c) ∈ X } and E is given by the formula u(a, x 1 ) = u(a, x 2 ). By the proof of theorem 3.5 we know that Y is a C F -definable set in C.

Note that X and Y are isomorphic. Indeed, for each b ∈ X assign the class cb of c ∈ C such that u(a, c) = b. This map is one-to-one and onto by construction. Now, suppose that you have D ⊂ M ∩ X definable in (M, a). By definition, D is F a s -definable in U ∩ X . The map given between X and Y is also F a s -definable; thus the image of D, now inside the quotient set, is also F a s -definable. This makes D definable inside N . Now we can check that, although we are working in a stable, non-superstable theory, the auxiliary structure we built is totally transcendental. Fact 3.11. M is saturated and its theory Th(M) has quantifier elimination and is totally transcendental.

Proof. Let N be just as in fact 3.10.

Since N can be seen as C with names for the elements of C F , then it is saturated and totally transcendental. The fact that being totally transcendental and saturation are preserved under taking reducts and interpretability, allow us to conclude that M is also saturated and totally transcendental.

Since M is saturated, for quantifier elimination it is enough to prove that it is also quantifier-free homogeneous. Indeed, if d 1 and d 2 are two finite tuples from M with the same quantifier-free type, then, seeing them as tuples from U, they have the same type over F . The homogeneity of U then provide us with an automorphism of U over F taking one to the other. As it fixes F , this function is also an automorphism of M when restricted to its domain.

Given that Th(M) is totally transcendental, let M 0 be its prime model over the empty set. This structure will play the role of the differential closure of F . Lemma 3.12.

M 0 ∩ C = C F Proof.
Let c ∈ M 0 ∩ C and consider p, the type of c over the empty set in the language of M. Since M 0 is prime, p is isolated by a formula φ(x).

By lemma 2.4, the set that this formula defines inside C is also defined by a formula in the language of rings and with parameters in F ∩ C = C F . However, being algebraically closed, C F is an elementary substructure of C (in the language of rings), and so φ(C) ∩ C F is not empty. This implies that, as φ is an isolating formula over F , it must be of the form x = c ′ for some c ′ ∈ C F .

The following fact tells us that the whole extension can be somehow interpreted, in a multi-sorted way, inside M 0 , as if it were an scaffolding built on its side. Lemma 3.13. There is a bijection between the set of definably closed subsets of M eq 0 and the set of definably closed ID-fields lying between F and K. Proof. Any d ∈ M eq 0 is of the form a ′ /E where E, by quantifier elimination, is a quantifier free ∅-definable equivalence relation in M. This means that, in U, we have that E is the intersection of X and some F -definable set E ′ in U. By stability of SCH p,1 , it can be assumed that E ′ is also an equivalence relation. By elimination of imaginaries in SCH p,1 , we know that a ′ /E ′ is interdefinable over F in U with some tuple e ∈ dcl(F, a ′ ). Note that, since there is c ∈ C ∩ M 0 = C F such that a ′ = u(a, c), then, e ∈ dcl(F, a) = K. Thus, d is interdefinable over F in U with a tuple in K.

Let now e ∈ K. Then, e = f (a) for some F -definable function f . Let E(x, y) be f (x) = f (y). The restriction of E is ∅-definable in M 0 and d = a/E ∈ M eq 0 . Clearly, d (seen as an element in U) is interdefinable over F with e.

Galois correspondence and a G-primitive element theorem.

Let K a strongly normal extension of F and G the algebraic group whose C-rational points are isomorphic to Gal(K/F ), as provided by theorem 3.5. As in the previous subsection, let M the scaffolding built for the extension K/F and M 0 its prime model over the empty set.

(i) K/L is strongly normal. (ii) G L is a C F -definable subgroup of G(C) and is isomorphic to Gal(K/L). (

iii) The correspondence L → G L between intermediate ID-fields and C Fdefinable subgroups of G is an injection. (iv) L/F is strongly normal if and only if G L is a normal subgroup of G(C).

In this case, G(C)/G L ∼ = Gal(L/F ).

Before proving the theorem, let us observe that definably closed intermediate fields of a strongly normal extension are finitely generated over the base field. More precisely: Proof. Consider L as a definably closed subset in M eq 0 and let p = tp(a/L). Since T h(M) is totally transcendental, there is a finite tuple b such that p is the unique non-forking extension over L of tp(a/F b). This b is the tuple of the canonical bases of each of the finitely many complete extensions of p to acl(L). We claim F b s = L.

The left to right containment is clear. On the other hand, if e ∈ L, let g(•) an F -definable function such that g(a) = e. Consider the formula φ(x, y) defined as g(x) = y and let dφ x (y) be the tp(a/ acl(L))-definition of φ over dcl(F, b). Clearly, dφ x (e ′ ) iff e = e ′ , and so e ∈ dcl(F, b).

Proof of theorem 3.15. Let L = F b s .

(ii) and (iii) are easy. For (i), observe that as K/F is strongly normal, conditions (1), (2) (by lemma 3.16) and (3) of the definition of strongly normal extensions immediately hold in K/L. Condition (4) requires an explanation:

The correspondence provided by lemma 3.13 allows us to see D = acl(L)∩ K as a subset of M eq 0 . Consider then, in M, the type tp(a/D). Since K = L a s , the canonical base of tp(a/ acl(L)) is contained inside D. By ω-stability of T h(M), we have that Cb(tp(a/ acl(L)) is interdefinable with d, a single finite tuple in M eq . It is easy to check that L d s = D.

Finally, for (iv), suppose that L/F is strongly normal and let N be the normalizer of Gal(K/L) in Gal(K/F ). As both Gal(K/L) and Gal(K/F ) are definable, so is N . Thus, by part (3) of the present theorem, N = G L ′ for some L ′ such that F < L ′ < L. We will prove that N = Gal(K/F ):

Let σ ∈ Gal(K/F ), an automorphism τ ∈ Gal(K/L) and d ∈ L. Since L/F is strongly normal, σ(d) ∈ L C and so τ σ(d) = σ(d). This implies that σ -1 τ σ(d) = d and thus we conclude that σ -1 τ σ ∈ Gal(K/L) and, moreover, σ ∈ N as τ ∈ Gal(K/L) is arbitrary.

Assume now that G L = Gal(K/L) is a normal subgroup of Gal(K/F ). Conditions (1), ( 2) and ( 4) from the definition of strongly normal extensions are clear for L/F . We need to prove (3): Let σ : L ֒→ U be an embedding of L into U over F . Because of saturation of U and quantifier elimination, there is σ an embedding of K into U over F such that σ = σ| L . Since K/F is strongly normal, σ can be seen as an element of Gal(K/F ) (lemma 3.6). Let d ∈ L, we need to show that σ(d) ∈ L C : Consider τ ∈ Gal(K/L) and observe that, as Gal(K/L) is normal in Gal(K/F ), we have that

σ-1 τ σ(d) = σ -1 τ σ(d) = d. That is, τ (σ(d)) = σ(d).
As τ is arbitrary, this implies that σ(d) belongs to the set fixed by Gal(K/L), which is precisely L C .

Finally, consider the restriction map

| L C : Gal(K/F ) → Gal(L/F ).
This map is onto because of saturation of U and quantifier elimination, and its kernel is Gal(K/L). This implies that G(C)/G L ∼ = Gal(L/F ).

Theorem 3.17. Let K/F be a strongly normal extension of ID-fields. Suppose F is relatively algebraically closed in U. Then, there is α ∈ G(K) such that K = F α , and for all σ ∈ Gal(K/F ), we have that (⋆) Let α = h(a, d). Observe first that α is interdefinable with a over F (because of the way h is defined) and so K = F α . Additionally, as a, d ∈ K, we have that α ∈ K. Finally, let σ ∈ Gal(K/F ) and pick c | ⌣F σ(a). Then tp(a, c/F ) = tp(c, σ(a)/F ) by stationarity. So

σ(α) = (µ(σ)) -1 • α.
h(c, σ(a)) • h(σ(a), d) = h(c, d).
But this, combined with (⋆) implies that

h(a, c) • h(c, σ(a)) • h(σ(a), d) = h(a, d). Now, h(a, c) • h(c, σ(a)) = µ(σ) and h(σ(a), d) = σ(α) (because σ(d) = d). So, we have µ(σ) • σ(α) = α.
Which is what was left to prove.

Iterative differential Galois extensions

In the previous section we introduced a class of ID-field extensions with well behaved Galois groups. Now, we will concentrate on the differential equations that under suitable conditions have good Galois theory.

[ǫ]/(ǫ m+1 ). View k (m) as a k-algebra under the natural map a → a + 0ǫ + • • • + 0ǫ m .
Definition 4.2. Let X be an algebraic variety over F and define, for any field k extending F the m th arc bundle of X over k, denoted A m X(k), as the set of k (m) -rational points of X. This can be seen as an actual algebraic variety by identifying points in k (m) with points in (k m+1 ). Now, if f : X → Y a (regular) map of algebraic varieties over F , then define

A m (f ) : A m X → A m Y
as the map which is given, on k-points, by evaluating f on X(k (m) ).

Fact 4.3. Let X, Y and Z be algebraic varieties over

F . (1) A m (X) × A m (Y ) is naturally isomorphic to A m (X × Y ).
(2) Suppose f : X → Y and g : Y → Z are regular maps defined over

F , then A m (g • f ) = A m (g) • A m (f )
. This is, A m is a functor from the category of algebraic varieties with regular maps over F to itself. ( 3) If (G, • ) is an algebraic group defined over F , then so is

(A m G, A m ( • )).
Proof. (1) and ( 2) are immediate consequences of the given definition of A, and (3) follows from those two. For instance, for associativity, consider the commutative diagram, (g 1 , g 2 , g 3 )

/ / _ (g 1 • g 2 , g 3 ) _ (g 1 , g 2 • g 3 ) / / (g 1 • g 2 ) • g 3 = g 1 • (g 2 • g 3 )
and then apply A m .

For n > m, the quotient map

k (n) → k (m) induces a projection ρ n,m : A n X → A m X.
Identifying A 0 with the identity functor we will write ρ n,0 as ρ n .

Definition 4.4. For a ∈ X(k), the n th arc space A n X a of X at a is defined as the fibre of ρ n : A n X → X over a. Additionally, define AX(k), the full arc bundle of X over k, as the inverse limit of (A i X(k)) i∈ω .

Observe that AX(k) can be identified with the k[[ǫ]]-rational points of X.

Let us consider now the case when F is a ID-field of positive characteristic. As before, assume U is a highly saturated model of SCH p,1 .

Lemma 4.5. Let X be an algebraic variety defined over

C F . If a ∈ X(U), then ∇ X (a) = (∂ 0 (a), ∂ 1 (a), . . .) ∈ AX(U) and in particular ∇ X,m (a) = (∂ 0 (a), . . . , ∂ m (a)) ∈ A m X(U) for any m.
Additionally, if Y is an algebraic variety, f : X → Y is a morphism and both are also defined over

C F , then A(f ) • ∇ X = ∇ Y • f .
Proof. Before starting, observe that ∇ X (a) is just another way of presenting D ∂ (a) once you identify, as suggested above, AX(U) and

X(U[[ǫ]]).
For the first part, recall that

D ∂ : U → U[[ǫ]] : x → ∞ i=0 ∂ i (x)ǫ i is a ring homorphism.
Then, working locally, if p(x) is one of the defining polynomials of X and a ∈ X(U), then p(D ∂ (a)) = 0. Which is another way of saying that ∇ X (a) ∈ AX(U).

For the second part, note that D ∂ is not only a ring homomorphism but a C-algebra homomorphism. Thus, again locally, if q(x) is a polynomial with constant coefficients, then D ∂ (q(x)) = q(D ∂ (x)).

Corollary 4.6. If (G, •) is an algebraic group defined over the constants of F , then (AG, A(•)) is also a group and ∇ G : G → A(G) is a group embedding.

Proof. Since (G, • ) is an inverse limit of algebraic groups, it is a proalgebraic group. For the second part, the previous lemma gives us that

∇ G • • = A( • ) • ∇ G×G .
Let G be an algebraic group defined over C F and consider the following exact sequence of groups:

{(e, 0)} / / A e (G) i / / A(G) π / / G / / {e},
where i is the natural inclusion and π the canonical projection. Note that s : G → A(G) : g → (g, 0, 0, . . .) is a group embedding and so a homomorphic section of π. Recall also that the existence of such a homomorphic section provides us with an isomorphism A(G) ∼ = A e (G) ⋊ G. Thus, we can identify G with its image under s. Let h : A(G) → A e (G) be the projection induced by this isomorphism. Given (g, u) ∈ A(G), we have that (g, u) = ((g, u) • s(g -1 )) • s(g), so h((g, u)) = (g, u) • s(g -1 ). Although h is not a group homomorphism, we have:

Fact 4.7. If h((g, u)) = h((l, v)) then h((g, u) -1 • (l, v)) = (e, 0).
Proof. Since h((g, u)) = h((l, v)), then, by definition, (g, u) • s(g -1 ) = (l, v) • s(l -1 ).

Reorganizing the equation, we get

(g, u) -1 • (l, v) = s(g -1 ) • s(l).
Now, applying h to both sides, we obtain Fact 4.9. If G is defined over C F , then Ker(ℓD) = G(C). Furthermore, if ℓD(x) = ℓD(y), then x -1 • y ∈ G(C).

h((g, u) -1 • (l, v)) = h(s(g -1 l)) = s(g -1 l) • s(l -1 g) = (e, 0).
Proof. If ℓD(g) = (e, 0), then ∇(g) • (g -1 , 0) = (e, 0). Thus ∇(g) = (g, 0), which implies that ∂ i (g) = 0 for any i. That is, g ∈ G(C).

The additional remark is a direct consequence of fact 4.7.

The logarithmic derivative in the (differential) characteristic zero case, defined as a map from G to L(G), the Lie algebra of G, is surjective. In our setting, that is not the case: Thus, Im(ℓD) is contained in the set {(x i ) : x 0 = 0 and, for i > 0, ∂ j (x i ) = i + j i (x i+j )}, which is clearly not equal to A e (G). where ℓD is defined as in the previous section and α ∈ AG e (F ) is an element contained in the image of ℓD.

By an iterative differential Galois extension of F for that given logarithmic differential equation we mean K = F a s , where ℓD(a) = α and C F = C K . Theorem 4.12 (Existence and Uniqueness of iterative differential Galois extensions). If G is an algebraic group defined over the constants of (F, ∂) and ℓD(x) = α is a (consistent) logarithmic differential equation over F , then there exists an iterative differential Galois extension of F for the given equation. Furthermore, any two such extensions are isomorphic over F as ID-fields.

Once again, we will depend on the use of an appropriate auxiliary structure in order to prove this. Let M be the two-sorted structure (X , C), where X is the set of solutions in U of the equation ℓD(x) = α, and the relations of M are those induced by F -definable sets in U. Lemma 4.13. M is saturated, its theory Th(M) has quantifier elimination, it is totally transcendental and, additionally,

M 0 ∩ C = C F .
Proof. Just as in the proof of fact 3.11, this depends on the bi-interpretability of an expansion of M by a constant and another simpler structure. Let a ′ be any solution of the given logarithmic differential equation and let N be the structure whose universe is C and whose relations are induced by the F a ′ -definable sets in U. We will see that N is bi-interpretable with (M, a ′ ):

Let Y = G(C). As a subset of N , we have that Y is definable. Observe that there is a one-to-one correspondence between X and Y . This is given by the fact that, for any b ∈ X , there is g ∈ G(C) such that g • a ′ = b (a corollary of fact 4.7). Note that such g is unique given b and a ′ . Let f : Y → X : g → g•a ′ . The fact that X and Y are isomorphic via this function is proved as in fact 3.10. This shows that X (and so M) is interpretable in N . The fact that N is interpreted in (M, a) is clear.

Note that N is, once again, totally transcendental and saturated, and thus the argument used to prove fact 3.11 applies. Hence, M is also saturated and totally transcendental. The same is true for proving that M has quantifier elimination.

The proof that M 0 ∩C = C F is exactly the same given for lemma 3.12.

We now go back to the proof of theorem 4.12:

Proof of theorem 4.12. (Existence) Let a ∈ M 0 ∩ X and K = F a s . It is not hard to see that C F = C K .

(Uniqueness) Let a ′ ∈ X be such that K ′ = F a ′ s is another iterative differential Galois extension of F for the given equation.

Consider M 1 prime over a ′ . Note that M 1 ∩ C = C F . Let a ′′ ∈ M 1 with the same type as a over the empty set. As M 1 ≺ M, there is g ∈ M 1 ∩ C = C F such that a ′′ • g = a ′ . This implies that dcl(F a ′ ) = dcl(F a ′′ ).

Finally, since tp(a) = tp(a ′′ ) in M, then tp(a/F ) = tp(a ′′ /F ) in U. This, by saturation, implies that dcl(F a) is isomorphic to dcl(F a ′′ ) = dcl(F a ′ ).

What goes around comes around

Section 3 introduced a class of extensions of differential fields with good Galois theory. Section 4 provided us with extensions of differential fields related to iterative logarithmic differential equations. In this section we will prove that, under certain conditions on the base field, these two notions coincide.

Theorem 5.1. If K is an iterative differential Galois extension of F for a given logarithmic differential equation ℓD(x) = α, then K/F is a strongly normal extension.

Proof. The first two conditions of the definition of strongly normal extensions are explicitly stated in our definition of ID-Galois extensions. For the third one, let K = F a s , and tp(a ′ /F ) = tp(a/F ). Since α ∈ AG e (F ) and lD(a) = α, we have that ℓD(a ′ ) = α, and this implies that a -1 a ′ ∈ G(C) by fact 4.9. So, a ′ = a • d for some d ∈ G(C) and thus a ′ ∈ K C . Finally, for the fourth condition, the argument goes exactly as in the proof of the first part of theorem 3.15.

Theorem 5.2. Suppose F is relatively algebraically closed in U and K/F is a strongly normal extension. Let G be the algebraic group over C F that is provided by theorem 3.5 whose set of C-rational points is isomorphic to Gal(K/F ). Then K/F is an iterative differential Galois extension for some logarithmic differential equation on G.

Proof. Let K/F be a strongly normal extension and G as in the statement. Let µ : Gal(K/F ) → G(C) witness the isomorphism.

Since F is relatively algebraically closed in U, the primitive element theorem provides us with a ∈ G(K) such that K = F a and, for any σ ∈ Gal(K/F ), we have that σ(a) = (µ(σ)) -1 • a.

Let α = ℓD(a) and note that α is fixed by any automorphism of U fixing F : let ξ ∈ Aut(U/F ); since K/F is strongly normal, lemma 3.6 tells us that ξ = ξ| K C ∈ Gal(K/F ), and so ξ(a) • a -1 = (µ(ξ)) -1 ∈ G(C) = Ker(ℓD). Thus, ξ(α) = ℓD(ξ(a)) = ℓD(a) = α.

Since F is relatively algebraically closed in U, we get that α ∈ AG e (F ). Consider the iterative logarithmic differential equation ℓD(x) = α. Let K ′ = dcl(F a ′ ) be the unique iterative differential Galois extension of F for the given equation. Note that, since ℓD(a) = ℓD(a ′ ), fact 4.9 tells us that there exists g ∈ G(C) such that a ′ = g -1 • a. Since g ∈ G(C) there is σ ∈ Gal(K/F ) such that µ(σ) = g. So, by the way a was chosen, a ′ = ((µ(σ)) -1 • a = σ(a), which implies that σ induces an isomorphism between K and K ′ . Thus K is isomorphic to a Galois differential extension of F for an appropriate logarithmic differential equation.

To conclude, let us prove, under the same assumption on the base field, that a desirable equality between the transcendence degree of a strongly normal extension and the dimension of its Galois group holds. To be precise: Theorem 5.3. Suppose F is relatively algebraically closed in U, the extension K/F is strongly normal, and G is an algebferaic group over C F such that G(C) is isomorphic to Gal(K/F ). Then, dim(G(C)) = tr. deg(K/F ).

Proof. By the previous result, we know that K is an iterative differential Galois extension of F for a consistent logarithmic differential equation ℓD(x) = α on G(U) with α in F . That is, there is a ∈ G(K) such that ℓD(a) = α and K = F a s .

First note that K = F (a) s . To prove this, it is enough to see that ∂ n (a) ∈ F (a) for all n ∈ ω. However, by the definition of the logarithmic derivative, we know that ∇(a) = α • (a, 0, 0, . . .), thus coordinate by coordinate, we obtain that ∂ n (a) is a rational function of a and the coefficients of α, which are all in F . This in particular implies that tr. deg(K/F ) = tr. deg(F (a)/F ).

Secondly, by fact 4.9, we know that for each a ′ , a ′′ ∈ X , where X is the solution set in G(U) of the equation, there is g ∈ G(C) such that g • a ′ = a ′′ . Since G(C) is precisely our isomorphic copy of Gal(K/F ), this implies that, in U, any a ′ ∈ X has the same type as a over F . Thus, tr. deg(K/F ) is in fact equal to tr. deg(X ).

Finally, observe that, after naming a ∈ X , there is a rational bijection between X and G(C) given by a ′ → a -1 •a ′ . So, tr. deg(K/F ) = tr. deg(X ) = tr. deg(G(C)) = dim(G(C)).

Definition 2 . 1 .

 21 Let R be an arbitrary ring. A sequence of maps ∂ = (∂ i : R → R) i∈ω is called a Hasse-Schmidt derivation if ∂ 0 = id R and the map

Lemma 3 . 4 .

 34 If K = F a s is an iterative strongly normal extension of F , then there exists a function defined over F , let us call it u(•, •), such that for every b with tp(b/F ) = tp(a/F ), there is c ∈ C such that u(a, c) = b.

  Proof of theorem 3.5. This is a modified version of the general argument for proving the definability of the binding group.Let Y = Z/E where Z = {c ∈ C : u(a, c) ∈ X } and E is an equivalence relation on Z defined by the formula u(a, x 1 ) = u(a, x 2 ). Because of elimination of imaginaries of the theory of algebraically closed fields and the pureness of C, Y is a type-definable set in C over dcl(a) ∩ C ⊂ C F . For b ∈ X and d ∈ Y , define f (b, d) = u(b, c) with c ∈ Y 0 such that c/E = d, and note that for any b 1 and b 2 ∈ X , there is only one d ∈ Y such that f (b 1 , d) = b 2 .

Definition 3 . 14 .

 314 Given L a definably closed subfield of K containing F , let G L = {g ∈ G(C) : g(c) = c for all c ∈ L}. Theorem 3.15. Let K/F be a strongly normal extension of ID-fields. If L is a definably closed intermediate ID-field in K/F , then:

Lemma 3 . 16 .

 316 If K/F is strongly normal and L is an intermediate definably closed ID-field, then L = F b s for some b = (b 1 , • • • , b m ).

Proof.

  Let b, c ∈ U with tp(b/F ) = tp(a/F ) = tp(c/F ) such that a | ⌣F b, a | ⌣F c and c | ⌣F b. Using the notation from the proof of 3.5, it is clear that h(a, b), h(a, c), h(c, b) ∈ G(U) and h(a, c) • h(c, b) = h(a, b). Replacing a, b, c by a, b, c plus finitely many derivations and pth-roots, we may assume h is rational. Since b is algebraically independent of a and c over F and F is relatively algebraically closed in U, we can find d ∈ F such that h(a, d), h(c, d), h(a, c) ∈ G(U), and h(a, c) • h(c, d) = h(a, d).

4. 1 .

 1 Arc bundles and the iterative logarithmic derivative. Definition 4.1. For a natural number m and an arbitrary field k, define k (m) as the ring k

Definition 4 . 8 .

 48 Define the iterative logarithmic derivative be the map ℓD : G(U) → AG e (U) : g → h(∇(g)).

Example 4 . 10 .

 410 Let G = G a , the additive group. Then A(G) = ∞ i=0 G a and ℓD(g) = (g, ∂ 1 (g), . . .) -(g, 0, • • • ) = (0, ∂ 1 (g), ∂ 2 (g), . . .).

4. 2 .Definition 4 . 11 .

 2411 Logarithmic differential equations and Galois extensions. Given an algebraic group defined over the constants of (F, ∂) a (non-trivial) definably closed ID-field of characteristic p with algebraically closed constant field, by a consistent logarithmic differential equation over F we mean something of the form ℓD(x) = α,


Jean-Baptiste Courbot
email: jean-baptiste.courbot@uha.fr.

Bruno Colicchio

Transformed Gaussian random fields for unsupervised image deconvolution

Keywords: Bayesian Deconvolution, Transformed Gaussian Markov Random Fields, Expectation-Maximization, Hamiltonian Monte Carlo

This paper deals with the problem of Bayesian deconvolution. Starting from the classical Gaussian Markov Random Fields (GMRF) prior, we present a broader model referred as transformed GMRF (TGRF) in which the latent field results from a nonlinear transformation of a GMRF. We propose a Bayesian inference method to estimate TGRF from an observed image with known parameters, and introduce methods inspired from expectation-maximization in order to jointly deconvolve and estimate the statistical model's parameter for both GMRF and TGRF. Numerical results allow to determine the best inference method among several possibilities on fully synthetic data, on a phantom image and on real fluorescence microscopy images.

I. INTRODUCTION

A. Generalities I N this paper, we investigate the problem of non-blind image deconvolution within a Bayesian context. The problem is ill-posed and has, on one hand, received a lot of attention with several kind of priors, unknowns, and image formation models (see e.g. the tools provided in [START_REF] Sage | DeconvolutionLab2: An open-source software for deconvolution microscopy[END_REF]). On the other hand, originating from the spatial statistics, Gaussian Markov Random Fields (GMRF) have been proven powerful to solve inference problems involving regression [START_REF] Chilès | Fifty years of kriging[END_REF] over a spatially organized domain -namely, 2D images or 3D volumes. Therefore, using GMRF as priors to solve the deconvolution problem seems natural, and direct inference of the latent variable is even possible under suitable conditions.

GMRF were recently used in a deconvolution / segmentation context, in which they were used to model large interpixel correlation within observations, hiding a discrete latent field [START_REF] Gangloff | Unsupervised image segmentation with Gaussian pairwise Markov fields[END_REF] [START_REF] Vacar | Unsupervised joint deconvolution and segmentation method for textured images: a Bayesian approach and an advanced sampling algorithm[END_REF]. In this paper, we are interested in the use of random fields as priors on the image to infer. This includes GMRF and extends to transformed GMRF -referred to as TGRF in the following (see Fig. 1).

TGRF have made various appearances in the literature, see e.g. [START_REF] Prates | Transformed Gaussian Markov random fields and spatial modeling of species abundance[END_REF] for an introduction, [START_REF] Victor De Oliveira | Bayesian prediction of transformed Gaussian random fields[END_REF] for inference or [START_REF] De | A note on the correlation structure of transformed Gaussian random fields[END_REF], [START_REF] Li | Stochastic representation and dimension reduction for non-Gaussian random fields: review and reflection[END_REF] for further statistical studies. Overall, they were studied either in the geostatistical context, or more theoretically. To our knowledge, neither GMRF or TGRF were used for unsupervised image deconvolution.

Akin to TGRF exists the even broader family of non-Gaussian [START_REF] Suuronen | Cauchy Markov random field priors for Bayesian inversion[END_REF], possibly non-stationary, random fields [START_REF] Fuglstad | Exploring a new class of non-stationary spatial gaussian random fields with varying local anisotropy[END_REF], Fig. 1: GMRF and TGRF realizations with a Gaussian or exponential correlation function. LogNRF are log-normal random fields with ϕ the exponential function, and LogitNRF are logit-nomal random fields with ϕ the sigmoid function. [START_REF] Montoya-Noguera | Simulation of non-stationary non-Gaussian random fields from sparse measurements using Bayesian compressive sampling and Karhunen-loève expansion[END_REF]. While having a larger modeling capacity, these models are in general difficult to sample and infer from, mainly because of intricate correlation structures. Thus, we focus in this paper on TGRF who appear as a tradeoff between modeling and inference capacities.

The remaining of this letter is as follows: the end of the introduction recalls results on GMRF and describes the inference model. Then Section II introduces the inference methods used for (un)supervised deconvolution, supported by numerical results in Section III on synthetic and real images.

B. A primer on GMRF

A GMRF X ∈ R S follows a multivariate Gaussian distribution of mean µ ∈ R S and covariance matrix Σ ∈ R S×S . We also refer to its inverse Σ -1 = Q. One of the key assumptions enabling inference with GMRF is that the image lies on a torus, making Σ and Q circulant matrices. Indeed, let us denote b Q , b Σ ∈ R S the basis of Q and Σ respectively:

• b Q = IDFT(1 ⊘ DFT(b Σ ))
, with DFT and IDFT the discrete Fourier transform and its inverse, and ⊘ denoting element-wise division.

• ∀x ∈ R S , Qx = IDFT(DFT(b Q )⊙DFT(x)) = x * b Q ,
with ⊙ the element-wise product and * the convolution operator. This implies that there is no need to compute or store neither Q nor Σ as their base is sufficient enable all computations through the Fourier domain. Note that the above, written for vectors within R S , generalizes to images or volumes, in which case the Q and Σ are block-circulant with circulant blocks. These helpful operations enable sampling and inference for GMRF, which will constitute the algorithmic basis of the methods presented in the paper.

C. Observation model and joint distribution

We assume the image formation model is described by:

y = Hz + ϵ (1)
where y ∈ R S is the observed image, H ∈ R S×S is the known convolution matrix, ϵ ∈ R S is a realization of a centered Gaussian noise (of covariance Σ ϵ ), and z ∈ R S is the realization of a random field. We also assume that z results from the transformation of a GMRF x, so that z = ϕ(x), with ϕ : R S -→ R S an invertible and derivable function. Examples of realizations of z are given in Fig. 1.

x being a GMRF, p x (x) ∼ N (x; µ, Σ) with its mean µ and covariance matrix Σ. The latter depicts correlation between any two sites a and b in the image, typically of the form:

Σ a,b = σ 2 x ρ(a, b; r) = σ 2 x exp - 1 r p ∥a -b∥ p t (2)
when using a Gaussian (p = 2) or exponential (p = 1) basis 1 for correlation. ∥ • ∥ t represents the distance on the torus, while σ x and r are the variance and correlation range. Thanks to the multivariate change of variable theorem, one can express the density of z through the density of x so the prior distribution is:

p(z) = J ϕ -1 (z) p x ϕ -1 (z) (3)
where J ϕ -1 is the Jacobian of ϕ -1 and | • | the determinant. Using in addition the likelihood yields the posterior distribution:

p(z|y) ∝ |Σ ϵ | -1 2 exp - 1 2 (y -Hz) ⊤ Σ -1 ϵ (y -Hz) (4)
× J ϕ -1 (z) |Σ| exp - 1 2 ϕ -1 (z) -µ ⊤ Σ -1 (ϕ -1 (z) -µ)
Preliminary experiments showed that such a distribution is numerically difficult to handle, because ϕ -1 may have a bounded support and diverging gradient near boundaries. So, we will instead perform inference on and compute gradient with respect to x; for which the posterior distribution is:

p Θ (x|y) ∝|Σ ϵ | -1 2 exp - 1 2 (y -Hϕ(x)) ⊤ Σ -1 ϵ (y -Hϕ(x)) × |Σ| -1 2 exp - 1 2 (x -µ) ⊤ Σ -1 (x -µ) (5)
in which the prior and likelihood terms are clearly identifiable. This posterior, as well as the joint distribution, is not Gaussian in the general case because ϕ is not assumed linear. Besides, the model is parametrized by the following unknowns:

Θ = {µ, σ z , r, Σ ϵ } (6)
Then, to deconvolve one must jointly estimate z = ϕ(x), together with Θ, knowing only y and H.

II. BAYESIAN INFERENCE

In this section we introduce the main methods to be used to infer x within the observation model detailed in Section I-C. At first, we describe how to estimate z in a supervised fashion (when Θ is known); and then we propose a method to estimate both Θ and z in an unsupervised fashion from y and H.

A. Supervised inference GMRF case. The convolution is linear, and the noise ϵ is Gaussian with covariance Σ ϵ , so when z is a GMRF realization 1 Note that other correlation basis exist, such as a Matérn kernel.

(ϕ = Id), its posterior distribution is also normal:

p Θ (z|y) = N (z; µ * , Q * ), with (7)
µ * = Q * -1 (Σ -1 µ + H ⊤ Σ -1 ϵ y) Q * = Σ -1 + H ⊤ Σ ϵ H.
This also implies that µ * is the MAP estimator.

TGRF. When z is not a GMRF, there is no direct estimator available so one must resort to approximate sampling of the posterior p Θ (x|y) [START_REF] Chilès | Fifty years of kriging[END_REF]. A popular method for sampling posteriors is based on the Metropolis-Hastings algorithm [START_REF] Chib | Understanding the Metropolis-Hastings algorithm[END_REF], and its more efficient extension coined Hamiltonian Monte Carlo (HMC) [START_REF] Betancourt | A Conceptual Introduction to Hamiltonian Monte Carlo[END_REF]. HMC iteratively repeats the following steps:

• Make a proposal x proposal based on the previous state using a leapfrog integrator, that make proposals based on the gradient of the target distribution,

• With probability min 1, p Θ (x proposal |y) p Θ (x previous |y) , keep the proposal in the chain. Note that one could consider at this stage variants of HMC under the Riemann manifold perspective as introduced in [START_REF] Girolami | Riemann manifold Langevin and Hamiltonian Monte Carlo methods[END_REF]. However, it involves heavy computations notably regarding the inversion of the Fisher information matrix, making such approach impractical in high dimension. Let us now specify a few practical points regarding the HMC sampling of (5).

Disturbance distribution. Most MH and HMC cookbooks propose a simple disturbance to sample the proposal from the current state, typically in the form of a white Gaussian perturbation. Because the target distribution ( 5) is highly correlated, this is not adequate, and we use instead the prior distribution of x to sample a proposal. This prior distribution is also used to evaluate the "momentum" of the auxiliary variables used in HMC.

Leapfrog parameters. The leapfrog integrator is based on a disturbance amplitude, and a number of steps. Choosing both may impact the HMC outcomes and has been the topic of several works (see e.g. [START_REF] Wu | Faster Hamiltonian Monte Carlo by learning leapfrog scale[END_REF]) while being case-dependent. To overcome these limitations, we use the no U-turn sampler (NUTS) described in [START_REF] Matthew D Hoffman | The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo[END_REF].

Convergence evaluation. NUTS allows to efficiently sample along the target distribution, but one HMC chain is not sufficient to assess convergence, so we run M HMC chains of length N . At fixed intervals, we estimate the potential scale reduction R of L = log p Θ (x|y) defined as [START_REF] Andrew | Bayesian data analysis[END_REF]Sec. 11.4]:

R = N -1 N + 1 N B W (8)
with W and B the within-and between-sequence variances. This quantity decreases to 1 when N → ∞. Once R is low enough, we assume the convergence is attained, stop, and use the average over M chains as an expectation estimation. The method we use for supervised inference is formalized in Alg. 1.

B. Unsupervised inference

In most applications, Θ is unknown, so it must also be estimated within the inference. A possible approach consists in writing a joint distribution for x, y, and Θ; which requires priors on these parameters, hyperparameters to tune these priors, and the ability to compute the gradient of log(p Θ (x|y)) w.r.t. each component of Θ if we were to use an HMC framework. On the other hand, given a realization of (x, y), estimating Θ through maximum likelihood and/or least-squares is a viable alternative, which is equivalent to sampling along a wellpositioned Dirac distribution. This supports the design of algorithms similar in spirit to EM [START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF] or SEM [START_REF] Celeux | Stochastic versions of the em algorithm: an experimental study in the mixture case[END_REF], for which the expectation/simulation of the latent x alternates with the reestimation of Θ. The latter is made with maximum-likelihood estimators when available (mean µ, variance σ z , covariance Σ ϵ ) and least squares for r. Because the expectations are are not exactly available, and because we do not use only maximum-likelihood estimators, we refer to these methods as quasi-EM (qEM) and quasi-SEM (qSEM). qEM is expected to be less stochastic than qSEM ; while qSEM may be more able to avoid local maxima, so both must be tested. Because we do not have theoretical convergence guarantees, we must resort to measuring the variations of Θ to numerically assert the qEM and qSEM convergence. Both methods are reported in Alg. 2. Summarizing, we propose for unsupervised inference to firstly estimate Θ with Alg. 2 then to estimate ẑ = ϕ(x) with x estimated from Alg. 1 using Θ. The Python code will be made available at github.com/courbot/tgrf.

III. NUMERICAL EXPERIMENTS A. Synthetic data

In a first experimental setup (denoted Exp. 1 afterwards), we sample y using a logit-normal random field (LogitNRF) Algorithm 2 Quasi-(S)EM algorithm for parameter estimation in GMRF and TGRF Require: y, H, parameters (M , N , R stop ) of Alg. 1 Ensure: Θ Initialization: start from a pair x (0) et Θ (0) . repeat (iteration t): if qEM then ◁ Expectation step x (t) = ÊΘ (t-1) [x|y] through [START_REF] De | A note on the correlation structure of transformed Gaussian random fields[END_REF] or Alg 1. else qSEM ◁ Sampling step Sample x (t) ∼ p Θ (t-1) (x|y) from [START_REF] De | A note on the correlation structure of transformed Gaussian random fields[END_REF] or from a single chain in Alg 1. Estimate Θ (t) from (y, x (t) ).

◁ Maximization step until {Θ (t) } is stable enough Θ is estimated by averaging the last elements of {Θ (t) }.

model, where ∀s, ϕ(x s ) = 1 1+exp(-xs) together with a Gaussian correlation function (see Fig. 1). The image is then blurred by an Airy pattern, parametrized by a Full-Width at Half Maximum (FWHM); and corrupted by an additive white Gaussian noise of variance σ ϵ . The latter's intensity is measured by the signalto-noise ratio defined as SNR = 20 log 10 (σ ϵ /std(x)). We evaluate the performance of Algs. 2 and 1 and under varying SNR and FWHM, using either a GMRF or a LogitNRF model, in the qSEM and qEM variants. Performances are evaluated in term of normalized mean absolute error (NMAE), that is the mean absolute error scaled by the standard deviation of the reference image x. The results are summarized in Figure 3a: • LogitNRF yields better results than GMRF. While this may seem unsurprising, this was not taken as granted because GMRF yield often interesting result, even when applied on non-GMRF images. This can be seen as a choice between using exact posterior and an approximate model (GMRF) or using an approximate posterior with an exact model (LogitNRF). The numerical results suggest the second option is the finest. • On average, qEM is equivalent to qSEM. Since computing parallel MCMC chains (Alg. 1) is time-consuming to repeat, we retain qSEM in the remaining of this section.

In a second experimental setup (denoted Exp. 2), we broaden the study to permit the comparison of log-normal random fields (LogNRF) and to use either Gaussian or exponential correlation functions (Eq. ( 2)). We study then a more intricate object mixing all the 6 models at hand (GMRF, LogitNRF, LogNRF; with Gaussian or exponential correlation functions) to form a cell phantom as depicted in the left column in Fig. 2. For comparison, we also performed deconvolution using total variation (TV) regularization terms, using the homotopy-based method described in [6, App. D]. We denote it as H-TV1 and H-TV2 depending on the exponent used in the TV regularization. We also performed deconvolution based on the recent Cauchy Markov Random Field (Cauchy MRF) prior described in [START_REF] Suuronen | Cauchy Markov random field priors for Bayesian inversion[END_REF] (using the isotropic first-order difference prior). To do so, we set λ = 0.03 as in the reference (no maximum-likelihood estimator is given) and estimate σ ϵ as in Alg. 2, while NUTS proposals are made from GMRF sampling.

The results are depicted in Fig. 2 and summarized in Fig. 3b, yielding the following observations:

• The exponential correlation function overfits the noise but capture more details in the image, leading to overall better performances than its Gaussian counterpart. In other words, the exponential correlation function handle well the Gaussian features but not the opposite. well on the phantom cell. Typical computation times on a laptop yield ∼15 minutes for non-GMRF estimations on 1002 pixels, for any kernel size.

B. Real images

We then apply the proposed method on widefield fluorescence microscopy images acquired at 450 nm, with a 40x / 0.65 NA objective (ORCA-fusion C14440 camera). We observed phytoliths (the silica remnants of combustion) of Miscanthus sinensis in Eukitt mounting medium. 2 These objects present a peculiar structure with two lobes [START_REF] Vigliaturo | Opaline phytoliths in miscanthus sinensis and its cyclone ash from a biomass-combustion facility[END_REF] and are mostly flat w.r.t. the depth of field so the approximation of 2D convolution holds. H is obtained numerically by observing 500nm calibration beads (FocalCheck™ test slice #1). The results depicted in Fig. 4 allow the following observations:

• As in simulated image, LogitNRF together with an exponential correlation function does capture the small features of the objects, at the price of fitting part of the image noise. The latter is better handled in its Gaussian counterpart, but small features are not depicted anymore. • Cauchy MRF depicts well sharp edges but is coarser on small features, while being sensible to the non-uniform background. Overall, one might add that such images present backgroundand object-induced non-stationnarities, which are not accounted for in our models.

IV. CONCLUSION This paper describes how to model and infer from TGRF in the context of deconvolution, knowing only the PSF and the observation. Numerical results suggest that the newly introduced models have a higher modeling capacity, meaning that they are more flexible than classical GMRF or even some non-Gaussian random fields in this context.

Future work on this topic should address the modeling issue of non-stationarity as well as the computational burden related to 3D deconvolution.

Algorithm 1

 1 Parallel NUTS sampling Require: y, H, Θ, threshold potential scale reduction R stop , length N of a NUTS chain, number of independent chains M . Ensure: Posterior mean estimator of p Θ (x|y) Initialization: sample x along the prior, or use an available first guess. repeat (repetition r): for Several parralel HMC chain instances 1 < m < M do Start from the previous state x (r-1,m) when available. Sample along p(x|y) using NUTS [13]. Store the log-posteriors and the last sequence point, x r,m . Compute the residual variance R (8) . until R < R stop ÊΘ [x|y] = 1 M m x r,m is the posterior mean estimator.

Fig. 2 :

 2 Fig. 2: Example results for Exp. 2. The first line corresponds to the dotted vertical axis in Fig. 3b, located at SNR=15 dB and FWHM=4.84 px; while the second and third lines depict lower SNR (5 dB, keeping a 4.84 px FWHM) or a wider FWHM (8.75 px, keeping a 15 dB SNR).

Fig. 3 :

 3 Fig.3: Average results on synthetic images (100 repetitions). The shaded regions depict the standard error of the mean and the dashed vertical line is shared within each experiment.

yFig. 4 :

 4 Fig. 4: Deconvolution of two fluorescence microscopy images, using a single color scale for each. LogNRF results are visually similar to those of LogitNRF and are not depicted.

  The CauchyMRF offers interesting results as it captures well flat-intensity regions as well as their borders. However, it also erases smaller details and so does not perform

	x	y	LogitNRF/Gau.	GMRF/exp.	LogNRF/exp. LogitNRF/exp.	H-TV1	H-TV2	Cauchy MRF
					• The LogitNRF handles well either flat or non-flat features,
					while the LogNRF does not, leading to better LogitNRF
					performances w.r.t. LogNRF.
					• TV-based methods yield expected results in terms of
					extreme flatness / squarish features (TV1) or over-smooth
					result (TV2), indicating that these methods do not handle
					well strong image corruption.

•

The authors would like to thank G. Trouvé and D. Kehrli (GRE, Université de Haute-Alsace, Mulhouse, France) for the preparation of the samples.

funding from ANR HORUS ANR-18-CE45-0010 and THTTM ANR-19-CE42-0004.

JBC and BC are within IRIMAS, UR 7499,


Lorenzo Sala

Nicolas Golse

Alexandre Joosten

Eric Vibert

Irene Vignon-Clementel

PARTIAL HEPATECTOMY HEMODYNAMICS DIGITAL TWIN: A SENSITIVITY ANALYSIS STUDY

Introduction

Liver surgery is one of the only curative treatments for primary or secondary liver tumours. Partial hepatectomy is a partial resection of the liver allowing for tumour removal. The main risk of hepatectomy is the occurrence of Post-Hepatectomy Liver Failure, which may occur due to post-operative portal hypertension (PHT) [1]. This abstract presents a global sensitivity analysis (SA) study of a mathematical model simulating the hemodynamics response to partial hepatectomy to better understand the main drivers of the clinical outputs of interest.

Materials and Methods

Recently a lumped-parameter model of the cardiovascular system was proposed to simulate the whole-body hemodynamics before and after a partial hepatectomy and evaluate the risk of PHT due to this surgery [2]. The input parameters of this mathematical model ℳ considered for the SA study proposed are:

• the heart elastances of the right atrium and left ventricle (E !,#$ , 𝐸 %,&' , 𝐸 (,)* , 𝐸 %,)* ); • the resistances to the flow within the portal vein, hepatic artery, hepatic vein, total digestive organs, and other organs (𝑅 +, , 𝑅 -( , 𝑅 -, , 𝑅 ./ , 𝑅 // , respectively); • the fraction of the total liver mass to be resected during the surgery (Hpx). Motivated by the clinical needs to evaluate the patient state during hepatectomy, the preoperative (pre-hpx) and post-operative (posthpx) quantities of interest for this study are: (i) the portal vein pressure (𝑃 +, *, (ii) the portocaval gradient (PCG) that is the pressure difference between the portal vein and the vena cava, (iii) the systemic arterial pressure (MAP), (iv) the cardiac output (CO), and (v) the blood flows in the hepatic artery (𝑄 -( ) and (vi) in the portal vein (𝑄 +, *. While performing the SA using the Sobol indices method, the model outputs required to be constrained to physiological ranges. Thus, an innovative approach [3] that exploits the features of the polynomial chaos expansion method has been applied to generate a metamodel ℳ +01 , reducing the overall computational cost to complete a physiological global SA.

Results

Fig. 1 shows the most sensitive parameters (red), the fairly sensitive parameters (pink) and the insensitive parameters (white) to the studied quantities of interest.

Discussion and Conclusions

The results suggest which parameters are more important for this hemodynamic model targeting surgical actions, identifying which measurements can be negligible (e.g.𝑄 -( ) and which require good accuracy (e.g. 𝑄 +, ) to provide solid predictions. Moreover, these outcomes give new insights on how to improve the calibration of some model parameters to be more patient-specific.

Figure 1 :

 1 Figure 1: Sensitive (red), fairly sensitive (pink) and insensitive (white) input parameters for each quantity of interest suggested by the SA performed with the surrogate model ℳ !"# .

Acknowledgements of funding support:

European Research Council under the European Union's Horizon 2020 research and innovation program (Grant no: 864313).