Raju Vaggu

Nagender Thadem

Manda Rajesh

René Grée

Saibal Das
email: saibal@iict.res.in.

Acylsilane Directed Rh-Catalyzed Arene C-H Alkylation with Maleimides and Visible-Light-Induced Siloxycarbene-Amide Cyclization: [3+2] Carbo-Annulation in Ru Catalysis

ABSTRACT:

We herein demonstrate the acylsilane directed Rh-catalyzed arene C-H bond alkylation with maleimides. The resulting derivatives were utilized in visible-light-induced intramolecular siloxycarbene-amide cyclization for the synthesis of new tricyclic γlactams. In Parallel, we also harnessed the same acylsilane and maleimide units through [3+2] carbo-annulation by using Ru-catalysis. A wide range of maleimides and aroylsilanes were used to establish the broadness of these transformations.

Acylsilanes are captivating chemical probes in organic chemistry [START_REF] Reviews | Acylsilanes: Valuable Organosilicon Reagents in Organic Synthesis[END_REF][START_REF] Tatsumi | Zinc-Catalyzed Synthesis of Acylsilanes Using Carboxylic Acids and a Silylborane in the Presence of Pivalic Anhydride[END_REF][START_REF] Schmink | Reversed-Polarity Synthesis of Diaryl Ketones via Palladium-Catalyzed Cross-Coupling of Acylsilanes[END_REF][START_REF] Shen | Prepared by C-H Bond Functionalization with Acylsilanes[END_REF][START_REF] Ye | Visible-Light-Induced Cycloaddition of a-Ketoacylsilanes with Imines: Facile Access to b-Lactams[END_REF][START_REF] Directed | Acylsilanes in Rhodium (III)-Catalyzed Directed Aromatic C-H Alkenylations and Siloxycarbene Reactions with C-C Double Bonds[END_REF] which are partaking in wide range of transformations, including cross couplings, aldol reactions, radical reactions, nucleophilic additions, C-H functionalizations, and annulation reactions. From a plethora of possible reactions of acylsilanes, the nucleophilic addition/1,2-Brook rearrangement is one of the most attractive transformations to construct C-C and C-X bonds through singlet nucleophilic siloxycarbene intermediates. [START_REF] Schmink | Reversed-Polarity Synthesis of Diaryl Ketones via Palladium-Catalyzed Cross-Coupling of Acylsilanes[END_REF][START_REF] Shen | Prepared by C-H Bond Functionalization with Acylsilanes[END_REF][START_REF] Ye | Visible-Light-Induced Cycloaddition of a-Ketoacylsilanes with Imines: Facile Access to b-Lactams[END_REF] This rearrangement offers chemists an expeditious path to probe in unearthed chemical space for various organic synthetic activities. Diverse elegant routes were established for such rearrangements, using visible-light, organo catalysts and transition-metal catalysts with a range of carbon and heteroatom trapping reagents (Scheme 1ai).

Heck type alkenylation

Rh, Ru (b) this work: Conversely, directing group (DG) assisted transition-metal catalyzed site selective C-H functionalization is a swift strategy to construct verity of complex acyclic and cyclic motifs in a single operation. [START_REF] Engle | Weak Coordination as a Powerful Means for Developing Broadly Useful C-H Functionalization Reactions[END_REF] These strategies not only avoid the pre-functionalization of the reactive centers but also fetch atom-and step-economy in the anticipated transformation with high functional group compatibility. Over the past few years, many splendid routes were established using a large spectrum of transition-metals (Pd, Rh, Ru, Ir, Cu, Co, Mn, Ni, etc.) and coupling partners. In this context, pendent chelating group assists the metal catalyst in activation of C-H bond through precoordination (concerted metalation-deprotonation). These chelating directing groups mainly nitrogen, oxygen, and carbon-based functional groups, which will also join in annulation reactions along with various ᴫ-congeners. In this paradigm, hefty series of DGs, including weakly coordinated carbonyls also illustrated in such C-H functionalization and annulation reactions. [START_REF] Engle | Weak Coordination as a Powerful Means for Developing Broadly Useful C-H Functionalization Reactions[END_REF] In stark contrast to the Brook rearrangement, the ortho C-H functionalization of arene acylsilanes is subtle, [START_REF] Directed | Acylsilanes in Rhodium (III)-Catalyzed Directed Aromatic C-H Alkenylations and Siloxycarbene Reactions with C-C Double Bonds[END_REF] because, these chemprobs have aldehyde trait which can endure supplementary transformations in given condition. The unscathed acylsilane is a potential reactive handle for later transformations which offers novel potentialities to this arsenal, as the acylsilane itself is a uniquely reactive center (vide supra). Using this weakly coordinated functional group as C-H directing group, some elegant functionalizations (alkenylation, amidation, allylation) and annulation (indene) transformations were documented via Rh, 6a,f Ir, 6b Ru 6c-e and Co 6g catalysis (Scheme 1aii). The Rh/Ru catalyzed C-H alkenylation (Heck type) 6a, 6c was reported using unsaturated carbonyl derivatives through β-hydride elimination, where both metals shown identical reactivity (Scheme 1b). However, the availability of acylsilane C-H functionalization strategies remains scanty. In this letter, we demonstrate the acylsilane directed Rh-catalyzed C-H alkylation with maleimides and subsequent visible-lightinduced siloxycarbene-amide cyclization affording tricyclic indeno-γ-lactams. Further, a stereoselective annulation was also achieved under Ru catalysis with the same reactive partners (Scheme 1c).

+ SiR 3 O SiR 3 O X O R 1 [1,2]-Brook's (i) (ii)

C-H functionalization C-H annulation

The easily available and flexible reactive maleimides have been recently dragged a great attention of synthetic organic Page 1 of 6 ACS Paragon Plus Environment Submitted to Organic Letters chemists, due to the omnipresent nature of these motifs in various fields of biology. 8 Specially, the succinimide and its fused derivatives are frequently present in natural products and biologically intriguing molecules (Figure 1). Maleimides are linchpin prototypes in DG assisted C-H activation reactions, which are most engaged one in three (viz alkenylating, alkylating and annulating reactions; depending on the kind of DG, metal catalyst and reaction conditions) coupling partners in C-H functionalization. [START_REF] Maleimides In C-H Activation ; Manoharan | Annulation and Alkenylation of Organic Molecules with Maleimides via Transition-Metal-Catalyzed C-H Bond Activation[END_REF] Utilizing these maleimides, we herein report, the synthesis of acylsilylated phensuximide derivatives as well as indano-succinimide adducts in expeditious way.

Initially we conducted the reaction using benzoyl silane (1a) and N-phenyl maleimide (2a) as test substrates. After extensive optimization studies, we found that the reaction of 1a with 2a in presence of 5 mol% [Cp*RhCl 2 ] 2 , with 60 mol% of Cu(OAc) 2 .H 2 O as an additive and 10 mol % AgSbF 6 silver salt in DCE solvent at 70 o C afforded adduct 3aa in 82% yield (for detailed optimization studies see Supporting Information). Having these optimized conditions in hand, we headed to examine the generality of the C-H alkylation path (Scheme 2). We first verified the substrate scope by using electronically distinct acylsilanes, thus the halogenated (F, Cl, Br and I) benzoyl silanes were well tolerated under the given conditions and afforded expected adducts 3ba-3ga in 68-74% yields irrespective of their positions on core aryl group (ortho and para). The exquisite iodo functional group reactivity is particularly noteworthy, as it is one of the least often selected functional group in C-H activation. The resulting adducts could be used for further functionalization with various cross coupling conditions for succinimide handled products. To our astonishment, the ortho and para iodo substrates 1f-g smoothly transformed to C-H alkylated adducts 3fa-ga in 68-70% yields. Subsequently, the electronically neutral -tert-Butyl and electronic rich alkoxy substitutions like methoxy, 3,4dimethoxy and 2,3-dimethoxy benzoyl silanes (1h-l) could survive with equal ease in the formation of corresponding adducts 3ha-la in 65-74% yields. We subsequently investigated the reaction feasibility of various N-substituted maleimide coupling partners, as exemplified in Scheme 3. The alkyl groups (ethyl, dimethyl, trimethyl, diisopropyl and n-butyl) on N-aryl maleimides 2b-f provided corresponding final products 3ab-af in 64-77% yields. The N-para-methoxy phenyl maleimide delivered 3ag in 71% yield and the structure was confirmed by X-ray crystallography. The halogenated N-aryl maleimides also showed equal reactivity in standard conditions. Thus, the F, Cl, Br and I substituted chemprobs 2h-k afforded 3ah-ak in good yields (72-74%). These functionalities could also be used in further extension reactions. The electron withdrawing CF 3 , Cl and NO 2 aryl maleimides worked similarly to their halogenated cousins (3al-am in 70-75%). We further extended our focus to test the viability of N-alkylated maleimides under title conditions. The N-Methyl maleimide 2n afforded the acylsilated phensuximide 3an in 78% yield. The other N-alkylated (Et, n-propyl, t Bu and cyclohexyl) maleimides showed analogous reactivity to Nmethyl maleimide (3ao-ar). To our delight, the N-unsubstituted maleimide 2s also reacted with 1a in equal manner. Setting a limitation, the other doubly activated alkenes (maleic anhydride, 1,4-benzoquinone) were found to be inapt in this transformation.

Scheme 2. Scope of acylsilanes a

Scheme 3. Scope of Maleimides a

In continuation, the meticulous search on the aforementioned 1,2-Brook rearrangement brings to light the meagerness in the intramolecular trapping of such reactions. [START_REF] Shen | Prepared by C-H Bond Functionalization with Acylsilanes[END_REF] Thus the utility of unscathed acylsilane functionality in concomitant reaction with pendent reactive handle is limited to certain section of molecules. [START_REF] Ye | Visible-Light-Induced Cycloaddition of a-Ketoacylsilanes with Imines: Facile Access to b-Lactams[END_REF] In order to exemplify such a possibility, we utilized the newly formed o-succinimide benzoyl silanes in siloxycarbene-amide cyclization through 1,2-silyl migration followed by a cyclization sequence towards pharmacologically significant tricyclic hydroxy γ-lactam derivatives 4 (Scheme 4). [START_REF] Wu | N-Heterocyclic carbene-catalyzed tandem aza-benzoin/Michael reactions: on site reversal of the reactivity of N-Boc imines[END_REF] Owing to the occurrence of two stereogenic centers along with a quaternary carbon, the synthesis of these privileged tricyclic compounds are challenging. 10a,b To achieve this synthesis in easy and simple manner, we conducted a reaction with 3aa under 455 nm visible-light (blue LEDs) (see SI for optimization studies) which provided the cyclization product 4a in 89%. The other functionalized succinimide derivatives (aryl-alkyl, halo, NO 2 and alkyl) also participated well in this light-mediated amide cyclization affording corresponding tricyclic products in good to excellent yields (4b-4h in 73-92%). The structure of the tricyclic γ-latam 4i was confirmed by X-ray crystal analysis. Moving further, the alteration in the reactivity (chemo-divergence and/or regio-divergence) of same substrates came from switching the metal catalysts, metal attached ligand. [START_REF] Lu | Divergent Functionalization of Indoles with Acryloyl Silanes via Rhodium-Catalyzed C-H Activation[END_REF] An analogy to Rh(III)-catalyst, the other metal catalysts, for instance, less expensive Ru(II)-catalyst often provide complementary reaction patterns. 11c We then quested towards carbo-annulation using same substrates.

Scheme 4. Visible-light-induced amide cyclization

In this line, we harnessed aroylsilanes and maleimides in Ru(II)-catalyzed carbo-annulation approach for biologically intriguing indano-succinimide motifs. [START_REF] Yang | DBU-Promoted Intramolecular Crossed Aldol Reaction: A Facile Access to Indane-Fused Pyrrolidine[END_REF] Initially, we synthesized compound 5a in 71% yield under Ru-catalysis (for optimization studies, see supporting information) as shown in Scheme 5. To ascertain the generality of this annulation, we screened electronically dissimilar substrates under established conditions. The N-aryl functionalized maleimides (with halogens and alkyls) proved to be good starting materials for this reaction with benzoyl silane (5b-e in 58-64% yields). Delightfully, the N-methylated and N-unsubstituted maleimides produce the corresponding 5f and 5g in fair to good yields (56-73%). The structure of 5g was confirmed by X-ray crystallography. Subsequently, iodo-substituted indanone 5h was also synthesized from aroylsilane and maleimide. Whereas, the ortho-methoxy substituted acylsilane afforded the indanol product 12b 5i in 57% yield only, perhaps due to the orthomethoxy coordination which might have promoted the 1,2-Brook and proto-demetallation sequence. It is noteworthy to mention that the reduced pyrodo-indanol (5i) was found as antihypertensive agent. [START_REF] Yang | DBU-Promoted Intramolecular Crossed Aldol Reaction: A Facile Access to Indane-Fused Pyrrolidine[END_REF] Lastly, we utilized the synthesized succinimide derivative for further transformation. First, we synthesized the aldehyde 6a from 3la in 71% yield. This aldehyde 6a is an intermediate for antihypertensive agent 12a (Scheme 6,eq i). We further deacylated the compound 3an for the synthesis of formylated phensuximide 6b in 79% yield (Scheme 6, eq ii). continuation, to verify the competitive nature of rhodium and ruthenium catalysis, we performed a reaction using both catalysts in single pot (Scheme 6, eq iii). After 6 h of the reaction we observed C-alkylation as the only product. In prolongation time we isolated 3aa in 45% along with annulated adduct 5a (17% yield). We further conducted the cyclization reaction using 3aa under Ru-catalysis no expected product 5a was observed (Scheme 6, eq iv). These results suggest that the C-metal interaction is crucial for annulation.

Based on these observations and literature precedence, we propose the tentative mechanism, as shown in Scheme 7. Initially the cationic metal catalyst activates the C-H bond through concerted-metalation-deprotonation process, which upon chelation, followed by insertion of maleimide affords the intermediate C. This intermediate deliver compound 3 through proto-derhodation of C-Rh bond. In the case of ruthenium, the less acidic C-Ru bond inserts into the acylsilane and provides the intermediate E, which upon desilyl-rutheniation gave carboannulated adduct 5. Scheme 7. Mechanistic proposal. Further, the Cu(OAc) 2 oxidizes the metal catalysts for the next cycle. The tricyclic hydroxy γ-lactams (4) were achieved through light-induced 1,2-rearrangemt followed by the siloxycarbene insertion into amide bond and silyl migration sequence.

In conclusion, we have unveiled the acylsilane directed Rhcatalyzed C-H bond alkylation with maleimides for the synthesis of ortho-acylsilylated phensuximide derivatives. The weakly coordinated acylsilane group triggers the C-H bond functionalization and intervenes in the concurrent transformations. The synthesized o-succinimide aroylsilanes proved to be remarkable substrates in visible-light-induced intramolecular siloxycarbene-amide cyclization towards indano-γ-lactams. In contrast to the Rh-catalyzed alkylation, we further pursued the stereo-selective [3+2] annulation of aroylsilanes and maleimides using Ru-catalysis to indenol succinimide fused derivatives. Efforts to expand the aroylsialne reactivities are ongoing in our laboratory.

ASSOCIATED CONTENT

8

 8

Figure 1 .

 1 Figure 1. Some bioactive succinimide and indanopyrrolidine derivativesMaleimides are linchpin prototypes in DG assisted C-H activation reactions, which are most engaged one in three (viz alkenylating, alkylating and annulating reactions; depending on the kind of DG, metal catalyst and reaction conditions) coupling partners in C-H functionalization.[START_REF] Maleimides In C-H Activation ; Manoharan | Annulation and Alkenylation of Organic Molecules with Maleimides via Transition-Metal-Catalyzed C-H Bond Activation[END_REF] Utilizing these maleimides, we herein report, the synthesis of acylsilylated phensuximide derivatives as well as indano-succinimide adducts in expeditious way.Initially we conducted the reaction using benzoyl silane (1a) and N-phenyl maleimide (2a) as test substrates. After extensive optimization studies, we found that the reaction of 1a with 2a in presence of 5 mol% [Cp*RhCl 2 ] 2 , with 60 mol% of Cu(OAc) 2 .H 2 O as an additive and 10 mol % AgSbF 6 silver salt in DCE solvent at 70 o C afforded adduct 3aa in 82% yield (for detailed optimization studies see Supporting Information). Having these optimized conditions in hand, we headed to examine the generality of the C-H alkylation path (Scheme 2).

  = 4-Br, 70% 3ea, X = 2-Br, 72% 3fa, X = 4-I, 68% 3ga, X = 2-I, a 1 (0.25 mmol), 2 (0.375 mmol),[Cp*RhCl 2 ] 2 (5 mol%), AgSbF 6 (10 mol %), Cu(OAc) 2 .H 2 O (60 mol%) in DCE (1.5 mL) b Isolated yield. [Cp*RhCl 2 ] 2 (5 mol%), AgSbF 6 (10 mol %), Cu(OAc) 2 .H 2 O (60 mol%) DCE, 70 o C, 8

  Reaction conditions: a 1 (0.25 mmol), 2 (0.375 mmol), [Ru(p-cymene)Cl 2 ] 2 (5 mol%), AgSbF 6 (10 mol %), Cu(OAc) 2 .H 2 O (1.2 equiv) in DCE (1.5 mL) b Isolated yield.

  ) 2 .H 2 O (1.2 equiv), DCE, 70 o C, 10 h, after 6 h: 3aa, 26%, 5a, Trace after 16 h: 3aa, 45%,

This contains optimization studies, experimental procedures, characterization of related compounds and X-ray crystal structures(CCDC Nos 2220583 (3ag), 2220857 (4i) and 2220855 (5g)).

ACS Paragon Plus EnvironmentSubmitted to Organic Letters

ACKNOWLEDGMENT V. R. and N. T. thank CSIR for fellowship. M. R. thank CSIR-IICT for financial support. R. G. and S. D. thanks the Indo-French "Joint Laboratory for Natural Products and Synthesis toward Affordable Health" program between CSIR-Indian Institute of Chemical Technology and University of Rennes for the platform to carry out the finding jointly. We sincerely thank the Department of Analytical and Structural Chemistry at CSIR-IICT for all the analytical support and especially Dr. Sridhar Balasubramanian for X-ray crystal data. We gratefully acknowledge the overall support by CSIR-IICT to perform the research (IICT/Pubs./2023/011) ACS Paragon Plus Environment Submitted to Organic Letters ACS Paragon Plus Environment Submitted to Organic Letters

Data Availability Statement

The data underlying this study are available in the published article and its Supporting Information.

Supporting Information Statement

" The Supporting Information is available free of charge at https://pubs.acs.org."

AUTHOR INFORMATION

Corresponding Author

Saibal Das-a Department of Organic Synthesis & Process Chemistry, CSIR-Indian Institute of Chemical Technology (CSIR-IICT), Hyderabad 500007, India. E-mail saibal@iict.res.in. b Academy of Scientific and Innovative Research (AcSIR), Ghaziabad, 201002, India.

Authors

Notes

The authors declare no competing financial interest.


D Henrion
email: henrion@laas.fr

J B Lasserre
email: lasserre@laas.fr

C Savorgnan
email: carlo.savorgnan@gmail.com

APPROXIMATE VOLUME AND INTEGRATION FOR BASIC SEMI-ALGEBRAIC SETS *

Keywords: Computational geometry, volume, integration, K-moment problem, semidefinite programming AMS subject classifications. 14P10, 11E25, 12D15, 90C25

Given a basic compact semi-algebraic set K ⊂ R n , we introduce a methodology that generates a sequence converging to the volume of K. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on K can be approximated as closely as desired, and so permits to approximate the integral on K of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed.

Introduction.

Computing the volume and/or integrating on a subset K ⊂ R n is a challenging problem with many important applications. One possibility is to use basic Monte Carlo techniques that generate points uniformly in a box containing K and then count the proportion of points falling into K. To the best of our knowledge, all other approximate (deterministic or randomized) or exact techniques deal with polytopes or convex bodies only. Similarly, powerful cubature formulas exist for numerical integration against a weight function on simple sets (like e.g. simplex, box), but not for arbitrary semi-algebraic sets.

The purpose of this paper is to introduce a deterministic technique that potentially applies to any basic compact semi-algebraic set K ⊂ R n . It is deterministic (no randomization) and differs from previous ones in the literature essentially dedicated to convex bodies (and more particularly, convex polytopes). Indeed, one treats the original problem as an infinite dimensional optimization (and even linear programming (LP)) problem whose unknown is the Lebesgue measure on K. Next, by restricting to finitely many of its moments, and using a certain characterization on the K-moment problem, one ends up in solving a hierarchy of semidefinite programming (SDP) problems whose size is parametrized by the number of moments considered; the dual LP has a simple interpretation and from this viewpoint, convexity of K does not help much. For a certain choice of the criterion to optimize, one obtains a monotone non increasing sequence of upper bounds on the volume of K. Convergence to the exact value invokes results on the K-moment problem by Putinar [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF]. Importantly, there is no convexity and not even connectedness assumption on K, as this plays no role in the K-moment problem. Alternatively, using a different characterization of the K-moment problem due to Krivine [START_REF] Krivine | Anneaux préordonnés[END_REF], one may solve a hierarchy of LP (instead of SDP) problems whose size is also parametrized by the number of moments. Our contribution is a new addition to the already very long list of applications of the moment approach (some of them described in e.g. Landau [START_REF] Landau | Moments in Mathematics[END_REF] and Lasserre [START_REF] Lasserre | A Semidefinite programming approach to the generalized problem of moments[END_REF]) and semidefinite programming [START_REF] Vandenberghe | Semidefinite programming[END_REF]. In principle, the method also permits to approximate any finite number of moments of the uniform distribution on K, and so provides a means to approximate the integral of a polynomial on K. Extension to integration against a weight function is also proposed.

Background. Computing or even approximating the volume of a convex body is hard theoretically and in practice as well. Even if K ⊂ R n is a convex polytope, exact computation of its volume or integration over K is a computational challenge. Computational complexity of these problems is discussed in e.g. Bollobás [START_REF] Bollobás | Volume estimates and rapid mixing[END_REF] and Dyer and Frieze [START_REF] Dyer | The complexity of computing the volume of a polyhedron[END_REF]. Any deterministic algorithm with polynomial time complexity that would compute an upper bound vol (K) and a lower bound vol (K) on vol (K) cannot yield an upper bound on the ratio vol (K)/vol (K) better than polynomial in the dimension n. Methods for exact volume computation use either triangulations or simplicial decompositions depending on whether the polytope has a half-space description or a vertex description. See e.g. Cohen and Hickey, [START_REF] Cohen | Two algorithms for determining volumes of convex polyhedra[END_REF], Lasserre [START_REF] Lasserre | An analytical expression and an algorithm for the volume of a convex polyhedron in R n[END_REF], Lawrence [START_REF] Lawrence | Polytope volume computation[END_REF] and see Büeler et al. [START_REF] Büeler | Exact volume computation for polytopes : A practical study[END_REF] for a comparison. Another set of methods which use generating functions are described in e.g. Barvinok [3] and Lasserre and Zeron [START_REF] Lasserre | A Laplace transform algorithm for the volume of a convex polytope[END_REF]. Concerning integration on simple sets (e.g. simplex, box) via cubature formulas, the interested reader is referred to Gautschi [START_REF] Gautschi | A survey of Gauss-Christoffel quadrature formulae[END_REF][START_REF] Gautschi | Numerical analysis: an introduction[END_REF] and Trefethren [START_REF] Trefethen | Is Gauss quadrature better than Clenshaw-Curtis?[END_REF].

A convex body K ⊂ R n is a compact convex subset with nonempty interior. A strong separation oracle answers either x ∈ K or x ∈ K, and in the latter case produces a hyperplane separating x from K. A negative result states that for every polynomial-time algorithm for computing the volume of a convex body K ⊂ R n given by a well-guaranteed separation oracle, there is a constant c > 0 such that vol (K)/vol (K) ≤ (cn/ log n) n cannot be guaranteed for n ≥ 2. However, Lovász [START_REF] Lovász | An Algorithmic Theory of Numbers, Graphs and Convexity[END_REF] proved that there is a polynomial-time algorithm that produces vol (K) and vol (K) satisfying vol (K)/vol (K) ≤ n n (n+1) n/2 , whereas Elekes [START_REF] Elekes | A geometric inequality and the complexity of measuring the volume[END_REF] proved that for 0 < ǫ < 2 there is no polynomial-time algorithm that produces vol (K) and vol (K) satisfying

vol (K)/vol (K) ≤ (2 -ǫ) n .
If one accepts randomized algorithms that fail with small probability, then the situation is more favorable. Indeed, the celebrated Dyer, Frieze and Kanan probabilistic approximation algorithm [START_REF] Dyer | A random polynomial-time algorithm for approximating the volume of convex bodies[END_REF] computes the volume to fixed arbitrary relative precision ǫ, in time polynomial in ǫ -1 . The latter algorithm uses approximation schemes based on rapidly mixing Markov chains and isoperimetric inequalities. See also hitand-run algorithms for sampling points according to a given distribution, described in e.g. Belisle [START_REF] Belisle | Slow hit-and-run sampling[END_REF], Belisle et al. [START_REF] Belisle | Hit-and-run algorithms for generating multivariate distributions[END_REF], and Smith [START_REF] Smith | Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions[END_REF] Contribution. This paper is concerned with computing (or rather approximating) the volume of a compact basic semi-algebraic set K ⊂ R n defined by

K := { x ∈ R n : g j (x) ≥ 0, j = 1, . . . , m } (1.1)
for some polynomials (g j ) m j=1 ⊂ R[x]. Hence K is possibly non-convex and nonconnected. Therefore, in view of the above discussion, this is quite a challenging problem.

(a) We present a numerical scheme that depends on a parameter p, a polynomial that is nonnegative on K (e.g. p ≡ 1). For each parameter p, it provides converging approximations of moments of the measure uniformly supported on K (with mass equal to vol (K)). For the choice p ≡ 1 one obtains a monotone non-increasing sequence of upper bounds that converges to vol (K).

(b) The way we see the problem dates back to the 19th century pioneer work in the one-dimensional case by Chebyshev [START_REF] Cebysev | Sur les valeurs limites des intégrales[END_REF], Markov [START_REF] Markov | Démonstration de certaines inégalités de M. Tchébychef[END_REF] and Stieltjes [START_REF] Stieltjes | Recherches sur les fractions continues[END_REF], where given n moments s k = b a t k f (t)dt, k = 0, . . . , n -1, and a < c < d < b, one wishes to approximate the integral d c f (t)dt and analyzes asymptotics as n → ∞; characterizing feasible sequence (s k ) is referred to as the Markov moment problem (and L-moment problem if in addition one requires 0 ≤ f ≤ L for some scalar L). For an historical account on this problem as well as other developments, the interested reader is referred to e.g. Krein [START_REF] Krein | Markov in the theory of limiting values of integrals and their future developments[END_REF], Krein and Nuldelman [START_REF] Krein | man, Markov Moment Problems and Extremal Problems. Ideas and Problems of P. L. Cebysev and A. A. Markov and their Further Development[END_REF], Karlin and Studden [START_REF] Karlin | Tchebycheff Systems with Applications in Analysis and Statistics[END_REF] and Putinar [START_REF] Putinar | Extremal solutions of the Two-Dimensional L-problem of moments: II[END_REF].

Our method combines a simple idea, easy to describe, with relatively recent powerful results on the K-moment problem described in e.g. [START_REF] Scheiderer | Positivity and sums of squares: a guide to recent results[END_REF][START_REF] Schmüdgen | The K-moment problem for compact semi-algebraic sets[END_REF][START_REF] Schweighofer | Optimization of polynomials on compact semialgebraic sets[END_REF]. It only requires knowledge of a set B (containing K) simple enough so that the moments of the Lebesgue measure on B can be obtained easily. For instance B := {x ∈ R n : x p ≤ a} with p = 2 (the scaled n-dimensional ball) or p = ∞ (the scaled n-dimensional box) and a ∈ R a given constant. Then computing vol (K) is equivalent to computing the mass of the Borel measure µ which is the restriction to K of the Lebesgue measure on B. This in turn is translated into an infinite dimensional LP problem P with parameter p (some polynomial nonnegative on K) and with the Borel measure µ as unknown. Then, from certain results on the K-moment problem and its dual theory of the representation of polynomials positive on K, problem P can be approximated by an appropriate hierarchy of semidefinite programs (SDP) whose size depends on the number d of moments of µ considered. One obtains approximations of the moments of µ which converge to the exact value as d → ∞. For the choice p ≡ 1 of the parameter p, one even obtains an non-increasing sequence of upper bounds converging to vol (K). Asymptotic convergence is ensured by invoking results of Putinar [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF] on the K-moment problem. Alternatively, one may replace the SDP hierarchy with an LP hierarchy and now invoke results of Krivine [START_REF] Krivine | Anneaux préordonnés[END_REF] for convergence.

Interestingly, the dual of each SDP relaxation defines a strenghtening of P * , the LP dual of P, and highlights why the problem of computing the volume is difficult. Indeed, one has to approximate from above the function f (= p on K and 0 on B \ K) by a polynomial h of bounded degree, so as to minimize the integral B (h -f )dx. From this viewpoint, convexity of K plays no particular role and so, does not help much.

(c) Let d ∈ N be fixed, arbitrary. One obtains an approximation of the moments of degree up to d of the measure µ on K, as closely as desired. Therefore, this technique also provides a sequence of approximations that converges to K qdx for any polynomial q of degree at most d (in contrast, Monte Carlo simulation is for a given q). Finally, we also propose a similar approximation scheme for integrating a polynomial on K against a nonnegative weight function w(x). The only required data are moments of the measure dν = wdx on a simple set B (e.g. box or simplex) containing K, which can be obtained by usual cubature formulas for integration. On the practical side, at each step d of the hierarchy, the computational workload is that of solving an SDP problem of increasing size. In principle, this can be done in time polynomial in the input size of the SDP problem, at given relative accuracy. However, in view of the present status and limitations of current SDP solvers, so far the method is restricted to problems of small dimension n if one wishes to obtain good approximations. The alternative LP hierarchy might be preferable for larger size problems, even if proved to be less efficient when used in other contexts where the moment approach applies, see e.g. [START_REF] Lasserre | Semidefinite programming vs. LP relaxations for polynomial programming[END_REF][START_REF] Lasserre | LP relaxations for the moment approach in some performance evaluation problems[END_REF].

Preliminary results on simple problems for which vol (K) is known show that indeed convexity plays no particular role. In addition, as for interpolation problems, the choice of the basis of polynomials is crucial from the viewpoint of numerical precision. This is illustrated on a trival example on the real line where, as expected, the basis of Chebyshev polynomials is far better than the usual monomial basis. In fact, it is conjectured that trigonometric polynomials would be probably the best choice. Finally, the choice of the parameter p is also very important and unfortunately, the choice of p ≡ 1 which guarantees a monotone convergence to vol (K) is not the best choice at all. Best results are obtained when p is negative outside K.

So far, for convex polytopes, this method is certainly not competitive with exact specific methods as those described in e.g. [START_REF] Büeler | Exact volume computation for polytopes : A practical study[END_REF]. It rather should be viewed as a relatively simple deterministic methodology that applies to a very general context for which even getting good bounds on vol (K) is very difficult, and for which the only alternative presently available seems to be brute force Monte Carlo.

2. Notation, definitions and preliminary results. Let R[x] be the ring of real polynomials in the variables x = (x 1 , . . . , x n ), and let Σ 2 [x] ⊂ R[x] be the subset of sums of squares (SOS) polynomials. Denote R[x] d ⊂ R[x] be the set of polynomials of degree at most d, which forms a vector space of dimension s(d)

= n+d d . If f ∈ R[x] d , write f (x) = α∈N n f α x α in
the usual canonical basis (x α ), and denote by f = (f α ) ∈ R s(d) its vector of coefficients. Similarly, denote by Σ 2 [x] d ⊂ Σ 2 [x] the subset of SOS polynomials of degree at most 2d.

Moment matrix. Let y = (y α ) be a sequence indexed in the canonical basis (x α ) of R[x], let L y : R[x] → R be the linear functional

f (= α f α x α ) → L y (f ) = α f α y α ,
and let M d (y) be the symmetric matrix with rows and columns indexed in the canonical basis (x α ), and defined by:

M d (y)(α, β) := L y (x α+β ) = y α+β , for every α, β ∈ N n d := {α ∈ N n : |α| (= i α i ) ≤ d}. A sequence y = (y α ) is said to have a representing finite Borel measure µ if y α = x α dµ for every α ∈ N n . A necessary (but not sufficient) condition is that M d (y)
0 for every d ∈ N. However, if in addition, |y α | ≤ M for some M and for every α ∈ N n , then y has a representing measure on [-1, 1] n .

Localizing matrix. Similarly, with y = (y α ) and g ∈ R[x] written as

x → g(x) = γ∈N n g γ x γ ,
let M d (g y) be the symmetric matrix with rows and columns indexed in the canonical basis (x α ), and defined by:

M d (g y)(α, β) := L y g(x) x α+β = γ g γ y α+β+γ ,
for every α, β ∈ N n d . A necessary (but not sufficient) condition for y to have a representing measure with support contained in the level set {x : g(x) ≥ 0} is that M d (g y) 0 for every d ∈ N.

2.1. Moment conditions and representation theorems. The following results from the K-moment problem and its dual theory of polynomials positive on K provide the rationale behind the hierarchy of SDP relaxations introduced in [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF], and potential applications in many different contexts. See e.g. [START_REF] Lasserre | A Semidefinite programming approach to the generalized problem of moments[END_REF] and the many references therein.

SOS-based representations. Let Q(g) ⊂ R[x] be the quadratic module generated by polynomials (g j ) m j=1 ⊂ R[x], that is, Q(g) :=    σ 0 + m j=1 σ j g j : (σ j ) m j=1 ⊂ Σ 2 [x]    . (2.1) Assumption 2.1. The set K ⊂ R n in (1.1) is compact and the quadratic poly- nomial x → a 2 -x 2 belongs to Q(g) for some given constant a ∈ R. Theorem 2.2 (Putinar's Positivstellensatz [36]). Let Assumption 2.1 hold. (a) If f ∈ R[x] is strictly positive on K, then f ∈ Q(g).
That is:

f = σ 0 + m j=1 σ j g j , (2.2
)

for some SOS polynomials (σ j ) m j=1 ⊂ Σ 2 [x]. (b) If y = (y α ) is such that for every d ∈ N, M d (y) 0; M d (g j y) 0, j = 1, . . . , m, (2.3)
then y has a representing finite Borel measure µ supported on K.

Given f ∈ R[x], or y = (y α ) ⊂ R, checking whether (2.
2) holds for SOS (σ j ) ⊂ Σ 2 [x] with a priori bounded degree, or checking whether (2.3) holds with d fixed, reduces to solving an SDP.

Another type of representation. Let K ⊆ B be as in (1.1) and assume for simplicity that the g j s have been scaled to satisfy 0 ≤ g j ≤ 1 on K, for every j = 1, . . . , m. In addition, assume that the family of polynomials (1, g 1 , . . . , g m ) generates the algebra R[x]. For every α ∈ N m , let g α and (1 -g) β denote the polynomials

x → g(x) α := g 1 (x) α1 • • • g m (x) αm , and x → (1 -g(x)) β := (1 -g 1 (x)) β1 • • • (1 -g m (x)) βm .
the following result is due to Krivine [START_REF] Krivine | Anneaux préordonnés[END_REF] but is explicit in e.g. Vasilescu [START_REF] Vasilescu | Spectral measures and moment problems, Spectral Theory and Its Applications[END_REF].

Theorem 2.3. (a) If f ∈ R[x] is strictly positive on K, then f = α,β∈N m c αβ g α (1 -g) β (2.4)
for finitely many nonnegative scalars

(c αβ ) ⊂ R + . (b) If y = (y α ) is such that L y (g α (1 -g) β ) ≥ 0, (2.5)
for every α, β ∈ N m , then y has a representing finite Borel measure µ supported on K.

Theorem 2.3 extends the well-known Hausdorff moment conditions on the hyper cube [0, 1] n , as well as Handelman representation [START_REF] Handelman | Representing polynomials by positive linear functions on compact convex polyhedra[END_REF] for convex polytopes K ⊂ R n . Observe that checking whether (2.4), resp. (2.5), holds with α, β bounded a priori, reduces to solving an LP in the variables (c αβ ), resp. (y α ).

A preliminary result. Given any two measures

µ 1 , µ 2 on a Borel σ-algebra B, the notation µ 1 ≤ µ 2 means µ 1 (C) ≤ µ 2 (C) for every C ∈ B.
Lemma 2.4. Let Assumption 2.1 hold and let y 1 = (y 1α ) and y 2 = (y 2α ) be two moment sequences with respective representing measures µ 1 and µ 2 on K. If

M d (y 2 -y 1 ) 0 ; M d (g j (y 2 -y 1 )) 0, j = 1, . . . , m, for every d ∈ N, then µ 1 ≤ µ 2 .
Proof. As M d (y 2y 1 ) 0 and M d (g j (y 2y 1 )) 0 for j = 1, . . . , m and d ∈ N, by Theorem 2.2, the sequence y 0 := y 2y 1 has a representing Borel measure µ 0 on K. From y 0α + y 1α = y 2α for every α ∈ N n , we conclude that

x α dµ 0 + x α dµ 1 = x α dµ 2 , ∀α ∈ N n ,
and as K is compact, by the Stone-Weierstrass theorem,

f dµ 0 + f dµ 1 = f dµ 2
for every continuous function f on K, which in turn implies µ 0 + µ 1 = µ 2 , i.e., the desired result µ 1 ≤ µ 2 .

3. Main result. We first introduce an infinite-dimensional LP problem P whose unique optimal solution is the restriction µ of the normalized Lebesgue measure on B (hence with µ(K) = vol (K)/2 n ) and whose dual has a clear interpretation. We then define a hierarchy of SDP problems (alternatively, a hierarchy of LP problems) to approximate any finite sequence of moments of µ, as closely as desired.

3.1. An infinite-dimensional linear program P. After possibly some normalization of the defining polynomials, assume with no loss of generality that K ⊂ B ⊆ [-1, 1] n with B a set over which integration w.r.t. the Lebesgue measure is easy. For instance, B is the box [-1, 1] n or B is the euclidean unit ball.

Let B be the Borel σ-algebra of Borel subsets of B, and let µ 2 be the Lebesgue measure on B, normalized so that 2 n µ 2 (B) = vol(B). Therefore, if vol (C) denotes the n-dimensional volume of C ∈ B, then µ 2 (C) = vol (C)/2 n for every C ∈ B.

Also, the notation µ 1 ≪ µ 2 means that µ 1 is absolutely continuous w.r.t. µ 2 , and L 1 (µ 2 ) is the set of all functions integrable w.r.t. µ 2 . By the Radon-Nikodym theorem, there exists a nonnegative measurable function f ∈ L 1 (µ 2 ) such that µ 1 (C) = C f dµ 2 for every C ∈ B, and f is called the Radon-Nikodym derivative of µ 1 w.r.t. µ 2 . In particular, µ 1 ≤ µ 2 obviously implies µ 1 ≪ µ 2 . For K ∈ B, let M (K) be the set of finite Borel measures on K.

Theorem 3.1. Let K ∈ B with K ⊆ B and let p ∈ R[x] be positive almost everywhere on K. Consider the following infinite-dimensional LP problem:

P : sup µ1 { p dµ 1 : µ 1 ≤ µ 2 ; µ 1 ∈ M (K) } (3.1)
with optimal value denoted sup P (and max P if the supremum is achieved).

Then the restriction µ * 1 of µ 2 to K is the unique optimal solution of P and max P =

pdµ * 1 = K pdµ 2 . In particular, if p ≡ 1 then max P = vol (K)/2 n . Proof. Let µ * 1 be the restriction of µ 2 to K (i.e. µ * 1 (C) = µ 2 (C ∩ K), ∀ C ∈ B). Observe that µ *
1 is a feasible solution of P. Next, let µ 1 be any feasible solution of P. As

µ 1 ≤ µ 2 then µ 1 (C ∩ K) ≤ µ 2 (C ∩ K) = µ * 1 (C ∩ K), ∀ C ∈ B,
and so, µ 1 ≤ µ * 1 because µ 1 and µ * 1 are supported on K. Therefore, as p ≥ 0 on K, pdµ 1 ≤ pdµ * 1 which proves that µ * 1 is an optimal solution of P. Next suppose that µ 1 = µ * 1 is another optimal solution of P. As

µ 1 ≤ µ * 1 then µ 1 ≪ µ *
1 and so, by the Radon-Nikodym theorem, there exists a nonnegative measurable function f ∈ L 1 (µ * 1 ) such that

µ 1 (C) = C dµ 1 = C f (x) dµ * 1 (x), ∀ C ∈ B ∩ K. Next, as µ 1 ≤ µ * 1 , µ * 1 -µ 1 =: µ 0 is a finite Borel measure on K which satisfies 0 ≤ µ 0 (C) = C (1 -f (x)) dµ * 1 (x), ∀ C ∈ B ∩ K,
and so 1 ≥ f (x) for almost all x ∈ K. But then, since

pdµ 1 = pdµ * 1 , 0 = pdµ 0 = K p(x)(1 -f (x)) dµ * 1 (x),
which (recalling p > 0 almost everywhere on K) implies that f (x) = 1 for almost-all x ∈ K. And so µ 1 = µ * 1 . 3.2. The dual of P. Let F be the Banach space of continuous functions on B (equipped with the sup norm) and F + its positive cone, i.e., the elements f ∈ F which are nonnegative on B. The dual of P reads:

P * : inf f ∈F+ { f dµ 2 : f ≥ p on K} (3.2)
with optimal value denoted inf P * (min P * is the infimum is achieved).

Hence, a minimizing sequence of P * aims at approximating from above the function f (= p on K and 0 on B \ K) by a sequence (f ℓ ) of continuous functions so as to minimize f ℓ dµ 2 .

Let x → d(x, K) be the euclidean distance to the set K and with ǫ ℓ > 0, let K ℓ := {x ∈ B : d(x, K) < ǫ ℓ } be an open bounded outer approximation of K, so that B \ K ℓ is closed (hence compact) with ǫ ℓ → 0 as ℓ → ∞. By Urysohn's Lemma [1, A4.2, p. 379], there exists a sequence (f ℓ ) ⊂ F + such that 0 ≤ f ℓ ≤ 1 on B, f ℓ = 0 on B \ K ℓ , and f ℓ = 1 on K. Therefore,

f ℓ dµ 2 = vol (K)/2 n + K ℓ \K f ℓ dµ 2 ,
and so f ℓ dµ 2 → vol (K)/2 n as ℓ → ∞. Hence, for the choice of the parameter p ≡ 1, vol (K)/2 n is the optimal value of both P and P * .

3.3.

A hierarchy of semidefinite relaxations for computing the volume of K. Let y 2 = (y 2α ) be the sequence of all moments of µ 2 . For example, if B = [-1, 1] n , then

y 2 α = 2 -n n j=1 2((1 + α j ) mod 2) 1 + α j , ∀ α ∈ N n .
Let K be a compact semi-algebraic set as in (1.1) and let r j = ⌈(deg g j )/2⌉, j = 1, . . . , m. Let p ∈ R[x] be a given polynomial positive almost everywhere on K, and let r 0 := ⌈(deg p)/2⌉. For d ≥ max j r j , consider the following semidefinite program:

Q d :          sup y1 L y1 (p) s.t. M d (y 1 ) 0 M d (y 2 -y 1 ) 0 M d-rj (g j y 1 ) 0, j = 1, . . . , m (3.3)
with optimal value denoted sup Q d (and max Q d if the supremum is achieved).

Observe that sup Q d ≥ max P for every d. Indeed, the sequence y * 1 of moments of the Borel measure µ * 1 (restriction of µ 2 to K and unique optimal solution of P) is a feasible solution of Q d for every d.

Theorem 3.2. Let Assumption 2.1 hold and consider the hierarchy of semidefinite programs (Q d ) in (3.3). Then:

(a) Q d has an optimal solution (i.e. sup Q d = max Q d ) and

max Q d ↓ K p dµ 2 , as d → ∞. (b) Let y 1 d = (y 1 d α ) be an optimal solution of Q d , then lim d→∞ y 1 d α = K x α dµ 2 , ∀α ∈ N n . ( 3
L y1 (x 2d i ) ] ≤ 1.
By [29, Lemma 1], this in turn implies that |y 1α | ≤ 1 for every α ∈ N n 2d , and so the feasible set of Q d is closed, bounded, hence compact, which in turn implies that Q d is solvable (i.e., has an optimal solution).

Let y 1 d be an optimal solution of Q d and by completing with zeros, make y 1 d an element of the unit ball B ∞ of l ∞ (the Banach space of bounded sequences, equipped with the sup-norm). As l ∞ is the topological dual of l 1 , by the Banach-Alaoglu Theorem, B ∞ is weak ⋆ compact, and even weak ⋆ sequentially compact; see e.g. Ash [START_REF] Ash | Real analysis and probability[END_REF]. Therefore, there exists y 1 * ∈ B ∞ and a subsequence {d k } ⊂ N such that y 1 d k → y 1 * as k → ∞, for the weak ⋆ topology σ(l ∞ , l 1 ). In particular, lim k→∞ y 1

d k α = y 1 * α , ∀ α ∈ N n . ( 3
max P ≤ max Q d k = L y d k 1 (p) ↓ L y * 1 (p) = pdµ 1 .
Next, as both µ 1 and µ 2 are supported on [-1, 1] n , and M d (y 2y 1 * ) 0 for every d, one has |y 2α -y * 1α | ≤ 1 for every α ∈ N n . Hence y 2y * 1 has a representing measure on [-1, 1] n . As in the proof of Lemma 2.41 , we conclude that µ 1 ≤ µ 2 . Therefore µ 1 is admissible for problem P, with value L y * 1 (p) = pdµ 1 ≥ max P. Therefore, µ 1 must be an optimal solution of P (hence unique) and by Theorem 3.1,

L y * 1 = pdµ 1 = K pdµ 2 .
As the converging subsequence {d k } was arbitrary, it follows that in fact the whole sequence y 1 d converges to y 1 * for the weak ⋆ topology σ(l ∞ , l 1 ). And so (3.4) holds. This proves (a) and (b).

Writing M d (y 1 ) = α A α y 1α , and M d-rj (g j y 1 ) = α B j α y 1α for appropriate real symmetric matrices (A α , B j α ), the dual of Q d reads:

Q * d :            inf X,Y,Zj M d (y 2 ), Y s.t. A α , Y -X - m j=1 B j α , Z j = p α X, Y, Z j 0,
where X, Y = trace (XY ) is the standard inner product of real symmetric matrices, and X 0 stands for X is positive semidefinite. This can be reformulated as:

Q * d :              inf h,σ0,...,σm h dµ 2 s.t. h -p = σ 0 + m j=1 σ j g j h ∈ Σ 2 [x] d , σ 0 ∈ Σ 2 [x] d , σ j ∈ Σ 2 [x] d-rj . (3.6)
The constraint of this semidefinite program states that the polynomial h-p is written in Putinar's form (2.2) and so h -p ≥ 0 on K. In addition, h ≥ 0 because it is a sum of squares.

This interpretation of Q * d also shows why computing vol (K) is difficult. Indeed, when p ≡ 1, to get a good upper bound on vol (K), one needs to obtain a good polynomial approximation h ∈ R[x] of the indicator function I K (x) on B. In general, high degree of h will be necessary to attenuate side effects on the boundary of B and K, a well-known issue in interpolation with polynomials.

Proposition 3.3. If K and B \ K have a nonempty interior, there is no duality gap, that is, both optimal values of Q d and Q * d are equal. In addition, Q * d has an optimal solution (h * , (σ * j )). Proof. Let µ 1 be the uniform distribution on K, i.e., the restriction of µ 2 to K, and let y 1 = (y 1α ) be its sequence of moments up to degree 2d. As K has nonempty interior, then clearly M d (y 1 ) ≻ 0 and M d-rj (g j y 1 ) ≻ 0 for every j = 1, . . . , m. If B \ K also has nonempty interior then

M d (y 2 -y 1 ) ≻ 0 because with f ∈ R[x] d with coefficient vector f , f , M d (y 2 -y 1 )f = B\K f (x) 2 dµ 2 , ∀f ∈ R[x] d .
Therefore Slater's condition holds for Q d and the result follows from a standard result of duality in semidefinite programming; see e.g. [START_REF] Vandenberghe | Semidefinite programming[END_REF].

Remark 3.4. Let f ∈ R[x] and suppose that one wants to approximate the integral J * := K f dµ 2 . Then for d sufficiently large, an optimal solution of Q d allows to approximate J * . Indeed,

J * = K f dµ 2 = f dµ 1 = L y1 * (f ) = α∈N n f α y 1 * α ,
where y 1 * is the moment sequence of µ 1 , the unique optimal solution of P (the restriction of µ 2 to K). And so, from (3.4), L y d 1 (f ) ≈ J * when d is sufficiently large. 3.4. A hierarchy of linear programs. Let K ⊂ B ⊆ [-1, 1] n be as in (1.1) and assume for simplicity that the g j s have been scaled to satisfy 0 ≤ g j ≤ 1 on K for every j = 1, . . . , m. In addition, assume that the family of polynomials (1, g 1 , . . . , g m ) generates the algebra R[x]. For d ∈ N, consider the following linear program:

L d :              sup y1 y 10 s.t. L y2-y1 n i=1 (1 + x i ) αi (1 -x i ) βi ≥ 0, α, β ∈ N n d L y1 (g α (1 -g) β ) ≥ 0, α, β ∈ N n d (3.7)
with optimal value denoted sup L d (and max L d if sup L d is finite). Notice that sup L d ≥ vol (K)/2 n for all d. Indeed, the sequence y * 1 of moments of the Borel measure µ * 1 (restriction of µ 2 to K and unique optimal solution of P) is a feasible solution of L d for every d. Proof. We first prove that L d has finite value. L d always has a feasible solution y 1 , namely the moment vector associated with the Borel measure µ 1 , the restriction of µ 2 to K, and so sup L d ≥ vol (K)/2 n . Next, from the constraint L y2-y1 (•) ≥ 0 with α = β = 0, we obtain y 10 ≤ y 20 ≤ 1. Hence sup L d ≤ 1 and therefore, the linear program L d has an optimal solution y 1 d . Fix γ ∈ N n and ǫ > 0, arbitrary. As

|x γ | ≤ 1 < 1 + ǫ on B (hence on K), by Theorem 2.3(a), 1 + ǫ ± x γ = α,β∈N m c γ αβ g α (1 -g) β ,
for some (c γ αβ ) ⊂ R + with |α|, |β| ≤ s γ . Hence, as soon as d ≥ s γ , applying L y1 d yields

(1 + ǫ) y 1 d 0 ± y 1 d γ = α,β∈N m c γ αβ L y1 g α (1 -g) β ≥ 0,
and so

∀γ ∈ N n : |y 1 d γ | ≤ (1 + ǫ) y 1 d 0 ≤ 1 + ǫ, ∀d ≥ s γ . (3.8)
Complete y 1 d with zeros to make it an element of R ∞ . Because of (3.8), using a standard diagonal element, there exists a subsequence (d k ) and an element y

1 * ∈ (1 + ǫ) B ∞ (
L y0 n i=1 (1 + x i ) αi (1 -x i ) βi ≥ 0, ∀α, β ∈ N n ,
and so by Theorem 2.3(b), y 0 is the moment vector of some Borel measure µ 0 supported on [-1, 1] n . As measures on compact sets are identified with their moments, and y 0α + y 1 * α = y 2α for every α ∈ N n , it follows that µ 0 + µ 1 = µ 2 , and so µ 1 ≤ µ 2 . Therefore, µ 1 is an admissible solution to P with parameter p ≡ 1, and with value µ 1 (K) = y 1 * 0 ≥ vol (K)/2 n . Hence, µ 1 is the unique optimal solution to P with value

µ 1 (K) = vol (K)/2 n .
Finally, by using (3.5) and following the same argument as in the proof of Theorem 3.2, one obtains the desired result (3.4).

Remark 3.4 also applies to the LP relaxations (3.7).

3.5. Integration against a weight function. With K ⊂ B as in (1.1) suppose now that one wishes to approximate the integral

J * := K f (x) w(x) dx, (3.9)
for some given nonnegative weight function w : R n → R, and where f ∈ R[x] d is some nonnegative polynomial. One makes the following assumption: Assumption 3.6. One knows the moments y 2 = (y 2α ) of the Borel measure dµ 2 = wdx on B, that is:

y 2α = B x α dµ 2 = B x α w(x) dx , α ∈ N n . (3.10)
Indeed, for many weight functions w, and given d ∈ N, one may compute the moments y 2 = (y 2α ) of µ 2 via cubature formula, exact up to degree d. In practice, one only knows finitely many moments of µ 2 , say up to degree d, fixed. Consider the hierarchy of semidefinite programs

Q d :          sup y1 L y1 (f ) s.t. M d (y 1 ) 0 M d (y 2 -y 1 ) 0 M d-rj (g j y 1 ) 0, j = 1, . . . , m (3.11)
with y 2 as in Assumption 3.6. Theorem 3.7. Let Assumption 2.1 and 3.6 hold and consider the hierarchy of semidefinite programs (Q d ) in (3.11) with y 2 as in (3.10). Then Q d is solvable and

max Q d ↓ J * as d → ∞.
The proof is almost a verbatim copy of that of Theorem 3.2.

Numerical experiments and discussion.

In this section we report some numerical experiments carried out with Matlab and the package GloptiPoly 3 for manipulating and solving generalized problems of moments [START_REF] Henrion | Gloptipoly 3: moments, optimization and semidefinite programming[END_REF]. The SDP problems were solved with SeDuMi 1.1R3 [START_REF] Pólik | SeDuMi: a package for conic optimization[END_REF]. Univariate Chebyshev polynomials were manipulated with the chebfun package [START_REF] Battles | An extension of Matlab to continuous functions and operators[END_REF]. The single-interval example below permit to visualize the numerical behavior of the algorithm. The folium example illustrates that, as expected, the non-convexity of K does not seem to penalize the moment approach. Finally, our experience reveals that the choice of alternative polynomial bases affects the quality of the approximations.

Single interval. Consider the elementary one-dimensional set

K = [0, 1 2 ] = {x ∈ R : g 1 (x) = x( 1 2 -x) ≥ 0} included in the unit interval B = [-1, 1]
. We want to approximate vol (K) = 1 2 . Moments of the Lebesgue measure µ 2 on B are given by y 2 = (2, 0, 2/3, 0, 2/5, 0, 2/7, . . .).

Here is a simple Matlab script using GloptiPoly 3 instructions to input and solve the SDP relaxation Q d of the LP moment problem P with p ≡ 1: >> d = 10; % degree >> mpol x0 x1 >> m0 = meas(x0); m1 = meas(x1); >> g1 = x1*(1/2-x1); >> dm = (1+(0:d))'; y2 = ((+1).^dm-(-1).^dm)./dm; >> y0 = mom(mmon(x0,d)); y1 = mom(mmon(x1,d)); >> P = msdp(max(mass(m1)), g1>=0, y0==y2-y1); % input moment problem >> msol(P); % solve SDP relaxation >> y1 = double(mvec(m1)); % retrieve moment vector The volume estimate is then the first entry in vector y1. Note in particular the use of the moment constraint y0==y2-y1 which ensures that moments y 0 of µ 0 will be substituted by linear combinations of moments y 1 of µ 1 (decision variables) and moments y 2 of µ 2 (given). Three sequences of approximations of vol [0, 1 2 ] obtained by solving SDP relaxations of increasing degree.

• the upper curve (in black) is a monotone non increasing sequence of upper bounds obtained by maximizing dµ 1 , the mass of µ 1 , using the objective function max(mass(m1)) in the above script; • the medium curve (in gray) is a sequence of approximations obtained by maximizing pdµ 1 with p := g 1 , using the objective function max(g1) in the above script; • the lower curve (in black) is a monotone non decreasing sequence of lower bounds on vol (K) obtained by computing upper bounds on the volume of B\K, using the objective function max(mass(m1)) and the support constraint g1<=0 in the above script. The volume estimate is then 2-y1(1). We observe a much faster convergence when maximizing g 1 dµ 1 instead of dµ 1 ; the upper and lower curves apparently exhibit slow convergence. To analyze these phenomena, we use solutions of the dual SDP problems, provided automatically by the primal-dual interior-point method implemented in the . We observe the characteristic oscillation phenomena near the boundary, typical of polynomial approximation problems [START_REF] Trefethen | Chebfun Version[END_REF]. The continuous function max(0, g 1 ) is easier to approximate than discontinuous indicator functions, and this partly explains the better convergence of the medium approximation on , 1} which significantly corrupt the quality of the volume approximation. To some extent, these oscillations can be reduced by using a Chebyshev polynomial basis instead of the standard power basis. Note that in order to input and solve SDP problems in the Chebyshev basis, we used our own implementation and the chebfun package since GloptiPoly 3 supports only the power basis. In Figure 4.5 we see that above degree 20 the quality of the bounds obtained with the power basis deteriorates, which suggests that the SDP solver encounters some numerical problems rather than convergence becoming slower (which is confirmed when changing to Chebyshev basis; see below). It seems that the SDP solver is not able to improve the bounds, most likely due to the symmetric Hankel structure of the moment matrices in the power basis: indeed, it is known that the conditioning (ratio of extreme singular values) of positive definite Hankel matrices is an exponential function of the matrix size [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF]. When the smallest singular values reach machine precision, the SDP solver is not able to optimize the objective function any further.

In Figures 4.6 and 4.7 one observes that the degree-100 polynomial approximation h(x) of the indicator function and its complement are tighter in the Chebyshev basis (black) than in the power basis (gray). Firstly, we observe that the degree-100 approximations in the power basis do not significantly differ from the degree-50 approximations in the same basis, represented in Figures 4.2 and 4.4. This is consistent with the very flat behavior of the right half of the upper and lower curves (in gray) in Figure 4.5. Secondly, some coefficients of h(x) in the power basis have large magnitude h(x) = 1.0019 + 3.6161x -29.948x 2 + • • • + 88123x 49 + 54985x 50 + • • • -1018.4x 99 + 26669x 100 with the Euclidean norm of the coefficient vector greater than 10 6 . In contrast, the polynomial h(x) obtained in the Chebyshev basis h(x) = 0.1862t 0 (x) + 0.093432t 1 (x) -0.30222t 2 (x) + • • • + 0.0055367t 49 (x) -0.020488t 50 (x) + • • • -0.0012267t 99 (x) + 0.0011190t 100 (x) has a coefficient vector of Euclidean norm around 0.57627, where t k (x) denotes the k-th Chebyshev polynomial. Thirdly, oscillations around points x = 0 and x = 1/2 did not disappear with the Chebyshev basis, but the peaks are much thinner than with the power basis. Finally, in the Chebyshev basis (black) and power basis (gray).

the oscillations near the interval ends x = -1 and x = 1 are almost suppressed, a well-known property of Chebyshev polynomials which have a denser root distribution near the interval ends.

From these simple observations, we conjecture that a polynomial basis with a dense root distribution near the boundary of the semi-algebraic sets K and B should ensure a better convergence of the hierarchy of volume estimates.

Finally, Figure 4.8 displays the CPU time required to solve the SDP problems (with SeDuMi, in the power basis in gray and in the Chebyshev basis in black) as a function of the degree, showing a polynomial dependence slightly slower than cubic in the power basis (due to the sparsity of moment matrices) and slightly faster than cubic in the Chebyshev basis. For example, solving the SDP problem of degree 100 takes about 2.5 seconds of CPU time on our standard desktop computer.

Bean. Consider

K = {x ∈ R 2 : g 1 (x) = x 1 (x 2 1 + x 2 2 ) -(x 4 1 + x 2 1 x 2 2 + x 4
2 ) ≥ 0} displayed in Figure 4.9, which is a surface delimited by an algebraic curve g 1 (x) = 0 of genus zero, hence rationally parametrizable. From the parametrization x 1 (t) = (1 + t 2 )/(1 + t 2 + t 4 ), x 2 (t) = tx 1 (t), t ∈ R, obtained with the algcurves package of Maple, we can calculate

vol (K) = K dx 1 dx 2 = R x 1 (t)dx 2 (t) = R (1-t)(1+t)(1+t 2 )(1+3t 2 +t 4 ) (1+t+t 2 ) 3 (1-t+t 2 ) 3 dt = 7 √ 3π
36 ≈ 1.0581 with the help of the int integration routine of Maple. Similarly, we can calculate symbolically the first moments of the Lebesgue measure µ 1 on K, namely y 100 = vol (K), y 110 = 23 42 vol (K), y 101 = 0, y 120 = 23 63 vol (K), y 111 = 0,

y 102 = 113 1008 vol (K) etc. Observe that K ⊆ B = [-1, 1] 2 .
On Figure 4.10 we represent a degree-20 positive polynomial approximation h of the indicator function I K on B obtained by solving an SDP problem with 231 unknown moments. We observe the typical oscillations near the boundary regions, but we can recognize the shape of Figure 4.9.

In Table 4.1 we give relative errors in percentage observed when solving successive SDP relaxations (in the power basis) of the LP moment problems of maximizing g 1 dµ 1 . Note that the error sequence is not monotonically decreasing since we do not maximize dµ 1 and a good approximation can be obtained with few moments. Above degree 16, the approximation stagnates around 4%, Most likely this is due to the use of the power basis, as already observed in the previous univariate examples. For example, at degree 20, one obtains the 6 first moment approximation equation ρ = sin(2θ). The surface is contained in the unit disk B, on which the Lebesgue measure has moments

y 2α = (1 + (-1) α1 )(1 + (-1) α2 )Γ( 1 2 (1 + α 1 ))Γ( 1 2 (1 + α 2 )) Γ( 1 2 (4 + α 1 + α 2 )) , ∀ α ∈ N 2 ,
where Γ denotes the gamma function. The area is vol (K) = 1 2 2π 0 sin 2 (2θ)dθ = 1 2 π and so, vol (K \ B) = πvol (K) = 1 2 π. In Table 4.2 we give relative errors in percentage observed when solving successive SDP relaxations (in the power basis) of the LP moment problems of maximizing g 1 dµ 1 . We observe that nonconvexity of K does not play any special role. The quality of estimates does not really improve for degrees greater than 20. Here too, an alternative polynomial basis with dense root distribution near the boundaries of K and B would certainly help. 5. Concluding Remarks. The methodology presented in this paper is general enough and applies to compact basic semi-algebraic sets which are neither necessarily convex nor connected. Its efficiency is related to the degree needed to obtain a good polynomial approximation of the indicator function of K (on a simple set that contains K) and from this viewpoint, convexity of K does not help much. On the other hand, the method is limited by the size of problems that SDP solvers presently available can handle. Moreover, the impact of the choice of the polynomial basis (e.g., Chebyshev or trigonometric) on the quality of the solution of the SDP relaxations deserves further investigation for a better understanding. Therefore, in view of the present status of SDP solvers and since in general high accuracy will require high degree, the method can provide good approximations for problems of small dimension (typically n = 2 or n = 3). However, if one is satisfied with cruder bounds then one may consider problems in higher dimensions.

. 4 )

 4 Proof. (a) and (b). Recall that B ⊆ [-1, 1] n . By definition of µ 2 , observe that |y 2α | ≤ 1 for every α ∈ N n 2d , and from M d (y 2y 1 ) 0, the diagonal elements y 2 2α -y 1 2α are nonnegative. Hence y 1 2α ≤ y 2 2α for every α ∈ N n d and therefore, max [ y 10 , max i=1,...,n

. 5 )

 5 Next let d ∈ N be fixed, arbitrary. From the pointwise convergence (3.5) we also obtain M d (y 1 * ) 0 and M d (y 2y 1 * ) 0. Similarly, M d-rj (g j y 1 * ) 0 for every j = 1, . . . , m. As d was arbitrary, by Theorem 2.2, y 1 * has a representing measure µ 1 supported on K ⊂ B. In particular, from (3.5), as k → ∞,

Theorem 3 . 5 .

 35 For the hierarchy of linear programs (L d ) in (3.7), the following holds: (a) L d has an optimal solution (i.e. sup L d = max L d ) and max L d ↓ vol (K)/2 n as d → ∞. (b) Let y 1 d be an optimal solution of L d . Then (3.4) holds.

  where B ∞ is the unit ball of l ∞ ) such that (3.5) holds. Now with α, β ∈ N m fixed, arbitrary, (3.5) yields L y1 * (g α (1 -g) β ) ≥ 0. Hence by Theorem 2.3(b), y 1 * has a representing measure µ 1 supported on K. Next, let y 0 := y 2y 1 * . Again, (3.5) yields:

Figure 4 .

 4 Fig.4.1. Three sequences of approximations of vol [0,1 2 ] obtained by solving SDP relaxations of increasing degree.

Fig. 4 . 2 . 2 ]

 422 Fig. 4.2. Positive polynomial approximation of degree 50 (solid) of the indicator function I [0, 1 2 ]

Fig. 4 . 3 .

 43 Fig. 4.3. Positive polynomial approximation of degree 50 (solid) of the positive piecewisepolynomial function max(0, g 1 ) on [-1, 1]. Polynomial g 1 is represented in dashed line.

Fig. 4 . 4 .

 44 Fig. 4.4. Positive polynomial approximation of degree 50 (solid) of the complementary indicator function 1 -I [0, 1 2 ] (dashed) on [-1, 1].

Figure 4 . 1 .

 41 On Figures 4.2

  and 4.4, one observes relatively large oscillations near the boundary points x ∈ {-1, 0, 1 2

Fig. 4 . 5 .

 45 Fig.4.5. Upper and lower bounds on vol [0,1 2 ] obtained by solving SDP relaxations in the Chebyshev basis (black) and power basis (gray).

Figure 4 .

 4 Figure 4.5 displays upper and lower bounds on the volume, computed up to degree 100, with the power basis (in gray) and with the Chebyshev basis (in black). Note that in order to input and solve SDP problems in the Chebyshev basis, we used our own implementation and the chebfun package since GloptiPoly 3 supports only the power basis. In Figure4.5 we see that above degree 20 the quality of the bounds obtained with the power basis deteriorates, which suggests that the SDP solver encounters some numerical problems rather than convergence becoming slower (which is confirmed when changing to Chebyshev basis; see below). It seems that the SDP solver is not able to improve the bounds, most likely due to the symmetric Hankel structure of the moment matrices in the power basis: indeed, it is known that the conditioning (ratio of extreme singular values) of positive definite Hankel matrices is an exponential function of the matrix size[START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF]. When the smallest singular values reach machine precision, the SDP solver is not able to optimize the objective function any further.In Figures 4.6 and 4.7 one observes that the degree-100 polynomial approximation h(x) of the indicator function and its complement are tighter in the Chebyshev basis (black) than in the power basis (gray). Firstly, we observe that the degree-100 approximations in the power basis do not significantly differ from the degree-50 approximations in the same basis, represented in Figures 4.2 and 4.4. This is consistent with the very flat behavior of the right half of the upper and lower curves (in gray) in Figure4.5. Secondly, some coefficients of h(x) in the power basis have large magnitude h(x) = 1.0019 + 3.6161x -29.948x 2 + • • • + 88123x 49 + 54985x 50 + • • • -1018.4x 99 + 26669x 100 with the Euclidean norm of the coefficient vector greater than 10 6 . In contrast, the polynomial h(x) obtained in the Chebyshev basis h(x) = 0.1862t 0 (x) + 0.093432t 1 (x) -0.30222t 2 (x) + • • • + 0.0055367t 49 (x) -0.020488t 50 (x) + • • • -0.0012267t 99 (x) + 0.0011190t 100 (x) has a coefficient vector of Euclidean norm around 0.57627, where t k (x) denotes the k-th Chebyshev polynomial. Thirdly, oscillations around points x = 0 and x = 1/2 did not disappear with the Chebyshev basis, but the peaks are much thinner than with the power basis. Finally,

Fig. 4 . 7 .

 47 Fig. 4.7. Positive polynomial approximation of degree 100 of the indicator function 1 -I [0, 1 2 ]

Fig. 4 . 8 . 2 Fig. 4 . 9 .

 48249 Fig. 4.8. CPU time required to solve the SDP relaxations (Chebyshev basis in black, power basis in gray) as a function of the degree.

Fig. 4 . 10 .

 410 Fig. 4.10. Positive polynomial approximation of degree 20 of the indicator function of the bean surface.

  degree

+ x 2 2 ) 3 + 4x 2 1 x 2 2 ≥ 2 Fig. 4 . 11 .

 2322411 Fig. 4.11. Folium surface.

  degree

Figure 4 .

 4 Figure 4.12 displays a degree-20 positive polynomial approximation h of the indicator function I K on B obtained by solving an SDP problem with 231 unknown moments. For visualization purposes, max(5/4, h) rather than h is displayed. Again typical oscillations occur near the boundary regions, but we can recognize the shape of Figure 4.11.

Fig. 4 . 12 .

 412 Fig. 4.12. Positive polynomial approximation of degree 20 of the indicator function of the folium surface.

  Positive polynomial approximation of degree 100 of the indicator function I [0,1 2 ] in the Chebyshev basis (black) and power basis (gray).

		1.4
		1.2
		1
	polynomial approximation	0.6 0.8
		0.4
		0.2
		-1 0	-0.8	-0.6	-0.4	-0.2	0	0.2	0.4	0.6	0.8	1
							x
	0 0.2 0.4 1 1.2 1.4 Fig. 4.6. -1 0.6 0.8 polynomial approximation	-0.8	-0.6	-0.4	-0.2	0	0.2	0.4	0.6	0.8	1
							x

Table 4 . 1

 41 Relative error when approximating the volume of the bean surface, as a function of the degree of the SDP relaxation.

Table 4 . 2

 42 Relative error when approximating the volume of the folium surface, as a function of the degree of the SDP relaxation.

If K ⊂ [-1, 1] n then in Lemma

2.4, the condition M d (y 2 -y 1 ) 0, ∀d ∈ N, is sufficient.

* This work was completed with the support of the (French) ANR grant NT05-3-41612. The first author was also supported by the research program No. MSM6840770038 of the Czech Ministry of Education and Grant 102/08/0186 of the Grant Agency of the Czech Republic.


Zeynep Serinyel

Maxence Lailliau

Guillaume Dayma

Philippe Dagaut

A detailed high-pressure oxidation study of di-isopropyl ether

Keywords: isopropyl ether, jet-stirred reactor, oxidation, chemical kinetics

Introduction

In an effort to understand the oxidation and pyrolysis behavior of ethers, systematic experimental and modeling studies have been being conducted within our team [START_REF] Serinyel | A pyrolysis study on C4-C8 symmetric ethers[END_REF][START_REF] Belhadj | Oxidation of di-n-propyl ether: Characterization of low-temperature products[END_REF][START_REF] Serinyel | A high pressure oxidation study of di-n-propyl ether[END_REF][START_REF] Belhadj | Oxidation of di-n-butyl ether: Experimental characterization of low-temperature products in JSR and RCM[END_REF][START_REF] Serinyel | An experimental chemical kinetic study of the oxidation of diethyl ether in a jet-stirred reactor and comprehensive modeling[END_REF][START_REF] Thion | A chemical kinetic study of the oxidation of dibutyl-ether in a jet-stirred reactor[END_REF]. Oxidation of di-iso-propyl ether (DIPE) has been studied in this context, in the same experimental conditions as diethyl, di-n-propyl and di-n-butyl ethers, in order to make a comparison possible. These previous studies consisted in symmetric unbranched ethers, which all showed an important reactivity at low temperatures (between 450 and 750 K).

DIPE has an interesting character as it is considered as an octane improver, due its branched nature [START_REF] Boot | Impact of fuel molecular structure on autoignition behavior -Design rules for future high performance gasolines[END_REF], high octane number and low solubility in water [START_REF] Arteconi | Emissions from Ethers and Organic Carbonate Fuel Additives: A Review[END_REF]. Its oxidation was previously studied in a jet-stirred reactor by Goldaniga et al. [START_REF] Goldaniga | Oxidation of oxygenated octane improvers: MTBE, ETBE, DIPE, and TAME[END_REF] in 1998, along with other octane improvers such as methyl and ethyl tertiary butyl ethers (MTBE, ETBE) and tertiary amyl-methyl ether (TAME), at high pressure. In that study, while TAME oxidation was reported in detail, only formaldehyde and iso-butene mole fraction profiles were provided for DIPE oxidation. More recently, Fan et al. [START_REF] Fan | Exploring the oxidation chemistry of diisopropyl ether: Jet-stirred reactor experiments and kinetic modeling[END_REF] studied DIPE oxidation in a jet-stirred reactor at atmospheric pressure and for a stoichiometric mixture at a constant residence time of 2 s. Also Hashimoto et al. reported flame extinction limits of DIPE as well as those of ETBE and TAME in a counterflow burner [START_REF] Hashimoto | Extinction limits and flame structures of ETBE, DIPE and TAME non-premixed flames[END_REF]. In terms of engine studies, DIPE was found to decrease the knocking tendency in an HCCI engine and to expand the operating range of HCCI combustion [START_REF] Uyumaz | The effects of diisopropyl ether on combustion, performance, emissions and operating range in a HCCI engine[END_REF].

The present study will complement our systematic investigation, bringing new detailed data at high pressure for this branched symmetric ether.

Experimental set-up and kinetic modeling

Experimental

Experiments were carried out in a fused silica jetstirred reactor settled inside a stainless-steel pressure resistant jacket. An electrical oven enabled to perform experiments up to 1280 K. The temperature within the reactor was continuously monitored by a Pt/Pt-Rh 10% (S-type) thermocouple located inside a thin wall fused silica tube to prevent catalytic reactions on the metallic wires. Initial fuel mole fraction was 0.1 % for all experiments. Experimental pressure was 10 atm and the residence time () was 0.7 s. The residence time was held constant by adjusting the flow rates at each temperature. The reactive mixtures were highly diluted by nitrogen to avoid high heat release inside the reactor. The liquid fuel was atomized by a nitrogen flow and vaporized through a heated chamber. Reactants were brought separately to the reactor to avoid premature reactions and then injected by 4 injectors providing stirring. Flow rates of the diluent and reactants were controlled by mass flowmeters. A low-pressure sonic probe was used to freeze the reactions and take samples of the reacting mixtures.

Online analyses were performed after sending the samples via a heated line to a Fourier transform infrared (FTIR) spectrometer for the quantification of H2O, CO, CO2, and CH2O. Samples were also stored at ca. 40 mbar in Pyrex bulbs for further analyses using gas chromatography (GC). Two gas chromatographs with a flame ionization detector (FID) were used: one equipped with a DB624 column to quantify oxygenated compounds and the other one with a CP-Al2O3/KCl column to quantify hydrocarbons. Identification of the products was done by GC/MS on a Shimadzu GC2010 Plus, with electron impact (70 eV) as the ionization mode. Mole fraction profiles of H2 were measured using a GC-TCD (thermal conductivity detector) equipped with a CP-CarboPLOT P7 column. The species quantified in this study include di-iso-propyl-ether (DIPE), H2, H2O, CO, CO2, ethylene, methane, ethane, propene, formaldehyde, acetaldehyde, acetone and traces of 1butene and acetylene. The carbon balance was checked for each sample and the deviation was found to be typically within ±10-15%.

Kinetic mechanism

We have recently included pyrolysis reactions in our kinetic reaction mechanism [START_REF] Serinyel | A pyrolysis study on C4-C8 symmetric ethers[END_REF], and the same mechanism was used in the present work. In the conditions of this study, and in contrast to the pyrolysis experiment [START_REF] Serinyel | A pyrolysis study on C4-C8 symmetric ethers[END_REF], hydrogen abstraction reactions from the fuel by OH and HO2 radicals will be important. The structure of DIPE and its radicals are illustrated in Fig 1 . H-abstraction reaction rate constants by OH radicals were estimated by analogy to the calculations of Zhou et al. [START_REF] Zhou | An ab initio/Rice-Ramsperger-Kassel-Marcus study of the hydrogen-abstraction reactions of methyl ethers, H3COCH3-x(CH3)x, x = 0-2, by [radical dot]OH; mechanism and kinetics[END_REF] on iso-propyl methyl ether for the -site. For the -site, which involves 12 equivalent C-H bonds, the estimation (fit) was based on the calculations reported by Zador [START_REF] Zádor | 7th US National Techical Meeting, Combust. Inst., Combust. Inst[END_REF], Zhou [START_REF] Zhou | Rate constants for hydrogen-abstraction by OH from nbutanol[END_REF], Truhlar [START_REF] Seal | Kinetics of the Hydrogen Abstraction from Carbon-3 of 1-Butanol by Hydroperoxyl Radical: Multi-Structural Variational Transition-State Calculations of a Reaction with 262 Conformations of the Transition State[END_REF] and co-workers' calculations on the site of 1-butanol as well as Droege and Tully's work on n-butane [START_REF] Droege | Hydrogen atom abstraction from alkanes by hydroxyl. 5. n-Butane[END_REF]. This fit is then slowed down by a factor of 2 in order to take into account the fact that the beta C-H bond is slightly stronger (around 102 kcal.mol -1 ) than that in the -position in 1-butanol, which itself is equivalent to a primary C-H bond in an alkane.

H-abstraction reaction rate constants by HO2 radicals were estimated by analogy with the calculations of Mendes and co-workers [START_REF] Mendes | Rate Constant Calculations of H-Atom Abstraction Reactions from Ethers by HȮ2 Radicals[END_REF] on isopropyl methyl ether, with no modification as the C-H sites are comparable.

O a O O dipe1 dipe2
The JSR simulations were carried out with the Perfectly Stirred Reactor (PSR) code of Chemkin II package [START_REF] Glarborg | PSR: A Fortran Program for Modeling Well-Stirred Reactors[END_REF]. Results are shown in the following section.

Results and discussion

Results and model comparison

Experimental fuel conversion for all experiments as well as the pyrolysis profile from [START_REF] Serinyel | A pyrolysis study on C4-C8 symmetric ethers[END_REF] are presented in Figure 2. Clearly no reactivity is observed at low temperature (T < 750 K). Experiments were extended to 500 K for the lean mixture, where no fuel conversion occurred. The reactivity of DIPE begins at temperatures higher than 770 K and full conversion is observed around 1050 K under present operating conditions. One can also observe a change of slope for the  = 4 mixture (and to a lesser extent for  = 2) while the lean and stoichiometric mixtures behave similarly with a slightly higher fuel conversion for the lean case. Obviously, under pyrolysis conditions, fuel conversion is much less at a given temperature: For example at 900 K, only 15% of the fuel is converted under pyrolytic conditions while this is > 60% in oxidation. Nevertheless, DIPE is fully converted regardless of the O2 content when the temperature reaches 1050 K. Experimental results (mole fraction profiles of reactant, intermediate and products) are presented in Figs 3456. One can see from these figures that modelexperiment comparison is generally quite good, often within 15-20%. Reactivity profiles and the intermediate distribution is substantially different between the lean ( = 0.5) and the richest mixture ( = 4). For example, no iso-propanol is observed at  = 0.5 while iso-propanol formation with a peak of around 300 ppm can be seen at  = 4 just below 1050 K. Meanwhile, an increased peak mole fraction of propene can also be observed at the same temperature.

These intermediates are products of the molecular elimination reaction, which was the most important flux consuming the fuel under pyrolysis conditions [START_REF] Serinyel | A pyrolysis study on C4-C8 symmetric ethers[END_REF]. In this sense, the richest mixture shows some similarity to pyrolysis. Note that while iso-propanol is exclusively produced via this pathway, propene is a common oxidation intermediate coming from other pathways as well. This is illustrated in Fig 7 . Other production pathways for propene include the dehydration of iso-propanol and beta-scission of the (primary) fuel radical dipe2 ⇌ iC3H7O + C3H6.

Reaction path analysis

A reaction pathway analysis showing the initial decomposition pathways is given in Fig. 8. For the  = 4 mixture: At 1000 K, the fuel mostly reacts via unimolecular decomposition while only 40% of the fuel consumption occurs through H-abstraction reactions producing dipe1 and dipe2. Among H-abstraction channels, abstraction by H atoms represent 20%, followed by CH3 radicals, which represents 15% and that by OH radicals, representing a relatively small portion of 5-6%. Iso-propanol one of the products of the molecular reaction, which finally leads to the formation of allyl radicals. Recombination of allyl and methyl radicals yield 1-butene. This intermediate is observed with a peak around 55 ppm and owes its formation to this pathway. Note that 1-butene is not observed at neither  = 0.5 nor at  = 1. Note that 1butene is observed but in trace amounts (< 10 ppm) during the oxidation of lean and stoichiometric mixtures. This could be due to the combination of two phenomena linked to its formation and consumption. For example at 1000 K, allyl radicals combine with CH3 radicals to a lesser extent when the mixture is lean (22%) than when it is rich (58%). Also, 1-butene consumption takes place via different pathways among which reactions with O atoms and OH radicals are the most important. At 1000 K and  = 0.5, given the abundance of these radicals 1-butene is consumed at a more important rate than when  = 4 (altogether accounting for 73% of its consumption at  = 0.5, as opposed to a total of 32% at  = 4). This may lead to the accumulation of 1-butene as the mixture gets richer.

On the other hand, at 900 K, temperature is not still high enough to overcome the barrier of the unimolecular reaction, representing only 4% of the reaction flux. The rest of the reaction flux corresponds to H-abstraction reactions: 33% by OH radicals, 38% by H atoms, 20% by CH3 radicals and 5% by HO2 radicals. The -radical (dipe1) goes through scission to form acetone and the iso-propyl radical. At lower temperatures of interest (when T < 900 K) this is how acetone is formed. The -radical (dipe2) mostly (78%) decomposes into the isopropoxy radical and propene, the remaining flux yields isopropyl vinyl ether and methyl radicals (iC3H7OC2H3 + CH3). It is to be noted that neither iC3H7OC2H3 nor a conjugated olefin of this ether an unsaturated conjugate ether were observed experimentally under the present conditions.

For the  = 0.5 mixture:

At any temperature, unimolecular decomposition is not significant. The fuel is consumed by hydrogen abstraction reactions. For this lean mixture, abstraction by OH radicals is more significant, representing a flux of 47% at 1000 K (as opposed to 5-6% for the  = 4 mixture) followed by abstraction by H atoms representing 37% of the flux. At 900 K, hydrogen abstraction by OH radicals represents 63%, while abstraction by H atoms represent about 25%. The fuel radicals undergo the same pathways with same branching ratios, as described under rich conditions.

A major part of acetaldehyde formation (60% under lean condition -80% under fuel-rich condition) happens as a result of the -scission of the dipe2 radical, which the model represents quite well at  = 0.5 and 1 but over-predicts by a maximum of 50% when the mixture is fuel-rich. This may be due to the uncertainties in the estimated rate parameters for the H-abstraction reactions (especially by H-atoms that is important for the fuel-rich mixture) as well as the uncertainty related with the -scission of the fuel radical, also estimated by analogy [START_REF] Sakai | A quantum chemical and kinetics modeling study on the autoignition mechanism of diethyl ether[END_REF][START_REF] Curran | Rate constant estimation for C1 to C4 alkyl and alkoxyl radical decomposition[END_REF]. A systematic theoretical study of the kinetics of these reactions would certainly be useful, not only for this fuel but also for the rest of the ether family as well as the alcohols.

Comparison with literature data

Figure 9 illustrates the JSR data at atmospheric pressure in [START_REF] Fan | Exploring the oxidation chemistry of diisopropyl ether: Jet-stirred reactor experiments and kinetic modeling[END_REF], where reactants products and some important intermediates are represented. The pressure in that experiment was 1 atm, and the initial fuel mole fraction was 1%, only stoichiometric mixture was investigated by the authors. As can be seen in Fig 9, the present model can represent these data well. Note that authors' model showed similar discrepancies with their data as well. Iso-propanol is formed at T > 800 K, which the present model under-estimates while the authors' model represents it well. It is useful to note that given the differences in the operating conditions, no direct comparison can be made between the data in Fig 9 and the present work. The initial fuel mole fraction was 1% in their experiment (10 times higher than the present study) and the residence time was almost three times longer (2s vs 0.7s), which altogether enhance reactivity. However, when the Fan et al. mechanism [START_REF] Fan | Exploring the oxidation chemistry of diisopropyl ether: Jet-stirred reactor experiments and kinetic modeling[END_REF] is used to test our pyrolysis data where the unimolecular reaction dominates, iso-propanol is over-estimated and hence the overall reactivity. This can be observed in Fig 9 . A more comprehensive comparison with our pyrolysis data and the literature model can be found in the supplementary material (Figures S1-S4). Fig. 10. Pyrolysis of DIPE [START_REF] Serinyel | A pyrolysis study on C4-C8 symmetric ethers[END_REF], dashed lines represent the literature mechanism by Fan et al. [START_REF] Fan | Exploring the oxidation chemistry of diisopropyl ether: Jet-stirred reactor experiments and kinetic modeling[END_REF]. Experimental conditions are: p = 10 atm, X 0,fuel = 0.1%,  = 0.7 s.

Comparison with other ethers

In our previous pyrolysis study [START_REF] Serinyel | A pyrolysis study on C4-C8 symmetric ethers[END_REF], DIPE was found to be the most reactive among DEE, DPE and DBE, given its 12 primary C-H sites available for the molecular reaction whereas the other ethers have 4 available C-H sites. It was also expected that DIPE would be less reactive than DEE, DPE and DBE under oxidative conditions. Figure 11 illustrates the fuel conversion for diethyl (DEE), di-n-propyl (DPE), dibutyl (DBE) and di-iso-propyl ethers when  = 2, at 10 atm and 0.7 s of residence time. At other equivalence ratios, the conclusion would be the same.

It is clear from this figure that among these ethers, the least reactive one is DIPE. The weakest C-H site in DIPE is the tertiary one with 2 equivalent bonds. The radical formed (dipe1) can add to molecular O2 and undergo ROO • ⇌ • QOOH type of isomerization, as seen in the following scheme.

Second addition to O2 forms the • OOQOOH radical, which cannot isomerize via a 6-membered transition state because of the quaternary C, and hence cannot form the corresponding ketohydroperoxides (KHP). Other possible internal H-transfers involve either 5-or 7-membered transition states, which are much less favorable. This in overall, reduces reactivity at low temperatures. At high temperatures, on the other hand, DIPE is less reactive than other ethers given that it has 12 equivalent, slightly enhanced (compared to n-alkanes) primary C-H bonds and only 2 weakened tertiary C-H sites. Its isomer, DPE, is much more reactive when T > 850 K, given its 4 equivalent weakened secondary C-H sites and only 6 primary C-H bonds, compared to DIPE.

Figure 11. Comparison of fuel reactivity between DEE [START_REF] Serinyel | An experimental chemical kinetic study of the oxidation of diethyl ether in a jet-stirred reactor and comprehensive modeling[END_REF], DPE [START_REF] Serinyel | A high pressure oxidation study of di-n-propyl ether[END_REF], DIPE (present study) and DBE [START_REF] Thion | A chemical kinetic study of the oxidation of dibutyl-ether in a jet-stirred reactor[END_REF] (lines are used to guide the eye).

Conclusions and perspective

In line with our previous studies, this work deals with the high-pressure oxidation of DIPE in a JSR, at 10 atm, with 0.1% initial fuel mole fraction, and at a constant residence time of 0.7 s. Four mixtures with equivalence ratios of 0.5-4 were investigated. Our results show that DIPE shows no reactivity at low temperatures and reactivity kicks off when the temperature is over 800 K. Therefore, no low temperature chemistry is included in the mechanism. While fuel decomposition is mainly due to hydrogen abstraction reactions (by H, OH and CH3, and HO2 (to a lesser extent), as the equivalence ratio increases the molecular reaction DIPE ⇌ iso-propanol + propene comes into picture. This can be seen with the  = 4 mixture where a quasi-plateau is observed in the fuel profile between 850-950 K where we can also observe iso-propanol being produced and accumulated. This is also the case when  = 2, but to a lesser extent.

As a perspective, and as the data on DIPE oxidation is quite scarce, data on global combustion properties such as ignition delay times can be envisaged, which would highlight a probably less significant pathway in this study. Also, a systematic theoretical study would be useful for metathesis reactions by H atoms and CH3 radicals as well as fuel radical decompositions, which are often based on rough estimations.

Figure 1 .

 1 Figure 1. Structure of di-iso-propyl ether

Figure 2 .

 2 Figure 2. Fuel conversion of DIPE at all equivalence ratios. Pyrolysis data is from [1].

Figure 3 .

 3 Figure 3. Mole fraction profiles for the  = 0.5 mixture, P = 10 atm, X 0,fuel = 0.1%, τ = 0.7 s (symbols: data, lines: simulations)

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Mole fraction profiles for the  = 1 mixture, P = 10 atm, X 0,fuel = 0.1%, τ = 0.7 s (symbols: data, lines: simulations)

Figure 7 .

 7 Figure 7. Comparison of propene and iso-propanol mole fraction profiles for the  = 0.5 and 4 mixtures as well as the pyrolysis experiment [1]. P = 10 atm,  = 0.7s.

Figure 8 .

 8 Figure 8. Reaction pathways for the  = 0.5 and 4 mixtures at 900 K (black) and 1000 K (blue), P = 10 atm,  = 0.7s.

Figure 9 .

 9 Figure 9. JSR data from literature [10] and comparison with the present model. Experimental conditions are: P = 1 atm,  = 1, X 0,fuel = 1%,  = 2 s.

Supplementary material

The kinetic mechanism and the experimental data are provided. A document with experiment/literature model comparison is also available.