Didier Henrion

Milan Korda

Martin Kružík

Rodolfo Ríos-Zertuche

Occupation measure relaxations in variational problems: the role of convexity

This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming reformulation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving around the convexity of the data, are simple and apply in very general settings that may be of arbitrary dimensions and may include pointwise and integral constraints, thereby considerably strengthening the existing results. Our conditions are also extended to optimal control problems. In addition, we demonstrate how these results can be applied in non-convex settings, showing that the occupation measure relaxation is at least as strong as the convexification using the convex envelope; in doing so, we prove that a certain weakening of the occupation measure relaxation is equivalent to the convex envelope. This opens the way to application of the occupation measure relaxation in situations where the convex envelope relaxation is known to be equivalent to the original problem, which includes problems in magnetism and elasticity.

Calculus of variations and linear formulations with occupation measures

Let Ω ⊆ R n be a connected, bounded domain with piecewise C 1 boundary ∂Ω, let Y ⊂ R m and Z ⊂ R m×n . Denote by σ the (n -1)-dimensional Hausdorff measure on the boundary ∂Ω. Let us denote the transpose of a matrix A by A .

Introduction

The moment-sum-of-squares (SOS) hierarchy is a mathematical technology that consists of two steps: 1) formulating a nonlinear problem (of calculus of variations, optimization or control) as a linear convex optimization problem on the cone of nonnegative measures; 2) solving approximately the infinite-dimensional linear problem on measures by a hierarchy of finite-dimensional convex (typically semidefinite) optimization problems of increasing size. This hierarchy builds on the duality between the cone of moments and the cone of nonnegative polynomials, and its convergence relies on SOS representations of nonnegative polynomials -see [START_REF] Henrion | The moment-SOS hierarchy -Lectures in probability, statistics, computational geometry, control and nonlinear PDEs[END_REF] for a recent overview. For calculus of variations and optimal control problems, the linear problem on measures in step 1 is referred to as the occupation measure relaxation.

Whereas the convergence analysis of the hierarchy of finite-dimensional convex optimization problems to the solution of the infinite-dimensional linear problem on measures (in step 2) is now well understood, an essential remaining difficulty consists of ensuring that there is no relaxation gap when formulating (in step 1) the original nonlinear problem as a linear problem on measures. Indeed, if there is a relaxation gap, it is currently not understood in full generality how the approximate solutions obtained by the hierarchy are related to the solutions to the original problem, which is undesirable. For optimal control or regions of attraction of nonlinear ordinary differential equations, the absence of a relaxation gap was ensured under convexity assumptions [START_REF] Vinter | Convex duality and nonlinear optimal control[END_REF], see [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF]Theorem 2.3] and [9, Assumption I], or also [10, Assumption 2]. For scalar hyperbolic conservation laws, the absence of a relaxation gap was ensured by introducing appropriate entropy inequalities [START_REF] Marx | A moment approach for entropy solutions to nonlinear hyperbolic PDEs[END_REF]. See [START_REF] Korda | Moments and convex optimization for analysis and control of nonlinear partial differential equations[END_REF]Remark 6] for a discussion on the question of the relaxation gap in the general case of polynomial partial differential equations (PDEs).

Any attempt to apply the moment-SOS hierarchy to solve nonlinear calculus of variations (calculus of variations) problems [START_REF] Dacorogna | Introduction to the calculus of variations[END_REF] is invariably faced with this key relaxation gap issue. In [START_REF] Chernyavsky | Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization[END_REF], no relaxation gap was proved for three very specific calculus of variations problems with an integrand which is quadratic convex both in the function and its gradient. Similarly, there is no relaxation gap when the integrand is separably convex in the function and the gradient [START_REF] Fantuzzi | Sharpness and non-sharpness of occupation measure bounds for integral variational problems[END_REF]. In [START_REF] Korda | The gap between a variational problem and its occupation measure relaxation[END_REF] the absence of a relaxation gap was proved for calculus of variations problems with PDE constraints under convexity assumption in the gradient when the dimension of the codomain is 1, with the help of techniques from geometric measure theory. This reference also provides a counterexample with a positive relaxation gap when the codomain has dimension greater than one, despite the Lagrangian density being convex in the gradient.

The main result of our paper, Theorem 3.1, provides a simple proof of the absence of a relaxation gap for calculus of variations in a very general dimension-independent setting for problems with the Lagrangian and constraints that are jointly convex in the function and its gradient, thereby considerably strengthening the existing results. Concretely, on the one hand, the result eliminates the need for a separability of the Lagrangian in [START_REF] Fantuzzi | Sharpness and non-sharpness of occupation measure bounds for integral variational problems[END_REF] and allows for a very broad class of constraints to be considered. On the other hand, the result eliminates the dimensionality dependence of [START_REF] Korda | The gap between a variational problem and its occupation measure relaxation[END_REF], at the price of stronger convexity assumptions.

Similarly to [START_REF] Fantuzzi | Sharpness and non-sharpness of occupation measure bounds for integral variational problems[END_REF], our proof technique works for a large class of measures that satisfy a Liouville condition for test functions that are affine in codomain variables. We analyze this weaker relaxation with affine rather than nonlinear test functions further and prove in Theorem 4.2 that when it is applied to a nonconvex problem its infimum is equal to the infimum of the original problem convexified by taking the classical convex envelope. Therefore, for nonconvex problems, the value of the occupation measure relaxation with nonlinear test functions is sandwiched between the infimum of the original problem and the infimum of the convexification using the convex envelope. Consequently, the occupation measure relaxation is at least as strong as the convexification using the convex envelope. In particular, whenever one can prove the absence of a relaxation gap for the latter, it holds for the former.

The convexity assumptions of our result hold for some problems in continuum physics, namely for static micromagnetics [START_REF] Desimone | Energy minimizers for large ferromagnetic bodies[END_REF], which we explore in detail in Section 6, and for some models of elasticity [START_REF] Kohn | The relaxation of a double-well energy[END_REF][START_REF] Raoult | Quasiconvex envelopes in nonlinear elasticity[END_REF].

This opens the way to application of the occupation measure relaxation in situations where the convex envelope relaxation is known to be equivalent to the original problem. For these problems, the use of Young measures (also called parametrized measures) is classical, see e.g. [START_REF] Young | Lectures on the calculus of variations and optimal control theory[END_REF][START_REF]Relaxation in optimization theory and variational calculus[END_REF][START_REF]Parametrized measures and variational principles[END_REF]. As already explained in [START_REF] Henrion | Optimal control problems with oscillations, concentrations and discontinuities[END_REF], occupation measures can be seen as an extension of Young measures allowing the application of numerical methods based on the moment-SOS hierarchy.

The paper is organized as follows. In Section 2 we state the calculus of variations problem to be solved, and we introduce the occupation measures we are using to reformulate it as a linear optimization problem. Section 3 contains our main result Theorem 3.1 about the absence of a relaxation gap between the calculus of variations problem and its linear formulation, under convexity assumptions. Non-convex problems are the subject of Section 4 where we show how our linear formulation relates with the classical convex envelope relaxation. Section 5 explains how our calculus of variations results can be extended to optimal control problems. Finally, calculus of variations applications in micromagnetics are described in Section 6.

Consider the calculus of variations problem

M := inf y∈W 1,p (Ω;Y ) Ω L(x, y(x), Dy(x)) dx (1)
subject to A i (x, y(x), Dy(x)) = 0, B i (x, y(x), Dy(x)) ≤ 0 a.e. x ∈ Ω, i ∈ I, (2)

A ∂,i (x, y(x)) = 0, B ∂,i (x, y(x)) ≤ 0 σ-a.e. x ∈ ∂Ω, i ∈ I, (3)
Ω C i (x, y(x), Dy(x)) dx ≤ 0, ∂Ω C ∂,i (x, y(x)) dσ(x) ≤ 0, i ∈ I, ( 4
)
where I is a possibly uncountable index set and W

Z = R 2×3 . Let γ : [0, 2π] → R 3 be a continuous, non-self-intersecting loop, γ(0) = γ(2π). let L(x, y, Dy) = ∂y ∂x1 × ∂y ∂x2
, that is, L is the area of the parallelogram spanned by the columns of Dy, known as the element of surface area. Let A ∂,1 (x, y) be the distance from y to the image γ([0, 2π]). Let all other constraint functions A i , B i , C i , A ∂,i , B ∂i , C ∂,i be zero. In this setting, M is the area of the minimal surface with boundary γ([0, 2π]).

Example 2.2 (Geodesics of a differential inclusion).

Let Ω be the interval [0, 1] and let V be a connected, open subset of Y = R n , and Z = R n . Fix two points y 0 and y 1 in V , as well as a Riemannian metric g :

R n × R n → R ≥0 . Let B 1 : Y × Z → R be a continuous function such that the sets X(y) = {z ∈ Z : B(y, z) ≤ 0} are nonempty for each y ∈ Y . Let L(x, y, z) = g(z, z), let A 1 (x,
y, z) be the distance from y to V , and A ∂,1 (x, y, z) be the distance from y to the set {y 0 , y 1 }. Then M is the minimal g-distance from y 0 to y 1 for curves joining these points and respecting the constraint γ(t) ∈ V and the differential inclusion γ (t) ∈ X(γ(t)), and the minimizers are the geodesics joining these two points and respecting the differential inclusion. When X(y) is a linear subspace of Z for each y, this setting is known as sub-Riemannian geometry.

Example 2.3 (Higher-order PDEs). Considering an uncountable index set in (1) permits higher-order PDEs to be considered as constraints in (1) through their weak reformulation. We demonstrate this on the Laplace equation. In the m = 1 case, to model a Laplace equation in the weak form, we can use the constraint

Ω ∇φ(x) ∇y(x) dx = 0, φ ∈ C 1 0 (Ω).
Observe that if y were a smooth function, integrating by parts we would obtain φ(x) ∆y(x) dx = 0 for all φ ∈ C 1 0 (Ω), whence the above conditions indeed give a weak analogue of the Laplace equation ∆y = 0. In order to model it as in (4), we set I to be C 1 0 (Ω), and for each

φ ∈ I = C 1 0 (Ω) we set C φ (x, y, z) = ∇φ(x) z.
Observe that the equality constraint results from the inequalities for φ and -φ, which are both contained in C 1 0 (Ω).

To reformulate calculus of variations problem (1) as a linear optimization problem, we now introduce two classes of measures. The first ones, the affinely relaxed occupation measures, form a larger class that contains the second ones, the relaxed occupation measures. Our main no-gap results will hold for the first class of measures (see Theorems 3.1 and 5.2), and hence will also hold for the second one (see Corollary 3.2 and Remark 5.1). The second class is more commonly found in the literature, and we chose to introduce the first class in order to present our results in greater generality. The total mass of µ equals the Lebesgue measure of Ω, that is,

µ(Ω × Y × Z) = |Ω|. (5)
The sets supp µ and supp µ ∂ are compact if p = +∞, or

Ω×Y ×Z y p + i z i p dµ(x, y, z) < +∞ and ∂Ω×Y y p dµ ∂ (x, y) < +∞ (6) if p ∈ [1, +∞). Here, z = (z 1 , . . . , z n ), z i ∈ R m . For all φ 1 ∈ C ∞ (Ω; R) and all φ 2 ∈ C ∞ (Ω; R m ) it holds Ω×Y ×Z ∂φ 1 ∂x (x) + y ∂φ 2 ∂x (x) + φ 2 (x) z dµ(x, y, z) = ∂Ω×Y φ 1 (x) + y φ 2 (x) n(x) dµ ∂ (x, y). ( 7
)
Here n denotes the exterior unit (row) vector normal to the boundary ∂Ω.

Remark 2.5. Observe that [START_REF] Evans | Measure theory and fine properties of functions[END_REF] amounts to the Liouville equation

Ω×Y ×Z ∂φ ∂x + ∂φ ∂y z dµ = ∂Ω×Y φ n dµ ∂ for a test function φ ∈ C ∞ (Ω × Y ; R) that is affine in y, that is, of the form φ(x, y) = φ 1 (x) + y φ 2 (x).
Remark 2.6. The assumption of Ω being connected can be dropped at the price of replacing ( 5) by the slightly stronger requirement that its x-marginal is equal to the Lebesgue measure on Ω.

When Ω is connected and its mass is equal to the volume of Ω (condition (5)), its x-marginal is automatically equal to the Lebesgue measure on Ω due to [START_REF] Evans | Measure theory and fine properties of functions[END_REF]

where n denotes the exterior unit vector normal to the boundary ∂Ω,

F p = φ ∈ C ∞ (Ω × Y ) : ∃ c > 0 such that ∂φ ∂x (x, y) ≤ c(1 + y p ), ∂φ ∂y (x, y) ≤ c(1 + y p-1 ), |φ(x, y)| ≤ c(1 + y p ) , p ∈ [1, ∞)
and

F ∞ = C ∞ (Ω × Y )
. By Lemma B.1, the integrals in (8) are well defined.

Consider the relaxations with affinely relaxed occupation measures (Definition 2.4),

M aff r := inf (µ,µ ∂ )∈O aff p Ω×Y ×Z L(x, y, z) dµ(x, y, z) (9) subject to supp µ ⊆ {(x, y, z) ∈ Ω × Y × Z | A i (x, y, z) = 0, B i (x, y, z) ≤ 0, i ∈ I}, supp µ ∂ ⊆ {(x, y) ∈ ∂Ω × Y | A ∂,i (x, y) = 0, B ∂,i (x, y) ≤ 0, i ∈ I}, Ω×Y ×Z C i (x, y, z) dµ(x, y, z) ≤ 0, i ∈ I, ∂Ω×Y C ∂,i (x, y) dµ ∂ (x, y) ≤ 0, i ∈ I,
and with relaxed occupation measures (Definition 2.7),

M r := inf (µ,µ ∂ )∈Op Ω×Y ×Z L(x, y, z) dµ(x, y, z) (10)
subject to [same constraints as in ( 9)].

Remark that the sole difference between ( 9) and ( 10) is the set over which the infimum is taken, namely, O aff p and O p , respectively. Observe also that problems ( 9) and ( 10) are linear optimization problems in the unknown occupation measures (µ, µ ∂ ), whereas the original calculus of variations problem (1) can be nonlinear in the unknown function y.

Convex case -main result

Our main result states that under convexity assumption there is no relaxation gap between the original calculus of variations problem and its linear reformulation with affinely relaxed occupation measure. Theorem 3.1. Suppose that for every i ∈ I and x ∈ Ω the following holds: L, B i and C i are convex in (y, z), and A i is affine in (y, z). Suppose also that for every i ∈ I and x ∈ ∂Ω the following holds: A ∂,i is affine in y, and B ∂,i and C ∂,i are convex in y. Then, if Y and Z are closed and convex, we have

M aff r = M.
Recall that these are defined in (1) and (9).

Proof. Clearly we have

M aff r ≤ M because every function y induces measures (µ, µ ∂ ) ∈ O aff p by letting Ω×Y ×Z φ(x, y, z) dµ(x, y, z) = Ω φ(x, y(x), Dy(x)) dx, φ ∈ C 1 (Ω × Y × Z), ∂Ω×Y φ(x, y) dµ ∂ (x, y) = ∂Ω φ(x, y(x)) dσ(x), φ ∈ C 1 (Ω × Y ).
The harder part is to prove that M aff r ≥ M . To this end, let (µ, µ ∂ ) be feasible in [START_REF] Gaitsgory | Linear programming analysis of deterministic infinite horizon optimal control problems with discounting[END_REF]. By Lemma A.1, the projection of µ onto Ω is Lebesgue measure dx. Disintegrate µ such that

µ(x, y, z) = ν(y, z | x) dx, ( 11
)
where ν is the conditional distribution of (y, z) given x; in particular, for every fixed x, ν(•, • | x) is a probability measure. Define

y(x) := Y ×Z y dν(y, z | x), z(x) := Y ×Z z dν(y, z | x) (12)
to be the conditional expectation (or centroid) of µ in the y and z variables given x. We note that y(x) ∈ R m and z(x) ∈ R m×n . By Jensen's inequality we have

Ω×Y ×Z L dµ = Ω Y ×Z L(x, y, z) dν(y, z | x) dx ≥ Ω L(x, y(x), z(x)) dx.
Now we shall prove that y(•) is weakly differentiable and Dy(x) = z(x). To prove weak differentiability of y(•), take φ 1 = 0 and

φ 2 = ψ ∈ C ∞ 0 (Ω; R m
), and use it as a test function in [START_REF] Evans | Measure theory and fine properties of functions[END_REF]. This gives

Ω×Y ×Z y ∂ψ ∂x + ψ(x) z dµ(x, y, z) = 0.
Using the disintegration of µ we obtain

Ω Y ×Z y ∂ψ ∂x + ψ(x) z dν(y, z | x)dx = 0.
Since the integrand is linear in (y, z), we obtain, using the definitions of y(•) and z(•),

Ω y(x) ∂ψ ∂x + ψ(x) z(x) dx = 0, ∀ψ ∈ C ∞ 0 (Ω).
This is nothing but the definition of weak differentiability of y(•) with weak derivative z(•). If p = +∞, supp µ is bounded, and then z(

•) is L ∞ , so it follows that y ∈ W 1,∞ . If p ∈ [1, +∞), then z(•) is in L p ;
indeed, by Jensen's inequality applied to the convex function • p , we have, writing z = (z 1 , . . . , z m ),

Ω×Y ×Z z i (x) p dx = Ω Y ×Z z i dν(y, z | x) p dx ≤ Ω Y ×Z z i p dν(y, z | x)dx = Ω×Y ×Z z i p dµ(x, y, z) < +∞,
by [START_REF] Desimone | Energy minimizers for large ferromagnetic bodies[END_REF]. A similar calculation shows that y(•) is in L p as well. It follows that y is in W 1,p . We also need to verify the constraints ( 2)-( 4). This follows from the support constraints on µ and µ ∂ and the fact that A i and A ∂,i are affine and B i , B ∂,i , C i and C ∂,i are convex in (y, z). Indeed, the support constraints imply

ψ(x)A i (x, y, z) dµ(x, y, z) = 0 ∀ψ ∈ C 0 (Ω).
Using the disintegration [START_REF] Henrion | The moment-SOS hierarchy -Lectures in probability, statistics, computational geometry, control and nonlinear PDEs[END_REF], and definition of y(•), z(•) and the fact that A i is affine, we obtain

ψ(x)A i (x, y(x), z(x)) dx = 0 ∀ψ ∈ C 0 (Ω)
and hence the constraint A i (x, y(x), z(x)) = 0 is satisfied a.e. in Ω, i ∈ I. For the inequality constraint B i (x, y(x), Dy(x)) ≤ 0, we observe that for any nonnegative ψ ∈ C 0 (Ω), we have

0 ≥ ψ(x)B i (x, y, z) dµ = Ω Y ×Z ψ(x)B i (x, y, z) dν(y, z | x) dx ≥ Ω ψ(x)B i (x, y(x), z(x)) dx,
where the first inequality follows from the support constraints on µ and the last one from Jensen's inequality. Since z(x) = Dy(x), this implies that B i (x, y(x), Dy(x)) ≤ 0 a.e. in Ω. Another application of Jensen's inequality gives, for all i ∈ I,

Ω C i (x, y(x), z(x)) dx ≤ Ω Y ×Z C i (x, y, z) dν(y, z | x) dx = Ω×Y ×Z C i (x, y, z) dµ(x, y, z) ≤ 0.
Let us now verify the boundary condition. To this end, disintegrate µ ∂ as

µ ∂ = ν ∂ (y | x)dσ(x) (13)
with σ being the Hausdorff measure on ∂Ω. Define

y ∂ (x) = y dν ∂ (y | x).
We shall show that y ∂ (•) is a Sobolev trace of y(•) (see [START_REF] Evans | Measure theory and fine properties of functions[END_REF]Definition,4.5]). To this end, take φ 1 = 0 and φ 2 = ψ ∈ C ∞ (Ω; R m ), so that plugging in [START_REF] Evans | Measure theory and fine properties of functions[END_REF] and using the fact that z(x) = Dy(x) almost everywhere in Ω gives

Ω y(x) ∂ψ ∂x + ψ(x)Dy(x) dx = ∂Ω ψ(x) y ∂ (x)n(x) dσ(x), ∀ψ ∈ C ∞ (R n ).
This is the Stokes theorem for the Sobolev functions which shows that y ∂ (•) is indeed a Sobolev trace of y(•) [START_REF] Evans | Measure theory and fine properties of functions[END_REF]Theorem 4.6]. Hence, in order to show that A ∂,i (x, y(x)) = 0 on ∂Ω it suffices to prove it for y ∂ . The second support constraint of (9) implies that

Ω×Y φ(x)A ∂,i (x, y) dµ ∂ (x, y) = 0 ∀φ ∈ C 0 (∂Ω).
Using the disintegration [START_REF] James | Frustration in ferromagnetic materials[END_REF], the definition of y ∂ and the fact that A ∂ is affine in y, it follows that

Ω×Y φ(x)A ∂,i (x, y ∂ (x)) dx = 0 ∀φ ∈ C 0 (∂Ω).
This implies that y ∂ (•) and hence y(•) satisfy the boundary condition σ a.e. in ∂Ω. For p = +∞, y(•) is Lipschitz and hence y(•) satisfies it in fact everywhere in ∂Ω. The boundary inequality constraint B ∂,i (x, y(x)) ≤ 0 follows by Jensen's inequality the same way as for B i (x, y(x), Dy(x)) ≤ 0, and similarly for the integral boundary constraint [START_REF] Chernyavsky | Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization[END_REF]. It remains to verify that y(x) ∈ Y for all x ∈ Ω. This, however, follows by the fact that Y is closed and convex and the last support constraint of (9).

Nonconvex case

We now turn to the case when the problem is not convex and show that the affinely relaxed occupation measures provide a lower bound at least as good as the the convex envelope relaxation and, in the absence of integral constraints, the two lower bounds coincide.

Convex underestimators

First, we make an observation comparing the value of the occupation measure relaxation to the value of the original problem with data replaced by its convex underestimators. Concretely, suppose that there exist convex sets Ŷ ⊃ Y and Ẑ ⊃ Z, functions L, Bi , Ĉi :

Ω× Ŷ × Ẑ → R such that for all (x, y, z) ∈ Ω × Y × Z satisfying A i (x, y, z) = 0, B i (x, y, z) ≤ 0, it holds L(x, y, z) ≤ L(x,
y, z), Bi (x, y, z) ≤ B i (x, y, z), Ĉi (x, y, z) ≤ C i (x, y, z). Suppose also that there exist functions L∂ , B∂,i , Ĉ∂,i : Ω × Ŷ → R such that for all (x, y) ∈ ∂Ω × Y satisfying A ∂,i (x, y) = 0, B ∂,i (x, y) ≤ 0, it holds L∂ (x, y) ≤ L ∂ (x, y), B∂ (x, y) ≤ B ∂,i (x, y) and Ĉ∂,i (x, y) ≤ C ∂,i (x, y).

Consider the problem

M under = inf y∈W 1,p (Ω, Ŷ ) Ω L(x, y(x), Dy(x))dx + ∂Ω L∂ (x, y(x)) dσ(x) (14)
subject to A i (x, y(x), Dy(x)) = 0, Bi (x, y(x), Dy(x

)) ≤ 0, x ∈ Ω, i ∈ I, A ∂,i (x, y(x)) = 0, B∂,i (x, y(x)) ≤ 0, x ∈ ∂Ω, i ∈ I, Ω Ĉi (x, y(x), Dy(x)) dx ≤ 0, ∂Ω Ĉ∂,i (x, y(x)) dx ≤ 0, i ∈ I.
We have the following simple corollary, where we for simplicity assume that A i and A ∂,i are affine, i.e., they do not need further convexification. This could be relaxed by replacing A i and A ∂,i by A 2 i and A 2 ∂,i and considering convex underestimators thereof.

Corollary 4.1. Suppose that A i is affine in (y, z) for every x ∈ Ω and A ∂,i is affine in y for every x ∈ ∂Ω. It holds M under ≤ M aff r ≤ M r ≤ M (defined, respectively, in (14), ( 9), [START_REF] Henrion | Convex computation of the region of attraction of polynomial control systems[END_REF], and (1)). In particular, if

M under = M , then M aff r = M r = M .
Proof. Let Mr denote the infimum of the occupation measure relaxation applied to the convexified problem [START_REF] Kohn | The relaxation of a double-well energy[END_REF]. Applying Corollary 3.2, we have Mr = M under . Given a pair of measures (µ, µ ∂ ) feasible in [START_REF] Gaitsgory | Linear programming analysis of deterministic infinite horizon optimal control problems with discounting[END_REF], we observe that these two measures are feasible in the occupation measure relaxation of the convexified problem since the feasible set of the latter is larger than the feasible set of the original problem. Since L ≤ L, it follows that Mr ≤ M aff r and as a result

M under ≤ M aff r ≤ M r ≤ M since always M aff r ≤ M r ≤ M .
Note that when M = M = M aff r = M r , one has two choices for a relaxation to be solved numerically: either the convexification [START_REF] Kohn | The relaxation of a double-well energy[END_REF] or the occupation measure relaxation [START_REF] Gaitsgory | Linear programming analysis of deterministic infinite horizon optimal control problems with discounting[END_REF]. Both are convex optimization problems and it depends on particular situation whether one or the other is more advantageous to solve. For example, the occupation measure relaxation may be preferable when no explicit description of the nonnegative convex underestimators used in ( 14) is available.

Equivalence with convex envelope

Now we significantly refine the previous observation and prove that in fact M aff r is equal to the infimum of a convexification of the problem based on the convex envelope of the Lagrangian and the constraints. As a result, under the assumptions introduced below, we obtain the following inequalities Convex envelope = Affinely relaxed OM ≤ Relaxed OM ≤ Original, where OM stands for "occupation measures." To set the stage for proving this, we let conv X denote the closed convex hull of a set X ⊂ R n . Let p = +∞, and let Y and Z be compact sets. For x ∈ Ω, let K(x) denote the set of points (y, z) ∈ Y × Z such that the constraints A i (x, y, z) = 0 and B i (x, y, z) ≤ 0 are satisfied. We set C i = 0, C i,∂ = 0, i.e., we work without integral constraints in this section, except for Examples 4.3 and 4.4, that will show that we would not be able to obtain the same results in the presence of such constraints.

Let L : Ω×conv Y ×conv Z → R∪{+∞} be the function whose restriction to {x}×conv K(x) is the convex envelope of L| {x}×K(x) ; in other words, for x ∈ Ω,

L(x, y, z) = sup{ϕ(y, z) | ϕ : conv Y × conv Z → R convex, ϕ(y , z ) ≤ L(x, y , z ) ∀ (y , z ) ∈ K(x)}. Analogously, let K ∂ (x) ⊆ Y be, for each x ∈ ∂Ω, the set of points y ∈ Y such that A ∂,i (x, y) = 0 and B ∂,i (x, y) ≤ 0. Let Âi (x) = conv(A i | {x}×Y ×Z ) -1 (0) and Âi = x∈Ω Âi (x); in other words, Âi is the convexification in the Y × Z directions of the set A -1 i (0). Similarly, let Â∂,i (x) = conv(A ∂,i | {x}×Y ) -1 (0) and Â∂,i = x∈∂Ω Â∂,i (x). Let M = inf y∈W 1,∞ (Ω,conv Y ) Ω L(x, y(x), Dy(x))dx (15)
subject to (y(x), Dy(x)) ∈ conv K(x), a.e. x ∈ Ω,

y(x) ∈ conv K ∂ (x), a.e. x ∈ ∂Ω.
The following result shows that the convexified problem [START_REF] Korda | Moments and convex optimization for analysis and control of nonlinear partial differential equations[END_REF] has the same infimum as the occupation measure relaxation [START_REF] Gaitsgory | Linear programming analysis of deterministic infinite horizon optimal control problems with discounting[END_REF] with affine test functions applied to the original problem with nonconvex data. Theorem 4.2. Assume that K(x) is closed for each x ∈ Ω and that the set-valued map K is continuous in the topology induced by the Hausdorff metric. If L is continuous in (y, z) for every x ∈ Ω, then it holds that M = M aff r .

Proof. Let us first show that M aff r ≤ M . Let y ∈ W 1,p (Ω, conv Y ) be viable for the optimization problem M . By Choquet's Theorem [START_REF] Lukeš | Choquet's theory and the Dirichlet problem[END_REF], we know that, for each x ∈ Ω where Dy(x) is defined, there is a probability measure ν x supported in K(x) such that (y(x), Dy(x)) = K(x) (y, z) dν x (y, z) and L(x, y(x), Dy(x)) = K(x) L(x, y, z) dν x (y, z). [START_REF] Korda | The gap between a variational problem and its occupation measure relaxation[END_REF] Let us show that the mapping x → ν x can be chosen so that it is measurable. Let M x be the set of measures ν on K(x) such that K(x) (y, z) dν(y, z) = (y(x), Dy(x)). Endow M x with the weak* topology with respect to C 0 (Y × Z; R). Let

I L (ν, x) = {x}×Y ×Z L dν and φ(x) = min ν∈Mx I L (ν, x) = L(x, y(x), Dy(x)).
With these definitions, I L is continuous and φ is measurable. By the Prokhorov theorem, M x is compact and metrizable. The inverse map Φ x = I L (•, x) -1 , that is, Φ x (r) = {ν ∈ M x : I L (ν, x) = r}, r ∈ R, is weakly measurable as well, since every open set U ⊂ M x can be written as a countable union of compact sets U = i K i , and Φ -1

x (U ) = i Φ -1 x (K i ) = i I L (K i , x
) is a countable union of continuous images of compact sets, each of which is compact, so the union is measurable. Then the set-valued map

H(x) = I L (•, x) -1 (φ(x)) = arg min ν∈Mx {x}×Y ×Z L dν
is measurable, as it is the composition of two measurable maps. The image H(x) is closed, so we may apply the Kuratowski-Ryll-Nardzewski Selection Theorem [START_REF] Charalambos | Infinite dimensional analysis: a hitchhiker's guide[END_REF]Th. 18.13] to get an appropriate measurable function x → ν x ∈ H(x).

Letting µ = ν x (y, z) dx we obtain a measure that is viable for M aff r ; to see that it satisfies (7) note that since ν x ∈ M x , for each x the averages (12) on Y and Z coincide almost everywhere with y(x) and z(x) = Dy(x), respectively, and ( 7) is linear in these variables, so the fact that the Liouville equation for affine test functions,

Ω ∂φ 1 ∂x (x) + y(x) ∂φ 2 ∂x (x) + φ 2 (x) Dy(x) dx = ∂Ω φ 1 (x) + y(x) φ 2 (x) n(x) dσ(x),
is verified for y(•) implies that ( 7) is verified for µ. The measure µ also satisfies

Ω×Y ×Z L(x, y, z) dµ(x, y, z) = Ω Y ×Z L(x, y, z) dν x (y, z) dx = Ω L(x, y(x), Dy(x)) dx
by [START_REF] Korda | The gap between a variational problem and its occupation measure relaxation[END_REF]. Arguing analogously in the boundary ∂Ω to obtain a measure µ ∂ supported in x∈∂Ω {x} × K ∂ (x) from y(•)| ∂Ω , this implies that M aff r ≤ M . Now we turn to showing that M ≤ M aff r . Let (µ, µ ∂ ) be viable for M aff r . Define y(x) and z(x) for x ∈ Ω as in [START_REF] Henrion | Optimal control problems with oscillations, concentrations and discontinuities[END_REF]; then the same arguments as in the proof of Theorem 3.1 show that y(•) is a W 1,p function on Ω whose weak derivative is Dy(x) = z(x) for almost every x ∈ Ω, as well as the fact that the average of y with respect to µ ∂ coincides with the trace y(•)| ∂Ω . Moreover, (y(x), Dy(x)) ∈ conv K(x) for almost every x ∈ Ω, since it is an average of points in (supp µ) ∩ ({x} × Y × Z) ⊆ K(x), and similarly y(x) ∈ conv K ∂ (x) for a.e. x ∈ ∂Ω. By the Jensen inequality, we have

Ω×Y ×Z L(x, y, z) dµ(x, y, z) dµ ≥ Ω×Y ×Z L(x, y, z) dµ(x, y, z) ≥ Ω L(x, y(x), Dy(x)) dx.
This shows that M ≤ M aff r .

Example 4.3 (With integral constraints we may have M < M aff r ; nonconvex Y ). Let Y ⊂ R 2 be the set containing only the four points (y 1 , y 2 ) with y i ∈ {-1, 1}. Let Ω = (0, 1) ⊂ R and Z = B(0, 1) ⊂ R 2 the closed unit disc. Thus K(x) = {(±1, ±1)} × B(0, 1) for all x ∈ (0, 1). Let, for (x, y, z) ∈ Ω × Y × Z and writing y = (y 1 , y 2 ),

L(x, y, z) = y 1 y 2 + z 2 , C(x, y, z) = -y 1 y 2 .
Observe that y 1 y 2 only takes the values ±1 on Y . The convexifications in (y, z) are equal to conv

Y = [-1, 1] 2 , conv Z = Z = B(0, 1), conv K(x) = [-1, 1] 2 × B(0, 1
), and, for x ∈ (0, 1) and (y, z) ∈ conv K(x),

L(x, y, z) = |y 1 + y 2 | -1 + z 2 , Ĉ(x, y, z) = |y 2 -y 1 | -1.
Consider the problem

M = inf y∈W 1,∞ (Ω,conv Y ) Ω L(x, y(x), Dy(x))dx (17)
subject to (y(x), Dy(x)) ∈ conv K(x), a.e. x ∈ Ω, Ω Ĉ(x, y(x), Dy(x)) dx ≤ 0.

This problem is analogous to [START_REF] Korda | Moments and convex optimization for analysis and control of nonlinear partial differential equations[END_REF] in the bulk, with the addition of the integral constraint. Let us compare it with the problem (9) of finding M aff r (with

L ∂ = A i = A ∂,i = B i = B ∂,i = C ∂,i = 0
). If y : (0, 1) → R 2 is a constant curve whose image is a point in the segment joining (-1, 1) to (1, -1), then the integral of L takes its minimal value of -1, and since the points in that segment that are close to the origin satisfy Ĉ(x, y, z) < 0, we conclude that M = -1. On the other hand, a measure µ viable in [START_REF] Gaitsgory | Linear programming analysis of deterministic infinite horizon optimal control problems with discounting[END_REF] we have

Ω×Y ×Z L(x, y, z) dµ(x, y, z) = Ω×Y ×Z y 1 y 2 + z 2 dµ(x, y, z) = Ω×Y ×Z -C(x, y, z) + z 2 dµ(x, y, z).
Thus in order to satisfy the constraint that the integral of C be ≤ 0, the integral of L must be ≥ 0. The integral of L achieves its minimal value of 0 by letting µ be uniformly distributed on Ω × Y × {0}. This means that M aff r = 0 > -1 = M .

Example 4.4 (With integral constraints we may have M < M aff r ; nonconvex Z). We now modify Example 4.3 slightly to show that a similar situation can occur when the nonconvexity appears in the set of allowable velocities Z.

Let Z ⊂ R 2 be the set containing only the four points (z 1 , z 2 ) with

y i ∈ {-1, 1}. Let Ω = (0, 1) ⊂ R and Y = B(0, 1) ⊂ R 2 the closed unit disc. Thus K(x) = B(0, 1) × {(±1, ±1)} for all x ∈ (0, 1). Let, for (x, y, z) ∈ Ω × Y × Z and writing z = (z 1 , z 2 ), L(x, y, z) = z 1 z 2 , C(x, y, z) = -z 1 z 2 .
Observe that z 1 z 2 only takes the values ±1 on Z. The convexifications in (y, z) are equal to conv

Y = Y = B(0, 1), conv Z = [-1, 1] 2 , conv K(x) = B(0, 1) × [-1, 1] 2
, and, for x ∈ (0, 1) and (y, z) ∈ conv K(x),

L(x, y, z) = |z 1 + z 2 | -1, Ĉ(x, y, z) = |z 2 -z 1 | -1.
Consider the problem [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF]. Let us compare it with the problem (9) of finding M aff r (with

L ∂ = A i = A ∂,i = B i = B ∂,i = C ∂,i = 0). If y : (0, 1) → R 2 is
a curve whose derivatives Dy(x) are, for almost every x ∈ (0, 1), contained in the segment joining (-1, 1) to (1, -1), then the integral of L takes its minimal value of -1, and since the points in that segment that are close to the origin satisfy Ĉ(x, y, z) < 0, we conclude that M = -1; for an explicit example, one may take the curve

y(x) = (x, -x), x ∈ (0, 1 2 ), ( 1 2 -x, -1 2 + x), x ∈ [ 1 2 , 1
). On the other hand, a measure µ viable in [START_REF] Gaitsgory | Linear programming analysis of deterministic infinite horizon optimal control problems with discounting[END_REF] we have

Ω×Y ×Z L(x, y, z) dµ(x, y, z) = Ω×Y ×Z z 1 z 2 dµ(x, y, z) = - Ω×Y ×Z C(x, y, z) dµ(x, y, z).
Thus in order to satisfy the constraint that the integral of C be ≤ 0, the integral of L must be ≥ 0. The integral of L achieves its minimal value of 0 by letting µ be uniformly distributed on Ω × {0} × Z. This means that M aff r = 0 > -1 = M .

Optimal control

This section extends our results to optimal control. We first formulate the results under as general assumptions as possible and then present their simplified versions. We shall denote the control inside Ω by u : Ω → U and on the boundary by u : ∂Ω → U ∂ . We shall assume that U and U ∂ are complete separable metric spaces (e.g., closed subsets of a Euclidean space). The function

π Ω×Y ×Z : Ω × Y × Z × U → Ω × Y × Z will denote the projection on Ω × Y × Z,. i.e., π Ω×Y ×Z (x, y, z, u) = (x, y, z). Similarly π ∂Ω×Y : ∂Ω × Y × U ∂ → ∂Ω × Y is defined by π ∂Ω×Y (x, y, u) = (x, y).
Let A i , B i : Ω×Y ×Z ×U → R and C i : Ω×Y ×Z → R be Borel measurable functions for all i in a (possibly uncountably-infinite) index set I. Similarly, let

A ∂,i , B ∂,i : ∂Ω×Y ×U ∂ → R and C ∂,i : ∂Ω × Y → R be Borel measurable functions for all i ∈ I. Let also L : Ω × Y × Z × U → R and L ∂ : ∂Ω × Y × U ∂ →
R be measurable and locally bounded functions.

Problem statement

For p ∈ [1, +∞], consider the optimal control problem

M oc = inf y : Ω→Y u : Ω→U u ∂ : ∂Ω→U ∂ Ω L(x, y(x), Dy(x), u(x))dx + ∂Ω L ∂ (x, y(x), u ∂ (x)) dσ(x) subject to y ∈ W 1,p (Ω; Y ), u, u ∂ are measurable, A i (x, y(x), Dy(x), u(x)) = 0, B i (x, y(x), Dy(x), u(x)) ≤ 0, x ∈ Ω, i ∈ I, A ∂,i (x, y(x), u(x)) = 0, B ∂,i (x, y(x), u ∂ (x)) ≤ 0, x ∈ ∂Ω, i ∈ I, Ω C i (x, y(x), Dy(x)) dx ≤ 0, ∂Ω C ∂,i (x, y(x)) dx ≤ 0, i ∈ I.
Consider also its relaxation, for p ∈ [1, +∞):

M oc r = inf µ∈M(Ω×Y ×Z×U ) µ ∂ ∈M(∂Ω×Y ×U ∂ ) Ω×Y ×Z×U
L(x, y, z, u)dµ(x, y, z, u)

+ ∂Ω×Y ×U ∂ L ∂ (x, y, u) dµ ∂ (x, y, u) subject to supp µ ⊆ i∈I A -1 i (0) ∩ B -1 i ((-∞, 0]), supp µ ∂ ⊆ i∈I A -1 ∂,i (0) ∩ B -1 ∂,i ((-∞, 0]), Ω×Y ×U C i (x, y, z) dµ(x, y, z, u) ≤ 0, ∂Ω×Y ×U ∂ C ∂,i (x, y) dµ ∂ (x, y, u) ≤ 0, i ∈ I, µ(Ω × Y × Z × U ) = |Ω|, Ω×Y ×Z×U y p + z p dµ(x, y, z, u) < +∞, ∂Ω×Y ×U ∂ y p dµ ∂ (x, y, u) < +∞, Ω×Y ×Z×U ∂φ1 ∂x + y ∂φ2 ∂x + φ 2 z dµ = ∂Ω×Y ×U ∂ (φ 1 + y φ 2 )n dµ ∂ ∀φ 1 ∈ C ∞ (Ω; R), φ 2 ∈ C ∞ (Ω; R m ).
For p = +∞, the constraints involving integrals of • p are replaced by the assumption that the supports of µ and µ ∂ be compact.

Remark 5.1. The measures involved in M oc r satisfy a Liouville equation for affine test functions only, that is, for functions φ(x, y) = φ 1 (x) + φ(x)y, and are thus analogous to the affinely relaxed occupation measures O aff p ; cf. Remark 2.5. A corollary to Theorem 5.2 analogous to Corollary 3.2 holds, although for brevity we do not state it explicitly.

General formulation

The functions L and L ∂ have close relatives that do not depend on U and U ∂ and that will play a role in our assumptions: let the functions L :

Ω × Y × Z → R and L∂ : ∂Ω × Y → R be defined by L(x, y, z) = inf{L(x, y, z, u) : u ∈ U, A i (x, y, z, u) = 0, B i (x, y, z, u) ≤ 0, i ∈ I} L∂ (x, y) = inf{L ∂ (x, y, u) : u ∈ U ∂ , A ∂,i (x, y, u) = 0, B ∂,i (x, y, u) ≤ 0, i ∈ I}.
Assume the following: OC1. (Fiberwise convexity of the constraints) We assume:

(a) π Ω×Y ×Z ( i∈I A -1 i (0) ∩ B -1 i ((-∞, 0])) ∩ {x} × Y × Z is nonempty and convex for all x ∈ Ω. (b) π ∂Ω×Y ( i∈I A -1 ∂,i (0)∩B -1 ∂,i ((-∞, 0]
))∩{x}×Y is nonempty and convex for all x ∈ ∂Ω and all i ∈ I. L(x,y,z,u) and arg min

(c) C i (x, y, z) is convex in (y,
u∈U ∂ A ∂,i (x,y,u)=0 B ∂,i (x,y,u)≤0 L ∂ (x, y, u),

Simplified formulation 1

Here is a simple situation that satisfies assumptions OC1-OC3 above. Let p = +∞, Ω ⊂ R n be as above, Y ⊆ R m and Z ⊆ R m×n be compact, convex sets with nonempty interior. Let U and U ∂ be compact and convex subsets of R d for some d ∈ IN. Assume that, for each x ∈ Ω, L(x, y, z, u), B i (x, y, z, u), and C i (x, y, z, u) are convex in (y, z, u), and that A i (x, y, z, u) is affine in (y, z, u). Then we are in the setting concerning our results below. Similar assumptions on the boundary Lagrangian density L ∂ and in the boundary constraints A ∂,i , B ∂,i , and C ∂,i are also possible within our framework.

Simplified formulation 2

Here is another situation that satisfies assumptions OC1-OC3 above. For simplicity we will ignore the boundary part of the problem. Let p = +∞, Ω ⊂ R n be as above, Y ⊆ R m and Z ⊆ R m×n be compact, convex sets with nonempty interior. Let U be compact subset of R n , and let f : Ω × Y × U → Z be a continuous function such that, for each x ∈ Ω, the image of the map (x, y, u)

→ (y, f (x, y, u)) is convex 1 . Let : Ω × Y × U → R be a continuous Lagrangian density such that ¯ (x, y, z) = inf u∈U f (x,u)=z (x, y, u)
is convex in (y, z). Consider the optimal control problem inf (y,u) Ω (x, y(x), u(x)) dx subject to Dy(x) = f (x, y(x), u(x)), a.e. x ∈ Ω,

a i (x, y(x), f (x, y(x), u(x)) = 0, i ∈ I, a.e. x ∈ Ω, b i (x, y(x), f (x, y(x), u(x))) ≤ 0, i ∈ I, a.e. x ∈ Ω, Ω c i (x, y(x), f (x, y(x), u(x))) dx ≤ 0, i ∈ I,
where a i , b i , c i : Ω × Y × Z → R are continuous and satisfy a i | -1 {x}×Y ×Z (0) is convex for each x ∈ Ω, and b i and c i are convex in (y, z) for fixed x ∈ Ω.

To model this situation in our setup above, we add a constraint of the form

A 0 (x, y, z, u) = dist(z, f (x, y, u))
so that the constraint A 0 = 0 means that the velocity z is controlled by u via z ∈ f (x, u). We let, for i ∈ I, i = 0,

A i (x, y, z, u) = a i (x, y, z), B i (x, y, z, u) = b i (x, y, z), C i (x, y, z, u) = c i (x, y, z),
and L(x, y, z, u) = (x, y, u) and L(x, y, z) = ¯ (x, y, z).

With these assumptions, one can check easily that OC1-OC3 hold.

Optimal control -main result

In the optimal control context outlined above, we have the following theorem.

Theorem 5.2. In the setting described above, with assumptions OC1-OC3, we have M oc = M oc r . Proof. To see that M oc ≥ M oc r , note that for every triad of functions (y, u, u ∂ ) that are minimization candidates in M oc , we can produce a pair of measures (µ, µ ∂ ) defined by

Ω×Y ×Z×U f (x, y, z, u)dµ(x, y, z, u) = Ω f (x, y(x), Dy(x), u(x)) dx, f ∈ C 0 (Ω × Y × Z × U ), ∂Ω×Y ×U ∂ f (x, y, u) dµ ∂ (x, y, u) = ∂Ω f (x, y(x), u ∂ (x)) dσ(x), f ∈ C 0 (∂Ω × Y × U ∂ );
then it is immediate that the pair (µ, µ ∂ ) is a minimization candidate in M oc r that satisfies all the relevant constraints, and also satisfies Then the same arguments as in the proof of Theorem 3.1 show that z(•) is in L p , that Dy(•) = z(•) weakly, that y(•) is in W 1,p , and that y ∂ (•) is the Sobolev trace of y(•).

Ω×Y ×Z×U L dµ+ ∂Ω×Y ×U ∂ L ∂ dµ ∂ = Ω L(x, y(x), Dy(x), u(x)) dx+ ∂Ω L ∂ (x, y(x), u ∂ (x)) dσ(x).
We pick measurable functions u(•) : Ω → U and u ∂ (•) : ∂Ω → U ∂ satisfying u(x) ∈ arg min u∈U Ai(x,y(x),z(x),u)=0 Bi(x,y(x),z(x),u)≤0 L(x, y(x), z(x), u),

x ∈ Ω, (18)

u ∂ (x) ∈ arg min u∈U ∂ A ∂,i (x,y ∂ (x),u)=0 B ∂,i (x,y ∂ (x),u)≤0 L ∂ (x, y ∂ (x), u), x ∈ ∂Ω. (19)
Let us explain why we can do this. The set-valued maps induced by the right-hand sides of ( 18) and ( 19) Ω ⇒ U and ∂Ω ⇒ U ∂ have nonempty and closed images by OC2 and have measurable graphs, so we may apply a Characterization Theorem [START_REF] Aubin | Set-valued analysis[END_REF]Th. 8.1.4] to conclude that they are weakly measurable. Then we may apply the Kuratowski-Ryll-Nardzewski Selection Theorem (see [START_REF] Charalambos | Infinite dimensional analysis: a hitchhiker's guide[END_REF]Th. 18.13] or [START_REF] Aubin | Set-valued analysis[END_REF]Th. 8.1.3]) to get the functions u(x) and u ∂ (x). The arguments used in the proof of Theorem 3.1 to prove that the constraints involving A i , B i , C i , A ∂,i , B ∂i , C ∂i are satisfied by the points (x, y(x), z(x)), x ∈ Ω, and (x, y(x)), x ∈ ∂Ω, carry through to the present situation with minimal modifications for the points (x, y(x), z(x), u(x)), x ∈ Ω, and (x, y(x), u ∂ (x)), x ∈ ∂Ω, using OC1.

Thus with this choice, the triad (y(•), u(•), u ∂ (•)) is a minimization candidate for M oc , and by Jensen's inequality and OC3 we have

M oc ≤ Ω L(x, y(x), z(x), u(x)) dx + Ω ∂ L ∂ (x, y(x), u ∂ (x)) dσ(x) = Ω L(x, y(x), z(x)) dx + Ω ∂ L∂ (x, y(x)) dσ(x) ≤ Ω×Y ×Z×U L(x, y, z) dµ(x, y, z, u) + Ω ∂ ×Y ×U ∂ L∂ (x, y) dµ ∂ (x, y, u)

Application to micromagnetics

In this section we describe how variational problems appearing in micromagnetics fit our convexity assumptions and hence can be solved with our linear formulation with occupation measures. We confine ourselves to the two-dimensional situation. However, results of this section hold also in the N -dimensional case, too, for any N ≥ 2. See e.g. [START_REF]Relaxation in ferromagnetism: the rigid case[END_REF]. In the classical theory of rigid ferromagnetic bodies, based mainly on [START_REF] Brown | [END_REF], a magnetization m : D → R 2 , describing the state of the body D ⊂ R 2 , depends on a position x ∈ D and has a given temperature-dependent magnitude |m(x)| = w(T ) for almost all x ∈ D, where w(T ) = 0 for T ≥ T c the so-called Curie point. Let us treat the case when the temperature is fixed below the Curie point and thus we shall assume that |m| = 1 almost everywhere in D.

In the so-called no-exchange formulation, the energy of a large rigid ferromagnetic body D ⊂ R 2 consists of three parts and the variational principle governing equilibrium configurations can be stated (see e.g. [START_REF] James | Frustration in ferromagnetic materials[END_REF] and references therein) as the problem of minimizing

E(m) = D f (m(x)) dx - D H(x) • m(x) dx + 1 2 D ∇u m (x) • m(x) dx (20)
with respect to m and u m , where f : S 1 → [0; +∞) is continuous function defined on the unit sphere in R 2 centered at the origin S 1 . Further, m :

D → R 2 satisfies |m(x)| = 1 a.e. x ∈ D, (21)
H : D → R 2 is an applied external magnetic field and u m : D → R is a potential of an induced magnetic field. The first term in ( 20) is an anisotropy energy with a density f which is supposed to be an even nonnegative function depending on material properties and exhibiting crystallographic symmetry. The second term is an interaction energy and the last term is a magnetostatic energy coupled with the magnetization field through the equation div

(-∇u m + mχ D ) = 0 in R 2 , (22)
where χ D : R 2 → {0, 1} is the characteristic function of D. Equation ( 22) is interpreted in the classical weak sense, i.e.,

R 2 ∇φ • (-∇u m + mχ D ) dx = 0 ∀ φ ∈ C ∞ 0 (R 2 ). (23)
Let us denote

A := {m ∈ L 2 (D; R 2 ); |m(x)| = 1 for almost all x ∈ D}.
Eventually, we are concerned with the problem of minimizing

E(m) = D f (m(x)) dx - D H(x) • m(x) dx + 1 2 D ∇u m (x) • m(x) dx (24)
with respect to m and u m , and subject to [START_REF]Relaxation in optimization theory and variational calculus[END_REF]. It was shown in [START_REF] James | Frustration in ferromagnetic materials[END_REF] that if N = 1 the infimum of ( 24) is not necessarily attained even if H = 0. The reason is that minimizing sequences of magnetizations oscillate between m1 and -m1 keeping the divergence of the magnetization small, so that the gradient of u m is negligible. This construction leads to a minimizing sequence that weakly converges to zero in L 2 (D; R 2 ) and at the same time it shows that inf m∈A E(m) = 0. However, no minimizer exists and m = 0 does not belong to A. Therefore it is desirable to construct the largest lower semicontinuous envelope of E from ( 24) and there are well known techniques how to do that [START_REF] Dacorogna | Introduction to the calculus of variations[END_REF][START_REF]Parametrized measures and variational principles[END_REF]. One uses parameterized Young measures, the other one extends f by infinity outside S 1 and then calculates the convex envelope of this newly defined function. Obviously, the convex envelope is finite only on the closed unit ball in R 2 . The relaxed problem is convex but not linear, since the stray-field energy, i.e. the last term in ( 24) is quadratic.

Formulation with Young measures

As A is not convex we cannot rely on direct methods [START_REF] Dacorogna | Introduction to the calculus of variations[END_REF] to prove the existence of a solution. In fact, the solution to [START_REF]Relaxation in ferromagnetism: the rigid case[END_REF] need not exist in A, cf. [START_REF] James | Frustration in ferromagnetic materials[END_REF] for the case H = 0. Therefore, it makes sense to look for a relaxation of the problem. It is known [START_REF]Relaxation in ferromagnetism: the rigid case[END_REF] that this can be done using Young measures [START_REF] Young | Lectures on the calculus of variations and optimal control theory[END_REF]. Moreover, Young measures competing in the minimization problem are supported on S 1 , i.e. on a compact set. The relaxed problem consists of minimizing

E Y (ν) := D S 1 f (A)ν x (dA) dx - D H(x) • m(x) dx + 1 2 D ∇u m (x) • m(x) dx, (25)
with respect to ν and u m , subject to [START_REF]Relaxation in optimization theory and variational calculus[END_REF] with m(x) = S 1 Aν x (dA) for a.a. x ∈ D and ν ∈ Ā := Y ∞ (D; S 1 ), where Y ∞ (D; S 1 ) denotes the space of Young measures such that ν x is a probability measure on S 1 for a.a. x ∈ D and the dependence of ν x on x is measurable. On the other hand, it was shown in [6, Thm. 3.4, Remark 3.7] that if f :

S 1 → [0, +∞) and H ∈ L 2 loc (D; R 2 ) then inf m∈A E(m) = min ν∈ Ā E Y (ν) = min m∈A * * E * * (m) (26)
where

A * * = {m ∈ L 2 (D; R 2 ); |m(x)| ≤ 1 for almost all x ∈ D} and E * * (m) = D f * * (m(x)) dx - D H(x) • m(x) dx + 1 2 D ∇u m (x) • m(x) dx, (27)
and f * * : R

2 → R, f * * = sup f ; f ≤ f , f is convex is the convex envelope of f : R 2 → R ∪ {+∞}, f (m) = f (m) if |m| = 1 +∞ otherwise.
Again, u m in ( 27) is calculated from [START_REF]Relaxation in optimization theory and variational calculus[END_REF] The functional E * * is sequentially weakly lower semicontinuous on A * * and it possesses a minimum on A * * . We again emphasize that both [START_REF] Vinter | Convex duality and nonlinear optimal control[END_REF] and ( 27) are convex but nonlinear. The formulation of these problems in terms of occupation measures leads to a linear problem that makes it attractive for numerical solution. Similarly, this applies to some problems in linear and nonlinear elasticity where the lower semicontinuous envelope of the original problem is convex; see [START_REF] Kohn | The relaxation of a double-well energy[END_REF][START_REF] Raoult | Quasiconvex envelopes in nonlinear elasticity[END_REF] for some examples.

Formulation with occupation measures

To formulate the problem [START_REF] Rudin | Real and complex analysis[END_REF] in terms of occupation measures following Definition 2.7), let Ω be a subset of R 2 containing D, and let Y = R 3 and Z = R 2×3 . Right away, we see that the problem [START_REF] Rudin | Real and complex analysis[END_REF] with constraints ( 21) and ( 23) can be modelled as min

Observe that the solutions of ( 24) are of the form y(x) = (m(x), u(x)), where m(x) is in the unit circle S 1 ⊂ R 2 for all x ∈ Ω, and u(x) ∈ R. In the formulation above, (y 1 , y 2 ) correspond to m(x) and y 3 corresponds to u(x), so that also z 3 corresponds to ∇u(x).

Let us now show how to put this in the form of our problem [START_REF] Henrion | Convex computation of the region of attraction of polynomial control systems[END_REF]. To get the integral in (28), let Observe that even if f is convex, Theorem 3.1 cannot be applied to (28)-(30) above because the constraint (29) is not convex. Instead, we can apply Theorem 4.2, which tells us that if we take the affine occupation relaxation [START_REF] Gaitsgory | Linear programming analysis of deterministic infinite horizon optimal control problems with discounting[END_REF], that is, if we minimize (28) over O aff 2 (rather than O 2 ; see (31) below), then we get the same result M as in the convexification [START_REF] Korda | Moments and convex optimization for analysis and control of nonlinear partial differential equations[END_REF]. The latter coincides with the problem of minimizing E * * as in (27) over A * * and, by [START_REF] Young | Lectures on the calculus of variations and optimal control theory[END_REF], with [START_REF] Rudin | Real and complex analysis[END_REF]. We have proved:

subject to (29) and (30), that is, the minimum in (31) is equal to the infimum in [START_REF] Rudin | Real and complex analysis[END_REF].

Remark 6.2. Since the occupation measure relaxation (28) is sandwiched between the affinely relaxed problem (31) and the original problem [START_REF] Rudin | Real and complex analysis[END_REF], it follows from Proposition 6.1 that these are all equal to each other.

Proof. This is true for p = +∞ because in that case the supports of the measures involved are bounded and the integrands are continuous. For p ∈ [1, +∞) we have, from (6), and, by the Hölder inequality with 1 p + 1 q = 1, so that p/q = p -1, and using (6), Ω×Y ×Z c q (1 + y p-1 ) q dµ(x, y, z)

1 q ≤ Ω×Y ×Z
c(1 + y p )dµ(x, y, z)

+ i Ω×Y ×Z
z i p dµ(x, y, z)

1 p × Ω×Y ×Z
c q dµ(x, y, z)

1 q

+

Ω×Y ×Z c q y p-1 ) q dµ(x, y, z) < +∞.

Definition 2 . 4 (

 24 Affinely relaxed occupation measures). Let p ∈ [1, +∞]. Let O aff p be the set of pairs (µ, µ ∂ ) consisting of positive Radon measures on Ω × Y × Z respectively ∂Ω × Y satisfying:

Corollary 3 . 2 .

 32 Under the assumptions of Theorem 3.1, we have M r = M aff r = M . Proof. Since O p ⊆ O aff p and O p contains the pairs of measures induced by functions in W 1,p , we have that M aff r ≤ M r ≤ M . Moreover, by Theorem 3.1 we know that M = M aff r , so M r must be equal to them as well.

  z), and (d) C ∂,i (x, y) is convex in y, for all x ∈ Ω and all i ∈ I. OC2. (Fiberwise minima are attained by the controls) The functions L and L ∂ reach their minima in U and U ∂ , respectively, once we fix (x, y, z); in other words, the sets arg min u∈U Ai(x,y,z,u)=0 Bi(x,y,z,u)≤0

  are nonempty and closed for all (x, y, z) ∈ Ω × Y × Z and (x, y) ∈ ∂Ω × Y , respectively. OC3. (Convexity of the Lagrangian densities) (a) L(x, y, z) is locally bounded and convex in (y, z) for each fixed x ∈ Ω. (b) L∂ (x, y) is locally bounded and convex in y for each fixed x ∈ ∂Ω.

  Let us see that M oc ≤ M oc r . Pick a minimization candidate (µ, µ ∂ ) for M oc r satisfying all the constraints of that problem. As in the proof of Theorem 3.1, disintegrateµ(x, y, z, u) = ν(y, z, u | x) dx, µ ∂ (x, y, u) = ν ∂ (y, u | x) dσ(x),and define y(x) = Y ×Z×U y dν(y, z, u | x), z(x) = Y ×Z×U z dν(y, z, u | x), y ∂ (x) = Y ×U ∂ y dν ∂ (y, u | x).

  , y, z, u) dµ(x, y, z, u) +Ω ∂ ×Y ×U ∂ L ∂ (x, y, u) dµ ∂ (x, y, u),which implies indeed that M oc ≤ M oc r .

  (µ,µ ∂ )∈O2 Ω×Y ×Z f (y 1 , y 2 ) -H(x) • (y 1 , y 2 ) + 1 2 (z 3 • (y 1 , y 2 )) dµ(x, y, z) (28) subject to supp µ ⊆ Ω × (S 1 × R) × Z ⊂ Ω × Y × Z, (29) Ω×Y ×Z ∇φ(x) • (-z 3 + (y 1 , y 2 )χ D (x)) dµ(x, y, z) = 0, φ ∈ C 1 0 (Ω).

L 2 (z 3 •

 23 (x, y, z) = f (y 1 , y 2 ) -H(x) • (y 1 , y 2 ) + 1 (y 1 , y 2 )), (x, y, z) ∈ Ω × Y × Z.To model the constraint (29), let A(x, (y 1 , y 2 , y 3 ), z) = y 2 1 + y 2 2 -1. To model the constraint (30), we use the trick described in Example 2.3: we use I = C 1 0 (Ω) as index set and we letC φ (x, y, z) = ∇φ(x) • (-z 3 + (y 1 , y 2 )χ D (x)), φ ∈ C 1 0 (Ω). All other functions L ∂ , A i,∂ , B i , B i,∂ , C i,∂ in[START_REF] Henrion | Convex computation of the region of attraction of polynomial control systems[END_REF] can be set to 0.

Proposition 6 . 1 . 2 (z 3 •

 6123 The problem[START_REF] Rudin | Real and complex analysis[END_REF] is equivalent to the (computationally tractable) problemmin (µ,µ ∂ )∈O aff 2 Ω×Y ×Z f (y 1 , y 2 ) -H(x) • (y 1 , y 2 ) + 1 (y 1 , y2)) dµ(x, y, z)

  ∂Ω×Y φ(x, y)n(x) dµ(x, y) ≤ ∂Ω×Y c(1 + y p ) n(x) dµ ∂ (x, y) = ∂Ω×Y c(1 + y p ) dµ ∂ (x, y) < +∞

  1,p (Ω; Y ) denotes the Sobolev space of measurable functions φ : Ω → Y that are weakly differentiable and such that both φ and its weak derivative Dφ are in L p . Considering constraints indexed by a possibly uncountable index set provides a significant modeling freedom, allowing one, for example, to model higher-order PDE constraints in the weak form as shown in Example 2.3.Example 2.1 (Plateau's problem). Let Ω be the unit disc in R 2 , Y = R 3 , and

  and Lemma A.1 in the appendix.

	Definition 2.7 (Relaxed occupation measures). Let p ∈ [1, +∞]. Let O p be the set of pairs
	(µ, µ ∂ ) ∈ O aff p that additionally satisfy:
	For all φ ∈ F p , it holds
	Ω×Y ×Z	∂φ ∂x	(x, y) +	∂φ ∂y	(x, y)z dµ(x, y, z) =	∂Ω×Y	φ(x, y)n(x) dµ ∂ (x, y),

This is true for example when f depends only on (x, u) and f ({x} × U ) is convex for each x ∈ Ω.

Acknowledgment. The work of supported by the Barrande project 44702QH and the M ŠMT ČR project 8J20FR019. M. Kružík was further supported by the GA ČR project 21-06569K. R. Ríos-Zertuche gratefully acknowledges the support of ANR-3IA Artificial and Natural Intelligence Toulouse Institute (ANITI) and the hospitality of LAAS-CNRS. The research of M. Korda is partly supported by AI Interdisciplinary Institute ANITI funding, through the French "Investing for the Future PIA3" program under the Grant agreement n • ANR-19-PI3A-0004. This research is also part of the programme DesCartes and is supported by the National Research Foundation, Prime Minister's Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme.

A Constancy lemma

The following is essentially a version of the what is known as the Constancy Theorem in Geometric Measure Theory, and it is a slight generalization of [START_REF] Korda | The gap between a variational problem and its occupation measure relaxation[END_REF]Lemma 2.9]. We use it in the proof of Theorem 3.1. for all compactly-supported φ ∈ C ∞ 0 (Ω; R). Then µ equals the n-dimensional Lebesgue measure on Ω.

Proof. Let R ⊂ Ω be a small parallelepiped, and let τ be a translation such that τ (R) ⊂ Ω. We will show that µ(R) = µ(τ (R)), and since this will be true for all R and all τ , µ must be a positive multiple of Lebesgue on Ω [START_REF] Rudin | Real and complex analysis[END_REF]Thm. 2.20]. Write τ as a finite composition of translations τ i in the directions of the axes x 1 , . . . , x n ,

set τ0 equal to the identity. We assume τ 1 , . . . , τ k have been chosen also in a such a way that the convex hull of τi-1 (R) ∪ τi (R) is contained in Ω for each i.

For each i = 1, . . . , k, let j i be such that τ i is a translation in direction x ji . Recall that χ τi(R) is the indicator function of the translated rectangle τi (R), and let

which is a compact set properly contained in Ω. Approximating with smooth, compactlysupported functions and using the Lebesgue dominated convergence theorem, we conclude that (32) is true for φ = ψ i , which means, for the j th i entry,

By induction we get

Thus µ = c dx for some c > 0. Since µ(Ω) = |Ω|, we conclude that c = 1. are integrable with respect to µ ∂ and µ, respectively.

B Technical lemma for the definition of occupation measures


Florent Di

Bartolo Xavier

Hautbois Lisaa

Alexandra David-Neel

Tackle Climate Change Through Art: A case study

Keywords: Datascape, Creative visualisation, Anthropocene, Climate Change, Visual music, Digital art, Music

This paper describes the creative process behind the creation of an audiovisual installation. Titled

The Great Acceleration (2021), this installation is built upon scientific data simulating the retreat of the Ross Ice Shelf from 2015 to 2100. The dataset is the result of a computer simulation based on a scientific model provided by researchers from the Laboratory for Sciences of Climate and Environment (LSCE). The paper details the methodology used to offer a sensitive experience of climate change through the analysis of a large amount of data and the creation of generative audiovisual compositions. It describes the artistic approach and how the dataset related to the Ross Ice Shelf activity has been processed in order to create visual compositions and shape sounds. The paper also explains the relationship between these images and sounds and how they react to external outputs such as the presence and movement of visitors within an exhibition space.

INTRODUCTION

This paper describes the creative process behind the creation of an audiovisual installation that has obtained the financial support of the Diagonale Paris-Saclay through its call for projects Expérimentation 2020.

Titled

The Great Acceleration (2021), this installation is built upon scientific data evoking the uncertain future of Antarctica's melting ice. The selected dataset relates more precisely to the transfiguration of the largest ice shelf in Antarctica: The Ross Ice Shelf from 2015 to 2100. The dataset is the result of a computer simulation based on a scientific model provided by researchers from the LSCE laboratory: a laboratory that studies past present and future climate from the analysis of natural archives and the development of numerical models.

The paper details the methodology used to offer a sensitive experience of climate change through the creation of an interactive installation. This methodology is built upon the analysis of a large amount of data and is used to create generative audiovisual compositions that can be seen at this address: http://vimeo.com/608419914/c27388c024. First, the artistic approach and the processing of the dataset related to the activity of the Ross Ice Shelf is defined. Then, we explain how images and sounds have been created and animated while using this dataset as raw and refined materials. Finally, we discuss the dynamic relationship between these images and sounds and how they react to external outputs such as the presence and movement of visitors within an exhibition space.

ARTISTIC APPROACH AND DATASET

The musical work

The musical part of the installation is based on a work of mixed music for cello and electroacoustic (9'16) by Xavier Hautbois composed on the theme of melting ice. In accordance with the slowness of the whole installation, the general atmosphere of the musical piece has a static and timeless aspect, despite the moments of tension carried by the instrument which plays the main role: the cello.

The soundtrack that carries the music is produced by virtual instruments. It is in perpetual and slow development.

Two recognisable melodic instruments emerge from it: the cello, playing the dominant role, and the celesta, staying back, which adorns the song of the cello in an almost concertante writing.

A light granular texture accompanies this ensemble in an autonomous way. This texture is built from natural sounds recalling flowing waters as well as ice cracking and breaking up. Although the sound background presents more or less strong tonal reminiscences, the cello melody is expressed in atonal and contemporary expressions.

Formally, the musical piece is made up of eight well-defined sequences. In each one of them, the cello exposes one or more modes of play: These bowing techniques create musical variations which affect the movements of the particle systems used to depict the evolution of the Ross Ice Shelf activity.

(i)

Data selection

The installation The Great Acceleration is built upon a dataset provided by researchers from the LSCE working on past and future interactions between climate and ice sheets.

The dataset is an output of the GRISLI ice sheet model: a model able to reproduce expected grounding line advances during glacial periods and subsequent retreats during terminations with reasonable glacial-interglacial ice volume changes [START_REF] Quiquet | The GRISLI ice sheet model (version 2.0): calibration and validation for multimillennial changes of the Antarctic ice sheet[END_REF]. The dataset represents with the help of a Cartesian coordinate system the possible retreat of the Ross Ice Shelf from 2015 to 2100.

The first step to develop an installation based on this data and the movements it contains was to give it a default representation by creating a program able to display the evolution of the Ross Ice Shelf activity. These first attempts of visualisation were used to verify that the data has been handled properly by being able to retrieve an accurate representation of the Ross Ice Shelf that could be compared to visualisations used by climate scientists.

These first images of the Ross Ice Shelf activity were also used to identify a specific parameter, an iconic variable that is used as the main material to create images and sound related to the Great Acceleration: a concept that depicts the dramatic acceleration in human enterprise and its impacts on the Earth system over the last two centuries [START_REF] Steffen | The trajectory of the Anthropocene: The Great Acceleration[END_REF].

Amongst the many parameters available, ice surface velocity has been selected due to its significant increase over the observed period of time. The increased velocity of glaciers and ice sheets due to climate change is one of the main causes of sea level rise. Ice velocity represents one of the most important parameters researchers used to study ice sheet and glacier dynamics. Rates of ice movement vary from one glacier to another and change with depth. For example, glacial motion gets smaller on small glaciers or in the centre of ice sheets.

In this case only the ice sheet surface velocity has been taken into account. The internal and sliding velocities of the Ross Ice Shelf have not been selected to shape the audiovisual compositions presented in this paper.

Data translation

The data selection process was followed by its treatment. Ice velocities were used to define the outlines of a constantly evolving territory. Ice surface velocities were originally related to a specific position on a Cartesian grid. Together, they draw the movements animating a territory of 16 km by 16 km which represents the Ross Ice Shelf. As such, they have been first used to create grayscale images of ice velocities at the ice shelf surface. Then, these digital images have been processed using blob detection methods in order to identify different regions inside each of them (Figure 1). This computational process serves a speculative inquiry as any process of speculative computing [START_REF] Drucker | Speculative Computing: Aesthetic Provocations in Humanities Computing[END_REF]. It aims to produce a useful aesthetic provocation that is the result of a subjective interpretation enacted by an interpreter and predicting the disappearance of the Ross Ice Shelf.

The regions identified show 5 different levels of intensity related to the ice shelf movement. They allow us to follow the evolution of the Ross Barrier and to observe the acceleration of Antarctic Ice Sheet mass loss over a long period of time. These regions are mainly defined by their curves, surface, position and the level of intensity they are associated with. Their curves represent the main element used to create the datascape of the installation The Great Acceleration.

THE CREATION OF GENERATIVE AUDIOVISUAL COMPOSITIONS

The lists of 2D vectors that define each region are at the core of the images that depict the activity of the Ross Ice Shelf. These images are built using multiple particle systems that redraw the curves of each region over time in a 3D environment (Figure 3). The third dimension of the 3D space is used to separate the regions between them based on the intensity of the movements they are representing.

With the colour code used to draw these lines, this spatial distancing recreates figurative representations of the Ross Ice Shelf that allows their viewers to observe the retreat of the ice barrier.

The curves of each region are used to modulate the sound composition using IanniX: a graphical open-source sequencer. They become velocity curves modifying in real-time musical parameters. Each curve is assigned to a specific sound source depending on its velocity level. Celesta is assigned to curve level 1, the harmonic background to levels 2 and 3, while the granular texture is assigned to levels 4 and 5. The perimeter of each curve influences the loudness of the sound assigned to it. Therefore, the granular texture becomes more present as the curves of levels 4 and 5 appear and grow stronger over time. The sound generator which controls this progression sets noticeable variations in the balance between voices. In the sequencer IanniX, a playhead slides on every velocity curve (Figure 2). Every curve is being played in a loop at a constant speed. The shape of these curves acts on band-pass filters in order to give an iridescent and changing colour to the sounds they are associated with. Depending on the angle of rotation of their playhead, the sound quality takes on darker or, on the contrary, brighter tones. Each velocity curve has its own length and shape. This characteristic permits the creation of complex filter overlays over time.

The whole installation tends to point out ecological issues through the lenses of an objective observer.

The retreat of the Ross Ice Shelf is presented as a slow process without much emphasis on ice calving or the rise of sea level. The installation tends to show the melting of ice shelves as the result of an unstoppable force difficult to see to the naked eye.

The installation builds a representation of this long process by giving it a figure that slowly emerges through time over a period of approximately 10 minutes covering 85 simulated years (Figure 3).

AUDIO-REACTIVE PARTICLE SYSTEMS

One of the key elements of the installation The Great Acceleration was to create a close relationship between a visual and a musical expression using custom written programs. The goal was to design a generative system able to simultaneously create images and sounds from the same refined data. In this case, "sounds" represent modulated versions of audio samples combined together inside a sound composition made of height musical sequences and linked to audio track settings. On the other hand, "images" represent animated visual compositions built mainly using particle systems. Images and sounds are shaped by the same ice velocity data. They also interact in many ways. Each mode of playing of the cello has a singular influence on the particle systems used to express the Ross Ice Shelf activity. This interaction between images and sounds is built upon a communication between two entities: a sound generator and an image generator exchanging information. The sound generator provides in real-time information about the number of the musical sequence being played, the amplitude of the sounds as well as the current mode of playing.

These last two pieces of information are used together to modify the strength of forces applied to particle systems in order to recreate well-defined movements inside a visual composition related not only to a specific mode of playing but also to sound intensity. The visual compositions respond to the sound composition by taking into account the transition from one mode of playing to another and the strength with which they are introduced and then played. The movements of the visual compositions tend to follow the rhythm being suggested by the current audio playing mode.

Short sounds introduce, depending on their intensity and related playing mode, almost invisible Brownian noise inside a visual composition or, on the contrary, sudden movements able to vigorously displace entire particle systems. Another example: the legato sounds of the cello related to the first playing mode create slow theatrical movements inside a visual composition that affect almost every particle by forcing them to follow a path specified by a force field. These new trajectories are defined by taking into account the actual position of each particle. They have the effect of momentarily blurring the lines used to represent the Ross Barrier.

DESTRUCTED FORM AND ARITHMETIC CONSTRAINTS

Destructured musical form

The musical form is "destructured" (Bootz & Hautbois 2020). Its processing is not the destruction of a preconceived form or its negation, but an extension of a formal, harmonic or acoustic work. A destructured musical form reveals sonic details which did not appear in the original work and which are changing its characteristics, much like a magnifying glass focusing on fragments of a drawing under observation.

The musical score is controlled by the sound generator which gives life to musical aspects of the work not present in its original and closed form: details underlined by the reorganisation of the musical form, by the application of filters activated in real-time or by the balance between voices.

In this case, the music explores new sound configurations of the initial work in order to replay endlessly, but always in a different way, through interactivity, the inexorable process of ice shelves retreating.

The destructuring of the musical form is orchestrated by performing different actions:

(i)

The differentiated treatment of the four sound sources extracted from the initial piece. (ii)

The application of filters and spatialisation effects based on the reading of velocity curves. (iii)

The interaction of the public with the installation. (iv)

The reorganisation of the musical form itself according to arithmetical constraints.

Arithmetic constraints

The original musical composition lasts 9 minutes and 16 seconds. As a multimedia installation theoretically has no end, the musical form had to be rethought in order to take into account repetitions of sequences, while avoiding this repetition being too frequent. As part of the installation, the musical composition is reorganised each time the cycle of eight sequences comes to an end.

These new forms of arrangement are not randomly chosen. They follow arithmetic constraints that allow 15 different combinations while maintaining a sense of musical order. Considering the musical sequences numbered from 1 to 8, the following rule has been applied: each sequence n can only be followed by the following sequence n+1 in the initial series (which constitutes the order of succession of the musical work) or else by the sequence located five steps forwards in the series n+5 (Table 1).

These new forms of arrangement between audio sequences don't affect the way the visual compositions are organised:

The visual compositions follow a fixed timeline. They start by drawing a landscape that depicts the Ross Ice Shelf in 2015 and slowly evolve through the years until reaching 2100 that marks the end of the simulation period.

Nevertheless, each new cycle is not identical to the previous one. Each cycle of musical sequences being different from the previous one, the visual sequences, which react to the music, evolve each time in a different way, while being generated from the same velocity data linked to a specific year.

The arithmetic constraints applied to reorder the audio sequences at the end of each cycle permit to always end a cycle with the audio sequences 4 or 8. This example shows how the rules that associate musical sequences with visual ones are defining a non-unilinear structure which allows building audiovisual momentum and not random chaos.

INTERACTIONS AND INTERACTIVITY

Interactions between visual compositions

Interactions and interactivity play an important role in the arrangement of visual compositions and evolution of the musical form. The main visual compositions are generated in real-time from velocity data related to a specific year and respond to different modes of playing. They represent individual images of the datascape drawn to show the evolution of the Ross Barrier using particle systems (Figure 3). This datascape is presented through the eyes of two different cameras. The first camera is used to display a panoramic view of the curves and revolves slowly around them. It works in tandem with a second virtual camera which is used to target specific curves of the datascape from another point of view. This camera is closer to the curves and moves towards them. It provides fragmentary and animated views from above. The visual sequences created using these two different cameras last between 10 and 15 seconds. They can be presented one after the other but can also be automatically swapped with three other types of visual sequences in relation to the musical sequence being played and its attributes. These three other types of visual sequences represent:

(i)
Icebergs that have broken off an ice shelf.

(ii) Meltwater. (iii)

Filtered views of the dataspace. They are built on knowledge acquired while working on audio-reactive point clouds (Di Bartolo 2019). The transition from one visual sequence to another is related to sound events. For example, when the cello ceases to be audible, another type of visual composition can appear. This composition represents icebergs moving slowly towards the camera (Figure 4). While this representation is still made of particles, it offers a counterpoint to the visual compositions displaying the datascape. Its stability contrasts with the slow and long movements generally applied to the datascape according to a specific musical playing mode. The third type of visual compositions gives birth to abstract forms. It has been created to generate meltwater representation from the inside of an ice shelf (Figure 5). The virtual camera used to capture this scene is inside the particle system used to define the visual composition. A turbulence field constrains the particle movements in order to give shape to meltwater slowly going through sheets of ice.

While these last two types of visual compositions are not directly created from ice surface velocity data (unlike the datascape) they evolve with it. The size of icebergs decreases over time while the movement of meltwater is accelerating throughout the entire simulation period. The shrinking size of icebergs shows they are doomed to fade. Icebergs act as contemporary vanitas. They remind us of the transience of life (Remaud 2020).

Experiencing interactivity

The fourth and last type of visual compositions is only being displayed in interaction with the viewers.

Several motion detectors are used to monitor the movements of the public inside the exhibition space. The data outputted by the different sensors is processed to assign their current location to four different zones of equal area. It allows the viewers to interact with the datascape based on its location. When the presence of a visitor is being detected, in one of the four zones, for the first time of a simulation year, the fourth type of visual composition is used to filter the datascape by slowly redrawing its largest region while masking every other visual element (Figure 6). This interactive behaviour creates a mapping between the simulation space and the exhibition space. The location of each sound in the quadraphonic space (defined by the installation speakers) is established according to the position of the velocity curves and the position of their respective playhead in the sequencer IanniX. The public is therefore surrounded by sounds, each having its own trajectory in space.

This fourth type of animated visual composition is, as the datascape, responsive to the musical composition and its different mode of playing. It also shows a progression throughout the simulated period using progressive colour changes.

The audio composition responds to the appearance of this fourth type of visual. The display of a single region is used to mute all the sounds which are not related to the selected region in the sound generator. This filtering process allows the viewers to listen closely to a single velocity curve. It produces a zoom effect by focusing on a musical detail of the electroacoustic piece which participates in the controlled and indeterminate destructuring of the audio work.

Figure 1 :

 1 Figure 1: Examples of different regions identified using blob detection methods. The curves' colour code ranges from blue (low velocity) to red (high velocity).

Figure 2 :

 2 Figure 2: Velocity curves in the graphic space of the IanniX sequencer.

Figure 3 :

 3 Figure 3: Example of the datascape made of different curves drawn using particle systems.

Figure 4 :

 4 Figure 4: Representation of icebergs that have broken off an ice shelf.

Figure 5 :

 5 Figure 5: Representation of meltwater.

Figure 6 :

 6 Figure 6: Example of a filtered view of the datascape displaying a single curve.

Table 1 :

 1 Repetitions of musical sequences

	1	2	3	4	5	6	7	8
	5	2	3	4	1	6	7	8
	1	2	3	8	5	6	7	4
	5	2	3	8	1	6	7	4
	1	2	7	4	5	6	3	8
	5	2	7	4	1	6	3	8
	1	2	7	8	5	6	3	4
	5	2	7	8	1	6	3	4
	1	6	7	4	5	2	3	8
	5	6	7	4	1	2	3	8
	1	6	7	8	5	2	3	4
	5	6	7	8	1	2	3	4
	1	6	3	4	5	2	7	8
	5	6	3	4	1	2	7	8
	1	6	3	8	5	2	7	4

ACKNOWLEDGEMENTS

We would like to thank the researchers of the LSCE laboratory (Laboratory for Sciences of Climate and Environment) and in particular Sylvie Charbit for providing the dataset related to the Ross Ice Shelf activity. Our thanks also go to Aurélien Quiquet and Christophe Dumas who ran the simulations at the origin of the dataset.