Beijing J Chen

Huazhong Z Shu

Hui Zhang

Gang Chen

Christine Toumoulin

Jean-Louis L Dillenseger

Limin M Luo

Quaternion Zernike moments and their invariants

Keywords: Color image, Quaternion, Zernike moment, Moment invariant. ---

and research institutions in France or abroad, or from public or private research centers.

Introduction

Moment invariants have been extensively used in pattern recognition [START_REF] Khotanzad | Invariant image recognition by Zernike moments[END_REF], [START_REF] Zhang | Aircraft recognition in infrared image using wavelet moment invariants[END_REF], scene matching [START_REF] Lin | Template matching using the parametric template vector with translation, rotation and scale invariance[END_REF] and object classification [START_REF] Yang | Classification of rotifers with machine vision by shape moment invariants[END_REF] owing to their image description capability and invariance property. However, they are mainly used to deal with the binary or gray-scale images. With the development of inexpensive digital camera, nowadays almost all images acquired are chromatic. Generally speaking, there are two approaches that are often used for color image processing. The first one consists of transforming the color image into gray-scale one, which may lose some significant color information. For example, if we use the average of three channels for graying the color image, it is impossible to identify the objects having the same shape but different color (red, green, blue). The second one decomposes the color image into three channels, and then calculates the moment invariants of these three channels separately [START_REF] Mindru | Moment invariants for recognition under changing viewpoint and illumination[END_REF]- [START_REF] Chan | A color image retrieval method based on color moment and color variance of adjacent pixels[END_REF]. Among the latter, Mindru et al. [START_REF] Mindru | Moment invariants for recognition under changing viewpoint and illumination[END_REF], [START_REF] Mindru | Color-based moment invariants for the viewpoint and illumination independent recognition of planar color patterns[END_REF] proposed the generalized color moments for color images, whose integral function is the product of the powers of the pixel coordinates and those of the intensities for one or more color channels. Based on these moments, they constructed a set of invariants to geometric transformation and photometric changes. Suk and Flusser [START_REF] Suk | Affine moment invariants of color images[END_REF] derived a set of affine geometric invariants for color images, which are based on the product of moments defined for different channels. Specially, they introduced the notion of common centroid for defining the central moments in order to achieve the translation invariance. Both methods are the generalization of conventional geometric moments, the difference between them is that the computation of Mindru's moments may refer to two or three color channels, and that of Suk's moments only relates to one channel. Since the kernel functions of geometric moments are not orthogonal, this may lead to information redundancy and low noise robustness.

The orthogonal Zernike polynomials were first introduced by Zernike in 1934 [START_REF] Zernike | Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode (Diffraction theory of the cut procedure and its improved form, the phase contrast method)[END_REF], Teague used them to define the orthogonal Zernike moments [START_REF] Teague | Image analysis via the general theory of moments[END_REF]. Since then, the Zernike moments have been applied to a number of computer vision problems [START_REF] Khotanzad | Invariant image recognition by Zernike moments[END_REF], [START_REF] Wang | Using Zernike moments for the illumination and geometry invariant classification multispectral texture[END_REF]- [START_REF] Chen | A Zernike moment phase-based descriptor for local image representation and matching[END_REF] because they have overall better performance than other moments [START_REF] Teh | On image analysis by the method of moments[END_REF]- [START_REF] Shu | Moment-based approaches in image Part 2: invariance[END_REF]. The purpose of this paper is to extend the conventional orthogonal Zernike moments to color image in a holistic manner. To that end, we will use the algebra of quaternions.

In recent years, quaternions have been utilized more and more in color image processing. Let f(x, y) be an RGB image, it can be represented by encoding three channels as a pure quaternion as follows ( , ) ( , ) ( , ) ( , )

R G B f x y f x y f x y f x y = + + i j k , (1)
where f R (x, y), f G (x, y) and f B (x, y) represent respectively the red, green and blue channel of the color images, and i, j and k are the complex operators. The main advantage of the use of quaternion-type representation is that a color image can be treated as a vector field [START_REF] Ell | Hypercomplex Fourier transforms of color images[END_REF]. The algebra of quaternions has been exploited in color image processing by Ell and Sangwine [START_REF] Ell | Hypercomplex Fourier transforms of color images[END_REF]- [START_REF] Sangwine | Fourier transforms of colour images using quaternion or hypercomplex, numbers[END_REF]. In 1992, Ell [START_REF] Ell | Hypercomplex spectral transforms[END_REF] introduced the quaternion Fourier transforms (QFTs) in his Ph. D dissertation. Sangwine [START_REF] Sangwine | Fourier transforms of colour images using quaternion or hypercomplex, numbers[END_REF] then applied them to color image. Since then, the QFTs have been successfully employed in color image registration [START_REF] Moxey | Color-grayscale image registration using hypercomplex phase correlation[END_REF], [START_REF] Feng | A subpixel color image registration algorithm using quaternion phase-only correlation[END_REF], watermarking [START_REF] Bas | Color image watermarking using quaternion Fourier transform[END_REF], [START_REF] Ma | Color image watermarking using local quaternion Fourier spectral analysis[END_REF], motion estimation [START_REF] Alexiadis | Estimation of motions in color image sequences using hypercomplex Fourier transforms[END_REF] and texture analysis [START_REF] Assefa | Local quaternion Fourier transform and color image texture analysis[END_REF]. Recently, the use of quaternion-based moment functions to color image has been investigated [START_REF] Chen | Color image analysis by quaternion Zernike moments[END_REF], [START_REF] Guo | Quaternion Fourier-Mellin moments for color image[END_REF]. We have introduced the notion of the quaternion Zernike moments (QZMs) and derived a set of invariants with respect to image translation and rotation [START_REF] Chen | Color image analysis by quaternion Zernike moments[END_REF]. By using the quaternion algebra, Guo et al. proposed the quaternion Fourier-Mellin moments (QFMMs) to deal with the color images and constructed a set of invariants to rotation, scale and translation (RST) transformation [START_REF] Guo | Quaternion Fourier-Mellin moments for color image[END_REF]. However, in both methods, the rotation invariance was achieved by taking the modulus of the quaternion moments, this may lead to two disadvantages: 1) the modulus loses the phase information which may be useful in some applications; 2) the modulus only provides one real-valued invariant.

In this paper, we propose a new approach to construct a set of QZM invariants with respect to RST transformation. The remaining of this paper is organized as follows. In Section 2, we first recall some preliminaries about the quaternions, and then present the definition of the QZMs. In Section 3, we derive a set the moment invariants with respect to RST transformation. Experimental results are provided in Section 4 to illustrate the performance of the proposed descriptors. Section 5 concludes the paper.

Quaternion moments

Quaternion number

Quaternions, a generalization of the complex numbers, were introduced by the mathematician Hamilton in 1843 [START_REF] Hamilton | Elements of quaternions[END_REF]. A quaternion consists of one real part and three imaginary parts as follows q = a + bi + cj + dk.

(2) where a, b, c and d are real numbers, and i, j and k are three imaginary units obeying the following rules

2 2 2 1 = = = - i j k , (3)
, , - - - ij = ji = k jk = kj = i ki = ik = j .
(4) The conjugate and modulus of a quaternion are respectively defined by

q * = a -bi -cj -dk, ( 5
) 2 2 2 2 q a b c d = + + + . ( 6
)

Quaternion Zernike moments

Let f(r, θ) be an RGB image defined in polar coordinates, we define the right-side QZMs of order n with repetition m as

[31] 1 2 , , 0 0 1 ( ) ( ) ( , ) , R m n m n m n Z f R r f r e rd dr π θ θ θ π + = ∫ ∫ -µ -µ -µ -µ
|m| ≤ n and n -|m| being even, [START_REF] Suk | Affine moment invariants of color images[END_REF] where µ is a unit pure quaternion chosen as ( ) / 3 = + + µ i j k in this paper, R n,m (r) is the real-valued radial polynomial given by

( ) / 2 2 , 0 ( 1)
( )! ( ) . ! ! ! 2 2 n m k n k n m k n k R r r n m n m k k k - - = - - =  +   -  - -         ∑ (8)
Since the radial Zernike polynomials are orthogonal, the color image f(r, θ) can be approximatively reconstructed from a finite number M of QZMs as follows , , 0 ( , ) ( ) ( )

M n R m n m n m n m n f r Z f R r e θ θ = =- ≈ ∑ ∑ µ µ µ µ . ( 9
)
Because the Zernike moments are defined in polar coordinates (r, θ) with |r| ≤ 1, the computation of Zernike moments requires a linear transformation of the image coordinates to a suitable domain inside a unit circle. Here we use the mapping transformation proposed by Chong et al. [START_REF] Chong | Translation invariants of Zernike moments[END_REF]. Based on this transformation, we have the following discrete approximation of (7): , , , ,

( ) ( ) ( , ) 2 1 1 1
x y

N N m R n m n m x y x y n Z f R r f x y e N θ π = = + = -∑∑ -µ -µ -µ -µ . ( 10
)
where N is the number of pixels along each axis of the image, the mapping transformation to the interior of the unit circle is given by

2 2 1 1 2 , 1 2 1 2 , 1 2 ( ) ( ) , tan , x y x y c y c r c x c c y c c x c θ -  + = + + + =   +   (11) with 1 2 2 1 , . 1 2 c c N = = - -
It can be seen from ( 4) that the multiplication of quaternions is not commutative, so we also define the left-side QZMs of order n with repetition m as

1 2 , , 0 0 1 ( ) ( ) ( , ) L m n m n m n Z f R r e f r rd dr π θ θ θ π + = ∫ ∫ -µ -µ -µ -µ , (12)
The corresponding inverse transform is

, , 0 ( , ) ( ) ( ) M n m L n m n m n m n f r e Z f R r θ θ = =- ≈ ∑ ∑ µ µ µ µ . ( 13
)

Relationship between the quaternion Zernike moments and the conventional Zernike moments for single channel of the RGB image

Substituting (1) into (7) and using ( 3) and (4), we obtain

1 2 , , 0 0 1 2 1 2 , , 0 0 0 0 1 2 , 0 0 1 ( )= ( )[ ( , ) ( , ) ( , ) ] 1 1 = ( ) ( , ) ( ) ( , ) 1 ( ) ( , )
= R m n m n m R G B m m n m R n m G m n m B n Z f R r f r f r f r e rd dr n n R r f r e rd dr R r f r e rd dr π θ θ π + + + + + + + + + ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ i j k i j k i -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ ( ) 1 2 1 2 , , 0 0 0 0 1 2 , 0 0 1 2 , , 0 0 1 1 ( ) ( , )(cos sin ) ( ) ( , )(cos sin ) 1 ( ) ( , )(cos sin ) 1 1 = ( ) ( , )cos ( ) n m R n m G n m B n m R n m R n R r f r m m rd dr R r f r m m rd dr n R r f r m m rd dr n n R r f r m rd dr R r f θ θ θ π π         + + + -      + + + -  ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ j k µ µ µ µ µ µ µ µ , , , , , , , , , , ( ) ( )
= Re( ( )) Im( ( )) Re( ( )) Im( ( )) 3 3
( ) Re( ( )) Im( ( )) 3 = , n m R n m R n m G n m G n m B n m B R R R R n m n m n m n m Z f Z f Z f Z f Z f Z f A B C D     + + + +     + + +         + +   + +     + + + i j k i j k i j i j k k i j k (14) where , , , , , , , , , , , , , , , , 1 [Im(
( )) Im( ( )) Im( ( ))], 3 1 Re( ( )) [Im( ( )) Im( ( ))], 3 1 Re( ( )) [Im( ( )) Im( ( ))], 3 1 Re( ( )) [Im( ( )) Im( ( ))]. 3 R n m n m R n m G n m B R n m n m R n m G n m B R n m n m G n m B n m R R n m n m B n m R n m G A Z f Z f Z f B Z f Z f Z f C Z f Z f Z f D Z f Z f Z f = - + + = + - = + - = + - (15)
Here Z n, m (f R ), Z n, m (f G ) and Z n, m (f B ) are respectively the conventional Zernike moments for the red channel, green channel and blue channel, Re(x) represents the real part of conventional complex number x, and Im(x) the imaginary part, that is, Re(a + bi) = a, Im(a + bi) = b. Equation ( 14) shows that the QZMs can be obtained from the conventional Zernike moments for single channel. Similarly, we have the following relationship for the left-side QZMs , , ,

L L L L L n m n m n m n m n m Z f A B C D = + + + i j k , , , ( )
( )) Im( ( )) Im( ( ))], 3 1 Re( ( )) [Im( ( )) Im( ( ))], 3 1 Re( ( )) [Im( ( )) Im( ( ))], 3 1 Re( ( )) [Im( ( )) Im( ( ))]. 3 L n m n m R n m G n m B L n m n m R n m B n m G L n m n m G n m R n m B L n m n m B n m G n m R A Z f Z f Z f B Z f Z f Z f C Z f Z f Z f D Z f Z f Z f = - + + = + - = + - = + - (16) where , , , , , , , , , , , , , , , , 1 [Im(
Comparing ( 14), ( 15) with ( 16), (

) L R n m n m Z f Z f - = - . 17), we have * , , ( ) ( ( )

RST invariants of quaternion Zernike moments

In our previous work [START_REF] Chen | Color image analysis by quaternion Zernike moments[END_REF], we have derived the translation and rotation invariants of QZMs. The rotation invariance was achieved by taking the modulus of QZMs, which leads to some drawbacks as noted in the Introduction. Here, we propose a new approach to construct the rotation invariants. We will also discuss the way to derive the scaling invariants. Before that, we give a brief description of translation invariants proposed in [START_REF] Chen | Color image analysis by quaternion Zernike moments[END_REF].

Translation invariants [31]

Suk and Flusser [START_REF] Suk | Affine moment invariants of color images[END_REF] have defined the common centroid (x c , y c ) of all three channels as follows

1,0 1,0 1,0 0,0 0,1 0,1 0,1 0,0 0,0 0,0 0,0 0,0 ( ( ) ( ) ( )) / , ( ( ) ( ) ( )) / , ( ) ( ) ( ), c R G B c R G B R G B x m f m f m f m y m f m f m f m m m f m f m f = + + = + + = + + (19)
where m 0, 0 (f R ), m 1, 0 (f R ) and m 0, 1 (f R ) are respectively the zero-order and first-order geometric moment for R channel,

m 0, 0 (f G ), m 1, 0 (f G ) and m 0, 1 (f G ) for G channel, and m 0, 0 (f B ), m 1, 0 (f B ) and m 0, 1 (f B ) for B channel.
Let the origin of the coordinate system be located at (x c , y c ), the central QZMs, which are invariant to image translation, can be obtained as follows

1 2 , , 0 0 1 ( ) ( ) ( , ) R m n m n m n Z f R r f r e rd dr π θ θ θ π + = ∫ ∫ -µ -µ -µ -µ , ( 20
)
where ( , ) r θ is the image pixel coordinate representation in polar form with the mapping transformation (11) by locating the origin at (x c , y c ).

Rotation invariants

Let f ′ be the rotated version of f, i.e., ( , )

( , ) f r f r θ θ α ′ =
-, where α denotes the rotation angle, then we have

1 2 , , 0 0 1 2 , 0 0 1 2 ( ) , 0 0 1 2 , 0 0 1 ( ) ( ) ( , ) 1 = ( ) ( , ) 1 = ( ) ( , ) 1 = ( ) ( , ) R m n m n m m n m m n m m m n m n Z f R r f
+ ′ ′ = + - + + ⋅ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ -µ , = ( ) . R m n m Z f e α -µ -µ -µ -µ (21)
Equation [START_REF] Ell | Hypercomplex Fourier transforms of color images[END_REF] shows that the modulus of ,

R n m

Z is invariant to rotation, which is the method proposed in [START_REF] Chen | Color image analysis by quaternion Zernike moments[END_REF] to achieve the rotation invariance. However, such a process loses the phase information which may be useful in some applications.

L m L n m n m Z f e Z f α ′ = -µ -µ -µ -µ . ( ( ) ( )
) 22
Theorem 1. Let * , , , , , ( ) ( ) ( ) ( )( ( )
) m R L R R n k n m k m n m k m f Z f Z f Z f Z f φ - = = - , |m| ≤ n, |m| ≤ k, n -|m| and k -|m| being even. ( 23
)
Then, , ( )

m n k f
φ is invariant to image rotation for any integer m and non-negative integer n and k.

Proof. Let f ′ be the rotated image of f with rotation angle α, using ( 21) and ( 22), we

( ) ( ) ( ) m R L n k n m k m R m m L n m k m R L m n m k m n k f Z f Z f Z f e e Z f Z f Z f f α α φ φ - - - ′ ′ ′ = = = = -µ µ -µ µ -µ µ -µ µ have , , , , , , , , ( ) ( ) ( ) ( ) ( )
The proof has been completed. 

Based on Theorem 1, when k = n, we have

Scaling invariants and combined RST invariants

In this subsection, we first derive a set of invariants with respect to image scale. Equation [START_REF] Chan | A color image retrieval method based on color moment and color variance of adjacent pixels[END_REF] shows that the radial polynomial R n,m (r) is symmetric with m, that is, R n,-m (r) = R n,m (r). So, the case m ≥ 0 is considered. Letting n = m+2l and using ( 8), ( 7) can be rewritten as

1 2 2( ) 2 , 0 0 0 1 2 2 0 0 0 2 1 ( 2 )! ( ) ( 1) ( , ) !( )!( )! 2 1 ( )! ( 1) ( , ) !( )!( )! 2 1 ( ) ( 1) l R k m l k m m l m k l l k m k m k l k m l m l k Z f r f r e rd dr k m l k l k m l m l k r f r e rd dr k m k l k m l m l k π θ π θ θ θ π θ θ π π + - - + = - + - = -   + + + - = -   + - -     + + + + = -   + -   + + + + = - ∑ ∫ ∫ ∑ ∫ ∫ µ µ µ µ µ µ µ µ 1 2 2 0 0 0 , 2 , 0 ! ( , ) !( )!( )! ( ), l m k m k l m R l k m k m k r f r e rd dr k m k l k c f π θ θ θ ψ + - = + = + - = ∑ ∫ ∫ ∑ µ µ µ µ ( 25
)
where

1 2 2 2 , 0 0 ( ) ( , ) , R m k m m k m f r f r e rd dr π θ ψ θ θ + - + = ∫ ∫ µ µ µ µ (26) , 2 1 ( )! ( 1) . !( )!( )! m l k l k m l m l k c k l k m k π - + + + + = - - + (27)
From ( 25), 2 , ( )

R m l m f ψ +
can also be expressed as a series of QZMs

2 , , 2 , 0 (
) ( ), l R m R m l m l k m k m k f d Z f ψ + + = = ∑ (28)
where , ( )

m m l i j D d = , with 0 ≤ j ≤ i ≤ l, is the inverse matrix of , ( )
m m l i j C c =
. The elements of m l D are given by [START_REF] Shu | A general method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes[END_REF] ,

!( )! , ( )!( 1)! m i j i m i d i j m i j π + = - + + + 0 ≤ j ≤ i ≤ l. ( 29
)
Let f ′′ and f be two images having the same content but scale (λ), that is, ( , )

( / , ) f r f r θ λ θ ′′ =
, using ( 25) and ( 28), the QZMs of the transformed image can be obtained as

1 2 2 2 , , 0 0 0 1 2 2 2 , 0 0 0 2 2 , 2 , 0 2 2 , , 2 , 0 0 2 2 ( )= ( / , ) = ( ) ( , ) = ( ) = ( ) = l R m m k m m l m l k k l m m k m l k k l m k m R l k m k m k l k m k m m R l k k t m t m k t m k l Z f c r f r e rd dr c r f r e rd dr c f c d Z f c π θ π θ λ θ θ λ λ θ θ λ ψ λ λ + - + = + - = + + + = + + + = = + + ′′ ∑ ∫ ∫ ∑ ∫ ∫ ∑ ∑∑ µ µ µ µ µ µ µ µ , , 2 , 0 ( ).
l m m R k k t m t m t k t d Z f + = = ∑∑ (30) Theorem 2. Let ( 2 2) 2 , , , 2 , 0 ( ) ( )
l l R m k m m R m l m f l k k t m t m t k t L f c d Z f -+ + + + = = = Γ ∑∑ , (31)
with 0,0 ( )

R f Z f Γ = . Then 2 , ( ) R m l m L f
+ is invariant to scaling for any non-negative integer m and l.

The proof is given in Appendix A.

. R R m n m n m L f L f e α ′ = -µ -µ -µ -µ ( ) ( )
The proof of Corollary 1 is very similar to [START_REF] Shu | Moment-based approaches in image Part 2: invariance[END_REF] and is thus omitted. Then, using Corollary 1, we have

Corollary 2. Let * , , , ( ) ( )( ( )
) m R R n k n m k m f L f L f ϕ = , m ≤ n, m ≤ k, n -m and k -m being even. (33)
Then, , ( )

m n k f
ϕ is invariant to both image rotation and scaling for any non-negative integer n, k and m.

The proof of Corollary 2 is very similar to that of Theorem 1 and is also omitted. Combining [START_REF] Shu | Moment-based approaches in image Part 1: basic features[END_REF] and [START_REF] Hamilton | Elements of quaternions[END_REF], the RST invariants of QZMs (QZMIs) can be obtained as follows

Corollary 3. Let * , , , ( ) ( )( ( )
) m R R n k n m k m f L f L f ϕ = , m ≤ n, m ≤ k, n -m and k -m being even. ( 34
)
where , ( )

R n m
L f is the scaling invariants defined in (31) using the central QZMs 2 , ( )

R m t m Z f + instead of 2 , ( ) R m t m Z f + . Then, , ( ) m n k f
ϕ is invariant to image RST transformation for any non-negative integer n, k and m.

Remark 1. By using the symmetric property of R n,m (r) with m, it can be easily proven that Corollary 3 is also true for m < 0.

Experimental results

In this section, several experiments are carried out to test the invariance property of the proposed descriptors to various geometric transformations, and their robustness to different kinds of noises. Note that the Zernike moments are calculated with the modified Kintner's method [START_REF] Chong | A comparative analysis of algorithm for fast computation of Zernike moments[END_REF] and images are mapped inside the unit circle before the moment computation.

A. Test of color image reconstruction capability

For this experiment, the standard Lena and Pepper images of size 256 × 256 were used. The reconstructed images using [START_REF] Zernike | Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode (Diffraction theory of the cut procedure and its improved form, the phase contrast method)[END_REF] with different values of M are shown in Fig. 1. Let f(x, y) be the original image and ˆ( , ) f x y be the reconstructed image, then the following normalized mean square error ε 2 [START_REF] Revaud | Improving Zernike moments comparison for optimal similarity and rotation angle retrieval[END_REF] is used to measure the accuracy of the reconstructed images

ε = = = = - = ∑∑ ∑∑ . ( 35
)
The reconstruction errors are also given in Fig. 1. The results show that the reconstructed images are very close to the original image.

B. Test of invariance to RST transformations

For the experiments presented in this subsection as well as in the subsection a), a set of thirteen images (Fig. 2) with size 96 × 72 has been chosen from the public Amsterdam Library of Object Images (ALOI) database [START_REF] Geusebroek | The Amsterdam library of object images[END_REF]. In order to contain the entire transformed image after transformation, the actual size of all the original images is 152 × 128 by adding some background pixels.

To test the invariance of the proposed QZMIs with respect to RST transformations, obj1 (Fig. 2(a)) was undergone different geometric transformations (Fig. 3). The proposed QZMIs defined in [START_REF] Chong | Translation invariants of Zernike moments[END_REF] of order from 1 to 4 were calculated for each image (the invariant of order zero is used to achieve the scale invariance in all the methods based on Zernike moments).

Table 1 shows the modulus of invariant values and that of σ/µ, where µ denotes the mean of invariants and σ the standard deviation. It can be seen from this table that excellent results have been obtained whatever the geometric transformations (σ/µ ≤ 1.369%).

C. Color object recognition

To further assess the performance of the proposed QZMIs to RST transformation and their robustness against noise, three object recognition procedures were conducted. a) Performance comparison of QZMIs and Zernike moment invariants based on RGB decomposition and graying in terms of recognition efficiency By proceeding in a similar way as that described in section 3 for color image, we can also derive the Zernike moment invariants with respect to RST transformation for single channel of the RGB image and the gray-scale image obtained with image graying. In both cases, the conventional Zernike moments are concerned. So, for image graying, a set of Zernike moment invariants denoted by GZMIs are extracted from the graying image of the RGB image; for RGB decomposition, those invariants for three channels are grouped in a whole set (denoted hereafter by DZMIs). In order to compare the proposed QZMIs with GZMIs and DZMIs in terms of recognition efficiency, we use almost the same number of invariants in the experiment: The GZMIs include 19 non-zero real-valued invariants with order from 2 to 6 (the invariants of order one equal zero for gray-scale image when the central moments are concerned, but this is not the case for both DZMIs and QZMIs). The DZMIs contain 18 real-valued invariants with order from 1 to 3. QZMIs include 20 real-valued numbers with order from 1 to 4.

We used thirteen images shown in Fig. 2 as the training set. To obtain the testing set, each image was translated with ∆x = 8, ∆y = 5 (other translation can also be chosen), rotated with angle α ∈ {0, 30,…, 300, 330}, and scaled with scaling factor λ ∈ {0.5, 0.75,…,1.75, 2.0}, forming a set of 1092 images, and a bilinear interpolation was used when required. This was followed by adding a white Gaussian noise with different standard deviations and a salt-and-pepper noise with different noise densities. Fig. 4 shows some examples of the transformed and corrupted images.

These three moment invariant sets are normalized according to the method presented by Suk and Flusser [START_REF] Suk | Affine moment invariants of color images[END_REF] 1/ ( )

r p p p I sign I I = , p = 1, 2,…, n. ( 36
)
where r is the number of moments in one term and n is the size of moment invariant set. Then, the minimum-Euclideandistance is used as the classifier.

The recognition results of three different invariant descriptors for both noise-free and noisy cases are shown in Table 2. The results show that: (1) the recognition results are quite good (100%) for each method in noise-free case; (2) the recognition rates decrease with the noise level going up. However, the proposed QZMIs perform much better than other types of invariants whatever the noise and the noise level. Although some images are highly corrupted, the recognition rates of QZMIs are still better than 92%. However, this is not the case for the other two methods; (3) the worst is GZMIs. The reason is that graying loses some color information, which may be important in object recognition. b) Performance comparison of the new rotation invariants and the Zernike moment rotation invariants presented in [START_REF] Chen | Color image analysis by quaternion Zernike moments[END_REF] To compare the new rotation invariants defined in (23) (denoted by QZMRIs) and the rotation invariants presented in our previous work [START_REF] Chen | Color image analysis by quaternion Zernike moments[END_REF] (denoted by QZMRIs ICPR ), fourteen images with size 128 × 128 (Fig. 5) selected from the public Columbia Object Image Library (COIL-100) database [START_REF] Nene | Columbia Object Image Library[END_REF] were used as the training set. The actual size of all the original images is 204 × 204. Then, each image was rotated with angle varying from 0° to 350° every 5°, forming a testing set of 1008 images. Moreover, these testing images were corrupted by a white Gaussian noise and a salt-and-pepper noise, respectively. A set of 20 QZMRIs ICPR , whose order from 0 to 7, were used for object recognition. We used two sets of QZMRIs. The first one (denoted by QZMRIs 21 ) includes 21 real-valued invariants of order from 0 to 4. The second one (denoted by QZMRIs 56 ) uses the same order as that of QZMRIs ICPR , thus includes 56 real-valued invariants. The same recognition procedure used in subsection a) is adopted.

Table 3 shows the recognition rates of these methods. It can be seen that the proposed new invariant descriptors have always better performance than QZMRIs ICPR . The reason is that the new method can get more real-valued invariants than our previous method when the same order of moments is used. When the number of moment invariants used in both methods are nearly the same, the new descriptors require lower order of moments, thus they are less sensitive to noise. c) Performance comparison of the QZMIs and the existing moment invariants

In order to compare our QZMIs with the geometric/photometric invariants (GPIs) reported in [START_REF] Mindru | Moment invariants for recognition under changing viewpoint and illumination[END_REF] which are invariant to affine geometric deformations and diagonal photometric transformations invariants, the affine color moment invariants (ACMIs) proposed in [START_REF] Suk | Affine moment invariants of color images[END_REF] and QFMM invariants (QFMMIs) presented in [START_REF] Guo | Quaternion Fourier-Mellin moments for color image[END_REF], the nearly same number of invariants is used in all the methods: The 21 GPIs are the invariants described in Table 1 of Ref. [START_REF] Mindru | Moment invariants for recognition under changing viewpoint and illumination[END_REF]; The 21 ACMIs are the invariants used in [START_REF] Suk | Affine moment invariants of color images[END_REF]; The 24 QFMMIs are the invariants shown in [START_REF] Kim | Invariant image watermark using Zernike moments[END_REF] of Ref. [START_REF] Guo | Quaternion Fourier-Mellin moments for color image[END_REF] with order from 0 to 4 except the invariant of order 2 with repetition 0 which is used to achieve the scaling invariance. Our 20 QZMIs are the same as those used in subsection a).

We also used the fourteen images shown in Fig. 5 as the training set. To obtain the testing set, each image was transformed with the same transformations as those presented in subsection a) except for the translation (the actual translation is ∆x = -11, ∆y = 9), forming a set of 1176 images. Then, to test the robustness against noise, these testing images were added a white Gaussian noise and a salt-and-pepper noise, respectively. The minimum-Euclidean-distance is used as classifier after the normalization with [START_REF] Chong | A comparative analysis of algorithm for fast computation of Zernike moments[END_REF] for the four types of invariants.

The recognition results using different moment invariants are summarized in Table 4. It can be observed from this table that: (1) the invariants based on quaternion moments (QZMIs and QFMMIs) are more robust against noise than those based on conventional moments (GPIs and ACMIs); (2) the invariants based on orthogonal Zernike moments (QZMIs) are better than those based on non-orthogonal moments (QFMMIs) in noisy case. It is in accordance with that reported in [START_REF] Teh | On image analysis by the method of moments[END_REF], where the authors pointed out that the orthogonal Zernike moments have on overall better performance than non-orthogonal ones.

D. Template matching

The objective of the last experiment is to verify the performance of our descriptors for real outdoor scene images after RST transformation. The template matching test was performed. For that purpose, two pictures (Fig. 6) were taken by digital camera (Panasonic DMC-FZ50) with different focus and varying position through a rotation of the camera. Then, nine circular areas with radius r = 11 in Fig. 6(a) were extracted as templates (numbered from 1 to 9). The scale factor between two images is obtained with the automatic scale selection [START_REF] Lindeberg | Feature detection with automatic scale selection[END_REF]. Then, the scaled template was shifted across the other image (Fig. 6(b)) and in each position the normalized invariants were calculated and compared with the invariants of the original templates. For more detail about the matching procedure, we refer to [START_REF] Flusser | Degraded image analysis: an invariant approach[END_REF]. The 'matched position' we consider corresponds to the location where the Euclidean distance d(f, g) reaches its minimum value, with f representing the template of the original image and g the template of the transformed image. The matching results obtained by different moment invariants are shown in Fig. 6(c),(d),(e) and (f). It can be observed form these figures that our QZMIs can match correctly for eight templates with a reasonable error (within one pixel), but the number of correctly matched templates is only three for GPIs (number 2, 5 and 9) and for ACMIs (number 3, 4 and 9), five for QFMMIs (number 4, 6, 7, 8 and 9). These results further validate the effectiveness of the proposed QZMIs.

Conclusion

In this paper, we have extended the conventional Zernike moment defined in gray-scale image to color image using the algebra of quaternions. The moment invariants with respect to RST transformations have been constructed. The advantages of the proposed QZMs and QZMIs over the existing descriptors are as follows: (1) the proposed QZMs are based on the orthogonal Zernike polynomials, thus they are more robust to noise; (2) the quaternion-type representation, treating a color image as a vector field, is used in the definition of moments; (3) the constructed QZMIs are quaternion-valued invariants instead of real-valued invariants, thus they provide more real-valued invariants. The results obtained from both simulation and real data show that the proposed descriptors are more robust to noise and have more discriminative power than the existing methods.

m l l R R m l m m l m Z f Z f Z f Z f C diag D Z f Z f λ λ λ + + + + + + + + ′′         ′′     =             ′′     L M M . (A1)
Applying [START_REF] Guo | Quaternion Fourier-Mellin moments for color image[END_REF] to the transformed image f ′′ , it can also be written in matrix form as , , (e) σ /µ (%)

( 2, 2, ( 2) ( 4)
R R m m m m R R m m m m m m m m l m l f f f l R R m l m m l m L f Z f L f Z f C diag D L f Z f + + -+ -+ -+ + ′′ ′′ ′′ + + ′′ ′′         ′′ ′′  l R m l m L f L f C diag L f diag D C Z f Z f diag D Z λ λ λ λ λ λ -+ -+ -+ + + + -+ -+ -+ + + + + + + + ′′     ′′   = Γ Γ Γ       ′′   × × L M L L M ( 2) ( 4) ( 2 2) , , 2 2) 2 , 2 , ( ) ( ) ( ) ( ) ( , , , ) ( ) ( )
m m m l R R m m m m R R m m m m m m l f f f l R R m l m m l m f Z f L f Z f L f C diag D Z f L f -+ -+ -+ + + + 2, 2, 2 , 2 , ( ) ( ) ( ) ( ) ( ) ( , , , ) . ( ) ( )

Reconstructed images

  that the rotation invariants constructed by taking the modulus of , ( ) R n m Z f correspond to a special case of Theorem 1. Note that each invariant , ( ) number, which includes four real-valued invariants (one real part and three imaginary parts) except , ( ) m n n f φ .

  l

Corollary 1 .

 1 Let f ′ be a rotated version of f with rotation angle α. It holds for any non-negative integers n and m that , ,

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .

 12345 Fig. 3. Geometric transformed images of Fig. 2(a) (∆x is the translation (in pixel) along the xaxis, ∆y the translation along the y-axis, λ the scaling factor, θ the rotation angle)

Fig. 6 . 3 Fig

 63 Images of the outdoor scene. (a) and (b) are the original image and the transformed image, (c), (d), (e) and (f) are the matched templates using GPIs, ACMIs, QFMMIs and QZMIs, respectivelyTable 1. Modulus of QZMIs of images shown in Fig.

Table 2 .

 2 Recognition rates (%) of different Zernike moment invariants in object recognition with RST transformation and noise

	1 1,1 ϕ	1.294e-05	1.297e-05	1.287e-05	1.294e-05	1.294e-05	1.293e-05	1.292e-05	1.291e-05	0.226
	ϕ	0 2,0	5.174	5.174	5.174	5.174	5.174	5.174			5.174	5.174	0.00
	ϕ	2 2,2	1.590e-06	1.589e-06	1.592e-06	1.590e-06	1.563e-06	1.589e-06	1.594e-06	1.591e-06	0.626
	ϕ	1 3,1	5.142e-05	5.155e-05	5.114e-05	5.142e-05	5.141e-05	5.140e-05	5.134e-05	5.132e-05	0.228
	ϕ	3 3,3	9.627e-10	9.735e-10	9.545e-10	9.627e-10	9.290e-10	9.650e-10	9.619e-10	9.610e-10	1.369
	ϕ	0 4,0	8.552	8.552	8.552	8.552	8.552	8.552			8.552	8.552	0.00
	ϕ	2 4,2	7.884e-06	7.875e-06	7.894e-06	7.884e-06	7.747e-06	7.877e-06	7.903e-06	7.886e-06	0.635
	ϕ	4 4,4	7.513e-11	7.524e-11	7.540e-11	7.513e-11	7.569e-11	7.535e-11	7.524e-11	7.504e-11	0.272
									GZMIs	DZMIs	QZMIs
						Noise-free			100.00	100.00	100.00
					Gaussian noise with STD = 1

  

Acknowledgement

The authors would like to thank the anonymous reviewers and Associate Editor for their insightful suggestions which helped improve the manuscript. This work was supported by the National Basic Research Program of China under Grant 2011CB707904, the NSFC under Grants 61073138, 60911130370, and 60972163, the Ningbo Natural Science Foundation under Grants 2010A610106 and 2009A610090, and the Natural Science Foundation of Jiangsu Province under Grants SBK200910055 and BK2010426.

Appendix A

Proof of Theorem 2. Equation ( 31) can be written in matrix form as

Based on the definition of Γ f , it can be easily verified that .

Substituting (A1) and (A3) into (A2), and using the identity , m m l l D C I = where I is the identity matrix, we obtain


Bernt Øksendal
email: oksendal@math.uio.no.

Agnès Sulem
email: agnes.sulem@inria.fr

Tusheng Zhang
email: tusheng.zhang@manchester.ac.uk

Singular control of SPDEs and backward SPDEs with reflection

Keywords: Stochastic partial differential equations (SPDEs), singular control of SPDEs, maximum principles, comparison theorem for SPDEs, reflected SPDEs, optimal stopping of SPDEs

In the first part, we consider general singular control problems for random fields given by a stochastic partial differential equation (SPDE). We show that under some conditions the optimal singular control can be identified with the solution of a coupled system of SPDE and a kind of reflected backward SPDE (RB-SPDE).In the second part, existence and uniqueness of solutions of RBSPDEs are established, which is of independent interest.

Mots-clés :

Équations aux Dérivés Partielles Stochastiques (EDPS), contrôle singulier, arrêt optimal 1 Introduction Let B t , t ≥ 0 be an m-dimensional Brownian motion on a filtered probability space (Ω, F , F t , P ). Let D be a bounded smooth domain in R d . Fix T > 0 and let φ(ω, x) be an

F T -measurable H = L 2 (D)-valued random variable. Let k : [0, T ] × D × R × R m → R
be a given measurable mapping and L(t, x) : [0, T ] × D → R a given continuous function. Consider the problem to find F t -adapted random fields u(t, x) ∈ R, Z(t, x) ∈ R m , η(t, x) ∈ R + left-continuous and increasing w.r.t. t, such that du(t, x) = -Au(t, x)dtk(t, x, u(t, x), Z(t, x))dt + Z(t, x)dB t , t ∈ (0, T ) -η(dt, x), t ∈ (0, T ), (1.1)

u(t, x) ≥ L(t, x), T 0 D (u(t, x) -L(t, x))η(dt, x) = 0, u(T, x) = φ(x) a.s, (1.2)
where A is a second order linear partial differential operator. This is a backward stochastic partial differential equation (BSPDE) with reflection.

The maximum principle method for solving a stochastic control problem for stochastic partial differential equations involves a BSPDE for the adjoint processes p(t, x), q(t, x).See [ØPZ].

The purpose of this paper is twofold: (i) We study a class of singular control problems for SPDEs and prove a maximum principle for the solution of such problems. This maximum principle leads to a kind of reflected backward stochastic partial differential equations. (ii) We study backward stochastic partial differential equations (BSPDEs) with reflection. This means that we solve the BSPDE with the constraint that the solution must stay in a pre-described region.

Singular control of SPDEs

Suppose the state equation is an SPDE of the form dY (t, x) = {AY (t, x) + b(t, x, Y (t, x))}dt + σ(t, x, Y (t, x))dB(t)

+ λ(t, x, Y (t, x))ξ(dt, x) ; (t, x) ∈ [0, T ] × D (2.1)

RR n°7791

Y (0, x) = y 0 (x) ; x ∈ D Y (t, x) = y 1 (t, x) ; (t, x) ∈ (0, T ) × ∂D.

(2.2)

Here A is a given linear second order partial differential operator.

The performance functional is given by

J(ξ) = E D T 0 f (t, x, Y (t, x))dtdx + D g(x, Y (T, x))dx + D T 0 h(t, x, Y (t, x))ξ(dt, x) , (2.3)
where f (t, x, y), g(x, y) and h(t, x, y) are bounded measurable functions which are differentiable in the argument y and continuous w.r.t. t.

We want to maximize J(ξ) over all ξ ∈ A, where A is a given family of adapted processes ξ(t, x), which are non-decreasing and left-continuous w.r.t. t for all x, ξ(0, x) = 0. We call A the set of admissible singular controls. Thus we want to find ξ * ∈ A (called an optimal control) such that

sup ξ∈A J(ξ) = J(ξ * )
Define the Hamiltonian H by H(t, x, y, p, q)(dt, ξ(dt, x)) = {f (t, x, y) + b(t, x, y)p + σ(t, x, y)q}dt + {λ(t, x, y)p + h(t, x, y)}ξ(dt, x).

(2.4)

To this Hamiltonian we associate the following backward SPDE (BSPDE) in the unknown process (p(t, x), q(t, x)):

dp(t, x) = -A * p(t, x)dt + ∂H ∂y (t, x, Y (t, x), p(t, x), q(t, x))(dt, ξ(dt, x)) + q(t, x)dB(t) ; (t, x) ∈ (0, T ) × D (2.5) with boundary/terminal values p(T, x) = ∂g ∂y (x, Y (T, x)) ; x ∈ D (2.6) p(t, x) = 0 ; (t, x) ∈ (0, T ) × ∂D.
(2.7)

Here A * denotes the adjoint of the operator A.

RR n°7791

Theorem 2.1 (Sufficient maximum principle for singular control of SPDE) Let ξ ∈ A with corresponding solutions Ŷ (t, x), p(t, x), q(t, x). Assume that

y → h(x, y) is concave (2.8) and (y, ξ) → H(t, x, y, p(t, x), q(t, x))(dt, ξ(dt, x)) is concave.
(2.9)

Assume that

E[ D ( T 0 {(Y ξ (t, x) -Ŷ (t, x)) 2 q2 (t, x) + p2 (t, x)(σ(t, x, Y ξ (t, x)) -σ(t, x, Ŷ (t, x)) 2 } dt)dx] < ∞, for all ξ ∈ A.
(2.10)

Moreover, assume that the following maximum condition holds:

ξ(dt, x) ∈ argmax ξ∈A H(t, x, Ŷ (t, x), p(t, x), q(t, x))(dt, ξ(dt, x)), (2.11)
i.e. {λ(t, x, Ŷ (t, x))p(t, x) + h(t, x, Ŷ (t, x))}ξ(dt, x)

≤ {λ(t, x, Ŷ (t, x))p(t, x) + h(t, x, Ŷ (t, x))} ξ(dt, x) for all ξ ∈ A. (2.12)

Then ξ is an optimal singular control.

Proof of Theorem 2.1 Choose ξ ∈ A and put Y = Y ξ . Then by (2.3) we can write

J(ξ) -J( ξ) = I 1 + I 2 + I 3 , (2.13)
where

I 1 = E T 0 D f (t, x, Y (t, x)) -f (t, x, Ŷ (t, x)) dxdt (2.14) I 2 = E D g(x, Y (T, x)) -g(x, Ŷ (T, x)) dx
(2.15)

I 3 = E T 0 D h(t, x, Y (t, x))ξ(dt, x) -h(t, x, Ŷ (t, x)) ξ(dt, x) . (2.16) RR n°7791
By our definition of H we have

I 1 = E T 0 D {H(t, x, Y (t, x), p(t, x), q(t, x))(dt, ξ(dt, x)) -H(t, x, Ŷ (t, x), p(t, x), q(t, x))(dt, ξ(dt, x)) - T 0 D {b(t, x, Y (t, x)) -b(t, x, Ŷ (t, x)}p(t, x)dxdt - T 0 D {σ(t, x, Y (t, x)) -σ(t, x, Ŷ (t, x))}q(t, x)dxdt - T 0 D p(t, x){λ(t, x, Y (t, x))ξ(dt, x) -λ(t, x, Ŷ (t, x)) ξ(dt, x)}dx - T 0 D {h(t, x, Y (t, x))ξ(dt, x) -h(t, x, Ŷ (t, x)) ξ(dt, x)}dx .
(2.17) By (2.10) and concavity of g we have, with Ỹ = Y -Ŷ ,

I 2 ≤ E D ∂g ∂y (x, Ŷ (T, x))(Y (T, x) -Ŷ (T, x))dx = E D p(T, x) Ỹ (T, x)dx = E D T 0 Ỹ (t, x)dp(t, x)dx + D T 0 p(t, x)d Ỹ (t, x)dx + D T 0 {σ(t, x, Y (t, x)) -σ(t, x, Ŷ (t, x))}q(t, x)dtdx = E D T 0 Ỹ (t, x) -A * p(t, x)dt - ∂H ∂y (t, x, Ŷ , p, q)(dt, ξ(dt, x)) dx + D T 0 p(t, x){A Ỹ (t, x) + b(t, x, Y (t, x)) -b(t, x, Ŷ (t, x))}dtdx + D T 0 p(t, x){λ(t, x, Y (t, x))ξ(dt, x) -λ(t, x, Ŷ (t, x)) ξ(dt, x)}dx + D T 0 {σ(t, x, Y (t, x)) -σ(t, x, Ŷ (t, x))}q(t, x)dtdx . (2.18)
Using integration by parts we get, since Ỹ (t, x) = p(t, x) = 0 for all

(t, x) ∈ (0, T ) × ∂D, D Ỹ (t, x)A * p(t, x)dx = D p(t, x)A Ỹ (t, x)dx.
(2.19)

Hence, combining (2.13)-(2.19) and concavity of H,

J(ξ) -J( ξ) ≤ E D T 0 {H(t, x, Y (t, x), p(t, x), q(t, x))(dt, ξ(dt, x)) -H(t, x, Ŷ (t, x), p(t, x), q(t, x))(dt, ξ(dt, x)) -Ỹ (t, x) ∂H ∂y (t, x, Ŷ , p, q)(dt, ξ(dt, x)) dx ≤ D T 0 ∇ ξ H(t, x, Ŷ (t, x), p(t, x), q(t, x))(ξ(dt, x) -ξ(dt, x))dx = E D T 0 {λ(t, x, Ŷ (t, x))p(t, x) + h(t, x, Ŷ (t, x))}(ξ(dt, x) -ξ(dt, x))dx ≤ 0 by (2.12).
This proves that ξ is optimal.

For ξ ∈ A we let V(ξ) denote the set of adapted processes ζ(t, x) of finite variation w.r.t. t such that there exists δ = δ(ξ) > 0 such that ξ + yζ ∈ A for all y ∈ [0, δ].

Proceeding as in [ØS] we prove the following useful result:

Lemma 2.2 The inequality (2.12) is equivalent to the following two variational inequalities:

λ(t,
{λ(s, x, Ŷ (s, x))p(s, x) + h(s, x, Ŷ (s, x))}ζ(ds, x) ≤ 0; (s, x) ∈ (0, T ) × D (2.22) for all ζ ∈ V( ξ).
In particular, this holds if we fix t ∈ (0, T ) and put

ζ(ds, x) = a(ω)δ t (ds)φ(x); (s, x, ω) ∈ (0, T ) × D × Ω,
where a(ω) ≥ 0 is F t -measurable and bounded, φ(x) ≥ 0 is bounded, deterministic and δ t (ds) denotes the Dirac measure at t. (ii). Conversely, suppose (2.20) and (2.21) hold. Since ξ(dt, x) ≥ 0 for all ξ ∈ A we see that (2.12) follows.

We may formulate what we have proved as follows:

Theorem 2.3 ( Sufficient maximum principle II) Suppose the conditions of Theorem 2.1 hold. Suppose ξ ∈ A, and that ξ together with its corresponding processes Y ξ (t, x), p ξ (t, x), q ξ (t, x) solve the coupled SPDE-RBSPDE system consisting of the SPDE (2.1)-(2.2) together with the reflected backward SPDE (RBSPDE) given by

dp ξ (t, x) = -A * p ξ (t, x) + ∂f ∂y (t, x, Y ξ (t, x)) + ∂b ∂y (t, x, Y ξ (t, x))p ξ (t, x) + ∂σ ∂y (t, x, Y ξ (t, x))q ξ (t, x) dt - ∂λ ∂y (t, x, Y ξ (t, x))p ξ (t, x) + ∂h ∂y (t, x, Y ξ (t, x)) ξ(dt, x) ; (t, x) ∈ [0, T ] × D λ(t, x, Y ξ (t, x))p ξ (t, x) + h(t, x, Y ξ (t, x)) ≤ 0 ; for all t, x, a.s. {λ(t, x, Y ξ (t, x))p ξ (t, x) + h(t, x, Y ξ (t, x))}ξ(dt, x) = 0 ; for all t, x, a.s. p(T, x) = ∂g ∂y (x, Y ξ (T, x)) ; x ∈ D p(t, x) = 0 ; (t, x) ∈ (0, T ) × ∂D.
Then ξ maximizes the performance functional J(ξ).

The concavity conditions of Theorem 2.1 are not always satisfied in applications, and it is of interest to have a maximum principle which does not need these assumptions. Moreover, it is useful to have a version which is of so called "necessary type". To this end, we first prove some auxiliary results:

RR n°7791 Lemma 2.4 Let ξ(dt, x) ∈ A and choose ζ(dt, x) ∈ V(ξ). Define the deriva- tive process Y(t, x) = lim y→0 + 1 y (Y ξ+yζ (t, x) -Y ξ (t, x)).
(2.26)

Then Y satisfies the SPDE dY(t, x) = AY(t, x)dt + Y(t, x)[ ∂b ∂y (t, x, Y (t, x))dt + ∂σ ∂y (t, x, Y (t, x))dB(t) + ∂λ ∂y (t, x, Y (t, x))ξ(dt, x)] + λ(t, x, Y (t, x))ζ(dt, x) ; (t, x) ∈ [0, T ] × D Y(t, x) = 0; (t, x) ∈ (0, T ) × ∂D Y(0, x) = 0 ; x ∈ D (2.27)
Proof. This follows from the equation (2.1)-( 2.2) for Y (t, x). We omit the details.

Lemma 2.5 Let ξ(dt, x) ∈ A and ζ(dt, x) ∈ V(ξ). Put η = ξ + yζ; y ∈ [0, δ(ξ)]. Assume that E[ D ( T 0 {(Y η (t, x) -Y ξ (t, x)) 2 q 2 (t, x) + p 2 (t, x)(σ(t, x, Y η (t, x)) -σ(t, x, Y ξ (t, x))) 2 }dt)dx] < ∞ for all y ∈ [0, δ(ξ)],
(2.28)

where (p(t, x), q(t, x)) is the solution of (2.5)-(2.7) corresponding to Y ξ (t, x). Then

lim y→0 + 1 y (J(ξ + yζ) -J(ξ)) = E[ D ( T 0 {λ(t, x, Y (t, x))p(t, x) + h(t, x, Y (t, x))}ζ(dt, x))dx].
(2.29)

Proof. By (2.3) and (2.26), we have

lim y→0 + 1 y (J(ξ + yζ) -J(ξ)) = E[ D { T 0 ∂f ∂y (t, x, Y (t, x))Y(t, x)dt + ∂g ∂y (x, Y (T, x))Y(T, x)}dx + D T 0 ∂h ∂y (t, x, Y (t, x))Y(t, x)ξ(dt, x)dx + D T 0 h(t, x, Y (t, x))ζ(dt, x)dx].
(2.30)

RR n°7791

By (2.4) and (2.27) we obtain

E[ D T 0 ∂f ∂y (t, x, Y (t, x))Y(t, x)dtdx] = E[ D ( T 0 Y(t, x){ ∂H ∂y (dt, ξ(dt, x)) -p(t, x) ∂b ∂y (t, x)dt -q(t, x) ∂σ ∂y (t, x)dt -(p(t, x) ∂λ ∂y (t, x) + ∂h ∂y (t, x))ξ(dt, x)})dx, (2.31)
where we have used the abbreviated notation

∂H ∂y (dt, ξ(dt, x)) = ∂H ∂y (t, x, Y (t, x), p(t, x), q(t, x))(dt, ξ(dt, x))
etc.

By the Itô formula and (2.5), (2.28) we see that

E[ D ∂g ∂y (x)Y(T, x)dx] = E[ D p(T, x)Y(T, x)dx] = E[ D ( T 0 {p(t, x)dY(t, x) + Y(t, x)dp(t, x)} + [p(•, x), Y(•, x)](T ))dx] = E[ D ( T 0 [p(t, x){AY(t, x)dt + Y(t, x) ∂b ∂y (t, x)dt + Y(t, x) ∂λ ∂y (t, x)ξ(dt, x) + λ(t, x)ζ(dt, x)} + Y(t, x){-A * p(t, x)dt - ∂H ∂y (dt, ξ(dt, x))} + Y(t, x) ∂σ ∂y (t, x)q(t, x)]dt)dx], (2.32)
where

[p(•, x), Y(•, x)](t) denotes the covariation process of p(•, x) and Y(•, x). Since p(t, x) = Y(t, x) = 0 for x ∈ ∂D, we deduce that D p(t, x)AY(t, x)dx = D A * p(t, x)Y(t, x)dx.
(2.33) Therefore, substituting (2.31) and (2.32) into (2.30), we get

lim y→0 + 1 y (J(ξ + yζ) -J(ξ)) = E[ D ( T 0 {λ(t, x)p(t, x) + h(t, s)}ζ(dt, x))dx].

RR n°7791

We can now state our necessary maximum principle:

Theorem 2.6 [Necessary maximum principle] (i) Suppose ξ * ∈ A is optimal, i.e. max ξ∈A J(ξ) = J(ξ * ).
(2.34)

Let Y * , (p * , q * ) be the corresponding solution of (2.1)-(2.2) and (2.5)-(2.7), respectively, and assume that (2.28) holds with ξ = ξ * . Then

λ(t, x, Y * (t, x))p * (t, x) + h(t, x, Y * (t, x)) ≤ 0 for all t, x ∈ [0, T ] × D, a.s. (2.35) and {λ(t, x, Y * (t, x))p * (t, x)+h(t, x, Y * (t, x))}ξ * (dt, x) = 0 for all t, x ∈ [0, T ]×D, a.s.
(2.36) (ii) Conversely, suppose that there exists ξ ∈ A such that the corresponding solutions Ŷ (t, x), (p(t, x), q(t, x)) of (2.1)-(2.2) and (2.5)-(2.7), respectively, satisfy λ(t, x, Ŷ (t, x))p(t, x) + h(t, x, Ŷ (t, x)) ≤ 0 for all t, x ∈ [0, T ] × D, a.s.

(2.37) and {λ(t, x, Ŷ (t, x))p(t, x)+h(t, x, Ŷ (t, x))} ξ(dt, x) = 0 for all t, x ∈ [0, T ]×D, a.s.

(2.38) Then ξ is a directional sub-stationary point for J(•), in the sense that

lim y→0 + 1 y (J( ξ + yζ) -J( ξ)) ≤ 0 for all ζ ∈ V( ξ). (2.39)
Proof. This is proved in a similar way as in Theorem 2.4 in [ØS]. For completeness we give the details:

(i) If ξ ∈ A is optimal, we get by Lemma 2.5

0 ≥ lim y→0 + 1 y (J(ξ + yζ) -J(ξ)) = E[ D T 0 {λ(t, x)p(t, x) + h(t, x)}ζ(dt, x)dx], for all ζ ∈ V(ξ). (2.40)
In particular, this holds if we choose ζ such that

ζ(ds, x) = a(ω)δ t (s)φ(x) (2.41) RR n°7791
for some fixed t ∈ [0, T ] and some bounded F t -measurable random variable a(ω) ≥ 0 and some bounded, deterministic φ(x) ≥ 0, where δ t (s) is Dirac measure at t. Then (2.40) gets the form

E[ D {λ(t, x)p(t, x) + h(t, x)}a(ω)φ(x)dx] ≤ 0.
Since this holds for all such a(ω), φ(x) we deduce that λ(t, x)p(t, x) + h(t, x) ≤ 0 for all t, x, a.s.

(2.42)

Next, if we choose ζ(dt, x) = ξ(dt, x) ∈ V(ξ), we get from (2.40) E[ D T 0 {λ(t, x)p(t, x) + h(t, x)}ξ(dt, x)dx] ≤ 0. (2.43)
On the other hand, we can also choose ζ(dt, x) = -ξ(dt, x) ∈ V(ξ), and this gives

E[ D T 0 {λ(t, x)p(t, x) + h(t, x)}ξ(dt, x)dx] ≥ 0.
( as claimed. This proves (i).

(ii) Conversely, suppose ξ ∈ A is as in (ii). Then (2.39) follows from Lemma 2.5.

Existence and Uniqueness

In this section, we will prove the main existence and uniqueness result for reflected backward stochastic partial differential equations. For notational RR n°7791 simplicity, we choose the operator A to be the Laplacian operator ∆. However, our methods work equally well for general second order differential operators like

A = 1 2 d i,j=1 ∂ ∂x i (a ij (x) ∂ ∂x j ),
where a = (a ij (x)) :

D → R d×d (d > 2
) is a measurable, symmetric matrixvalued function which satisfies the uniform elliptic condition

λ|z| 2 ≤ d i,j=1 a ij (x)z i z j ≤ Λ|z| 2 , ∀ z ∈ R d and x ∈ D
for some constant λ, Λ > 0 First we will establish a comparison theorem for BSPDEs, which is of independent interest. Consider two backward SPDEs:

du 1 (t, x) = -∆u 1 (t)dt -b 1 (t, u 1 (t, x), Z 1 (t, x))dt + Z 1 (t, x)dB t , t ∈ (0, T ) u 1 (T, x) = φ 1 (x) a.s.
(3.1)

du 2 (t, x) = -∆u 2 (t)dt -b 2 (t, u 2 (t, x), Z 2 (t, x))dt + Z 2 (t, x)dB t , t ∈ (0, T ) u 2 (T, x) = φ 2 (x) a.s. (3.2)
From now on, if u(t, x) is a function of (t, x), we write u(t) for the function u(t, •).

The following result is a comparison theorem for backward stochastic partial differential equations.

Theorem 3.1 (Comparison theorem for BSPDEs) Suppose φ 1 (x) ≤ φ 2 (x) and b 1 (t, u, z) ≤ b 2 (t, u, z). Then we have u 1 (t, x) ≤ u 2 (t, x), x ∈ D, a.e. for every t ∈ [0, T ].

Proof. For n ≥ 1, define functions ψ n (z), f n (x) as follows (see [START_REF] Donai-Martin | White noise driven SPDEs with reflection[END_REF]).

ψ n (z) =    0 if z ≤ 0, 2nz if 0 ≤ z ≤ 1 n , 2 if z > 1 n .
(3.3)

f n (x) = 0 if x ≤ 0, x 0 dy y 0 ψ n (z)dz if x > 0.
(3.4)

We have

f ′ n (x) =    0 if x ≤ 0, nx 2 if x ≤ 1 n , 2x -1 n if x > 1 n .
(3.5)

RR n°7791 Also f n (x) ↑ (x + ) 2 as n → ∞. For h ∈ K := L 2 (D), set F n (h) = D f n (h(x))dx.
F n has the following derivatives for h 1 , h 2 ∈ K,

F ′ n (h)(h 1 ) = D f ′ n (h(x))h 1 (x)dx, (3.6)
F ′′ n (h)(h 1 , h 2 ) = D f ′′ n (h(x))h 1 (x)h 2 (x)dx. (3.7)
Applying Ito's formula we get

F n (u 1 (t) -u 2 (t)) = F n (φ 1 -φ 2 ) + T t F ′ n (u 1 (s) -u 2 (s))(∆(u 1 (s) -u 2 (s)))ds + T t F ′ n (u 1 (s) -u 2 (s))(b 1 (s, u 1 (s), Z 1 (s)) -b 2 (s, u 2 (s), Z 2 (s)))ds - T t F ′ n (u 1 (s) -u 2 (s))(Z 1 (s) -Z 2 (s))dB s - 1 2 T t F ′′ n (u 1 (s) -u 2 (s))(Z 1 (s) -Z 2 (s), Z 1 (s) -Z 2 (s))ds =: I 1 n + I 2 n + I 3 n + I 4 n + I 5 n , (3.8)
where,

I 2 n = T t F ′ n (u 1 (s) -u 2 (s))(∆(u 1 (s) -u 2 (s)))ds = T t D f ′ n (u 1 (s, x) -u 2 (s, x))(∆(u 1 (s, x) -u 2 (s, x)))dxds = - T t f ′′ n (u 1 (s, x) -u 2 (s, x))|∇(u 1 (s, x) -u 2 (s, x)| 2 dxds ≤ 0,(3.9) I 5 n = -n T t D χ {0≤u 1 (s,x)-u 2 (s,x)≤ 1 n } (u 1 (s, x) -u 2 (s, x))|Z 1 (s, x) -Z 2 (s, x)| 2 dxds - T t D χ {u 1 (s,x)-u 2 (s,x)> 1 n } |Z 1 (s, x) -Z 2 (s, x)| 2 dxds.
(3.10)

RR n°7791

For I 3 n , we have

I 3 n = T t D f ′ n (u 1 (s, x) -u 2 (s, x))(b 1 (s, u 1 (s, x), Z 1 (s, x)) -b 2 (s, u 2 (s, x), Z 2 (s, x)))dxds = T t D f ′ n (u 1 (s, x) -u 2 (s, x))(b 1 (s, u 1 (s, x), Z 1 (s, x)) -b 2 (s, u 1 (s, x), Z 1 (s, x)))dxds + T t D f ′ n (u 1 (s, x) -u 2 (s, x))(b 2 (s, u 1 (s, x), Z 1 (s, x)) -b 2 (s, u 2 (s, x), Z 1 (s, x)))dxds + T t D f ′ n (u 1 (s, x) -u 2 (s, x))(b 2 (s, u 2 (s, x), Z 1 (s, x)) -b 2 (s, u 2 (s, x), Z 2 (s, x)))dxds ≤ T t D f ′ n (u 1 (s, x) -u 2 (s, x))(b 2 (s, u 2 (s, x), Z 1 (s, x)) -b 2 (s, u 2 (s, x), Z 2 (s, x)))dxds +C T t D ((u 1 (s, x) -u 2 (s, x) + ) 2 dxds := I 3 n,1 + I 3 n,2 , (3.11)
where the Lipschiz condition of b and the assumption b 1 ≤ b 2 have been used.

RR n°7791

I 3 n,1 can be estimated as follows:

I 3 n,1 ≤ C T t D f ′ n (u 1 (s, x) -u 2 (s, x))|Z 1 (s, x) -Z 2 (s, x)|dxds = C T t D χ {0≤u 1 (s,x)-u 2 (s,x)≤ 1 n } n(u 1 (s, x) -u 2 (s, x)) 2 |Z 1 (s, x) -Z 2 (s, x)|dxds +C T t D χ {u 1 (s,x)-u 2 (s,x)> 1 n } [2(u 1 (s, x) -u 2 (s, x)) - 1 n ]|Z 1 (s, x) -Z 2 (s, x)|dxds ≤ C T t D χ {u 1 (s,x)-u 2 (s,x)> 1 n } (2(u 1 (s, x) -u 2 (s, x)) - 1 n ) 2 dxds + T t D χ {u 1 (s,x)-u 2 (s,x)> 1 n } |Z 1 (s, x) -Z 2 (s, x)| 2 dxds + 1 4 C 2 T t D χ {0≤u 1 (s,x)-u 2 (s,x)≤ 1 n } n(u 1 (s, x) -u 2 (s, x)) 3 dxds + T t D χ {0≤u 1 (s,x)-u 2 (s,x)≤ 1 n } n(u 1 (s, x) -u 2 (s, x)) 2 |Z 1 (s, x) -Z 2 (s, x)| 2 dxds ≤ C ′ T t D ((u 1 (s, x) -u 2 (s, x)) + ) 2 dxds + T t D χ {u 1 (s,x)-u 2 (s,x)> 1 n } |Z 1 (s, x) -Z 2 (s, x)| 2 dxds + T t D χ {0≤u 1 (s,x)-u 2 (s,x)≤ 1 n } n(u 1 (s, x) -u 2 (s, x)) 2 |Z 1 (s, x) -Z 2 (s, x)| 2 dxds (3.12)
(3.10),(3.11) and (3.12) imply that

I 3 n + I 5 n ≤ C T t D ((u 1 (s, x) -u 2 (s, x)) + ) 2 dxds (3.13)
Thus it follows from (3.8), (3.9) and (3.13) that

F n (u 1 (t) -u 2 (t)) ≤ F n (φ 1 -φ 2 ) + C T t D ((u 1 (s, x) -u 2 (s, x)) + ) 2 dxds - T t F ′ n (u 1 (s) -u 2 (s))(Z 1 (s) -Z 2 (s))dB s (3.14)
Take expectation and let n → ∞ to get

E[ D ((u 1 (t, x) -u 2 (t, x)) + ) 2 dx] ≤ T t dsE[ D ((u 1 (s, x) -u 2 (s, x)) + ) 2 dx] (3.15) RR n°7791
Gronwall's inequality yields that

E[ D ((u 1 (t, x) -u 2 (t, x)) + ) 2 dx] = 0, (3.16)
which completes the proof of the theorem.

Remark. After this paper was written we became aware of the paper [MYZ], where a similar comparison theorem is proved. However, the theorems are not identical and the proofs are quite different.

We now proceed to prove existence and uniqueness of the reflected BSPDEs. Let V = W 1,2 0 (D) be the Sobolev space of order one with the usual norm || • ||. Consider the reflected backward stochastic partial differential equation:

du(t) = -∆u(t)dt -b(t, u(t, x), Z(t, x))dt + Z(t, x)dB t , t ∈ (0, T ) (3.17) -η(dt, x), t ∈ (0, T ), (3.18) u(t, x) ≥ L(t, x), T 0 D (u(t, x) -L(t, x))η(dt, x)dx = 0, u(T, x) = φ(x) a.s. (3.19) Theorem 3.2 Assume that E[|φ| 2 K ] < ∞. and that |b(s, u 1 , z 1 ) -b(s, u 1 , z 1 )| ≤ C(|u 1 -u 2 | + |z 1 -z 2 |).
Let L(t, x) be a measurable function which is differentiable in t and twice differentiable in x such that

T 0 D L ′ (t, x) 2 dxdt < ∞, T 0 D |∆L(t, x)| 2 dxdt < ∞.
Then there exists a unique

K × L 2 (D, R m ) × K-valued progressively measur- able process (u(t, x), Z(t, x), η(t, x)) such that (i) E[ T 0 ||u(t)|| 2 V dt] < ∞, E[ T 0 |Z(t)| 2 L 2 (D,R m ) dt] < ∞.
(ii) η is a K-valued continuous process, non-negative and nondecreasing in t and η(0, x) = 0.

(iii) u(t, x) = φ(x) + T t ∆u(t, x)ds + T t b(s, u(s, x), Z(s, x))ds - T t Z(s, x)dB s +η(T, x) -η(t, x); 0 ≤ t ≤ T, (iv) u(t, x) ≥ L(t, x) a.e. x ∈ D, ∀t ∈ [0, T ]. (v) T 0 D (u(t, x) -L(t, x))η(dt, x)dx = 0 (3.20) RR n°7791
where u(t) stands for the K-valued continuous process u(t, •) and (iii) is understood as an equation in the dual space V * of V .

For the proof of the theorem, we introduce the penalized BSPDEs:

du n (t) = -∆u n (t)dt -b(t, u n (t, x), Z n (t, x))dt + Z n (t, x)dB t -n(u n (t, x) -L(t, x)) -dt, t ∈ (0, T ) (3.21) u n (T, x) = φ(x) a.s. (3.22)
According to [ØPZ], the solution (u n , Z n ) of the above equation exists and is unique. We are going to show that the sequence (u n , Z n ) has a limit, which will be a solution of the equation (3.20). First we need some a priori estimates.

Lemma 3.3 Let (u n , Z n ) be the solution of equation (3.21). We have

sup n E[sup t |u n (t)| 2 K ] < ∞, (3.23) sup n E[ T 0 ||u n (t)|| 2 V ] < ∞, (3.24) sup n E[ T 0 |Z n (t)| 2 L 2 (D,R m ) ] < ∞. (3.25) Proof. Take a function f (t, x) ∈ C 2,2 0 ([-1, T +1]×D) satisfying f (t, x) ≥ L(t, x). Applying Itô's formula, it follows that |u n (t) -f (t)| 2 K = |φ -f (T )| 2 K + 2 T t < u n (s) -f (s), ∆u n (s) > ds +2 T t < u n (s) -f (s), b(s, u n (s), Z n (s)) > ds -2 T t < u n (s) -f (s), Z n (s) > dB s +2n T t < u n (s) -f (s), (u n (s) -L(s)) -> ds - T t |Z n (s)| 2 L 2 (D,R m ) ds +2 T t < u n (s) -f (s), f ′ (s) > ds, a.s. (3.26)
where <, > denotes the inner product in K. Now we estimate each of the terms on the right hand side.

2 T t < u n (s) -f (s), ∆u n (s) > ds = -2 T t ||u n (s)|| 2 V ds + 2 T t < ∂f (s) ∂x , ∂u n (s) ∂x > ds ≤ - T t ||u n (s)|| 2 V ds + T t ||f (s)|| 2 V ds (3.27) 2 T t < u n (s) -f (s), b(s, u n (s), Z n (s)) > ds = 2 T t < u n (s) -f (s), b(s, u n (s), Z n (s)) -b(s, f (s), Z n (s)) > ds +2 T t < u n (s) -f (s), b(s, f (s), Z n (s)) -b(s, f (s), 0) > ds +2 T t < u n (s) -f (s), b(s, f (s), 0) > ds ≤ C T t |u n (s) -f (s)| 2 H ds + 1 2 T t |Z n (s)| 2 L 2 (D,R m ) ds +C T t |b(s, f (s), 0)| 2 H ds (3.28) 2n T t < u n (s) -f (s), (u n (s) -L(s)) -> ds = 2n T t D (u n (s, x) -f (s, x))χ {u n (s,x)≤L(s,x)} (L(s, x) -u n (s, x))dsdx ≤ 0 (3.29)
Substituting (3.27),(3.28) and (3.29) into (3.26) we obtain

|u n (t) -f (t)| 2 K + T t ||u n (s)|| 2 V ds + 1 2 T t |Z n (s)| 2 L 2 (D,R m ) ds ≤ |φ -f (T )| 2 K + C T t |u n (s) -f (s)| 2 K ds + C T t |b(s, f (s), 0)| 2 K ds + T t ||f (s)|| 2 V ds -2 T t < u n (s) -f (s), Z n (s) > dB s (3.30)
Take expectation and use the Gronwall inequality to obtain

sup n sup t E[|u n (t)| 2 K ] < ∞ (3.31) RR n°7791 sup n E[ T 0 ||u n (t)|| 2 V ] < ∞ (3.32) sup n E[ T 0 |Z n (t)| 2 L 2 (D,R m ) dt] < ∞ (3.33)
By virtue of (3.33), (3.31) can be further strengthened to (3.23). Indeed, by Burkholder inequality,

E 2 sup v≤t≤T | T t < u n (s) -f (s), Z n (s) > dB s | ≤ CE ( T v |u n (s) -f (s)| 2 K |Z n (s)| 2 L 2 (D,R m ) ds) 1 2 ≤ CE sup v≤s≤T (|u n (s) -f (s)| K )( T v |Z n (s)| 2 L 2 (D,R m ) ds) 1 2 ≤ 1 2 E sup v≤s≤T (|u n (s) -f (s)| 2 K ) + CE T v |Z n (s)| 2 L 2 (D,R m ) ds (3.34)
With (3.34), taking superum over t ∈ [v, T ] on both sides of (3.26) we obtain (3.23).

We need the following estimates.

Lemma 3.4 Suppose the conditions in Theorem 3.2 hold. Then there is a constant C such that

E[ T 0 D ((u n (t, x) -L(t, x)) -) 2 dxdt] ≤ C n 2 . (3.35)
Proof. Let f m be defined as in the proof of Theorem 3.1. Then

f m (x) ↑ (x + ) 2 and f ′ m (x) ↑ 2x + as m → ∞. For h ∈ K, set G m (h) = D f m (-h(x))dx.
It is easy to see that for

h 1 , h 2 ∈ K, G ′ m (h)(h 1 ) = - D f ′ m (-h(x))h 1 (x)dx, (3.36)
G ′′ n (h)(h 1 , h 2 ) = D f ′′ m (-h(x))h 1 (x)h 2 (x)dx.
(3.37)

RR n°7791

Applying Itô's formula we get

G m (u n (t) -L(t)) = G m (φ -L(T )) + T t G ′ m (u n (s) -L(s))(∆u n (s)))ds + T t G ′ m (u n (s) -L(s))(b(s, u n (s), Z n (s)))ds +n T t G ′ m (u n (s) -L(s))((u n (s) -L(s)) -)ds + T t G ′ m (u n (s) -L(s))(L ′ (s))ds - T t G ′ m (u n (s) -L(s))(Z n (s))dB s - 1 2 T t G ′′ m (Z n (s), Z n (s))ds =: I 1 m + I 2 m + I 3 m + I 4 m + I 5 m + I 6 m + I 7 m . (3.38)
Now,

I 2 m = T t G ′ m (u n (s) -L(s))(∆u n (s)))ds = - T t D f ′ m (L(s, x) -u n (s, x))(∆(u n (s, x) -L(s, x)))dxds - T t D f ′ m (L(s, x) -u n (s, x))(∆L(s, x))dxds ≤ - T t D f ′′ m (L(s, x) -u n (s, x))|∇(u n (s, x) -L(s, x))| 2 dxds + 1 4 n T t D f ′ m (L(s, x) -u n (s, x)) 2 xds + C n T t D (∆L(s, x)) 2 dxds, (3.39)
I 3 m = - T t D f ′ m (L(s, x) -u n (s, x))b(s, u n (s, x), Z n (s, x))dxds ≤ 1 4 n T t D f ′ m (L(s, x) -u n (s, x)) 2 ds + C n T t D (b(s, u n (s, x), Z n (s, x))) 2 dxds, (3.40)
RR n°7791

I 5 m = - T t D f ′ m (L(s, x) -u n (s, x))(L ′ (s, x))dxds ≤ 1 4 n T t D f ′ m (L(s, x) -u n (s, x)) 2 ds + C n T t D (L ′ (s, x)) 2 dxds. ( 3
E[G m (u n (t) -L(t))] ≤ E[G m (φ -L(T ))] + 3 4 n T t D f ′ m (L(s, x) -u n (s, x)) 2 ds + C n E[ T t D (L ′ (s, x)) 2 dxds] + C n E[ T t D (∆L(s, x)) 2 dxds] + C n E[ T t D (b(s, u n (s, x), Z n (s, x))) 2 dxds] -nE[ T t D f ′ m (L(s, x) -u n (s, x))((u n (s, x) -L(s, x)) -)ds]. (3.42)
Letting m → ∞ we conclude that

E[ D ((u n (t, x) -L(t, x)) -) 2 dx] ≤ 3 4 nE[ T t D ((u n (s, x) -L(s, x)) -) 2 dxds] -nE[ T t D ((u n (s, x) -L(s, x)) -) 2 dxds] + C ′ n , (3.43)
where the Lipschiz condition of b and Lemma 3.3 have been used. In particular we have

E[ T t D ((u n (s, x) -L(s, x)) -) 2 dxds] ≤ C ′ n 2 .
(3.44) Lemma 3.5 Let (u n , Z n ) be the solution of equation (3.21). We have

lim n,m→∞ E[ sup 0≤t≤T |u n (t) -u m (t)| 2 K ] = 0, (3.45) RR n°7791 lim n,m→∞ E[ T 0 ||u n (t) -u m (t)|| 2 V dt] = 0. (3.46) lim n,m→∞ E[ T 0 |Z n (t) -Z m (t)| 2 L 2 (D,R m ) dt] = 0. (3.47)
Proof. Applying Itô's formula, it follows that

|u n (t) -u m (t)| 2 K = 2 T t < u n (s) -u m (s), ∆(u n (s) -u m (s)) > ds +2 T t < u n (s) -u m (s), b(s, u n (s), Z n (s)) -b(s, u m (s), Z m (s)) > ds -2 T t < u n (s) -u m (s), Z n (s) -Z m (s) > dB s +2 T t < u n (s) -u m (s), n(u n (s) -L(s)) --m(u m (s) -L(s)) -> ds - T t |Z n (s) -Z m (s)| 2 L 2 (D,R m ) ds (3.48)
Now we estimate each of the terms on the right side.

2 T t < u n (s) -u m (s), ∆(u n (s) -u m (s)) > ds = -2 T t ||u n (s) -u m (s)|| 2 V ds. (3.49)
By the Lipschitz continuity of b and the inequality ab ≤ εa 2 + C ε b 2 , one has

2 T t < u n (s) -u m (s), b(s, u n (s), Z n (s)) -b(s, u m (s), Z m (s)) > ds ≤ C T t |u n (s) -u m (s)| 2 K ds + 1 2 T t |Z n (s) -Z m (s)| 2 L 2 (D,R m ) ds. (3.50) RR n°7791
In view of (3.44),

2E[ T t < u n (s) -u m (s), n(u n (s) -L(s)) --m(u m (s) -L(s)) -> ds] = 2nE[ T t < u n (s) -L(s), (u n (s) -L(s)) -> ds] +2mE[ T t < L(s) -u n (s), (u m (s) -L(s)) -> ds] +2mE[ T t < u m (s) -L(s), (u m (s) -L(s)) -> ds] +2nE[ T t < L(s) -u m (s), (u n (s) -L(s)) -> ds] ≤ 2mE[ T t < L(s) -u n (s), (u m (s) -L(s)) -> ds] +2nE[ T t < L(s) -u m (s), (u n (s) -L(s)) -> ds] ≤ 2mE[ T t D (u n (s, x) -L(s, x)) -(u m (s, x) -L(s, x)) -dxds] +2nE[ T t D (u m (s, x) -L(s, x)) -(u n (s, x) -L(s, x)) -dxds] ≤ 2m(E[ T t D ((u n (s, x) -L(s, x)) -) 2 dxds]) 1 2 (E[ T t D ((u m (s, x) -L(s, x)) -) 2 dxds]) 1 2 +2n(E[ T t D ((u n (s, x) -L(s, x)) -) 2 dxds]) 1 2 (E[ T t D ((u m (s, x) -L(s, x)) -) 2 dxds]) 1 2 ≤ C ′ ( 1 n + 1 m ).
(3.51)

It follows from (3.48) and (3.49) that

E[|u n (t) -u m (t)| 2 K ] + 1 2 E[ T t |Z n (s) -Z m (s)| 2 L 2 (D,R m ) ds] +E[ T t ||u n (s) -u m (s)|| 2 V ds] ≤ C T t E[|u n (s) -u m (s)| 2 K ]ds + C ′ ( 1 n + 1 m ). (3.52)
Application of the Gronwall inequality yields

lim n,m→∞ {E[|u n (t)-u m (t)| 2 K ]+ 1 2 E[ T t |Z n (s)-Z m (s)| 2 L 2 (D,R m ) ds]} = 0, (3.53) RR n°7791 lim n,m→∞ E[ T t ||u n (s) -u m (s)|| 2 V ds] = 0. ( 3
E[ sup 0≤t≤T |u n (t) -u m (t)| 2 K ] = 0. (3.55)
The proof is complete.

Proof of Theorem 3.2. From Lemma 3.5 we know that (u n , Z n ), n ≥ 1, forms a Cauchy sequence. Denote by u(t, x), Z(t, x) the limit of u n and Z n . Put ηn (t, x) = n(u n (t, x) -L(t, x)) - Lemma 3.4 implies that ηn (t, x) admits a non-negative weak limit, denoted by η(t, x), in the following Hilbert space:

K = {h; h is a K-valued adapted process such that E[ T 0 |h(s)| 2 K ds] < ∞} with inner product < h 1 , h 2 > K = E[ T 0 D h 1 (t, x)h 2 (t, x)dtdx].
Set η(t, x) = t 0 η(s, x)ds. Then η is a continuous K-valued process which is increasing in t. Keeping Lemma 3.5 in mind and letting n → ∞ in (3.21) we obtain

u(t, x) = φ(x) + T t ∆u(t, x)ds + T t b(s, u(s, x), Z(s, x))ds - T t Z(s, x)dB s +η(T, x) -η(t, x); 0 ≤ t ≤ T. (3.56) Recall from Lemma 3.4 that E[ T t D ((u n (s, x) -L(s, x)) -) 2 dxds] ≤ C ′ 1 n 2
By the Fatou Lemma, this implies that E[ T t D ((u(s, x)-L(s, x)) -) 2 dxds] = 0. In view of the continuity of u in t, we conclude u(t, x) ≥ L(t, x) a.e. in

RR n°7791

x, for every t ≥ 0. Combining the strong convergence of u n and the weak convergence of ηn , we also have

E[ T 0 D (u(s, x) -L(s, x))η(dt, x)dx] = E[ T 0 D (u(s, x) -L(s, x))η(t, x)dtdx] ≤ lim n→∞ E[ T 0 D (u n (s, x) -L(s, x))η n (t, x)dtdx] ≤ 0 (3.57) Hence, T 0 D (u(s, x) -L(s, x))η(dt, x)dx = 0, a.s.
We have shown that (u, Z, η) is a solution to the reflected BSPDE (3.17).

Uniqueness. Let (u 1 , Z 1 , η 1 ), (u 2 , Z 2 , η 2 ) be two such solutions to equation (3.20). By Itô's formula, we have

|u 1 (t) -u 2 (t)| 2 K = 2 T t < u 1 (s) -u 2 (s), ∆(u 1 (s) -u 2 (s)) > ds +2 T t < u 1 (s) -u 2 (s), b(s, u 1 (s), Z 1 (s)) -b(s, u 2 (s), Z 2 (s)) > ds -2 T t < u 1 (s) -u 2 (s), Z 1 (s) -Z 2 (s) > dB s +2 T t < u 1 (s) -u 2 (s), η 1 (ds) -η 2 (ds) > - T t |Z 1 (s) -Z 2 (s)| 2 L 2 (D,R m ) ds (3.58)
Similar to the proof of Lemma 3.5, we have

2 T t < u 1 (s) -u 2 (s), ∆(u 1 (s) -u 2 (s)) > ds ≤ 0, (3.59) and 2 T t < u 1 (s) -u 2 (s), b(s, u 1 (s), Z 1 (s)) -b(s, u 2 (s), Z 2 (s)) > ds ≤ C T t |u 1 (s) -u 2 (s)| 2 K ds + 1 2 T t |Z 1 (s) -Z 2 (s)| 2 L 2 (D,R m ) ds (3.60) RR n°7791
On the other hand, 4 Link to optimal stopping

2E[ T t < u 1 (s) -u 2 (s), η 1 (ds) -η 2 (ds) >] = 2E[ T t D (u 1 (s, x) -L(s, x))η 1 (ds, x)dx] -2E[
In this section, we provide a link between the solution of a reflected backward stochastic partial differential equation and an optimal stopping problem. Let u(t, x) be the solution of the following reflected BSPDE. Let S t,T be the set of all stopping times τ satisfying t ≤ τ ≤ T . For τ ∈ S t,T , define R t (τ, x) = Here, and in the following we will use the simplified notation P t k(s, x) = (P t k(s, •))(x) etc.

Theorem 4.1 u(t, x) is the value function of the the optimal stopping problem associated with R t (τ, x), i.e., Combining this with (4.7) we obtain the theorem.

u

  On considère des problèmes de contrôle singulier d'Équations aux Dérivés Partielles Stochastiques (EDPS) et l'on prouve un principe du maximum stochastique pour le le contrôle optimal singulier, faisant intervenir un système couplé d'EDPS et d'EDPS réfléchies. Des résultats d'existence et d'unicité pour les EDPS réfléchie sont démontrés dans la deuxième partie du papier.

  (s, x) -L(s, x))η 2 (ds, x)dx] (s, x) -L(s, x))η 2 (ds, x)dx] (s, x) -L(s, x))η 1 (ds, x)dx] inequality, this implies u 1 = u 2 , Z 1 = Z 2which further gives η 1 = η 2 from the equation they satisfy.

  x)η(t, x); 0 ≤ t ≤ T, u(t, x) ≥ L(t, x), T 0 D (u(s, x) -L(s, x))η(dt, x)dx = 0 a.s. (4.1) RR n°7791

  t k(s, x)ds + P τ -t L(τ, x)χ {τ <T } + P τ -t φ(x)χ {τ =T } , where k(s, •) = k(s, •, u(s, •), Z(s, •)) and P t denotes the semigroup generated by the Laplacian operator 1 2 ∆, i.e.P t f (x) = (2πt) -d/2 R d f (y)exp(-|y -x| 2 2t )dy; f ∈ L 1 (R d ).

  Since η(s, x) is increasing in s and u(s, x) ≥ L(s, x) for s < T , it follows thatu(t, x) ≥ R t (τ, x) -From the property of η, it is not increasing on the interval [t, τt ]. Therefore, τt t P s-t η(ds, x) = 0. Thus we have from (4.4) that

	As τ is arbitrary, we obtain
		u(t, x) ≥ ess sup τ ∈S t,T	E[R t (τ, x)|F t ]	(4.7)
	Now, define
		τt = inf{s ∈ [t, T )|u(s) = L(s)} ∧ T
	u(t, x)
	τ
	+	P s-t η(ds, x); 0 ≤ t ≤ τ.	(4.4)
	t
			τ
			P s-t (Z(s, x))dB s	(4.5)
		t
	Take conditional expectation with respect to F t on both sides to get
		u(t, x)
	RR n°7791

(t, x) = esssup τ ∈S t,T E[R t (τ, x)|F t ] (4.2)

Proof. Observe that u admits the following mild representation:

u(t, x) = P T -t φ(x) + T t P s-t (k(s, u(s, x), Z(s, x)))ds -T t P s-t (Z(s, x))dB s + T t P s-t η(ds, x); 0 ≤ t ≤ T. (4.3)

This implies that for any stopping time τ with t ≤ τ ≤ T , we have

u(t, x) = P τ -t (u(τ, x)) + τ t P s-t (k(s, x))ds -τ t P s-t (Z(s, x))dB s ≥ E[R t (τ, x)|F t ] -E[ τ t P s-t (Z(s, x))dB s |F t ] = E[R t (τ, x)|F t ]. (

4

.6) = P τt-t (u(τ t ), x) + τt t P s-t (k(s, x))ds -τt t P s-t (Z(s, x))dB s = R t (τ t , x) -τt t P s-t (Z(s, x))dB s . (4.8) Taking conditional expectation yields that u(t, x) = E[R t (τ t , x)|F t ]

RR n°7791

The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [228087].

PARIS -ROCQUENCOURT


Loic Labrousse

Gaëlle Prouteau

Anne-Céline Ganzhorn

L Labrousse1

G Prouteau3

-C A Ganzhorn1

Continental exhumation triggered by partial melting at ultrahigh pressure

Partial melting textures, observed in most continental crust buried in ultrahigh-pressure (UHP) conditions, have mostly been related to their retrograde evolution during exhumation in collisional orogens. Analysis of leucosomes from the Western Gneiss Region (WGR, Norway) UHP and HP domains in the Caledonides show a wide scatter of their chemistries, from early ones close to trondhjemites restricted to UHP domains, to granites in late occurrences or associated with HP domains. Nearly trondhjemitic compositions compare with hydrous melts produced in felsic systems at high pressure (>2 GPa) and moderate temperature (<900 °C). Partial melting experiments at higher temperatures or in dry conditions produce granitic glasses similar to late leucosomes from the WGR. Comparison of pressuretemperature paths for Caledonian eclogites with melting and dehydration reactions for the surrounding gneiss suggest that (1) the continental crust remained partially hydrated during its subduction to ultrahigh pressure, and (2) partial melting reactions producing the trondhjemitic melts started as soon as the WGR rocks reached their hydrated solidus, at the peak pressure recorded by the eclogites. The limited partial melting degree at the peak conditions induced weakening of the continental crust, decoupling from the lithospheric root, and initiation of exhumation.

INTRODUCTION

Partial melting of the continental crust, coeval with thermal re-equilibration, is inferred to induce ductile flow of orogens [START_REF] Teyssier | Gneiss domes and orogeny[END_REF]. To evaluate the rheological impact of partial melting on the continental crust during burial and exhumation, the timing of partial melting initiation needs to be constrained, since the softening effect of partial melting is efficient for very low melt contents [START_REF] Rosenberg | Experimental deformation of partially melted granite revisited: Implications for the continental crust[END_REF]. Attempts to date partial melting via accessory mineral dating in leucosomes only provided constraints on crystallization stages [START_REF] Kylander-Clark | Slow exhumation of UHP terranes: Titanite and rutile ages of the Western Gneiss Region[END_REF]. This study intends to correlate the natural diversity of leucosomes in the high-pressure (HP) and ultrahigh-pressure (UHP) domains in the Western Gneiss Region (WGR, Norway) with their timing relative to the metamorphic and tectonic evolution of the eclogites and surrounding gneiss. The composition of leucosomes is compared with published experimental data and new experiments performed on a natural sample from the WGR. We conclude that the initiation of partial melting coincides with maximum burial of the Baltica continent, and we discuss the impact of partial melting on continental subduction.

PARTIAL MELTING AT ULTRAHIGH-PRESSURE CONDITIONS Natural Evidence from Ultrahigh-Pressure Domains

Migmatites have been described in different large-size UHP continental crust units in Dabie Shan (China; Faure et al., 2003), Kokchetav (Russia;[START_REF] Hermann | Multiple zircon growth during fast exhumation of diamondiferous, deeply subducted continental crust (Kokchetav Massif, Kazakhstan[END_REF], or Caledonides [START_REF] Labrousse | Pressure-temperature-time deformation history of the exhumation of ultra-high pressure rocks in the Western Gneiss Region, Norway[END_REF][START_REF] Lang | Partial melting of metapelites at ultrahigh-pressure conditions, Greenland Caledonides[END_REF]. In the Hercynides, trondhjemites and leucogranulites have also been interpreted as the result of partial melting of crust in HP conditions [START_REF] Nicollet | Petrology of HP trondhjemitic levels associated with eclogites and amphibolites of leptyno-amphibolitic series of French Massif Central[END_REF][START_REF] Kotkova | Anatexis during high-pressure crustal metamorphism: Evidence from garnet-whole-rock REE relationships and zircon-rutile Ti-Zr thermometry in leucogranulites from the Bohemian Massif[END_REF]. Comparison of P-T paths for UHP units with undersaturated solidus for continental crust material [START_REF] Auzanneau | Experimental evidence of decompression melting during exhumation of subducted continental crust[END_REF] shows partial melting evidence associated with decompression. Nevertheless, indirect evidence attests that silicate melts can be present at peak conditions. In Dabie Shan, zircon ages from granite dikes overlap with the age for peak metamorphism [START_REF] Wallis | Role of partial melting in the evolution of the Sulu (eastern China) ultrahigh-pressure terrane[END_REF]. In Kokchetav, zircons grown from Zr-saturated melts present at UHP conditions [START_REF] Hermann | Multiple zircon growth during fast exhumation of diamondiferous, deeply subducted continental crust (Kokchetav Massif, Kazakhstan[END_REF] were found in K, Na, Si, and light rare earth element-poor gneiss possibly affected by melt extraction. In the WGR, high initial Sr signatures in the Opxbearing eclogites [START_REF] Griffin | REE, Rb-Sr and Sm-Nd studies of Norwegian eclogites[END_REF] were interpreted as due to interaction with the surrounding gneiss at peak conditions. In addition, polyphase solid-inclusions, Sm-Nd, and Sr data from a peridotite body, in the northern WGR, suggest that it was infiltrated by crustalderived melts at pressures above 3.4 GPa and temperature ∼800 °C [START_REF] Vrijmoed | An alternative model for ultrahigh pressure in the Svartberget Fe-Ti garnet-peridotite, Western Gneiss Region[END_REF]. The above evidence therefore suggests that partial melting of buried crustal units can occur at their peak pressure.

Constraints for the Western Gneiss Region Pressure-Temperature History

A wealth of experiments have explored the melting behavior of felsic rocks at crustal to mantle pressures [START_REF] Auzanneau | Experimental evidence of decompression melting during exhumation of subducted continental crust[END_REF][START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF][START_REF] Huang | Phase relations of S-type granite with H 2 O to 35 kbar: Muscovite granite from Harney Peak[END_REF][START_REF] Schmidt | Melting and dissolution of subducting crust at high pressures: The key role of white mica[END_REF]. Melting curves determined from these experiments can be compared to the P-T paths for the WGR (Fig. 1). Low-pressure eclogites from the Sunnfjord area [START_REF] Labrousse | Pressure-temperature-time deformation history of the exhumation of ultra-high pressure rocks in the Western Gneiss Region, Norway[END_REF], where rare partial melting textures were described, yield peak conditions in the subsolidus domain. Nordfjord eclogites [START_REF] Carswell | Ultrahigh-pressure metamorphism in the Western Gneiss Region of the Norwegian Caledonides[END_REF][START_REF] Hacker | Ascent of the ultrahigh-pressure Western Gneiss Region[END_REF][START_REF] Labrousse | Pressure-temperature-time deformation history of the exhumation of ultra-high pressure rocks in the Western Gneiss Region, Norway[END_REF], included in the southernmost migmatites, equilibrated at conditions overlapping the different experimental estimates of the wet solidus curve for continental rocks. Peak conditions estimates for the northernmost UHP eclogites [START_REF] Hacker | High-temperature deformation during continentalmargin subduction and exhumation: The ultrahigh-pressure Western Gneiss Region of Norway[END_REF][START_REF] Terry | Kyanite eclogite thermobarometry and evidence for thrusting of UHP over HP metamorphic rocks, Nordøyane, Western Gneiss Region[END_REF] are bracketed by the wet solidus curves and the undersaturated curve for metapelites. It seems therefore possible that gneiss surrounding the eclogites did melt at UHP conditions if the water budget allowed them for fluid-present partial melting. Pressure-temperature diagram for the Western Gneiss Region (WGR) ultrahigh-pressure (UHP) and high-pressure (HP) domains (CC00: . [START_REF] Carswell | Ultrahigh-pressure metamorphism in the Western Gneiss Region of the Norwegian Caledonides[END_REF]L04: Labrousse et al., 2004;TR00: Terry et al., 2000;H07: Hacker, 2007) compared to hydrated solidus for granites (HW81: [START_REF] Huang | Phase relations of S-type granite with H 2 O to 35 kbar: Muscovite granite from Harney Peak[END_REF] and metapelites (HS08: [START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF]N94: Nichols et al., 1994;A06: Auzanneau et al., 2006). Computed dehydration curves for our starting material: Domino software (de Capitani and Petrakakis, 2010), [START_REF] Holland | An internally consistent thermodynamic data set for phases of petrological interest[END_REF] database. Curves labeled 1-3 are presented in Figure 3.

LEUCOSOMES IN THE WGR AND THEIR CHRONOLOGY

Nine natural leucosomes were collected throughout the eclogite province in the WGR from Nordfjord to Bud (location of sampling and complete chemical analysis is available in Table DR2 in the GSA Data Repository 1 ). Samples were selected in association with HP eclogites (sample NO10-16) or UHP eclogites (all others) and with all the possible relative chronologies with regard to the deformation in the gneiss. All samples come from gneiss wrapping the eclogite lenses, and therefore crystallized during or after the Caledonian HP event. Leucosomes transposed in the gneiss foliation are considered as early; leucosomes occurring in patches or veins crosscutting the foliation are considered as late and coeval with retrogression in the amphibolite facies. In an An-Ab-Or projection (Fig. 2A), their compositions define a remarkable trend parallel to the Ab-Or join from nearly trondhjemitic to granitic compositions. Among leucosomes associated with UHP eclogites, trondhjemitic compositions are found in the northernmost UHP localities (Svartberget), and early leucosomes are systematically richer in albite than their late counterparts, on the same outcrop. The leucosome associated with HP eclogites (NO10-16) is close to the latest and southernmost sample associated with UHP eclogites (NO10-27). One noteworthy sample (NO10-51) collected in the crystallization shadow of a meter-scale mafic lens in the Svartberget UHP locality shows a unique tonalitic composition out of the regional trend, with MgO + FeO up to 7.9 wt%. This tonalite is clearly distinct from other leucosomes and probably derived from a different source. The linear trend in granitic compositions, unprobably connected to any sampling bias, could result from variation of the partial melting conditions of a unique source. To test this hypothesis, these leucosome compositions are compared to published experimental data and results from new experiments performed in a pertinent system. DR2 (see footnote 1). UHP-ultrahigh-pressure; HPhigh-pressure. B: Temperature and wt% water added in our experiments (e.g., 800/5 denotes 800 °C and 5 wt% added water). Experimental melt compositions produced in hydrous metapelitic system with [START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF] or without [START_REF] Schmidt | Melting and dissolution of subducting crust at high pressures: The key role of white mica[END_REF][START_REF] Auzanneau | Experimental evidence of decompression melting during exhumation of subducted continental crust[END_REF] added water. Dashed line represents the evolution with pressure of melts produced from partial melting of hydrous basalt [START_REF] Prouteau | Evidence for mantle metasomatism by hydrous silicic melts derived from subducted oceanic crust[END_REF]. mpmetapelite; mb-metabasalt.

EXPERIMENTAL CONSTRAINTS

Literature Data

When plotted in the An-Ab-Or projection (Fig. 2B), experimental melt compositions produced in metasedimentary systems in the pressure range 2.2-4.5 GPa and temperature range 750-1050 °C [START_REF] Auzanneau | Experimental evidence of decompression melting during exhumation of subducted continental crust[END_REF][START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF][START_REF] Schmidt | Melting and dissolution of subducting crust at high pressures: The key role of white mica[END_REF] evolve from trondhjemitic to granitic with An contents not exceeding 10 wt%. Experiments without added water [START_REF] Auzanneau | Experimental evidence of decompression melting during exhumation of subducted continental crust[END_REF][START_REF] Schmidt | Melting and dissolution of subducting crust at high pressures: The key role of white mica[END_REF] cluster in the granitic field, while melts produced in hydrated conditions (H 2 O <6 wt%) define a trend parallel to the Ab-Or join [START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF]. This evolution is correlated to increasing melting temperature from 750 °C for trondhjemites to 1050 °C for the potassic end member. This trend is remarkably similar to the natural one, but the temperature range is not realistic in this geological context (Fig. 1). Temperatures exceeding 950 °C, necessary for the production of the more potassic glasses, were not reached in the WGR.

Experiments on a WGR Protolith

In order to estimate the impact of protolith composition and partial melting conditions on melt compositions for the WGR rocks, new piston-cylinder experiments were performed on a natural sample considered as representative of the regional bulk composition (NOG004-1, Table DR1). The starting material NOG004-1, a granitic two-mica gneiss, was reacted at 2.5 GPa with added H 2 O at 800 and 850 °C. These experimental conditions (Table DR1) correspond to the maximum temperature reached by the WGR rocks. Hydrous silicate glass (quenched liquid) is present in all experimental charges (Table DR1). Phases identified in addition to glass (Fig. DR1) are mainly garnet, biotite, and kyanite. Plagioclase is also present at 800 °C in the less hydrated run, and orthopyroxene is stable at 850 °C.

The composition of glass produced at 800 °C and 5% added water lies close to the granitetrondhjemite boundary in the An-Ab-Or projection (Fig. 2B), like glasses obtained from metapelites [START_REF] Auzanneau | Experimental evidence of decompression melting during exhumation of subducted continental crust[END_REF] in water-undersaturated experiments at comparable P and T, or derived from wet metapelites at higher pressures and temperatures ∼800 °C [START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF]. Glasses produced at higher temperature (850 °C) or higher water content (10 wt%) plot close to the starting material in this projection and compare well with glasses obtained from nearly fluid-absent partial melting experiments on metapelites [START_REF] Schmidt | Melting and dissolution of subducting crust at high pressures: The key role of white mica[END_REF]. They lie in the high-temperature part (>950 °C) of the compositional trend for melts derived from metapelites at HP and hydrated conditions [START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF]. These preliminary results therefore show that the natural trend observed in the WGR leucosomes is similar to the trend in experimental melts from a unique protolith and with realistic temperatures in presence of free water. Limited discrepancies in Mg# and TiO 2 contents between natural and experimental compositions can be explained by a difference in fO 2 , which controls Fe-Ti oxides crystallization [START_REF] Conrad | Water-saturated and -undersaturated melting of metaluminous and peraluminous crustal compositions at 10 kbar: Evidence for the origin of silicic magmas in the Taupo volcanic zone, New Zealand, and other occurrences[END_REF][START_REF] Prouteau | Experimental constraints on the origin of the 1991 Pinatubo dacite[END_REF]. The question of what part of the initial melt is actually represented by the final leucosome remains a subject of debate [START_REF] Kriegsman | Partial melting, partial melt extraction and partial back reaction in anatectic migmatites[END_REF]. However, the similarity between the compositions of experimental and natural data presented here suggests that leucosomes can preserve their genuine composition along the P-T path, with limited interaction with the surrounding gneiss, as opposed to their important metasomatic effect on ultramafic inclusions [START_REF] Vrijmoed | An alternative model for ultrahigh pressure in the Svartberget Fe-Ti garnet-peridotite, Western Gneiss Region[END_REF]. Among all experimental results for metapelites, only glasses from experiments at low temperatures (750-800 °C) and pressures above 2.5 GPa in water-present conditions show trondhjemitic compositions [START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF], suggesting that the nearly trondhjemitic leucosomes observed in the WGR may be produced at temperatures below 800 °C, i.e., close to the peak pressure recorded by the eclogites. This also implies that a significant fraction of free fluid was transported by the continental crust down to UHP conditions. Other evidence suggests free water in the continental crust subducting at mantle depth. If not the result of complex metasomatic reactions with the associated mafic body, the occurrence of a tonalitic leucosome implies that mafic material may have also sporadically melted during the Caledonian burial and exhumation event. Such a tonalitic melt can be produced by fluid-present melting of basalt in the 2-3 GPa range and temperatures below 950 °C (e.g., [START_REF] Prouteau | Evidence for mantle metasomatism by hydrous silicic melts derived from subducted oceanic crust[END_REF]. As temperatures reported along the P-T paths for the WGR never exceed 850 °C, a free-fluid phase must have been present in this pressure range. The free water present in the source may come from the breakdown of hydrous phases stable below the wet solidus [START_REF] Schmidt | Melting and dissolution of subducting crust at high pressures: The key role of white mica[END_REF]. Destabilization curves for paragonite, glaucophane, and chloritoid, as computed by Domino software (de Capitani and Petrakakis, 2010) for the NOG004-1 composition limit a free-fluid domain above 700 °C in the amphibolite facies, shifting to 600 °C in the jadeite stability field (Fig. 1). Above 2.5 GPa, the dehydration curve is crossed at lower temperature than the solidus. Therefore, a gneiss subducting along a thermal gradient lower than 8 °C/km would release ∼1% H 2 O just before reaching the partial melting field.

DISCUSSION: EVIDENCE FOR PARTIAL MELTING AT ULTRAHIGH PRESSURE AND RHEOLOGICAL IMPLICATIONS

To be subducted to mantle depths, the less dense portions of the continental lithosphere (i.e., the upper crust, with upward buoyancy) must remain coupled to the denser levels (i.e., the lower crust and the lithospheric mantle). In contrast, they must be decoupled to allow exhumation. The peak metamorphism therefore corresponds to the triggering of a decoupling mechanism, such as slab breakoff [START_REF] Von Blanckenburg | Slab breakoff: A model for syncollisional magmatism and tectonics in the Alps: Tectonics[END_REF] or metamorphic weakening [START_REF] Austrheim | Processing of crust in the root of the Caledonian continental collision zone: The role of eclogitization[END_REF][START_REF] Jolivet | Softening triggered by eclogitization, the first step toward exhumation during continental subduction[END_REF][START_REF] Labrousse | Initiation of crustal-scale thrusts triggered by metamorphic reactions at depth: Insights from a comparison between the Himalayas and Scandinavian Caledonides: Tectonics[END_REF]. Partial melting is considered as a possible weakening mechanism for the continental crust during thermal relaxation of a thickened crust [START_REF] Vanderhaeghe | Partial melting and flow of orogens[END_REF] or during decompression of subducted continental slices [START_REF] Wallis | Role of partial melting in the evolution of the Sulu (eastern China) ultrahigh-pressure terrane[END_REF], but rarely mentioned as a mechanism for the very initiation of the exhumation of continental crust at ultrahigh pressure [START_REF] Hermann | Multiple zircon growth during fast exhumation of diamondiferous, deeply subducted continental crust (Kokchetav Massif, Kazakhstan[END_REF]. The dehydration curve and the wet solidus for continental crust can be superimposed on the crustal stratigraphy and thermal structure of a collisional orogen (Fig. 3) compatible with P-T data from the WGR (Fig. 1) deduced from thermal models adapted for the Scandian continental subduction at its paroxysm [START_REF] Kylander-Clark | Slow subduction of a thick ultrahigh-pressure terrane: Tectonics[END_REF] and compatible with geometries and P-T paths predicted for continental subduction from fully coupled thermomechanical modeling with implemented partial melting [START_REF] Li | Influence of tectonic overpressure on P-T paths of HP-UHP rocks in continental collision zones: Thermomechanical modelling[END_REF]. Their relative positions suggest that the two boundaries are successively crossed by the upper continental crust between 2.5 and 3.5 GPa. The top of the upper crust evolves from a hydroxyl-bearing mineral state to a partially molten state at 2.3 GPa, while the base of the upper crust dehydrates in the subsolidus domain between 2.5 and 3.5 GPa. Even if partial melting at the peak pressure conditions may have remained limited compared to later decompression melting, the first silicate melt may have drastically lowered the effective viscosity of the continental crust [START_REF] Rosenberg | Experimental deformation of partially melted granite revisited: Implications for the continental crust[END_REF] and acted as a decoupling mechanism, triggering the exhumation of the upward-buoyant crust. The lowviscosity migmatites can also incorporate lumps of peridotite from meter to kilometer scale [START_REF] Brueckner | A general model for the intrusion and evolution of 'mantle' garnet peridotites in high-pressure and ultra-high-pressure metamorphic terranes[END_REF] and drag dense eclogite lenses or dry unreacted parts of the lower crust toward the surface without significant temperature change.

The long-term (>10 Ma?) persistence of such a partially molten channel as a steady-state geometry could also explain the scatter of eclogites with contrasted P-T records (Fig. 1) as well as the poor correspondence between UHP eclogite occurrences and structures in the gneiss. Indeed, the limits of UHP domains remain diffuse [START_REF] Hacker | High-temperature deformation during continentalmargin subduction and exhumation: The ultrahigh-pressure Western Gneiss Region of Norway[END_REF].

CONCLUSIONS

The chemical study of natural leucosomes from the WGR displays a linear trend from early trondhjemites to late granites, with occasional tonalite associated with a mafic lens. The products of partial melting experiments performed on a natural sample representative of the WGR gneiss at the maximum temperature recorded during their retrogression partly reproduce the natural trend. We deduce from this and published experiments in a similar system [START_REF] Hermann | Sediment melts at sub-arc depths: An experimental study[END_REF] that (1) leucosomes trapped in the gneiss preserved their genuine composition, and (2) first melts nearly trondhjemitic were produced at temperatures as low as 750 °C and pressures higher than 2.5 GPa, i.e., at the peak conditions recorded by eclogites. Onset of water-present partial melting and maximum burial of the continental slab would then be coeval, the release of free water by breakdown of hydroxylbearing minerals occurring at pressures as high as 2.5 GPa according to thermochemical modeling. A tonalitic leucosome even testifies to possible limited melting of eclogites themselves in hydrated conditions. Liquids produced by water-present melting at the peak conditions are sufficient to weaken the continental crust, decouple it from the down-going slab, and favor its ascent by buoyancy forces, explaining the apparent coincidence of peak P-T conditions recorded by the eclogites and the wet solidus for continental crust

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