David Defour
email: david.defour@univ-perp.fr

FP-ANR: A representation format to handle floating-point cancellation at run-time

When dealing with floating-point numbers, there are several sources of error which can drastically reduce the numerical quality of computed results. One of those error sources is the loss of significance or cancellation, which occurs during for example, the subtraction of two nearly equal numbers. In this article, we propose a representation format named Floating-Point Adaptive Noise Reduction (FP-ANR). This format embeds cancellation information directly into the floating-point representation format thanks to a dedicated pattern. With this format, insignificant trailing bits lost during cancellation are removed from every manipulated floating-point number. The immediate consequence is that it increases the numerical confidence of computed values. The proposed representation format corresponds to a simple and efficient implementation of significance arithmetic based and compatible with the IEEE Standard 754 standard.

Introduction

Floating-point numbers, which are normalized by the IEEE Standard 754 standard [START_REF]IEEE Standard for Floating-Point Arithmetic[END_REF], correspond to a bounded discretization of real numbers. Therefore, a floating-point number corresponds to the representation of an exact number combined with errors due to discretization, accumulation of rounding errors or cancellation. In other words, a floatingpoint number embeds useful information along with noise linked to those errors.

When numerical noise becomes dominant, for example during catastrophic cancellation, there are no more useful bits of information in the representated numbers. Unfortunately, the occurence of this situation is undetectable just by looking at the representation. This is due to the fact that with the widely used IEEE Standard 754 representation format, there is no way of distinguishing useful numerical information from noise. This problem has been identified and addressed since the late 1950s with significance arithmetic [START_REF] Goldstein | Significance arithmetic on a digital computer[END_REF]. Significance arithmetic addressed these issues by tailoring the number of digits to their needs.

Significance arithmetics is regaining interest thanks to the Unum proposal [START_REF] Gustafson | The end of numerical error[END_REF] or indirectly through numerous problems encountered with exascale computers and the lack of confidence in numerical results [START_REF] Collange | Numerical reproducibility for the parallel reduction on multi-and many-core architectures[END_REF]. If the Unum system is based on real problems, the proposed solution is subject to criticism for numerous reasons as pointed out by W. Kahan [START_REF] Kahan | A critique of John L. Gustafson's. the end of error -Unum computation and his a radical approach to computation with real numbers[END_REF]. On the other hand, indirect solutions based on software solutions to detect cancellation [START_REF] Denis | Verificarlo: Checking floating point accuracy through monte carlo arithmetic[END_REF], [START_REF] Jézéquel | CADNA: a library for estimating round-off error propagation[END_REF], or avoiding rounding errors [START_REF] Collange | Numerical reproducibility for the parallel reduction on multi-and many-core architectures[END_REF] are not meant to be efficient nor effective for real time execution.

This article proposes a new way to represent significant information in floating-point numbers. The solution consists of an altered IEEE Standard 754 representation format of the mantissa. That information is stored using a simple pattern that replaces insignificant digits. This makes such number representation almost as accurate as original IEEE Standard 754 numbers. Therefore, the proposed solution corresponds to a simple, efficient and IEEE Standard 754 compliant implementation of significance arithmetic.

Preliminaries

Floating-point numbers are approximations of real numbers. The concept of approximation is associated with the concept of errors. Digits of a floating-point representation number can be split into two parts; a significant and an insignificant part. This section provides some background on IEEE Standard 754 floating-point arithmetic, errors and significance arithmetic.

The IEEE Standard 754 standard

The current version of the floating-point standard, the IEEE Standard 754 [-2008] [START_REF]IEEE Standard for Floating-Point Arithmetic[END_REF] published in August 2008, includes the original binary formats along with three new basic formats (one binary and two decimal).

Definition 1 (Floating-Point Numbers). A IEEE Standard 754 representation format is a "set of representations of numerical values and symbols" made of finite numbers, two infinities and two kinds of NaN (Not A Number). The set of finite numbers are described by a set of three integers (s,m,e) corresponding respectively to the sign, the mantissa and the exponent. The numerical value associated with this representation is

(-1) s × m × b e .
Values that can be represented are determined by the base or radix b (2 or 10), the number (p) of digits in the mantissa and the exponent parameter emax such that:

0 ≤ m ≤ b p -1 and 1 -emax ≤ e + p -1 ≤ emax
It should be pointed out that the number e + p -1 is called the "exponent" in some literature.

The value Zero is represented with a 0 mantissa and a sign bit specifying a positive or negative zero.

In the case of binary formats, representation of finite numbers is made unique by choosing the smallest representable exponent. Numbers with an exponent in the normal range have the leading bit set to 1. It corresponds to an implicit bit as it is not present in the memory encoding, allowing the memory format to have one more bit of precision. This extra bit is not present for subnormal numbers which have an exponent outside the normal exponent range.

For example, the IEEE Standard 754 double precision format (or binary64) is represented with 64 bits which are split into 1 sign bit, p = 52 bits of mantissa and e = 11 bits of exponent, whereas single precision format (or binary32) is represented with 32 bits split into 1 sign bit, p = 23 bit of mantissa and e = 8 bits of exponent.

Floating-Point Errors

Floating-point numbers representation format differs by their radix and the number of bits used for their encoding. The 2008 revision of the IEEE Standard 754 defines formats for radix 2, ranging from 16 to 128 bits. For each of these formats, the number of bits that represent the exponent and the mantissa is fixed. Therefore, the floatingpoint representation of numerical value have to be either rounded or padded with zeros in the least significant digits of the mantissa. This means that by construction, FP numbers embed errors in their representation. These errors can be separated into three groups: data uncertainty, rounding and cancellation.

2.2.1. Uncertainty. Uncertainty [START_REF] Denker | Uncertainty as applied to measurements and calculations[END_REF] in data is linked to initial input values produced by measurements, experimentations using physical sensors, or numerical model such as polynomial approximation [START_REF] Funaro | Polynomial approximation of differential equations[END_REF]. For example, a physical sensor producing the twenty digit value x = 12345.678901234567890 with a process exhibiting an uncertainty of U = 10 -5 , corresponds to a real value in the interval

[x • (1 -U ); x • (1 + U )] = [12345.555; 12345.802].
This translates into 5 significant digits, the rest of the information corresponds solely to noise or insignificant digits. As floating-point numbers are of a fixed size, noise is kept in the representation of those numbers and remains present in all computation that follows. As those extra digits do not carry any numerical meaning, it may lead to an overconfidence in the numerical quality of the result.

Rounding.

Because floating-point numbers have a limited number of digits, they cannot represent real numbers accurately. When there are more digits than the format allows, the number is rounded and the leftovers are omitted. The standard defines five rounding rules, two rounding to the nearest (ties to even, ties away from zero) and three directed rounding (toward 0, -∞, +∞). Floating-point operations in IEEE Standard 754 satisfy:

f l(a • b) = (a • b) • (1 + ) | | ≤ u • ∈ {+, -, ×, /}
Where u = b/2•b -p depends on the radix b and the precision p , and f l() denote the result of a floating-point computation.

2.2.3. Cancellation. Cancellation occurs when two nearby quantities are subtracted and the most significant digits cancel each other. Cancellations are very common but when many digits are lost, the effect can be severe as the number of informative digits is reduced. In that case, this results in catastrophic cancellation that has a dramatical impact on the sequel of the computation. For example, let x = 1.5 × 2 0 and y = 1.0 × 2 26 be two floating-point numbers stored in binary32 format. Then the sequence of operations r = f l(f l(x + y) -y) produces the result r = 0.0 which has no correct digits, as the correct real result should be 1.5. This is due to the catastrophic cancellation which occurred during the subtraction. Such cancellations cannot be detected without additional examination of the source and destination of data elements, leaving no trace of the fact that r = 0.0 was completely incorrect.

Such sequences are used for example in numerical algorithms that compute errors such as the 2sum algorithm [START_REF] Møller | Quasi double-precision in floating point addition[END_REF], [START_REF] Knuth | The Art of Computer Programming[END_REF].

Significance arithmetic

Significance arithmetic [START_REF] Goldstein | Significance arithmetic on a digital computer[END_REF], [START_REF] Gray | Normalized floating-point arithmetic with an index of significance[END_REF], [START_REF] Bond | Significant digits in computation with approximate numbers[END_REF] brings a solution to the problem of representing an approximation of the error along with floating-point numbers. It relies on the concept of significant and insignificant digits.

Definition 2 (Significant and insignificant digits). Significant digits of a number are digits that carry meaning contributing to a number. The number of significant digits for a p-digits number X is represented by α X and the number of insignificant digits p -α X .

Significance arithmetic sets two methods to calculate a bound for the propagated and generated error called normalized significance and unnormalized significance. The normalized significance always keeps the floating-point number normalized and provides an index of significance. The unnormalized significance does not normalize floating-point numbers and uses the count of digits remaining after leading zeros as an indication of their significance.

The normalized method allows as many digits as possible of a number to be retained. This requires an added index that defines the number of significant digits. There exists software implementations of significance arithmetic such as for the FORSIG [START_REF] Hyman | Forsig: an extension of fortran with significance arithmetic[END_REF] library written in Fortran, or Python [START_REF] Johansson | Basic implementation of significance arithmetic[END_REF]. With the unnormalized method [START_REF] Ashenhurst | Unnormalized floating point arithmetic[END_REF], only digits considered significant are retained.

The integration of a specific pattern in the mantissa to categorize significant and insignificant digit has already been proposed for decimal computer in the BCD format [START_REF] Langdon | Method and means for tracking digit significance in arithmetic operations executed on decimal computers[END_REF]. It relies upon unused bit patterns in the BCD format which are bit-field 1010 and 1011 corresponding to respectively digits 10 and 11. More recently, Gustafson [START_REF] Gustafson | The end of numerical error[END_REF] extended significance arithmetic by proposing the Unum representation format which is able to represent exact and approximate numbers with varying mantissa and exponent field length.

Even though significance arithmetic offers an approximation of the error, it is not suitable for every numerical problem related to the management of error. In particular, significance arithmetic is not meant for self-correcting numerical algorithm.

A format to embed cancellation information

The proposed representation format: Floating-Point Adaptive Noise Reduction (FP-ANR) is detailed in this section. It allows the user to split the mantissa in two: the significant and insignificant part. Insignificant digits, or noise, can come from initial uncertainty, or cancellation generated during computation. This format corresponds to an implementation of significance arithmetics based on the existing IEEE Standard 754 format. In this article, we will consider the radix-2 arithmetic, where bit or digit will refer to the same notion.

The representation format

Our goal is to propose a non-intrusive solution while being able to keep track of uncertainty due to cancellation. By non-intrusive, we mean that the proposed solution must be compatible with existing floating-point representation format without exhibiting a large overhead. This discards any solutions relying on shadow memory, or extra fields.

The proposed format, named FP-ANR, is based on a modification of the mantissa that integrates information on cancellation. The modification of the mantissa consists in replacing uninformative bits, or bits lost during cancellation, by a given pattern. This pattern must be self-detectable to avoid using extra fields as in the Unum. There are two possible patterns: First, a 1 followed by as many 0s as needed, second a 0 followed by as many 1s as needed. With any one of these solutions, one can easily deduce the number of cancelled bits by scanning the mantissa from right to left to detect the first 1 (or 0 respectively). The assembly instruction that performs this operation is usually named Count Trailing Zero/One. The rest of this article will focus on the first pattern (1 followed by 0s). The first 1 encountered from right to left in the mantissa will be called the significant flag.

With FP-ANR, one bit of the mantissa is used to represent the significant flag. It means that a number with a p-bit mantissa will have at most p -1 informative bits which is 1 bit less than the corresponding IEEE Standard 754 representation format which FP-ANR is built upon. For example, the value 1.0 which corresponds to the binary32 IEEE Standard 754 representation number The rightmost bit equal to 1 and corresponding to the significant flag, indicates the position between significant and insignificant bit in the mantissa. In other words, this representation corresponds to the floating-point number 1.0 accurate up to 23 bits. Alternatively, the FP-ANR representation string 0 01111111 00000000000010000000000 corresponds to the floating-point number 1.0 as well, but accurate to 13 bits. This slight modification affects the set of finite numbers as defined by the IEEE Standard 754 standard including normal and subnormal numbers. The representation format of special values which includes infinities, NaN and 0 remains unchanged as no significant flag is embedded.

The major difference between the IEEE Standard 754 representation format and the FP-ANR format is that IEEE Standard 754 can manipulate exact values such as 1.0 whereas FP-ANR deals solely with approximation (except for 0). This is a drawback as discussed in section 2.3, which is why the proposed format cannot be considered as a universal format.

Managing uncertainties

With FP-ANR, uncertainty is integrated directly in the mantissa. For example, let us consider a physical process which produces the value x = 1234.56 with an uncertainty U = 10 -5 and its representations (Table 1). With the IEEE Standard 754, there is no direct solution to integrate the information on uncertainty in the representation number. It is still possible to circumvent this problem by using interval arithmetic, but this will requires at least 2 numbers. With FPANR, the information on uncertainty is integrated by evaluating the number of significant digit, which corresponds to |log 2 (U )| = 16 bits.

As we can observe, with the IEEE Standard 754 format the value will be translated directly into its binary format where the last 8 insignificant bits correspond to noise. Whereas with FP-ANR we can distinguish significant and insignificant bits.

When all signicant bits are lost we can keep track of that information, which is not the case with the other representation format. In that case we have information on the order of magnitude of the insignificance. This concept is similar to the concept of informatical zero represented by @.0 in CADNA [START_REF] Jézéquel | CADNA: a library for estimating round-off error propagation[END_REF]. For example, let us consider the following number where all bits of the mantissa are set to 0 and only the implicit bit is set to 1. 0 01111111 00000000000000000000000 This representation number means that there are no significant bits in the mantissa. However, there is still useful embedded information which is the order of magnitude of the error stored in the exponent. This information can be used in further computation involving such a number: For example, in an addition to discard bits of weight less than the one corresponding to insignificant bit. It can potentially avoid a division by zero resulting from an unwanted catastrophic cancellation where all bits are lost.

Addition of FP-ANR

As we have seen in section 2.2.3, the least significant bits of the mantissa are usually uninformative as they corresponds to noise due to cancelation or discretization. The information on insignificant bit has to be propagated during operations. This can be done by updating the position of the significant flag found in the result of an addition between two FP-ANR as follow.

Let A, B and R be three FP-ANR numbers with respectively α A , α B and α R significant bits. The number of significant bits α R of the results R = A•B with • ∈ {+, is determined by:

α R = exp R -M AX((exp A -α A ), (exp B -α B ))
where exp X corresponds to the exponents of the FP-ANR number X with X ∈ {A, B, R}. One can notice that the quantities (exp A -α A ) and (exp B -α B ) correspond to the absolute error.

Multiplication and division of FP-ANR

Propagation of significant information during multiplication corresponds to the simplest case. The number of significant bits resulting from a multiplication between two FP-ANR numbers is estimated as follows:

Let X with X ∈ A, B, R be a FP-ANR representation of the number x with x ∈ {a, b, r} respectively, with α X significant bits. The number α R of significant bits in the results

R = A • B can be approximated by α R = M IN (α A , α B ).
This approximation corresponds to the minimal number of bit lost during cancellation and does not consider the accumulation of it.

We chose to not consider the accumulation of uncertainty as its estimation would grow too fast during uncertainty propagation. This differs from interval arithmetic that will always overestimate the error. For comparison purposes, lets consider the case of accumulation of uncertainty. The number of significant bit α X corresponds to an error e X in X is such that

X = x • (1 + e X ) with |e X | ≤ 2 -α X .
|e r | ≤ 2 -α A + 2 -α B + 2 -α A -α B .
The error e r is maximal when α A = α B . A more accurate approximation that considers the accumulation of uncertainty for the multiplication is

α R = M IN (α A , α B ) -2 when α A = α B M IN (α A , α B ) -1 when α A = α B (1)
For similar reasons, we decided to not consider the accumulation of uncertainty for the division R = A/B. The number α R of significant bits in the results R = A/B is set to α R = M IN (α A , α B ). This differs from a solution considering the accumulation of uncertainty as follows.

The error for the division can be expressed as R =

|e r | ≤ 2 -α A + 2 -α B + 2 -α A -α B .
Therefore, equation 1 corresponds to a more accurate approximation that consider the accumulation of uncertainty for the division.

Other operations using FP-ANR

We can consider the propagation of uncertainty in the case of more complex operations as well (e.g. exponential, logarithms or trigonometric function). Such functions have already been considered in previous work on significant arithmetic [START_REF] Goldstein | Significance arithmetic on a digital computer[END_REF].

Let R and X be FP-ANR representations of the number r and x respectively, with α R and α X significant bits. We would like to estimate the number of significant bits α R when R = f (X) with f a function of X.

The number of significant digits can be approximated using results on the propagation of uncertainty. It can be done by looking at the extremum on the interval of values corresponding to the initial uncertainty interval

[X • (1 - e X ); X • (1 + e X )] with e X the error in X such that |e X | ≤ 2 -α X .
This uncertainty can be estimated using a first-order Taylor series expansion. It consists in replacing the function f by its local tangent:

f (X • (1 + e X )) = f (X) + f (X) • X • e X + o(X • e X )
with o(x) a function which quickly tends toward 0. Therefore, the uncertainty in the result R can be estimated by:

e R ≈ |f (X) • X • e X |

This estimation is valid only if the function is considered quasi-linear and quasi-Gaussian on the interval

[X • (1 - e X ); X • (1 + e X )]
. This corresponds to an estimation of the number of significant bits of the result α R :

α R = log 2 f (X) f (X) • X • e X Table 2. APPROXIMATION OF THE NUMBER OF SIGNIFICANT DIGITS α R FOR SOME FUNCTIONS f (X) . R = f (X) f (X) α R ≈ √ X -1 2• √ X α X + log 2 |2| = α X + 1 exp(X) exp(X) α X + log 2 |1/X| = α X -log 2 |X| ln(X) 1/X α X + log 2 |ln(X)| sin(X) cos(X) α X + log 2 sin(X) X•cos(X) cos(X) -sin(x) α X + log 2 cos(X) X•sin(X)
Combining this equation with the estimation of the number of significant digits α X = -log 2 |e X |, we get:

α R -α X ≈ log 2 f (X) f (X) • X
where log 2 is approximated using the exponent part of its floating-point representation format. Table 2 summarizes some of these approximation of the number of significant digits α R for some functions f (X).

Rounding in FP-ANR

It should be noticed that the presence of the significant flag is independent of the rounding problem. Therefore, we propose to use similar rounding strategies with FP-ANR as done with the IEEE Standard 754 format. The only difference being the bit position, where rounding will be done. With FP-ANR, rounding is operated on the last bit of the significant part, whereas it is done on the last bit of the mantissa for the IEEE Standard 754 representation format. However, the exact impact of rounding remains to be evaluated.

FP-ANR and the Table Maker's Dilemma

In addition to the propagation of the significant flag, there is another problem regarding elementary functions: the Table Maker's Dilemma [START_REF] Lefèvre | Towards correctly rounded transcendentals[END_REF]. The Table Maker's Dilemma corresponds to the problem of computing approximations of elementary functions with enough bits to ensure correct rounding. This problem is known to be difficult with the IEEE Standard 754 representation format since there is no bound on the number of bits required for every function and every format.

With FP-ANR, the Table Maker's Dilemma is circumvented as follows. One can set a target accuracy t function of the number of significant bits α X of the input number X mandatory to evaluate the results of an elementary function. For example, one can set t = 2 • α X . If rounding can be done, then the process ends. If rouding is not possible, this corresponds to a hard to round case meaning that we are not sure of the last bit in the significant part. This uncertainty due to the Table Maker's Dilemma can be integrated in the FP-ANR format by left-shifting the significant flag one position. This way reproducibility and portability of the results provided by correct rounding is preserved.

Interaction between FP-ANR and IEEE Standard 754

One major advantage of FP-ANR is that it is compatible with the IEEE Standard 754 representation format. As with any format, compatibility can be assured thanks to conversion. Conversion between those two formats is straightforward as only the mantissa must be modified. From FP-ANR to IEEE Standard 754 format, this can be done by replacing the significant flag with a 0. From IEEE Standard 754 to FP-ANR format, this can be done by replacing the right-most bit of the mantissa by the significant flag (a 1 in the last position).

In addition to conversion, one can notice that the IEEE Standard 754 operators can process FP-ANR numbers. This will not lead to a crash or irrelevant results: it merely modifies the meaning of the insignificant bits. Nevertheless, it should not be considered as a serious issue as those bits correspond to noise. However when FP-ANR operators process IEEE Standard 754 numbers, the situation becomes more problematic, as the meaning of the resulting number depends on the position of the last bit set to 1.

Implementations

Software implementation

In this section, we describe a simplified software emulation of the proposed format. This section focuses on basic operations on the FP-ANR format related to the IEEE Standard 754 binary32 format.

Two C++ classes have been implemented to deal with single and double precision formats. These two classes are based on the header file of the CADNA library [START_REF] Jézéquel | CADNA: a library for estimating round-off error propagation[END_REF], where the code related to stochastic arithmetic is replaced with operations on significance arithmetic. This library can advantageously replace the IEEE Standard 754 formats (float, double) and major operations for those formats. It is available for download at http://perso.univ-perp.fr/david.defour/ One can notice that the biggest advantage of the FP-ANR over other solutions that require extra memory (shadow memory or extra fields), is that it could be easily integrated in a compiler pass. Indeed, memory allocation, bit manipulations (such as extraction of exponent, sign,...), tricky pointer manipulation are straightforward with the proposed format. However, such implementations is out of the scope for this article and will be developed in future work.

Conversion.

Programs in Listing 1 rely on the ieee754.h header file provided by many Linux distributions. This header file defines the type ieee754 float that eases access to the bitfield of floating-point numbers. The two functions convert a number between binary32 and the FP-ANR format by managing the significant flag according to the rules defined in section 3.8. / / Remove t h e s i g n i f i c a n t f l a g d . i e e e . s i g n i f i c a n d ˆ= 1<<c ; } * p = 22-c ; r e t u r n ( d . f ) ; } 4.1.2. Operations. We wrote a set of operations over FP-ANR numbers. Listing 2 describes how information on cancellation is propagated during addition and multiplication.

Listing 2. Functions to perform addition and multiplication over FP-ANR format f l o a t FpAnrAdd ( f l o a t a1 , f l o a t a2 ){ f l o a t r e s ; i n t e1 , e2 , e r ; i n t p1 , p2 ; r e s = F p A n r 2 F l o a t ( a1 , &p1 ) + F p A n r 2 F l o a t ( a2 , &p2 ) ;

f r e x p ( a1 , &e1 ) ; f r e x p ( a2 , &e2 ) ; f r e x p ( r e s , &e r ) ; r e t u r n F l o a t 2 F p A n r ( r e s , e r-MAX( ( e1-p1 ) , ( e2-p2 ) ) ) ; } f l o a t FpAnrMul ( f l o a t a1 , f l o a t a2 ){ f l o a t r e s ; i n t p1 , p2 ; r e s = F p A n r 2 F l o a t ( a1 , &p1 ) * F p A n r 2 F l o a t ( a2 , &p2 ) ; r e t u r n F l o a t 2 F p A n r ( r e s , MIN( p1 , p2 ) ) ; } One can notice that this simplified version implements truncation as the rounding mode. There are two solutions to implement other roundings. The first and easiest solution consists of adding a given quantity to the mantissa followed by a truncation. However, this solution is subject to the double rounding problem [START_REF] Martin-Dorel | Some issues related to double rounding[END_REF]. The second solution consists of allowing the hardware to perform the rounding at the right position in the mantissa. It can be done by right shifting the mantissa so that the least significant bit of the

Hardware implementation

Hardware implementation of the FP-ANR is more straightforward and simpler than the software solution. As the FP-ANR and the IEEE Standard 754 format are similar, FP-ANR can rely on the existing IEEE Standard 754 hardware implementation. The only difference is the introduction of the necessary hardware to manage the position of the significant flag and rounding. This requires the introduction of a trailing zero count at the input and mantissa shifters for the rounding. This operation can be done with a priority enforcer/encoder corresponding to a chain of elements with a ripple signal, scanning bit of the mantissa from right to left. The ripple signal signifies that "nothing before it" is valid and it could be replaced with a tree of OR gates to split the mantissa between its significant and insignificant part. This could be done using a carry lookahead implementation. Figure 4.2 exhibits a simple implementation of this operation based on a tree of OR gates.

Example

Cancellation can affect the convergence and accuracy of iterative numerical algorithms. As an example, let us consider Archimedes formulae to compute the approximation of π. The iteration is given by:

t 0 = 1 √ 3 , t i+1 = t 2 i + 1 -1 t i , π ≈ 6 × 2 i × t i
We have implemented this equation using IEEE-754 double precision and the 64 bits FP-ANR format. Results for iteration up to i = 27 are given in Table 3. One can notice that the accuracy of the approximations based on the IEEE-754 format is increasing up to the 13th iteration, and is slowly decreasing till the 26th iteration. Moreover, starting from the 27th iteration, a problematic Not-a-Number appears. The rightmost columns of Table 3 are reporting for the FP-ANR format both the value and the number of bits which are considered. This valuable information helps to avoid invalid operation resulting in a NaN.

On this example, one can notice that with the IEEE-754, there is no way to determine when the result is wrong and how wrong it is. Whereas with FP-ANR, the number of bit that could potentially be considered valid is known at each iterations and invalid operations can be avoided.

Comparisons with Other Methods

Performance

We have tested the overhead for the addition, multiplication and division of the proposed format compared to hardcoded IEEE Standard 754 operations and CADNA [START_REF] Eberhart | High performance numerical validation using stochastic arithmetic[END_REF] operations on an 2,4 Ghz Intel Core i5, with LLVM version 8.1.0. Results are reported in table 4. These results correspond to the implementation of the prototype library available at http://perso.univ-perp.fr/david.defour/. One can notice that the overhead of the FP-ANR over hardcoded operations range between 8.5 for the multiplication and 21 for the addition. If this overhead is higher than the one of CADNA, we should recall that the FP-ANR is intended to be implemented in hardware and therefore available at no cost.

Comparison with Unum

Recently [START_REF] Gustafson | The end of numerical error[END_REF], Gustafson proposed a modified version of significance arithmetic with an extra field (unum field) which indicates if a number is exact. However, according to William Kahan, the principal architect of IEEE 754-1985, this format presents several drawbacks [START_REF] Kahan | A critique of John L. Gustafson's. the end of error -Unum computation and his a radical approach to computation with real numbers[END_REF]. Among them, he states:

• The Unum computation does not always deliver correct results.

• The Unums can be expensive in terms of time and power consumption. • The bit length of Unum format can change during computation, which make its hardware implementation harder than with fixed-size format especially regarding memory allocation, de-allocation and accesses. The last two points are serious issues that the FP-ANR format does not exhibit. However, the Unum possesses some properties that the FP-ANR does not, such as being able to handle exact numbers.

Comparison with Stochastic arithmetic

Stochastic arithmetic provides an estimation of the numerical confidence of computed results. The CESTAC method formalizes a simplified version of discrete stochastic arithmetic using randomized rounding for each floatingpoint operation. This method is implemented using C++ overloaded operators in the CADNA library [START_REF] Jézéquel | CADNA: a library for estimating round-off error propagation[END_REF]. This library detects the number of significant digits with a high degree of confidence. It also detects instability such as cancellation, branching instability and mathematical instability. It consists of replacing each floating-point number by a set of 3 floating-point numbers plus an integer, on which stochastic operation are performed. Thanks to those extra fields, such systems provide a tighter bound than the FP-ANR format. However, similarly to the Unum format, those extra fields manipulated with the CADNA hinder memory management and performance.

Comparison with Monte-Carlo arithmetic

Another alternative to estimate numerical quality of computed result can be achieved by using the Monte-Carlo arithmetic suggested by Parker [START_REF] Parker | Monte Carlo arithmetic: exploiting randomness in floating-point arithmetic[END_REF]. Monte-Carlo arithmetic gathers rounding and catastrophic cancellation errors by applying randomization on input and output operands at a given virtual precision. A recent implementation of this solution has been proposed with Verificarlo [START_REF] Denis | Verificarlo: Checking floating point accuracy through monte carlo arithmetic[END_REF]. Verificarlo implement a LLVM pass which replaces every floating-point operation automatically with the Monte Carlo Arithmetic.

Even though Verificarlo is implemented directly as a compiler pass, which makes it very efficient, the large number of execution samples necessary to collect qualitative results remains a major drawback. The solution proposed by the authors consists of running those numerous execution in parallel. Although this solution reduces the global execution time, it does not reduce the total amount of work to gather this information.

Conclusions and Perspectives

New representation formats for floating-point numbers were introduced in the IEEE Standard 754 [-2008] revision. This is an attempt to adapt the format to the real need of applications. However, dealing with various formats require a numerical analysis of the program, which is a tedious task that can be solely executed by the expert. Some recent work has been proposed to automate this analysis and/or the benefit of formats changes.

In this article, we have presented a solution that brings the significance arithmetic up-to-date, and makes it compatible with the IEEE Standard 754 [-2008]. Significance arithmetic is a concept that adds information on significant digits to each floating-point number. It can provide information on cancellation errors, and if sufficiently accurate, on rounding error. It consists of a representation format with rules for the propagation of error.

The proposed solution is a simple pattern embedded in the mantissa of floating-point numbers. This pattern is self-sufficient and does not require extra fields or memory. This solution presents numerous advantages as it is a simple concept to understand, simple to implement and proves to be memory efficient. Tests on a preliminary version shows that the cost for the detection in software of the proposed pattern is higher compared to other solutions. However, the simplicity of the solution suggests that the performance could be improved using hardware support similar to the management of the rounding modes (e.g. specific instructions or execution flag).

If implemented in hardware, this solution can definitely help developers gain confidence in their code by providing an estimation of the number of significance digits at no cost or help achieve reproducibility. However, it is not meant to solve all problems related to floating-point arithmetic. Significance arithmetics is similar to the interval arithmetic, produces over-pessimistic bound as results and is unable to solve the loss of correlation between variables. For example, error computation as used in compensated algorithm cannot be evaluated with the significance arithmetic, whereas it works perfectly with the IEEE Standard 754 floating-point arithmetic. For these reasons, we suggest that the FP-ANR format should be used as a complement to the traditional IEEE Standard 754 floatingpoint arithmetic.

  The error for the multiplication R = A • B corresponds to R = (a • b) • (1 + e a + e b + e a • e b ) and the error term e r = e a + e b + e a • e b is such that

  1+e b . This formula can be rewritten by expressing the denominator term for the error as an infinite series R = a b • (1 + e a ) • (1 -e b + e 2 b + ...). Since the error e b is required by the number format to be less than 1, e 2 b and all the higher order terms can be neglected. The error term for the division e r = e a -e b -e a • e b is such that

Listing 1 .

 1 Functions to convert between FP-ANR and binary32 format# i n c l u d e <i e e e 7 5 4 . h> / / C o n v e r t a b i n a r y 3 2 number f / / t o a p b i t s FP-ANR number f l o a t F l o a t 2 F p A n r ( f l o a t f , i n t p ){ union i e e e 7 5 4 f l o a t d ; d . f = f ; p r e c = MIN ( 2 2 , p ) ; d . i e e e . s i g n i f i c a n d &= ( 0 x7FFFFF<<(23-p ) ) ; / / S e t t h e s i g n i f i c a n t f l a g d . i e e e . s i g n i f i c a n d |= 1<<(22-p ) ; r e t u r n d . f ; } / / C o n v e r t a p b i t s FP-ANR number / / t o a b i n a r y 3 2 number f l o a t F p A n r 2 F l o a t ( f l o a t f , i n t * p ){ union i e e e 7 5 4 f l o a t d ; i n t c ; d . f = t h i s ->v a l u e ; i f ( d . i e e e . s i g n i f i c a n d ! = 0 ) { c = c o u n t t r a i l i n g z e r o s ( d . i e e e . s i g n i f i c a n d ) ;

Figure 1 .

 1 Figure 1. Generation of the significant flag from a mantissa in FP-ANR format based on a tree of OR gate.

Table 1 .

 1 BINARY REPRESENTATION OF THE VALUE 1234.56 WITH AN UNCERTAINTY 10 -3 %

	binary32	0 10001001 00110100101000111101100
	FP-ANR	0 10001001 00110100101000110000000
	0 01111111 00000000000000000000000 will be represented
	in the FP-ANR format by
	0 01111111 00000000000000000000001

Table 3 .

 3 COMPARISON BETWEEN THE IEEE-754 DOUBLE PRECISION FORMAT AND THE FP-ANR FORMAT FOR THE COMPUTATION OF π DECIMALS USING ARCHIMEDES FORMULAE. BOLD NUMBERS CORRESPOND TO VALID DECIMALS.

	Iter.	IEEE-754	FP-ANR (value ; α R )
	0	3.464101615137755e+00	3.464101615137754e+00	52
	1	3.215390309173475e+00	3.215390309173465e+00	49
	2	3.159659942097510e+00	3.159659942097420e+00	47
	3	3.146086215131467e+00	3.146086215131277e+00	45
	4	3.142714599645573e+00	3.142714599644023e+00	43
	5	3.141873049979866e+00	3.141873049977221e+00	41
	6	3.141662747055068e+00	3.141662747046212e+00	39
	7	3.141610176599522e+00	3.141610176535323e+00	37
	8	3.141597034323337e+00	3.141597034060396e+00	35
	9	3.141593748816856e+00	3.141593747306615e+00	33
	10	3.141592927873633e+00	3.141592921689153e+00	31
	11	3.141592725622592e+00	3.141592703759670e+00	29
	12	3.141592671741545e+00	3.141592592000961e+00	27
	13	3.141592618900886e+00	3.141592383384705e+00	25
	14	3.141592671741545e+00	3.141592025756836e+00	23
	15	3.141591935881973e+00	3.141588211059570e+00	21
	16	3.141592671741545e+00	3.141571044921875e+00	19
	17	3.141581007579364e+00	3.141540527343750e+00	17
	18	3.141592671741545e+00	3.141357421875000e+00	15
	19	3.141406154737622e+00	3.140625000000000e+00	13
	20	3.140543492401100e+00	3.136718750000000e+00	11
	21	3.140006864690968e+00	3.125000000000000e+00	9
	22	3.134945375658852e+00	3.093750000000000e+00	7
	23	3.140006864690968e+00	3.000000000000000e+00	5
	24	3.224515243534819e+00	3.000000000000000e+00	3
	25	2.791117213058638e+00	2.000000000000000e+00	1
	26	0.000000000000000e+00	ERR(2ˆ1)	0
	27	NaN	ERR(2ˆ1)	0

Table 4 .

 4 EXECUTION TIME OF COMMON OPERATIONS IN THE FP-ANR AND THE CADNA FORMAT NORMALIZED WITH THE IEEE STANDARD 754 OPERATIONS.

		double	float
	Operations	FP-ANR	CADNA	FP-ANR	CADNA
	Addition	11	7.5	11.3	15
	Multiplication	4.7	3.5	3.96	4.0
	Division	6.1	5.0	7.8	14.2


Philippe Laurenc ¸ot
email: laurenco@math.univ-toulouse.fr

MASS-CONSERVING SOLUTIONS TO COAGULATION-FRAGMENTATION EQUATIONS WITH NON-INTEGRABLE FRAGMENT DISTRIBUTION FUNCTION

Keywords: 1991 Mathematics Subject Classification, 45K05, Key words and phrases, coagulation -multiple fragmentation -non-integrable fragment distribution -conservation of matter

Existence of mass-conserving weak solutions to the coagulation-fragmentation equation is established when the fragmentation mechanism produces an infinite number of fragments after splitting. The coagulation kernel is assumed to increase at most linearly for large sizes and no assumption is made on the growth of the overall fragmentation rate for large sizes. However, they are both required to vanish for small sizes at a rate which is prescribed by the (non-integrable) singularity of the fragment distribution.

Introduction

A mean-field description of the dynamics of a system of particles varying their sizes by pairwise coalescence and multiple fragmentation is provided by the coagulationfragmentation equation

∂ t f (t, x) = Cf (t, x) + F f (t, x) , (t, x) ∈ (0, ∞) 2 , (1.1a)
where the coagulation and fragmentation reaction terms are given by Cf (x) := 1 2

x 0 K(x -y, y)f (y)f (x -y) dy - for x ∈ (0, ∞). We supplement (1.1) with an initial condition

f (0, x) = f in (x) , x ∈ (0, ∞) . (1.2)
In (1.1), f = f (t, x) denotes the size distribution function of the particles of size x ∈ (0, ∞) at time t > 0 and K is the coagulation kernel which describes how likely particles of respective sizes x and y merge. The first term in the coagulation term (1.1b) accounts for the formation of particles of size x resulting from the coalescence of two particles of respective sizes y ∈ (0, x) and x -y, while the second term describes the disappearance of particles of size x as they merge with other particles of arbitrary size. The first term in the fragmentation term (1.1c) involves the overall fragmentation rate a(x) and accounts for the loss of particles of size x due to breakup while the second term describes the contribution to particles of size x of the fragments resulting from the splitting of a particle of arbitrary size y > x, the distribution of fragments being given according to the daughter distribution function b(x, y). Since no matter is lost during fragmentation events, b is required to satisfy y 0 xb(x, y) dx = y , y > 0 .

(1.3)

As the merging of two particles of respective sizes x and y results in a particle of size x + y, it is then expected that conservation of matter holds true during time evolution, that is,

∞ 0 xf (t, x) dx = ̺ := ∞ 0 xf in (x)dx , t ≥ 0 , (1.4)
provided ̺ is finite. Since the pioneering works by Melzak [START_REF] Melzak | A scalar transport equation[END_REF], McLeod [START_REF] Mcleod | On the scalar transport equation[END_REF], and Stewart [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF], the existence of weak solutions to (1.1)-(1.2) is investigated in several papers under various assumptions on the coagulation and fragmentation coefficients K, a, and b and the initial condition f in [START_REF] Banasiak | Global strict solutions to continuous coagulation-fragmentation equations with strong fragmentation[END_REF][START_REF]Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates[END_REF][START_REF] Banasiak | Strong fragmentation and coagulation with powerlaw rates[END_REF][START_REF] Dubovskiȋ | Existence, uniqueness and mass conservation for the coagulation-fragmentation equation[END_REF][START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF][START_REF] Giri | The continuous coagulation equation with multiple fragmentation[END_REF][START_REF] Giri | Weak solutions to the continuous coagulation equation with multiple fragmentation[END_REF][START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF][START_REF] Laurenc | From the discrete to the continuous coagulationfragmentation equations[END_REF], see also [START_REF] Ball | The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation[END_REF][START_REF] Da Costa | Existence and uniqueness of density conserving solutions to the coagulationfragmentation equations with strong fragmentation[END_REF][START_REF]The discrete coagulation equations with multiple fragmentation[END_REF][START_REF] Leyvraz | Singularities in the kinetics of coagulation processes[END_REF][START_REF] White | A global existence theorem for Smoluchowski's coagulation equations[END_REF] for the discrete coagulation-fragmentation equation. Assuming in particular the initial condition f in to be a non-negative function in L 1 ((0, ∞), xdx), the solutions to (1.1)-(1.2) constructed in the above mentioned references are either mass-conserving, i.e. satisfy (1.4), or not, according to the growth of the coagulation kernel K for large sizes or the behaviour of the overall fragmentation rate a for small sizes. Indeed, it is by now well-known that, if the growth of the coagulation kernel for large sizes is sufficiently fast (such as K(x, y) ≥ K 0 (xy) λ/2 for some λ > 1 and K 0 ), then a runaway growth takes place in the system of particles and a giant particle (or particle of infinite size) is formed in finite time, a phenomenon usually referred to as gelation [START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF][START_REF] Escobedo | Gelation in coagulation and fragmentation models[END_REF][START_REF] Hendriks | Coagulation equations with gelation[END_REF][START_REF] Leyvraz | Existence and properties of post-gel solutions for the kinetic equations of coagulation[END_REF][START_REF] Leyvraz | Singularities in the kinetics of coagulation processes[END_REF]. Since all particles accounted for by the size distribution have finite size, the occurrence of gelation results in a loss of matter in the dynamical behaviour of (1.1). A somewhat opposite phenomenon takes place when the overall fragmentation rate a blows up as x → 0 (such as a(x) = a 0 x γ for some γ < 0 and a 0 > 0). In that case, the smaller the particles, the faster they split, leading to the instantaneous appearance of dust (or particles of size zero) and again a loss of matter takes place, usually referred to as the shattering transition [START_REF] Arlotti | Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss[END_REF][START_REF] Filippov | On the distribution of the sizes of particles which undergo splitting[END_REF][START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF]. We shall exclude the occurrence of these phenomena in the forthcoming analysis and focus on a different feature of the fragmentation mechanism, namely the possibility that an infinite number of fragments is produced during breakup. Specifically, a common assumption on the daughter distribution function b in the above mentioned references is its integrability, that is,

n 0 (y) := y 0 b(x, y) dx < ∞ for almost every y > 0 , (1.5)
which amounts to require that the splitting of a particle of size y produces n 0 (y) daughter particles, a particularly important case being the so-called binary fragmentation corresponding to n 0 (y) = 2 for all y > 0. Clearly, the integrability property (1.5) fails to be true when b is given by [START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF] b ν (x, y) = (ν + 2)

x ν y ν+1 , 0 < x < y , ν ∈ (-2, -1] . (1.6)
Observe that the restriction ν > -2 guarantees that b ν satisfies (1.3).

As far as we know, the existence of weak solutions to the coagulation-fragmentation equation (1.1)-(1.2) when the daughter distribution function b is given by (1.6) and the coagulation kernel is unbounded has not been considered so far, except the particular case K(x, y) = xy, a(x) = x, and ν = -1 which is handled in [START_REF] Laurenc | Absence of gelation and self-similar behavior for a coagulation-fragmentation equation[END_REF] by an approach exploiting fully the specific structure of the coefficients. The purpose of this note is to fill this gap, at least for coagulation kernels growing at most linearly for large sizes. In fact, the main difficulty to be overcome is the following: due to the non-integrability of b ν , the natural functional setting, which is the space L 1 ((0, ∞), (1 + x)dx), can no longer be used. As we shall see below, it might be replaced by the smaller space L 1 ((0, ∞), (x m + x)dx) for some m ∈ (0, 1) such that

y 0 x m b(x, y) dx = (ν + 2)y m 1 0 z m+ν dz < ∞ , y > 0 ,
that is, m > -1 -ν ≥ 0. This choice requires however the coagulation kernel K and the overall fragmentation rate a to vanish in an appropriate way for small sizes. More precisely, we assume that there are K 0 > 0 and m 0 ∈ (-1 -ν, 1) such that

0 ≤ K(x, y) = K(y, x) ≤ K 0 (2 + x + y) , (x, y) ∈ (0, ∞) 2 , (1.7a)
and, for all R > 0,

L R := sup (x,y)∈(0,R) 2 K(x, y) min{x, y} m 0 < ∞ . (1.7b)
We also assume that, for all R > 0, there is

A R > 0 such that a(x) ≤ A R x m 0 +ν+1 , x ∈ (0, R) . (1.8)
Roughly speaking, K and a are required to vanish faster for small sizes as the singularity of b ν . For instance, the coagulation kernel K and the homogeneous overall fragmentation rate a given by

K(x, y) = x α y β + x β y α , a(x) = x γ , (x, y) ∈ (0, ∞) 2 , (1.9)
with -1 -ν < α ≤ β ≤ 1 -α and γ > 0 satisfy (1.7) and (1.8), respectively, for any

m 0 ∈ (-1 -ν, max{α, γ -1 -ν}].
Observe in particular that no growth constraint is required on a for large sizes. Before stating our results, let us introduce some notation: given m ∈ R, we define the space X m and its positive cone X + m by X m := L 1 ((0, ∞), x m dx) , X + m := {f ∈ X m : f ≥ 0 a.e. in (0, ∞)} , respectively, and denote the space X m endowed with its weak topology by X m,w . We also put

M m (f ) := ∞ 0 x m f (x) dx , f ∈ X m .
We begin with the existence of mass-conserving weak solutions to (1.1)-(1.2) when the initial condition f in lies in X + m 0 ∩ X 1 for some m 0 ∈ (-ν -1, 1) and decays in a suitable way for large sizes.

Theorem 1.1. Let ν ∈ (-2, -1]. Assume that the coagulation kernel K and the overall fragmentation rate a satisfy (1.7) and (1.8), respectively, and that the daughter distribution function b = b ν is given by (1.6). Given an initial condition

f in ∈ X + m 0 ∩ X 1 such that ∞ 0 x ln(ln(x + 5))f in (x) dx < ∞ , (1.10)
there exists at least one weak solution

f ∈ C([0, ∞); X m 0 ,w ∩ X 1,w ) on [0, ∞) to (1.1)-(1.2) such that M 1 (f (t)) = M 1 (f in ) , t ≥ 0 . (1.11) Moreover, if f in ∈ X m for some m > 1, then f ∈ L ∞ (0, T ; X m ) for all T > 0.
When b = b ν and ν > -1, the existence of mass-conserving weak solutions on [0, ∞) to (1.1)-(1.2) is already known for f in ∈ X + 0 ∩ X 1 , the assumptions (1.7b) and (1.8) being replaced by the local boundedness of K and a [START_REF] Laurenc | From the discrete to the continuous coagulationfragmentation equations[END_REF][START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF]. In particular, it does not require the additional integrability condition (1.10) on f in and it is yet unclear whether Theorem 1.1 is valid under the sole assumption f in ∈ X + 0 ∩ X 1 . In fact, the condition (1.10) can be replaced by f in ∈ L 1 ((0, ∞), w(x)dx) for any weight function w enjoying the properties listed in Lemma 2.4 below, which includes weights involving multiple iterates of the logarithm function. We nevertheless restrict our analysis to the specific choice of the weight function in (1.10) as we have not yet identified an "optimal" class of initial conditions for which Theorem 1.1 is true.

The proof of Theorem 1.1 proceeds along the lines of the existence proofs performed in [START_REF] Laurenc | From the discrete to the continuous coagulationfragmentation equations[END_REF][START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF], the main differences lying in the control of the behaviour for small sizes and its consequences on the behaviour for large sizes. It relies on a weak compactness method in X m 0 , the starting point being the construction of a truncated version of (1.1) for which well-posedness can be established by a classical Banach fixed point argument. The compactness approach involves five steps: we first show that the additional integrability assumption (1.10) is preserved throughout time evolution and additionally provides a control on the behaviour of the fragmentation term for large sizes whatever the growth of the overall fragmentation rate a. We next turn to the behaviour for small sizes and prove boundedness in X m 0 , the outcome of the previous step being used to control the contribution of the fragmentation gain term. The third and fourth steps are more classical and devoted to the uniform integrability and the time equicontinuity in X m 0 , respectively. In the last step, we gather the outcome of the four previous ones to show the convergence of the solutions of the truncated problem as the truncation parameter increases without bound and that the limit thus obtained is a weak solution on [0, ∞) to (1.1)-(1.2) which satisfies the conservation of matter (1.11). We finally use the same argument as in the first step to prove the stability of X m for m > 1.

Remark 1.2. For simplicity, the analysis carried out in this paper is restricted to the daughter distribution functions given by (1.6). The proof of Theorem 1.1 can however be adapted to encompass a broader class of daughter distribution functions b featuring a non-integrable singularity for small fragment sizes, see [START_REF] Banasiak | Analytic methods for coagulationfragmentation models[END_REF]. In the same vein, existence of weak solutions (not necessarily mass-conserving) can be proved by a similar approach when the coagulation kernel K features a faster growth than (1.7a) and we refer to [START_REF] Banasiak | Analytic methods for coagulationfragmentation models[END_REF] for a more complete account.

We supplement the existence result with a uniqueness result, which is however only valid for a smaller class of coagulation kernels K and initial conditions f in . Theorem 1.3. Let ν ∈ (-2, -1] and δ ∈ (0, 1). Assume that the coagulation kernel K and the overall fragmentation rate a satisfy (1.7) and (1.8), respectively, and that the daughter distribution function b = b ν is given by (1.6). Assume also that there is

K 1 > 0 such that K(x, y) ≤ K 1 x m 0 y , (x, y) ∈ (0, 1) × (1, ∞) .
(1.12)

Given an initial condition f in ∈ X + m 0 ∩ X 2+δ , there is a unique mass-conserving weak solution f on [0, ∞) to (1.1)-(1.2) such that f ∈ L ∞ (0, T ; X 2+δ ) for all T > 0.
Observing that any function f in ∈ X + m 0 ∩ X 2+δ satisfies (1.10), the existence part of Theorem 1.3 readily follows from Theorem 1.1. As Theorem 1.1, Theorem 1.3 applies to (1.1)-(1.2) when the coagulation kernel K and the overall fragmentation rate a are given by (1.9) with -1 -ν < α ≤ β ≤ 1 -α, γ > 0, and m 0 ∈ (-1 -ν, max{α, γ -1 -ν}]. We also mention that, when b = b ν and ν > -1, Theorem 1.3 is valid without the assumption (1.12) on K and for f in ∈ X + 0 ∩ X 2 [START_REF] Laurenc | On coalescence equations and related models[END_REF][START_REF] Norris | Smoluchowski's coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent[END_REF].

As in [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Laurenc | On coalescence equations and related models[END_REF][START_REF] Norris | Smoluchowski's coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent[END_REF][START_REF]A uniqueness theorem for the coagulation-fragmentation equation[END_REF], the uniqueness proof relies on a control of the distance between two solutions in a weighted L 1 -space, the delicate point being the choice of an appropriate weight which turns out to be ξ(x) := max{x m 0 , x 1+δ }, x > 0, here. The superlinearity of ξ for large sizes compensates the sublinearity of ξ for small sizes which gives a positive contribution of the fragmentation term.

Existence

Throughout this section, the parameter ν ∈ (-2, -1] is fixed and we assume that the coagulation kernel K and the overall fragmentation rate a satisfy (1.7) and (1.8), respectively, while the daughter distribution function b = b ν is given by (1.6). Also, f in is a function in X + m 0 ∩ X 1 enjoying the additional integrability property (1.10) and we set ̺ := M 1 (f in ).

Preliminaries. Let us first begin with the definition of a weak solution to

(1.1)-(1.2). Definition 2.1. Let T ∈ (0, T ]. A weak solution on [0, T ) to (1.1)-(1.2) is a non- negative function f ∈ C([0, ∞); X m 0 ,w ∩ X 1,w ) on [0, ∞) such that, for all t ∈ (0, T ) and ϑ ∈ Θ m 0 , ∞ 0 (f (t, x) -f in (x))ϑ(x) dx = 1 2 ∞ 0 ∞ 0 K(x, y)χ ϑ (x, y)f (t, x)f (t, y) dydx - ∞ 0 a(x)N ϑ (x)f (t, x) dx , (2.1)
where

χ ϑ (x, y) := χ(x + y) -χ(x) -χ(y) , (x, y) ∈ (0, ∞) 2 , N ϑ (y) := ϑ(y) - y 0 ϑ(x)b ν (x, y) dx , y > 0 ,
and

Θ m 0 := {ϑ ∈ C m 0 ([0, ∞)) ∩ L ∞ (0, ∞) : ϑ(0) = 0} . Observe that, for ϑ ∈ Θ m 0 and (x, y) ∈ (0, ∞) 2 , |χ ϑ (x, y)| ≤ 2 ϑ C m 0 min{x, y} m 0 and |N ϑ (y)| ≤ 1 + ν + 2 ν + m 0 + 1 ϑ C m 0 y m 0 ,
so that the right-hand side of (2.1) is well-defined. We next define a particular class of convex functions on [0, ∞) which proves useful in the forthcoming analysis.

Definition 2.2. A non-negative and convex function

ϕ ∈ C ∞ ([0, ∞)) belongs to the class C V P,∞ if it satisfies the following properties: (a) ϕ(0) = ϕ ′ (0) = 0 and ϕ ′ is concave; (b) lim r→∞ ϕ ′ (r) = lim r→∞ ϕ(r)/r = ∞; (c) for p ∈ (1, 2), S p (ϕ) := sup r≥0 ϕ(r) r p < ∞ .
For instance, r → (r + 1) ln(r + 1) -r lies in C V P,∞ . Functions in the class C V P,∞ enjoy several properties which we list now.

Lemma 2.3. Let ϕ ∈ C V P,∞ . Then (a) r → ϕ(r)/r is concave on [0, ∞); (b) for r ≥ 0, 0 ≤ ϕ(r) ≤ rϕ ′ (r) ≤ 2ϕ(r) ; (c) for (r, s) ∈ [0, ∞) 2 , sϕ ′ (r) ≤ ϕ(r) + ϕ(s) , (2.2)
and

0 ≤ ϕ(r + s) -ϕ(r) -ϕ(s) ≤ 2 sϕ(r) + rϕ(s) r + s , (2.3)
0 ≤ ϕ(r + s) -ϕ(r) -ϕ(s) ≤ ϕ ′′ (0)rs .

(2.4)

Proof. Since ϕ(r + s) -ϕ(r) -ϕ(s) = r 0 s 0 ϕ ′′ (r * + s * ) ds * dr * , (r, s) ∈ [0, ∞) 2 ,
the inequality (2.4) readily follows from the concavity of ϕ ′ . The other properties listed in Lemma 2.3 are consequences of the convexity of ϕ and the concavity of ϕ ′ and are proved, for instance, in [19, Proposition 14].

We finally collect some properties of the weight involved in the assumption (1.10).

Lemma 2.4. Define W (x) := x ln(ln(x+5))-x ln(ln(5)) for x ≥ 0. Then W ∈ C V P,∞ and, for all m ∈ [0, 1),

lim x→∞ xW ′ (x) -W (x) x m = ∞ .
2.2. Approximation. Let j ≥ 2 be an integer and set

K j (x, y) := K(x, y)1 (0,j) (x + y) , a j (x) := a(x)1 (0,j) (x) , (2.5)
and

f in j (x) := f in (x)1 (0,j) (x) (2.6)
for (x, y) ∈ (0, ∞) 2 . We denote the coagulation and fragmentation operators with K j and a j instead of K and a by C j and F j , respectively. Simple computations along with (1.7b) and the boundedness of a j resulting from (1.8) show that both C j and F j are locally Lipschitz continuous from L 1 ((0, j), x m 0 dx) into itself [START_REF] Banasiak | Analytic methods for coagulationfragmentation models[END_REF]. Thanks to these properties, we are in a position to apply the Banach fixed point theorem to prove that there is a unique non-negative function

f j ∈ C 1 ([0, ∞), L 1 ((0, j), x m 0 dx)) solving ∂ t f j (t, x) = C j f j (t, x) + F j f j (t, x) , (t, x) ∈ (0, ∞) × (0, j) , (2.7a
)

f j (0, x) = f in j (x) , x ∈ (0, j) . (2.7b)
Introducing the space Θ m 0 ,j := {ϑ ∈ C m 0 ([0, j]) : ϑ(0) = 0}, it readily follows from (2.5) and (2.7) that f j satisfies the following weak formulation of (2.7): for all ϑ ∈ Θ m 0 ,j and t > 0, d dt

j 0 ϑ(x)f j (t, x) dx = 1 2 j 0 j-x 0 K(x, y)χ ϑ (x, y)f j (t, x)f j (t, y) dydx - j 0 a(x)N ϑ (x)f j (t, x) dx , (2.8)
the functions χ ϑ and N ϑ being defined in Definition 2.1. Choosing ϑ(x) = x, x ∈ (0, j), in (2.8) readily gives the conservation of matter

j 0 xf j (t, x) dx = j 0 xf in (x) dx , t ≥ 0 . Extending f j to [0, ∞) × (j, ∞) by zero (i.e. f j (t, x) = 0 for (t, x) ∈ [0, ∞) × (j, ∞)),
the previous identity reads

M 1 (f j (t)) = M 1 (f in j ) ≤ ̺ = M 1 (f in ) , t ≥ 0 .
(2.9)

In addition, introducing

C 0 := M m 0 (f in ) + ∞ 0 W (x)f in (x) dx < ∞ , (2.10)
which is finite according to the integrability properties of f in and in particular (1.10), we infer from (2.6), (2.10), and the non-negativity of W that

M m 0 (f in j ) + ∞ 0 W (x)f in j (x) dx ≤ C 0 . (2.11)
We now investigate the weak compactness features of the sequence (f j ) j≥2 . In the sequel, C and C i , i ≥ 1, denote positive constants depending on K, a, ν, m 0 , f in , ̺, and C 0 . Dependence upon additional parameters is indicated explicitly.

Moment Estimates.

We start with a control on the behaviour for large sizes.

Lemma 2.5. Let T > 0. There is C 1 (T ) > 0 depending on T such that, for t ∈ [0, T ], ∞ 0 W (x)f j (t, x) dx + t 0 ∞ 0 a(x)[xW ′ (x) -W (x)]f j (s, x) dxds ≤ C 1 (T ) .
Proof. On the one hand, since W ∈ C V P,∞ by Lemma 2.4, we infer from (1.7a), (2.3), and (2.4) that,

K(x, y)χ W (x, y) ≤ 2K 0 W ′′ (0)xy + 2K 0 [xW (y) + yW (x)]
(2.12)

for (x, y) ∈ (0, ∞) 2 . On the other hand, the function W 1 : x → W (x)/x is concave according to Lemma 2.3 (a) and it follows from (1.6) that, for y > 0,

N W (y) = y 0 [W 1 (y) -W 1 (x)]xb ν (x, y) dx ≥ y 0 W ′ 1 (y)x(y -x)b ν (x, y) dx ≥ yW ′ (y) -W (y) y 2 y 0 x(y -x)b ν (x, y) dx = yW ′ (y) -W (y) ν + 3 . (2.13)
Combining (2.8) with ϑ = W , (2.12), and (2.13) gives, for t > 0,

d dt ∞ 0 W (x)f j (t, x) dx ≤ K 0 j 0 j-x 0 [W ′′ (0)xy + xW (y) + yW (x)] f j (t, x)f j (t, y) dydx - 1 ν + 3 j 0 a(y)[yW ′ (y) -W (y)]f j (t, y) dy .
We further deduce from Lemma 2.3 (b) and (2.9) that d dt

∞ 0 W (x)f j (t, x) dx + 1 2 ∞ 0 a(y)[yW ′ (y) -W (y)]f j (t, y) dy ≤ K 0 W ′′ (0)̺ 2 + 2K 0 ̺ ∞ 0 W (x)f j (t, x) dx ,
Integrating the previous differential inequality and using (2.11) complete the proof.

Exploiting the properties of W for large sizes along with the outcome of Lemma 2.5 provides additional information on the fragmentation term.

Corollary 2.6. Let T > 0 and m ∈ (0, 1). There is C 2 (m, T ) > 0 depending on m and T such that

T 0 P m,j (s)ds ≤ C 2 (m, T ) , P m,j (s) := ∞ 1 x m a(x)f j (s, x) dx , s ∈ [0, T ] .
Proof. Owing to Lemma 2.4, there is x m > 1 such that xW ′ (x) -W (x) ≥ x m for x > x m . Therefore, by (1.8), (2.9), Lemma 2.3 (b), and Lemma 2.5,

T 0 ∞ 1 x m a(x)f j (s, x) dxds ≤ T 0 xm 1 x m a(x)f j (s, x) dxds + T 0 ∞ xm x m a(x)f j (s, x) dxds ≤ A xm x m 0 +ν+1 m T 0 xm 1 x m f j (s, x) dxds + T 0 ∞ xm [xW ′ (x) -W (x)]a(x)f j (s, x) dxds ≤ A xm x m 0 +ν+1 m ̺T + C 1 (T ) ,
and the proof is complete.

We now study the behaviour for small sizes.

Lemma 2.7. Let T > 0. There is C 3 (T ) > 0 depending on T such that

M m 0 (f j (t)) ≤ C 3 (T ) , t ∈ [0, T ] .
Proof. Setting ϑ 0 (x) := min{x, x m 0 } for x > 0, we observe that

χ ϑ 0 (x, y) ≤ 0 , (x, y) ∈ (0, ∞) 2 ,
and

N ϑ 0 (y) = - 1 -m 0 ν + m 0 + 1 y m 0 , y ∈ (0, 1) , N ϑ 0 (y) = - 1 -m 0 ν + m 0 + 1 1 y ν+1 ≥ - 1 -m 0 ν + m 0 + 1 y m 0 , y ∈ (1, ∞) .
We then infer from (1.8) with R = 1 and (2.8) with ϑ = ϑ 0 that

d dt M m 0 (f j (t)) ≤ 1 -m 0 ν + m 0 + 1 ∞ 0 y m 0 a(y)f j (t, y) dy ≤ A 1 ν + m 0 + 1 1 0 y m 0 f j (t, y) dy + P m 0 ,j (t) ν + m 0 + 1 ≤ A 1 ν + m 0 + 1 M m 0 (f j (t)) + P m 0 ,j (t) ν + m 0 + 1 .
We integrate the previous differential inequality and complete the proof with the help of Corollary 2.6.

2.4. Uniform Integrability. The previously established estimates guarantee that there is no unlimited growth for small sizes nor escape towards large sizes during the evolution of (f j ) j≥2 . To achieve weak compactness in X m 0 , it remains to prevent concentration near a finite size. For that purpose, we recall that, since f in ∈ X m 0 , a refined version of the de la Vallée Poussin theorem ensures that there is Φ ∈ C V P,∞ depending only on f in such that

I := ∞ 0 x m 0 Φ(f in (x)) dx < ∞ , (2.14)
see [START_REF] Banasiak | Analytic methods for coagulationfragmentation models[END_REF][START_REF] De | Sur l'intégrale de Lebesgue[END_REF][START_REF] Hoàn | Etude de la classe des opérateur m-accrétifs de L 1 (Ω) et accrétif dans L ∞ (Ω)[END_REF][START_REF] Ph | Weak compactness techniques and coagulation equations[END_REF].

Lemma 2.8. Let T > 0 and R > 1. There is C 4 (T, R) > 0 depending on T and R such that

R 0 x m 0 Φ(f j (t, x)) dx ≤ C 4 (T, R) , t ∈ [0, T ] ,
the function Φ being defined in (2.14).

Proof. We combine arguments from [START_REF] Laurenc | From the discrete to the continuous coagulationfragmentation equations[END_REF] with the subadditivity of x → x m 0 . Let T > 0, R > 1, and t ∈ [0, T ]. On the one hand,

I 1,j (R, t) := R 0 x m 0 Φ ′ (f j (t, x))C j f j (t, x) dx ≤ 1 2 R 0 x 0 x m 0 K(x -y, y)Φ ′ (f j (t, x))f j (t, x -y)f j (t, y)dydx ,
and, by Fubini's theorem,

I 1,j (R, t) ≤ 1 2 R 0 R-y 0 (x + y) m 0 K(x, y)Φ ′ (f j (t, x + y))f j (t, x)f j (t, y)dxdy .
Owing to the subadditivity of x → x m 0 and the symmetry of K, we further obtain

I 1,j (R, t) ≤ 1 2 R 0 R-y 0 x m 0 K(x, y)Φ ′ (f j (t, x + y))f j (t, x)f j (t, y)dxdy + 1 2 R 0 R-y 0 y m 0 K(x, y)Φ ′ (f j (t, x + y))f j (t, x)f j (t, y)dxdy ≤ 1 2 R 0 R-x 0 y m 0 K(x, y)Φ ′ (f j (t, x + y))f j (t, x)f j (t, y)dydx + 1 2 R 0 R-x 0 y m 0 K(x, y)Φ ′ (f j (t, x + y))f j (t, x)f j (t, y)dydx ≤ R 0 R-x 0 y m 0 K(x, y)Φ ′ (f j (t, x + y))f j (t, x)f j (t, y)dydx .
Since Φ ∈ C V P,∞ , we infer from (2.2) with r = f j (t, x + y) and s = f j (t, y) that

I 1,j (R, t) ≤ R 0 R-x 0 y m 0 K(x, y) [Φ(f j (t, x + y)) + Φ(f j (t, y))] f j (t, x)dydx ≤ R 0 R x (y -x) m 0 K(x, y -x)Φ(f j (t, y))f j (t, x)dydx + R 0 R-x 0 y m 0 K(x, y)Φ(f j (t, y))f j (t, x)dydx .
We finally use (1.7b) to conclude that

I 1,j (R, t) ≤ 2L R M m 0 (f j (t)) R 0 y m 0 Φ(f j (t, y)) dy . (2.15)
On the other hand, we fix p 0 > 1 satisfying

1 < p 0 < m 0 + 1 -ν ≤ 1 + m 0 -ν < 2 ,
which is possible as m 0 + 1 > -ν. Using again Fubini's theorem,

I 2,j (R, t) := R 0 x m 0 Φ ′ (f j (t, x))F j f j (t, x) dx ≤ R 0 ∞ x x m 0 a(y)Φ ′ (f j (t, x))b ν (x, y)f j (t, y)dydx ≤ ∞ 0 a(y)f j (t, y) min{y,R} 0 x m 0 b ν (x, y)Φ ′ (f j (t, x)) dxdy .
Since Φ ∈ C V P,∞ , it follows from (1.6), Definition 2.2 (c) with p = p 0 , and (2.2) with r = f j (t, x) and s = x ν that

I 2,j (R, t) ≤ (ν + 2) ∞ 0 a(y)y -ν-1 f j (t, y) min{y,R} 0 x m 0 [Φ(x ν ) + Φ(f j (t, x))] dxdy ≤ (ν + 2)S p 0 (Φ) ∞ 0 a(y)y -ν-1 f j (t, y) y 0 x m 0 +νp 0 dxdy + (ν + 2) ∞ 0 a(y)y -ν-1 f j (t, y) R 0 x m 0 Φ(f j (t, x)) dxdy ≤ (ν + 2)S p 0 (Φ) m 0 + νp 0 + 1 ∞ 0 a(y)y m 0 +ν(p 0 -1) f j (t, y) dy + (ν + 2) ∞ 0 a(y)y -ν-1 f j (t, y) dy R 0 x m 0 Φ(f j (t, x)) dx .
Since m 0 + ν(p 0 -1) and -ν -1 both belong to [0, 1) and m 0 + νp 0 + 1 > 0, we deduce from (1.8) and Lemma 2.7 that

I 2,j (R, t) ≤ C A 1 1 0 y 2m 0 +1+νp 0 f j (t, y) dy + P m 0 +ν(p 0 -1) (t) + A 1 1 0 y m 0 f j (t, y) dy + P -ν-1,j (t) R 0 x m 0 Φ(f j (t, x)) dx ≤ C A 1 C 3 (T ) + P m 0 +ν(p 0 -1) (t) + [A 1 C 3 (T ) + P -ν-1,j (t)] R 0 x m 0 Φ(f j (t, x)) dx .
Combining the previous inequality with (2.7), (2.15), and Lemma 2.7 gives

d dt R 0 x m 0 Φ(f j (t, x)) dx = I 1,j (R, t) + I 2,j (R, t) ≤ [(2L R + A 1 )C 3 (T ) + (ν + 2)P -ν-1,j (t)] R 0 x m 0 Φ(f j (t, x)) dx + C A 1 C 3 (T ) + P m 0 +ν(p 0 -1) (t) .
Owing to Corollary 2.6, we are in a position to apply Gronwall's lemma to complete the proof.

Owing to (2.9), Lemma 2.8, and the superlinearity of Φ at infinity, see Definition 2.2 (b) , we deduce from Dunford-Pettis' theorem that, for all T > 0, there is a weakly compact subset K(T ) of X m 0 depending on T such that

f j (t) ∈ K(T ) , t ∈ [0, T ] . (2.16)
2.5. Time Equicontinuity. Having established the (weak) compactness of the sequence (f j ) j≥2 with respect to the size variable x, we now focus on the time variable. Our aim being to apply a variant of Arzelà-Ascoli's theorem, we study time equicontinuity properties of (f j ) j≥2 in the next result.

Lemma 2.9. Let T > 0 and R > 1. There are C 5 (T, R) > 0 depending on T and R and C 6 (T ) > 0 depending on T such that, for t 1 ∈ [0, T ) and

t 2 ∈ (t 1 , T ], R 0 x m 0 |f j (t 2 , x) -f j (t 1 , x)| dx ≤ C 5 (R, T )(t 2 -t 1 ) + C 6 (T )R (m 0 -1)/2 .
Proof. Let t ∈ [0, T ]. First, by Fubini's theorem,

I 3,j (R, t) := R 0 x m 0 |C j f j (t, x)| dx ≤ 1 2 R 0 R-x 0 (x + y) m 0 K(x, y)f j (t, x)f j (t, y) dydx + R 0 R 0 x m 0 K(x, y)f j (t, x)f j (t, y) dydx + R 0 ∞ R x m 0 K(x, y)f j (t, x)f j (t, y) dydx .
Owing to (1.7) and the elementary inequality (x + y) m 0 ≤ 2 max{x, y} m 0 , (x, y) ∈ (0, ∞) 2 , we further obtain

I 3,j (R, t) ≤ 2L R R 0 R 0 x m 0 y m 0 f j (t, x)f j (t, y) dydx + K 0 R 0 ∞ R x m 0 (2 + x + y)f j (t, x)f j (t, y) dydx ≤ 2L R M m 0 (f j (t)) 2 + 4K 0 M m 0 (f j (t))M 1 (f j (t)) ,
hence, thanks to (2.9) and Lemma 2.7,

I 3,j (R, t) ≤ 2L R C 3 (T ) 2 + 4̺K 0 C 3 (T ) .
(2.17)

Next, using once more Fubini's theorem,

I 4,j (R, t) := R 0 x m 0 |F j f j (t, x)| dx ≤ R 0 x m 0 a(x)f j (t, x) dx + R 0 a(y)f j (t, y) y 0 x m 0 b ν (x, y) dxdy + ∞ R a(y)f j (t, y) R 0
x m 0 b ν (x, y) dxdy .

We now infer from (1.6), (1.8), and Lemma 2.7 that

I 4,j (R, t) ≤ A R R m 0 +ν+1 M m 0 (f j (t)) + ν + 2 m 0 + ν + 1 R 0 a(y)y m 0 f j (t, y) dy + ν + 2 m 0 + ν + 1 R m 0 +ν+1 ∞ R a(y)y -ν-1 f j (t, y) dy ≤ CA R R m 0 +ν+1 C 3 (T ) + CR (m 0 -1)/2 ∞ R a(y)y (m 0 +1)/2 f j (t, y) dy , hence I 4,j (R, t) ≤ CA R R m 0 +ν+1 C 3 (T ) + CR (m 0 -1)/2 P (m 0 +1)/2,j (t) .
(2.18) It then follows from (2.7), (2.17), and (2.18) that R 0

x m 0 |∂ t f j (t, x)| dx ≤ I 3,j (R, t) + I 4,j (R, t) ≤ C 5 (R, T ) + CR (m 0 -1)/2 P (m 0 +1)/2,j (t) .
Consequently, since (m 0 + 1)/2 ∈ (0, 1), Corollary 2.6 entails

R 0 x m 0 |f j (t 2 , x) -f j (t 1 , x)| dx ≤ t 2 t 1 R 0 x m 0 |∂ t f j (t, x)| dxdt ≤ C 5 (R, T )(t 2 -t 1 ) + C 6 (T )R (m 0 -1)/2 ,
and the proof is complete.

Combining (2.9) and Lemma 2.9 allows us to improve the equicontinuity with respect to time of the sequence (f j ) j≥2 . and, for t 1 ∈ [0, T ) and

t 2 ∈ (t 1 , T ], ∞ 0 x m 0 |f j (t 2 , x) -f j (t 1 , x)| dx ≤ ω(T, t 2 -t 1 ) .
Proof. Let 0 ≤ t 1 < t 2 ≤ T and R > 1. By (2.9) and Lemma 2.9,

∞ 0 x m 0 |f j (t 2 , x) -f j (t 1 , x)| dx ≤ R 0 x m 0 |f j (t 2 , x) -f j (t 1 , x)| dx + R m 0 -1 ∞ R x[f j (t 1 , x) + f j (t 2 , x)] dx ≤ C 5 (R, T )(t 2 -t 1 ) + C 6 (T )R (m 0 -1)/2 + 2̺R m 0 -1 .
Introducing

ω(T, s) := inf R>1 C 5 (R, T )s + C 6 (T )R (m 0 -1)/2 + 2̺R m 0 -1 , s ≥ 0 ,
we deduce from the previous inequality which is valid for all R > 1 that

∞ 0 x m 0 |f j (t 2 , x) -f j (t 1 , x)| dx ≤ ω(T, t 2 -t 1 )
and observe that ω(T, •) enjoys the property (2.19) as m 0 < 1.

2.6. Convergence. According to a variant of Arzelà-Ascoli's theorem [START_REF] Vrabie | C 0 -semigroups and applications[END_REF]Theorem A.3.1], it readily follows from (2.16) and Corollary 2.10 that the sequence (f j ) j≥2 is relatively compact in C([0, T ]; X m 0 ,w ) for all T > 0. Using a diagonal process, we construct a subsequence of (f j ) j≥2 (not relabeled) and f ∈ C([0, ∞); X m 0 ,w ) such that, for all T > 0,

f j -→ f in C([0, T ]; X m 0 ,w ) .
(2.20) A first consequence of (2.20) is that f (t) is a non-negative function for all t ≥ 0 and that f (0) = f in . It further follows from Lemma 2.5, the superlinearity of W for large sizes, and (2.20) that the latter can be improved to

f j -→ f in C([0, T ]; X 1,w ) . (2.21)
A straightforward consequence of (2.9) and (2.21) is the mass conservation

M 1 (f (t)) = M 1 (f in ) , t ≥
sup j≥2 T 0 ∞ R a(x)f j (s, x) dxds = 0 ,
and thereby allows us to perform the limit j → ∞ in the fragmentation term. We have thus completed the proof of Theorem 1.1, except for the propagation of moments of higher order which is proved in Proposition 2.11 in the next section.

2.7. Higher Moments. We supplement the above analysis with the study of the evolution of algebraic moments of any order. Let f be the mass-conserving weak solution on [0, ∞) to (1.1) constructed in the previous section.

Proposition 2.11. Consider m > 1 and assume further that f in ∈ X m . Then f ∈ L ∞ (0, T ; X m ) for all T > 0.

Proof. Let T > 0, t ∈ (0, T ), and j ≥ 2. Setting ϑ m (x) := x m for x ∈ (0, ∞), it readily follows from (1.6) that

N ϑm (y) = m -1 m + ν + 1 y m ≥ 0 , y > 0 ,
while there is C 7 (m) > 0 depending only on m such that

(x + y)χ ϑm (x, y) ≤ C 7 (m) (x m y + xy m ) , (x, y) ∈ (0, ∞) 2 , by [7, Lemma 2.3 (ii)]. Consequently, by (1.7), χ ϑm (x, y)K(x, y) ≤ 2L 1 min{x, y} m 0 max{x, y} m ≤ 2L 1 (xy) m 0 , (x, y) ∈ (0, 1) 2 ,
and

χ ϑm (x, y)K(x, y) ≤ K 0 (2 + x + y)χ ϑm (x, y) ≤ 3K 0 (x + y)χ ϑm (x, y) ≤ 3K 0 C 7 (m) (x m y + xy m ) , (x, y) ∈ (0, ∞) 2 \ (0, 1) 2 .
We then infer from (2.8) with ϑ = ϑ m and the previous inequalities that

d dt M m (f j (t)) ≤ L 1 M m 0 (f j (t)) 2 + 3K 0 C 7 (m)M 1 (f j (t))M m (f j (t)) .
Recalling (2.9) and Lemma 2.7, we end up with

d dt M m (f j (t)) ≤ L 1 C 3 (T ) 2 + 3K 0 C 7 (m)̺M m (f j (t)) ,
and integrate the previous differential inequality to deduce that

M m (f j (t)) ≤ e 3K 0 C 7 (m)̺t M m (f in j ) + L 1 C 3 (T ) 2 3K 0 C 7 (m)̺ , t ∈ [0, T ] . (2.22)
Since M m (f in j ) ≤ M m (f in ) < ∞, Lemma 2.9 readily follows from (2.22) after letting j → ∞ with the help of (2.21) and Fatou's lemma.

Uniqueness

Proof of Theorem 1.3. Consider two weak solutions f 1 and f 2 on [0, ∞) to (1.1)-(1.2) on [0, ∞) enjoying the properties listed in Theorem 1.3. We set F := f 1 -f 2 , σ := sign(f 1 -f 2 ), ξ(x) := max{x m 0 , x 1+δ }, and Ξ(t, x, y) := ξ(x + y)σ(t, x + y) -ξ(x)σ(t, x) -ξ(y)σ(t, y) for (t, x, y) ∈ (0, ∞) 3 . Arguing as in the proof of [13, Theorem 2.9] and [START_REF]A uniqueness theorem for the coagulation-fragmentation equation[END_REF] with Ξ 0 (x, y) := ξ(x + y) -ξ(x) + ξ(y) , (x, y) ∈ (0, ∞) 2 . On the one hand, we infer from (1.7), (1.12), and the subadditivity of x → x m 0 and x → x δ that:

(c1) for (x, y) ∈ (0, 1) 2 , K(x, y)Ξ 0 (x, y) ≤ L 1 min{x, y} m 0 [(x + y) m 0 -x m 0 + y m 0 ] ≤ 2L 1 min{x, y} m 0 y m 0 ≤ 2L 1 ξ(x)y m 0 ;

(c2) for (x, y) ∈ (0, 1) × (1, ∞), K(x, y)Ξ 0 (x, y) ≤ K 1 x m 0 y x 1+δ + (1 + δ)y(x + y) δ -x m 0 + y 1+δ ≤ K 1 x m 0 y (1 + δ)2 δ y 1+δ + y 1+δ Collecting the estimates in (c1)-(c4) and (f1)-(f2), we infer from (3.1) that there is a positive constant κ > 0 depending only on L 1 , K 1 , δ, K 0 , ν, m 0 , and a such that d dt Since M m 0 (f i ) and M 2+δ (f i ) both belong to L ∞ (0, T ) for i ∈ {1, 2} and f 1 (0) = f 2 (0) = f in , we use Gronwall's lemma to complete the proof.

∞ 0 K

 0 (x, y)f (x)f (y) dy (1.1b) and F f (x) := -a(x)f (x) + ∞ x a(y)b(x, y)f (y) dy (1.1c)

Corollary 2 . 10 .

 210 Let T > 0. There is a function ω(T, •) : [0, ∞) → [0, ∞) depending on T such that lim s→0 ω(T, s) = 0 ,(2.19)

  y)Ξ(t, x, y)(f 1 + f 2 )(t, y)F (t, x) dydx+ )σ(t, x)b ν (x, y) dx -ξ(y)σ(t, y) dy . Since Ξ(t, x, y)F (t, x) = ξ(x + y)σ(t, x + y)F (t, x) -ξ(x)|F (t, x)| -ξ(y)σ(t, y)F (t, x) ≤ ξ(x + y)|F (t, x)| -ξ(x)|F (t, x)| + ξ(y)|F (t, x)| and F (t, y) y 0 ξ(x)σ(t, x)b ν (x, y) dx -ξ(y)σ(t, y) ≤ y 0 ξ(x)b ν (x, y) dx|F (t, y)| -ξ(y)|F (t, y)| ≤ -N ξ (y)|F (t, y)| , , y)Ξ 0 (x, y)(f 1 + f 2 )(t, y)|F (t, x)| dydx-∞ 0 a(y)N ξ (y)|F (t, y)| dy , (3.1)

≤ K 1 1 +

 1 (1 + δ)2 δ ξ(x)y 2+δ ;(c3) for (x, y) ∈ (1, ∞) × (0, 1), K(x, y)Ξ 0 (x, y)≤ K 0 (2 + x + y) (1 + δ)y(x + y) δ + y m 0 ≤ 4K 0 x (1 + δ)2 δ x δ y + x δ y m 0 ≤ 4K 0 1 + (1 + δ)2 δ ξ(x)y m 0 ; (c4) for (x, y) ∈ (1, ∞) 2 , K(x, y)Ξ 0 (x, y) ≤ K 0 (2 + x + y) (1 + δ)y(x + y) δ + y 1+δ ≤ 2K 0 (x + y) (1 + δ)yx δ + (2 + δ)y 1+δ ≤ 2(2 + δ)K 0 x 1+δ y + xy 1+δ + x δ y 2 + y 2+δ ≤ 8(2 + δ)K 0 ξ(x)y 2+δ .On the other hand, owing to (1.6) and (1.8), (f1) for y ∈ (0, 1),-a(y)N ξ (y) = 1 -m 0 ν + 1 + m 0 y m 0 a(y) ≤ A 1 ν + 1 + m 0 ξ(y) ; (f2) for y > 1, -N ξ (y) = ν + 2 ν + 1 + m 0 y -ν-1 + ν + 2 ν + 2 + δ y 1+δ -y -ν-1 -y 1+δ = (ν + 2)(1 + δ -m 0 ) (ν + 1 + m 0 )(ν + 2 + δ) y -ν-1 -δ ν + 2 + δ y 1+δ = δ ν + 2 + δ y -ν-1 Y 2+δ+ν -y 2+δ+ν , with Y 2+δ+ν := (ν + 2)(1 + δ -m 0 ) δ(ν + 1 + m 0 ) . Either y ≥ max{1, Y } and -a(y)N ξ (y) ≤ δ ν + 2 + δ a(y)y -ν-1 Y 2+δ+ν -y 2+δ+ν ≤ 0 , or y ∈ (1, max{1, Y }) and -a(y)N ξ (y) ≤ δ ν + 2 + δ a(y)y -ν-1 Y 2+δ+ν ≤ A Y Y 2+δ+ν y m 0 ≤ A Y Y 2+δ+ν ξ(y) .

  ) y m 0 + y 2+δ (f 1 + f 2 )(t, y)|F (t, x)| dydx + κ ∞ 0 ξ(y)|F (t, y)| dy ≤ κ [1 + M m 0 ((f 1 + f 2 )(t)) + M 2+δ ((f 1 + f 2 )(t))] ∞ 0 ξ(y)|F (t, y)| dy .

  0 .

	Owing to (1.7), (1.8), Corollary 2.6, (2.20), and (2.21), it is by now a standard
	argument to pass to the limit in (2.8) and deduce that f is a weak solution on [0, ∞)
	to (1.1) in the sense of Definition 2.1, see [6, 23, 31] for instance. It is worth pointing
	out that the behaviour of the fragmentation term for large sizes is controlled by
	Corollary 2.6 which guarantees that, for all T > 0,
	lim R→∞


Faiza Ajmi

Sarah Ben Othman

Zgaya Hayfa

Slim Biau

Hammadi

Hayfa Zgaya Biau

Slim Hammadi

Scheduling Approch to Control the Execution of the Patient Pathway Workflow in the Emergency Department

The majority of developed countries are interested in enhancing their healthcare information system in order to expect the overcrowding situations and improve the patient quality of care. In this paper, we focus on the patient pathway workflow modeling using the business process management notation (BPMN) graphical language in the adult emergency department (AED). The goal of our work is to identify the dysfunction situations and improve the performance indicators such as the waiting time. The idea is to optimize this criterion using a real-time scheduling algorithm applied simultaneously on each decision point (gateway) of the patient pathway workflow model. The objective of our algorithm is to assign a dynamic priority to the patients according to their health state evolution. In addition, taking into account the uncertainty of patient arrival an adequate a real time scheduling algorithm is proposed. Our approach is tested on a set of real database from the AED of the regional university hospital (RUH) of Lille (in the north of France). We conducted interviews, performed healthcare tasks analysis, and validated results with the AED medical staff. The goal is to have the workflow model as close as possible to the real functioning of the AED. The simulation results show that the average waiting time of the patients at the AED drop by 11% thanks to our approach.

Introduction

The patient medical treatment may require a number of various interactions between several services (MRI, biological tests, surgical operations, etc.). At each step of the care process, each patient is associated to a care activity and to a medical resource allocation. However, the functioning of this care process always faces to various problem such as: the lack of coordination between the various services, the complexity of the patient pathway management, financial resources limitations, the unpredictability of the arrival time of patients and their pathologies. Consequently, the health organizations systems are more and more aware of the need to use their resources as efficiently as possible, which requires health authorities to increase emphasis on process optimization in order to control and minimize operating costs and improve the quality services levels. Indeed, because the lack of medical information circulation leads to delays resulting in poor quality of care and increased cost, a connected Workflow system should be set up [1].

In fact, in this article, we use a Workflow model to represent patient pathway with different bottleneck points generating long Waiting Time (WT)s. We propose then to develop a Real-Time Scheduling Algorithm (RTSA) applied in parallel way in each decision point of patient pathway Workflow in order to orchestrate them and reduce the WT during the execution of the workflow instances.

Our paper is organized as follows: a state of the art will be presented in the second section. The third section describes the patient pathway Workflow model. The proposed RTSA is presented in sections 4, followed by simulations and results in the section 5. A conclusion and perspectives are presented at the end of this paper.

State of the art

In the literature, the application of Workflow model in the hospital system has been successful, which rendered the health system a very active area of research [2], [3], [4]. The majority of this research is about providing an efficient health care system for the patient and ensuring its continuity for the next generation [2], [5], [6], [7]. In fact, many research works focus on assessing the ED performance. In [START_REF] Salmon | A structured literature review of simulation modelling applied to Emergency Departments: Current patterns and emerging trends[END_REF], [START_REF] Et | Patient flow within UK emergency departments: a systematic review of the use of computer simulation modelling methods[END_REF] authors present a state of the art about the computer simulation approaches used for understanding the causes of ED overcrowding.

In [START_REF] Harzi | Scheduling Patients in Emergency Department by Considering Material Resources[END_REF] authors presented a mixed integer linear programming (MILP) approach to minimize the total waiting time of patient's in the emergency department. Their model is not applicable on a large population size and does not take into account the uncertainty aspect of patient arrival.

Real-time optimization approaches have been developed in order to take rapid decisions especially when an unexpected event occurs in the system such as the random arrival of patients. A state of the art in real-time optimization methods is presented in [START_REF] Jaillet | Risk and Optimization in an Uncertain World[END_REF], the computation time is the most important criterion for these methods. For example, appointment-planning optimization in radiotherapy presents a major difficulty because of uncertainty, which is the main characteristic of patient arrival. In this context, the majority of researchers usually propose off-line provisional schedule approaches but the proposed solutions can explode in terms of deadlines and costs. In [START_REF] Erdogan | [END_REF], authors propose a real time approach in order to calculate the best appointment for each patient arrival with a small size problem. In [13], authors propose a Markov decision process to solve the same problem. They obtain an appropriate solution for several instances generated randomly. However, they do not take into account all information generated from the beginning of the resolution process. This information is important to optimize in real-time patient pathway, especially in overcrowding situations.Table 1. Summary of the related research on patient scheduling in the healthcare systempresents a comparison between the different works cited above.

The originality of our paper over existing works is the take into account new patient arrival and the health status evolution of the ongoing treating patients. Our objective is to adapt the best solution according to what really happen in the AED.

Patient Pathway workflow model

The workflow model, implemented using BPMN language (Business Process Management Notation) and presented in Figure 1, represents the patient pathway in the AED. This model has been realize thanks to observations made during several visits over a period of 6 months. The analysis done by the researchers and AED staff medical conducted us to construct a Workflow models.

An instance of this model triggers with a single start-event representing the patient arrival and ends with 6 possible end-events representing the different exit-ways of the patient. This workflow model includes several decision points. At each point, a patient real-time scheduling is needed.

Decision variables

Yk The time that pathway workflow instances assignment is made for patient k ; Zko The starting time for the treatment task Pk,o,h.

A solution to the AED workflow model is a feasible schedule of patients' pathway workflow instances assignment and health care task-resource allocations. The goal is to assign each patient to an appropriate pathway workflow instance and schedule the health care treatment tasks. The object here is to minimize the weighted response time between arrival and pathway workflow instances assignment and the objective of the task-resource allocation is to minimize the total care time of all patients. The weights ωk are selected according to priority class of the patients, it is penalized in the objective and this accounts for the priority class assigned during triage. These weights are dynamic and are updated according to the evolution of patient pathology gravity.

Waiting Time before the first consultation

() k k k kK WT w Y b     Length Of Stay () kk kK LOS c Y    Completion time k c { }, { ,.,.} k kn kn k c Max Y p n o   
After all the treatment tasks   ,.,.

k o are completed for a patient k, the pathway workflow instances assignment to this patient is released.

The goal of our RTSA proposed algorithm is to reduce the average waiting time of the patients. So, when two patients have the same priority level and respectively waiting in the WS and WP rooms, the switch function allows to take care firstly of the patient in the WP. This solution reduces the waiting time in the WP, called primary waiting time. The RTSA algorithm uses the Priority function, which calculates the priority order of each patient in the AED, basing on the application of the three dynamic rules applied on each patient (Table 32).

R1 R2 R3 ( ) / k k s Y b r  [( ) / ]* k k s k Y b r w    ( ) , k Max w t k 

Simulation and results

In order to validate our algorithm we did interviews with the AED medical staff and observations during visits in the RUH of Lille. We defined a sample of 20 patients with different gravity degrees (Table 3). For simulation we consider that the treatment is achieved by a single doctor in a single box available at that time and two types of waiting rooms: WS (secondary waiting room) and WP (primary waiting room). We are here faced to a resource constraints scheduling problem. These patients pathway represented by the Workflow with different decision points are controlled by the RSTA during its execution.

This table shows that the WT in the real treatment process functioning (8th column) of these 20 patients (according to visits) is often high compared to the WT with RTSA (10th column). The real functioning of the AED does not take into account the patient consultation time; it considers only the severity of the patients in the WP, that is to say, the patient with the greatest severity is scheduled first. However, if there are two patients who have the same gravity the priority is given to the one who is in the WP and if they have the same WT then sorting is done randomly. This reflects the inconvenient of this functioning and explains an excessive total WT. While applying our RTSA approach in each decision point of the Workflow model the average WT of patient at AED drop of 11% (Figure 2). The originality and effectiveness of our RTSA is shown at the application of the rule R2, thanks to this rule the 3 patients (P7, P10, P2) where their health care state evolution dynamically changed respectively from 3-5-3 to 5-6-7. The RTSA takes into account this evolution of gravity and the algorithm has updated the order in patients' treatment sequence (13th column).

Discussion

Thanks to our workflow model and our dynamically (re)-orchestrate algorithm of workflow instances (RTSA), the patient wait time rate drops by 11% during the execution of workflow. The results demonstrate the utility of the proposed methodologies to optimize the using of human and material resources to treat patients as quickly as possible while to take into account the evolution of patient health care state gravity. Therefore, the main cause of overcrowding situation is that the demand exceeds the supply generated by the functioning nature of this department. The situation becomes worse, when several patients arrive at the same time to the AED. In fact, in order to anticipate and avoid the overcrowding situation in the AED, we need to develop in a future work a decision support system provided with a computerized resolution architecture based on the multiagent system allowing communication, coordination and negotiation between the different actors of the AED to ensure collaborative optimization that generates good actions to adopt.

Conclusion and prospects

In this paper, we have modeled the AED patient pathway using a processoriented approach (Workflow). We have chosen the Waiting Time as the most crucial Performance Indicator. In order to control the workflow patient pathway during its execution we have developed a real-time scheduling algorithm (RTSA) at each decision point in the workflow model. The originality of this work is to take into account the evolution of patient health care state gravity at each decision point thanks to the dynamic rules driven RTSA. To evaluate our approach, a simulation of the WT rate before and after scheduling is done and shows a decrease of 11% in the WT within the AED.

Fig. 1 . 4 . 1 .

 141 Fig.1. Global patient pathway Workflow model in the AED using BPMN modeling language

Fig. 2 .

 2 Fig. 2. Algorithm 1: RTSA

Fig. 3 .

 3 Fig. 3. Comparison of WT with and without RTSA

Table 3 .

 3 Priority rules

Table 4 .

 4 Comparison of WT with and without RTSA

Table 5 .

 5 Comparison of WT with and without RTSA

Acknowledgments

This work was supported and funded by the Federative Research Structures Technologies for Health and Drugs (SFR-TSM): http://sfr-tsm.ec-lille.fr/