Alejandro Ribés
email: alejandro.ribes@edf.fr

Théophile Terraz
email: theophile.terraz@inria.fr

Yvan Fournier
email: yvan.fournier@edf.fr

Bertrand Iooss
email: bertrand.ioss@edf.fr

Bruno Raffin
email: bruno.raffin@inria.fr

Unlocking Large Scale Uncertainty Quantification with In Transit Iterative Statistics

Multi-run numerical simulations using supercomputers are increasingly used by physicists and engineers for dealing with input data and model uncertainties. Most of the time, the input parameters of a simulation are modeled as random variables, then simulations are run a (possibly large) number of times with input parameters varied according to a specific design of experiments. Uncertainty quantification for numerical simulations is a hard computational problem, currently bounded by the large size of the produced results. This book chapter is about using in situ techniques to enable large scale uncertainty quantification studies. We provide a comprehensive description of Melissa, a file avoiding, adaptive, fault-tolerant, and elastic framework that computes in transit statistical quantities of interest. Melissa currently implements the on-the-fly computation of the statistics necessary for the realization of large scale uncertainty quantification studies: moment-based statistics (mean, standard deviation, higher orders), quantiles, Sobol' indices, and threshold exceedance.

Introduction

A numerical simulation is a calculation that is run on a computer following a program that implements a mathematical model for a physical system. Nowadays, engineers and scientists often use numerical simulation as a tool, and its use in industries or scientific laboratories is very broad. From the mathematical point of view, discretized differential equations are numerically solved, often using a mesh as spatial support. Popular methods include Finite Difference, Finite Volumes, Finite Elements, or particle-based methods. From the computer science point of view, a numerical simulation consists of a workflow of actions. First, the engineer or scientist prepares the mesh or other spatial discretization such as particles, and she/he defines the initial and boundary conditions. Second, the calculations are run in a computer and generate results that are written to files. Finally, the results are analyzed.

Chapter 1 of this book already presented an introduction to in situ techniques for computational science. We would like to insist on the fact that engineers and scientists are so used to the classical worklow presented above, that they currently do not realize that a major change is occurring in current computer systems. The sizes of the simulations are strongly increasing, and writing files is becoming cumbersome and time-consuming. This bottleneck, in numerous cases, limits the size of the simulations. When executing multiple simulation runs, this problem becomes even more critical.

Multiple simulation runs (sometimes several thousands) are required to compute sound statistics in the context of uncertainty quantification [START_REF] Smith | Uncertainty quantification[END_REF]. Taking uncertainties into account when dealing with complex numerical simulations is necessary to assess their robustness, as well as to answer tighter regulatory processes (security, safety, environmental control, health impacts, etc.). Many attempts at treating uncertainty in industrial applications (e.g. automotive and aerospace engine design, nuclear safety, agronomy, renewable energy production) have involved different mathematical approaches from many scientific domains as in metrology, structural reliability, variational analysis, design of experiments, machine learning and global sensitivity analysis [START_REF]Uncertainty in industrial practice[END_REF]. As an example, in aeronautic and nuclear industries, uncertainty quantification approaches are applied on numerical models simulating non destructive testing procedures, in order to evaluate probability of detection curves [START_REF] Gratiet | Model assisted probability of detection curves: New statistical tools and progressive methodology[END_REF]. Such curves allow the operators to evaluate the performance of their non destructive tests for the detection of harmful defects of the inspected structure, which is particularly important for the system safety.

Current practice consists of performing multiple executions of a classical workflow. All the necessary instances with different sets of input parameters are run, and the results are stored to disk, often called ensemble data, to later read them back from disk to compute statistics. In this context, we are confronted with two important problems. First, the amount of storage needed may quickly become overwhelming, with the associated long read time that makes statistic computing time-consuming. To avoid this pitfall, scientists reduce their study size by running low-resolution simulations or down-sampling output data in space and time. Second, humans should be able to somehow navigate through the complexity of these large simulation results. Down-sampling output data in space and time, extracting probes, or concentrating on specific features of the ensemble are usually performed to reduce complexity; this introduces strong dependences on a priori ideas related to the behavior of the ensemble.

Novel approaches are required. In situ and in transit processing emerged as a solution to perform data analysis starting as soon as the results are available in the memory of the simulation. The goal is to reduce the data to store to disk and to avoid the time penalty to write and then read back the raw data set as required by the classical post hoc analysis approach. In recent works, we proposed the Melissa framework for the on-line data aggregation of high-resolution ensemble runs [START_REF] Terraz | Melissa: Large scale in transit sensitivity analysis avoiding intermediate files[END_REF][START_REF] Ribés | Large scale in transit computation of quantiles for ensemble runs[END_REF]. As soon as each available simulation provides the results produced to a set of staging nodes, these nodes process them to update the statistics on a first-come-first-served basis thanks to one-pass algorithms. This in transit processing mode enables us to fully avoid storage of intermediate data on disks. Furthermore, this new approach allows the computation of ubiquitous statistics: we compute multidimensional and time-varying statistics, i.e. everywhere in space and time; instead of providing a down-sampled subset, for a limited sample of probes or concentrating on specific features of the ensemble, as usually done.

In the context of this book, we would like to remark that this chapter does not present any in situ visualization system but an example of how in situ techniques can be used for the statistical analysis of large quantities of data; which is defined in Chapter 1 as "use cases beyond exploratory analysis". As a matter of fact, uncertainty quantification for numerical simulations is a hard computational problem, currently bounded by the large size of the produced results. This chapter is about using in situ techniques to unleash large scale uncertainty quantification (UQ) studies.

In the following Section 2 presents the general methodology for dealing with UQ studies; Section 3 introduces the iterative statistics necessary to perform on-line UQ; Section 4 briefly describes the MELISSA platform which implements the statistics introduced in Section 3; Section 5 uses a fluid mechanics example to illustrate the realization of a large scale UQ study; finally, a short conclusion and a bibliography section end up the chapter.

Uncertainty Management Methodology

Introduction

A general framework has been proposed in order to deal with various uncertainties that arise in numerical simulations [START_REF]Uncertainty in industrial practice[END_REF][START_REF] Baudin | Open TURNS: An industrial software for uncertainty quantification in simulation[END_REF]. The uncertainty management generic methodology is schematized in Figure 1. Based on a probabilistic modeling of the model input variables, it consists of the following steps:

• Step A: specify the random inputs X, the deterministic inputs d, the numerical model G (analytical, complex computer code or experimental process), the variable of interest (model output) Y and the quantity of interest on the output (central dispersion, its distribution, probability to exceed a threshold, . . . ). The fundamental relation writes:

Y = G(X, d) = G(X), with X = (X 1 , . . . , X p ) ∈ R p .
• Step B: quantify the sources of uncertainty. This step consists in modeling the joint probability density function (pdf) of the random input vector by direct methods (e.g. statistical fitting, expert judgment). • Step B': quantify the sources of uncertainty by indirect methods using some real observations of the model outputs. The calibration process aims to estimate the values or the pdf of the inputs while the validation process aims to model the bias between the model and the real system. • Step C: propagate uncertainties to estimate the quantity of interest. With respect to this quantity, the computational resources and the CPU time cost of a single model run, various methods will be applied as linear-based analytical formula, geometrical approximations, Monte Carlo sampling strategies, metamodel-based techniques.

• Step C': analyze the sensitivity of the quantity of interest to the inputs in order to identify the most influential inputs (useful if one wants to reduce the output uncertainty) and to rank the uncertainty sources.

Figure 1 The methodology of uncertainty management in numerical simulation.

Quantiles of Simulation Outputs

Quantiles are important order statistics for outlier detection or computation of non parametric prediction and tolerance intervals. Then, in the context of uncertainty quantification analysis of computer models, quantile estimation is one of the key steps. Low or high-order quantiles are often required, especially in industrial safety studies [START_REF]Uncertainty in industrial practice[END_REF][START_REF] Morio | Estimation of rare event probabilities in complex aerospace and other systems[END_REF][START_REF] Iooss | Advanced methodology for uncertainty propagation in computer experiments with large number of inputs[END_REF].

Standard approaches deal with the problem of quantile estimation of scalar outputs [START_REF] Glynn | Importance sampling for monte carlo estimation of quantiles[END_REF][START_REF] Hesterberg | Control variates for probability and quantile estimation[END_REF][START_REF] Cannamela | Controlled stratification for quantile estimation[END_REF]. Let us consider a N-sample (Y 1 , . . . ,Y N ) of independent and identically distributed random variables from an unknown distribution f Y (y). We look for an estimator qα of the α-quantile q α defined by:

P(Y ≤ q α ) = α , (1)
which is sometimes written as

q α = inf{y|P(Y ≤ y) ≥ α} . (2)
However, simulation models most often return spatial fields varying over time. For example, recent studies have considered quantiles of one-dimensional functional outputs (temporal curves) [START_REF] Popelin | Visualization tools for uncertainty and sensitivity analyses on thermalhydraulic transients[END_REF][START_REF] Ribés | Visualizing statistical analysis of curves datasets in Paraview[END_REF][START_REF] Nanty | Uncertainty quantification for functional dependent random variables[END_REF][START_REF] Ribés | A Visual Sensitivity Analysis for Parameter-Augmented Ensembles of Curves[END_REF], that demonstrates users' interest in computing these functional quantiles.

Sensitivity Analysis via Sobol' Indices

Sensitivity studies are an important application of uncertainty quantification in numerical simulation [START_REF] Smith | Uncertainty quantification[END_REF]. The objective of such studies can be broadly viewed as quantifying the relative contributions of individual input parameters to a simulation model, and determining how variations in parameters affect the outcomes of the simulations. In this context, multi-run studies treat simulations as black boxes that produce outputs when a set of parameters is fixed (Figure 2). Global sensitivity analysis is an ensemble of techniques that deal with a probabilistic representation of the input parameters [START_REF]Sensitivity analysis[END_REF][START_REF] Iooss | A review on global sensitivity analysis methods[END_REF] to consider their overall variation range.

G Y X 1 X 2 ... X p
Figure 2 A simple solver G taking p input parameters X 1 to X p , and computing a scalar output Y.

Variance-based sensitivity measures, also called Sobol' indices [START_REF] Sobol | Sensitivity estimates for non linear mathematical models[END_REF], are popular among methods for global sensitivity analysis because they can deal with nonlinear responses. They decompose the variance of the output, Y , of the simulation into fractions, which can be attributed to random input parameters or sets of random inputs.

For example, with only two input parameters X 1 and X 2 and one output Y (Figure 2), Sobol' indices might show that 60% of the output variance is caused by the variance in the first parameter, 30% by the variance in the second, and 10% due to interactions between both. These percentages are directly interpreted as measures of sensitivity. If we consider p input parameters, the Sobol' indices can identify parameters that do not influence or influence very slightly the output, leading to model reduction or simplification. Sobol' indices can also measure the effect of interactions in non-additive systems.

Mathematically, the first and second order Sobol' indices [START_REF] Sobol | Sensitivity estimates for non linear mathematical models[END_REF] are defined by:

S i = Var(E[Y |X i ]) Var(Y ) , S i j = Var(E[Y |X i X j ]) Var(Y ) -S i -S j , (3)
where X 1 , . . . , X p are p independent random variables. In Eq. ( 3), S i represents the first order sensitivity index of X i while S i j represents the effect of the interaction between X i and X j . Higher-order interaction indices (S i jk , . . . , S 1...p ) can be similarly defined. The total Sobol' indices express the overall sensitivity of an input variable X i :

ST i = S i + j i S i j + j i,k i, j<k S i jk + . . . + S 1...p . (4)
The previous formula applies for a scalar output Y . Some authors have proposed the generalization of the concept of Sobol' indices for multidimensional and functional data [START_REF] Gamboa | Sensitivity analysis for multidimensional and functional outputs[END_REF] by synthesizing all the sensitivity information of the multidimensional output in a single sensitivity value. Few authors have considered the estimation of Sobol' indices at each output cell (see [START_REF] Marrel | Sensitivity analysis of spatial and/or temporal phenomena[END_REF] for an overview on this subject) and this estimation has always been applied to small models. Applications of these techniques on environmental assessment can be found for example in [START_REF] Higdon | Computer model calibration using highdimensional output[END_REF][START_REF] Marrel | Global sensitivity analysis for models with spatially dependent outputs[END_REF] for spatial outputs and in [START_REF] Marrel | Development of a surrogate model and sensitivity analysis for spatio-temporal numerical simulators[END_REF][START_REF] Marrel | Sensitivity analysis with dependence and variance-based measures for spatio-temporal numerical simulators[END_REF] for spatio-temporal outputs. All these works have shown that obtaining temporal/spatial/spatio-temporal sensitivity maps leads to powerful information for the analysts. Indeed, the parameter effects are localized in time or space, and can be easily examined in relation with the studied physical phenomena.

In Transit Statistics

Computing statistics from N samples classically requires O(N) memory space to store these samples. But if the statistics can be computed in one-pass (also called iterative, on-line or even parallel [START_REF] Pébay | Formulas for robust, one-pass parallel computation of covariances and arbitraryorder statistical moments[END_REF]), i.e. if the current value can be updated as soon as a new sample is available, the memory requirement goes down to O(1) space. With this approach, not only simulation results do not need to be saved, but they can be consumed in any order, loosening synchronization constraints on the simulation executions.

There also exist multi-pass algorithms for the computation of statistics, where P passes are necessary. In this case, the on-line processing system would need to have access to the same data P times thus forcing the data to be stored till the last pass is finished. This is not feasible for large scale use cases. These approaches, along with the classical ones requiring O(N) memory space, are avoided in on-line applications.

Moment-Based Statistics: Mean, Std, Higher Orders

One-pass variance algorithms were proposed in [START_REF] Welford | Note on a method for calculating corrected sums of squares and products[END_REF][START_REF] Chan | Updating formulae and a pairwise algorithm for computing sample variances[END_REF][START_REF] Finch | Incremental calculation of weighted mean and variance[END_REF]. Numerically stable, one-pass formulas for arbitrary centered statistical moments and co-moments are presented in [START_REF] Bennett | Numerically stable, single-pass, parallel statistics algorithms[END_REF][START_REF] Pébay | Formulas for robust, one-pass parallel computation of covariances and arbitraryorder statistical moments[END_REF]. Reference Pébay et al. [START_REF] Pébay | Formulas for robust, one-pass parallel computation of covariances and arbitraryorder statistical moments[END_REF] also contains update formulas for higher order moments (skewness, kurtosis and more). These works set the base for a module of parallel statistics in the VTK scientific visualization toolkit [START_REF] Pébay | Design and performance of a scalable, parallel statistics toolkit[END_REF]. In this context, the one-pass algorithms enables to compute partial results in parallel before performing a reduction to get the final result. These iterative statistics were used for computing large scale parallel statistics for a single simulation run either from raw data files [START_REF] Bennett | Feature-based statistical analysis of combustion simulation data[END_REF], compressed data files [START_REF] Lakshminarasimhan | Isabela-qa: query-driven analytics with isabela-compressed extreme-scale scientific data[END_REF] or in situ [START_REF] Bennett | Combining in-situ and in-transit processing to enable extreme-scale scientific analysis[END_REF]. More recently Lampitella et al. [START_REF] Lampitella | Note on a formula for one-pass, parallel computations of arbitrary-order, weighted, multivariate central moments[END_REF] proposed a general update formula for the computation of arbitrary-order, weighted, multivariate central moments.

In Melissa, we iteratively compute the moments of a random variable Y as

µ (k), S (Y ) = µ (k), S = µ (k), S 1 + 1 n y k -µ (k), S 1 (5)
for k = 1, 2, 3 and 4, where S = S 1 ∪ {y} and n = card(S). Then from these moments, we compute the mean, variance, skewness and kurtosis:

Mean = µ (1) , Variance = n n -1 (µ (2) -µ 2 (1) ) , Skewness = µ (3) -3µ (1) µ (2) + 2µ 3 (1)
(µ (2) -µ 2 (1) ) 1.5 ,

Kurtosis = µ (4) -4µ (1) µ (3) + 6µ 2 (1) µ (2) -3µ 4 (1)
(µ (3) -3µ (1) µ (2) + 2µ 3 (1) ) 2 .

Sobol' Indices

The information contained in this section can be found in the previously published article [START_REF] Terraz | Melissa: Large scale in transit sensitivity analysis avoiding intermediate files[END_REF]. We include it here for self-completeness. Thus the reader can find the description, in this book chapter, of all iterative methods currently implemented in Melissa.

In order to compute Sobol' indices, we use the so-called pick-freeze scheme that uses two random independent and identically distributed samples of the model inputs [START_REF] Sobol | Sensitivity estimates for non linear mathematical models[END_REF][START_REF] Homma | Importance measures in global sensitivity analysis of non linear models[END_REF][START_REF] Janon | Asymptotic normality and efficiency of two sobol index estimators[END_REF]. One-pass iterative Sobol' indices formulas directly derive from the iterative variance (presented in Section 3.1) and iterative covariance [START_REF] Pébay | Formulas for robust, one-pass parallel computation of covariances and arbitraryorder statistical moments[END_REF]. Note that another iterative computation of Sobol' indices has been introduced in [START_REF] Gilquin | Recursive estimation procedure of sobol' indices based on replicated designs[END_REF] for the case of a scalar output.

Our goal is to compute in transit the Sobol' indices of each input parameter X i (Figure . 2). We explain below the so-called pick-freeze scheme that uses two random independent and identically distributed samples of the model inputs [START_REF] Sobol | Sensitivity estimates for non linear mathematical models[END_REF][START_REF] Homma | Importance measures in global sensitivity analysis of non linear models[END_REF][START_REF] Janon | Asymptotic normality and efficiency of two sobol index estimators[END_REF].

We first define the p variable input parameters of our study as a random vector, with a given probabilistic law for each parameter. We then randomly draw two times n sets of p parameters, to obtain two matrices A and B of size n × p (each row is a set of parameters for one simulation):

A = a 1,1 • • • a 1,p . . . . . . . . . a n,1 • • • a n,p ; B = b 1,1 • • • b 1,p . . . . . . . . . b n,1 • • • b n,p
.

For each k ∈ [1, p] we define the matrix C k , which is equal to the matrix A but with its column k replaced by column k of B. Each row of each matrix is a set of input parameters:

C k = a 1,1 • • • a 1,k-1 b 1,k a 1,k+1 • • • a 1,p . . . . . . . . . . . . . . . a i,1 • • • a i,k-1 b i,k a n,k+1 • • • a i,p . . . . . . . . . . . . . . . a n,1 • • • a n,k-1 b n,k a n,k+1 • • • a n,p
.

Then, a study consists in running the n × (p + 2) simulations defined by the matrices A, B and C k for k ∈ [1, p]. For each matrix M with n rows, ∀i ∈ [1, n], let M i be the i th row of M, and M [:i] the matrix of size i × p built from the i first lines of M. For example :

C k i = a i,1 • • • a i,k-1 b i,k a i,k+1 • • • a i,p and C k [:i] = a 1,1 • • • a 1,k-1 b 1,k a 1,k+1 • • • a 1,p . . . . . . . . . . . . . . . a i,1 • • • a i,k-1 b i,k a n,k+1 • • • a i,p . Let Y A i be the result of G(A i ), and Y A ∈ R n the vector built from component i of Y A i , ∀i ∈ [1, n].
We define Y B i and Y B in the same way, as well as Y C k i and Y C k ∀k ∈ [1, p]. Let Var(x) be the unbiased variance estimator, and Cov(x, y) the unbiased covariance estimator, as defined in [START_REF] Pébay | Formulas for robust, one-pass parallel computation of covariances and arbitraryorder statistical moments[END_REF]. First order Sobol' indices S k can be estimated by the following formula, called Martinez estimator [START_REF] Baudin | Numerical stability of sobol' indices estimation formula[END_REF]:

S k ( f , A, B) = Cov(Y B ,Y C k ) Var(Y B ) Var(Y C k ) , (6)
while total order Sobol' indices ST k are estimated by:

ST k ( f , A, B) = 1 - Cov(Y A ,Y C k ) Var(Y A ) Var(Y C k ) . ( 7
)
Since variances and covariances can be updated iteratively, first order and total Sobol' indices can be computed from these formulas. The covariance update formula between two random variables X and Y writes [START_REF] Pébay | Formulas for robust, one-pass parallel computation of covariances and arbitraryorder statistical moments[END_REF]:

Cov S = Cov S 1 + n -1 n x -µ (1), S 1 (X) y -µ (1), S 1 (Y ) (8)
where S = S 1 ∪ {x, y}, n = card(S) and the mean update µ (1) comes from Eq. ( 5). There are many other estimators than those of Eqs ( 6) and (7) (see for example [START_REF] Prieur | Variance-based sensitivity analysis: Theory and estimation algorithms[END_REF]) relying on the matrices A, B and C k to compute the variance and the covariance with different formulas. We use the Martinez estimator because it provides an asymptotic confidence interval [START_REF] Baudin | Numerical stability of sobol' indices estimation formula[END_REF], which is very simple to express, and is easy to compute in an iterative fashion. In addition, it has been shown to be unbiased and one of the most numerically stable estimator.

Order Statistics: Quantiles

The classical estimator of the α-quantile y α of the random variable Y is the empirical quantile, based on the notion of order statistics [START_REF] David | Order statistics, third edn[END_REF]. Essentially, we associate with the independent and identically distributed sample (Y 1 , . . . ,Y N ) the ordered sample (Y (1) , . . . ,Y (N ) ) in which Y (1) ≤ . . . ≤ Y (N ) . The empirical estimator then writes:

qα = Y ( αN +1) , (9)
where x is the integer part of x.

For an iterative statistical estimation, the Robbins-Monro estimator [START_REF] Robbins | A stochastic approximation method[END_REF] consists in updating the quantile estimate q α (n) at each new observation Y n+1 with the following rule:

q α (n + 1) = q α (n) - C n γ 1 Y n+1≤qα (n) -α , (10)
with n = 1 . . . N, q α (1) = Y 1 an independent realization of Y , qα = q α (N), 1 x the indicator function, C a strictly positive constant and γ ∈]0, 1] the step of the gradient descent of the stochastic algorithm. Under several hypotheses with γ ∈]0.5, 1], this algorithm has been shown to be consistent and asymptotically normal. A fine tuning of the constant C is important; several numerical tests have shown that a value of C of the order of the dispersion of Y (for example its standard deviation or an interquantile interval) would be satisfactory [START_REF] Iooss | Estimation itérative en propagation d'incertitudes : réglage robuste de l'algorithme de Robbins-Monro[END_REF][START_REF] Iooss | Robust tuning of Robbins-Monro algorithm for quantile estimation in iterative uncertainty quantification[END_REF].

Asymptotically (where N is large), a value γ = 1 is known to be optimal. However, in practical studies, N is often not large. For example, in nuclear safety studies (see for example [START_REF] Cannamela | Controlled stratification for quantile estimation[END_REF][START_REF] Iooss | Advanced methodology for uncertainty propagation in computer experiments with large number of inputs[END_REF]), α = 0.95 and N is in the order of several hundreds of simulated values. In this case and as we look for γ values that can work for different distributions of Y (which are unknown in practice), we propose to define γ as a function of n. Indeed, one can observe that a good γ value for a certain type of probability distribution produces bad results for another type of distribution (for example, γ = 0.6 gives good results for a normal distribution and incorrect quantile estimates for a uniform one). We then use the following heuristic formula for γ:

γ(n) = 0.1 + 0.9 n -1 N -1 . ( 11
)
The idea is to have strong mixing properties at the beginning of the algorithm (with small γ), then to slow down the potential variation of the quantile estimation all along the iterations of the algorithm. Slightly different linear profiles can be proposed as γ(n) = 0.5[1+(n-1)/(N -1)]. Several tests on simple analytical functions (where the true quantile can be known) have been performed in order to calibrate and validate these γ-profiles [START_REF] Ribés | Large scale in transit computation of quantiles for ensemble runs[END_REF][START_REF] Iooss | Estimation itérative en propagation d'incertitudes : réglage robuste de l'algorithme de Robbins-Monro[END_REF]. Other algorithmic developments are currently under study to further improve the robustness of the Robbins-Monro estimate.

Probability of Threshold Exceedance

If y crit is a safety output value, a classical failure probability estimation problem occurs:

p f = P(Y > y crit ) , (12)
well-known in structural reliability [START_REF] Lemaire | Structural reliability[END_REF][START_REF] Bect | Sequential design of computer experiments for the estimation of a probability of failure[END_REF]. For this issue, without loss of generality, we can turn to:

p f = P(Y < 0) .
Computing a failure probability can be seen as a direct problem of uncertainty propagation [START_REF] Morio | Estimation of rare event probabilities in complex aerospace and other systems[END_REF].

If the the failure domain is defined by

D f = {x ∈ χ ⊆ R p | G(x) ≤ 0}
, the probability that the event -failure occurs is given by

p f = P(G(X) ≤ 0) = ∫ D f f X (x)dx = ∫ χ 1 G(x)≤0 f X (x)dx = E[1 G(X)≤0 ] , (13)
where f X is the joint probability density function of X. One of the goals of a structural reliability study is to provide an estimate of p f and the uncertainty involved. The complexity of models and large potential number of input variables means that, in general, we cannot calculate the exact probability of failure. The evaluation of the integral in formula [START_REF] Fournier | Optimizing code_saturne computations on petascale systems[END_REF] is the subject of numerous mathematical techniques, laid out in an abundant array of international scientific literature [START_REF]Uncertainty in industrial practice[END_REF][START_REF] Smith | Uncertainty quantification[END_REF]. The use of Monte Carlo simulation methods is the most common. The naive Monte Carlo estimator is

pf = 1 n n i=1 1 {G(x (i) )≤0} , (14)
where the x (i) are n independent and identically distributed random vectors simulated according to f X . This is an unbiased estimate of the quantity of interest, for which it is possible to control the precision via its variance and the provision of a confidence interval (thanks to the central limit theorem).

Of course, as this estimation is based on an expectation, the trivial iterative algorithm for the mean (see Section 3.1) applies to compute p f on the fly.

The Melissa Framework

Melissa (Modular External Library for In Situ Statistical Analysis) proposes a new approach to compute statistics at large scale by avoiding to store the intermediate results produced by the multiple parallel simulation runs. The key enabler is the use of iterative formulations for the required statistics. This allows for updating statistics on-the-fly each time new simulation results are available. To manage the simulation runs as well as the in transit computation of iterative statistics, we developed a full framework built around an elastic and fault tolerant parallel client/server architecture. The benefits of this framework are multiple:

• Storage saving: no intermediate files are generated. Melissa fully avoids storage of intermediate data on disks. • Time saving: simulations run faster when sending data to the server than when writing their results to disk. Statistics are computed while simulations are running, saving the time of post hoc statistic computing that, in addition, requires time to read back simulation results once all are performed. • Ubiquitous: performance and scalability gains enable computing ubiquitous multidimensional and time varying statistics, i.e. everywhere in space and time, instead of providing statistics for a limited sample of probes as usually done with post hoc approaches to reduce the amount of temporary data storage. • Adaptive: simulations can be defined, started or interrupted on-line according to past runs behavior or the statistics already computed. • Elasticity: Melissa enables the dynamic adaptation of compute resource usage according to availability. Simulations are independent and connect dynamically to the parallel server when they start. They are submitted as independent jobs to the batch scheduler. Thus, the number of concurrently running simulations providing data to the server can vary during the course of a study to adapt to the availability of compute resources. • Fault tolerance: Melissa's asynchronous client/server architecture supports a simple yet robust fault tolerance mechanism. Only some lightweight bookkeeping and a few heartbeats are required to detect issues and restart the server or the simulations, with limited loss of intermediate results.

Melissa Architecture

Melissa is an open source framework 1 that relies on a three tier architecture (Figure 3). The Melissa Server aggregates the simulation results and updates iterative statistics as soon as a new result is available. The Melissa clients are the parallel simulations, providing their outputs to the server. Melissa Launcher interacts with 1 https://melissa-sa.github.io the batch scheduler and server, for creating, launching, and supervising the server and clients. We present in this section an overview of the Melissa architecture. Please refer to [START_REF] Terraz | Melissa: Large scale in transit sensitivity analysis avoiding intermediate files[END_REF] for more details.

Melissa Server

Melissa Server is parallel and runs on several nodes. The number of nodes required for the server is driven by 1) memory needs 2) data pressure. The amount of memory needed for each computed statistic field is of same order as the size of the output field of one simulation (number of timesteps × the number of cells or points in the mesh). The number of server nodes should also be large enough to process incoming data without stalling the simulations.

Dynamic Connection to Melissa Server

When a simulation starts, it dynamically connects to the Melissa Server. Each simulation process opens individual communication channels to each necessary server process for enabling a N ×M data redistribution. Every time new results are available, simulation processes push the results toward Melissa Server. Melissa is designed to keep intrusion into the simulation code minimal. Melissa provides 3 functions to integrate in the simulation code through a dynamic library. The first function (Initialize) allocates internal structures and connects the simulation to the server. At each timestep, the second function (Send) sends the simulation data to its corresponding Melissa Server processes. The third function (Finalize) disconnects the simulation and releases the allocated structures.

Melissa Launcher

Melissa Launcher takes care of generating the parameter sets, requesting the batch scheduler to start the server and the clients, and track the progress of the running server and clients jobs. It first submits to the batch scheduler a job for the Melissa Server. Then, the launcher retrieves the server node addresses (the server is parallelized on several nodes) and submits the simulation jobs. Each simulation is submitted to the batch scheduler as a standalone job, making Melissa very elastic, i.e. capable of adapting to a varying compute resource availability. Simulations can be submitted all at once or at a more regulated pace depending on the machine policy for job submissions.

Fault Tolerance

The Melissa asynchronous client/server architecture leverages the iterative statistics computations to support a simple yet robust fault tolerance mechanism. Melissa supports detection and recovery from failures (including straggler issues) of Melissa Server and simulations, through heartbeats and server checkpointing. Melissa Launcher communicates with the server and the batch scheduler to detect simulation or server faults. As every simulation runs in a separate job, the failure of one simulation does not impact the ongoing study: Melissa launcher simply restarts it and the server discard already processed messages. Please refer to [START_REF] Terraz | Melissa: Large scale in transit sensitivity analysis avoiding intermediate files[END_REF] for a complete description of this fault tolerance system.

An Illustrative Example 5.1 A Large Scale Study

We used Code_Saturne [START_REF] Archambeau | Code_saturne: a finite volume code for the computation of turbulent incompressible flows[END_REF], an open-source computational fluid dynamics tool designed to solve the Navier-Stokes equations, with a focus on incompressible or dilatable flows and advanced turbulence modeling. Code_Saturne relies on a finite volume discretization and allows the use of various mesh types, using an unstructured polyhedral cell model, allowing hybrid and non-conforming meshes. The parallelization [START_REF] Fournier | Optimizing code_saturne computations on petascale systems[END_REF] is based on a classical domain partitioning using MPI, with an optional second (local) level using OpenMP.

Use Case

Varying parameters for the upper and lower inlet:

• dye concentration, • width of the injection, • duration of the injection, Figure 4 Use case: water flows from the left, between the tube bundle, and exits to the right with 6 varying parameters.

We validated our implementation on a fluid mechanics demonstration case simulating a water flow in a tube bundle (Figure 4). The mesh is composed of 6002400 hexahedra. We generate a multi-run sensitivity study by simulating the injection of a tracer or dye along the inlet, with 2 independent injection surfaces, each defined by three varying parameters (Figure 4). The solved scalar field representing dye concentration could be replaced by temperature or concentration of chemical compounds in actual industrial studies.

To initialize our multi-run study, we first ran a single 1000 timesteps simulation, to obtain a steady flow. Each simulation consists of 100 time steps starting from this steady flow, with different parameter sets.

This study ran a total of 80 000 simulations for computing ubiquitous variances, covariances, Sobol' indices, and the 5 th , 25 th , 50 th , 75 th and 95 th percentiles on the 6M hexahedra and 100 timesteps. Sobol' index computations rely on the pick-freeze method that requires to run groups of simulations with non-independent parameter sets. These correlated simulations are not used for the quantiles that are computed from the remaining independent 20 000 simulations.

The study took a total 260 000 CPU hours for the simulations and 11 112 CPU hours for the server (4% of the total CPU time). Melissa Server processed on-line a cumulated total of 288 TB of data coming from the simulations.

Ubiquitous Statistic Interpretation

In this section we interpret the ubiquitous statistics computed during the experiments. By ubiquitous statistics we mean the statistics of multidimensional and time varying physical quantities, i.e. statistics everywhere in space and time. Using ParaView we have chosen a timestep and performed a slice on a mid-plane of the mesh presented in Figure 4. This slice is aligned with the direction of the fluid. The chosen timestep belongs to the last temporal part of the simulation (80 th timestep over 100). This operation reduces the ubiquitions statistics to 2D spatial maps, thus allow us to generate the images of this section. We remark that this is just a choice for illustration purposes, any other visualisation pipeline can be applied to the ubiquitous statistics.

Quantiles

Figure 5 presents six spatial maps extracted from the ubiquitous quantiles. We concentrate in percentiles because they are easily interpreted. We recall that a percentile is not a per cent but a type of quantile: a 10-quantile. Our system can also calculate 4-quantiles, the so-called quartiles that are popular measures in statistics, or any other kind of quantile.

On the four top panels of Figure 5, Figure 5a, 5b, 5c and 5d, we present the 75 th , 95 th , 25 th , 5 th percentiles, respectively. On the two bottom panels the interpercentile ranges containing 50% and 90% of the samples are shown. Interpercentile ranges are easily computed from percentiles by substraction: the 50% interpercentile range corresponds to the 75 th percentile minus the 25 th percentile; the 90% interpercentile range corresponds to the 95 th percentile minus the 5 th percentile. In Figure 5 each column shows an interpercentile map on the bottom and the percentile maps that served for its calculation above it. Looking at these maps an analyst can deduce several things:

1. Extreme high percentile maps such as 95 th , Figure 5b, give an idea of the distribution of the upper bounds of all simulations. In our use case, we can asses which spatial areas contain low quantities of contaminant. Indeed areas coloured in blue necessarily contain low contaminant concentrations for any simulation in the multi-run study. Extreme low percentiles maps, such as 5 th has also a direct interpretation in the opposite sense. 2. Interpercentile-range maps such as Figure 5e or 5f are maps that show the spatial variability of statistical dispersion. Indeed, scalar interpercentile ranges are non-parametric measures of statistical dispersion, which means that no a priori knowledge about the distribution of the data is needed. This characteristic makes these ranges both general and robust. Visualising a map of such a measure of dispersion allows to understand how the data distribution is spatially concentrated. In our use case the low percentile maps used to calculate the interpercentile maps are mainly close to zero for all cells of the mesh, which makes these maps resemble the higher percentiles maps. However, this is in general not true. Moreover, if we visually compare Figure 5a and 5e we realise that both maps are different. The maps shown in Figure 5 are static and 2D but we recall that we calculate ubiquitous percentiles, thus 3D and time dependent data is available. Figure 6 shows the temporal evolution of a probe positioned in the mesh using ParaView. At a specific location, a temporal evolution of all computed quantiles can be performed. In Figure 6b this evolution is plotted for the 95 th , 75 th , 25 th and 5 th percentiles. The vertical line indicates the position of the current time step (80 th time step). This figure clearly shows how the output variability of the ensemble study depends on time. Indeed, all simulations contain no dye for the first 15 time-steps, which is the time the dye takes to propagate from the top injector to the spatial location of the probe. After this point, we observe a moment where the variability of the dye concentration is the highest before a general decrease.

Figure 6b can be seen as the evolution of a Tukey boxplot [START_REF] Tukey | Exploratory data analysis[END_REF] over time. In fact, the 25 th and 75 th percentiles correspond to the 1 st and 3 r d quartiles thus delimit the central box of the plot, while the 5 th and 95 th quantiles can be a choice for the whiskers. Using this analogy, we can easily observe that the dispersion of the dye concentration on the whole ensemble moves over time. Furthermore, the distribution of this quantity is not symmetrical and its asymmetry is evolving over time.

Finally, Figure 7 shows a different representation of the evolution of the dye concentration at a fixed probe (the same than in Figure 6). At different regularly sampled time steps, the quantile functions of the concentration values are plotted (as a function of the order of the quantiles, between 0% and 100%). One can first observe zero-valued quantile functions for the first time steps (time steps 4 and 14). Indeed, at the probe, the dye concentration is zero during the first times of the injection. Then, from time step 24 to time step 44, all the values of the quantile functions regularly increase. It means that the dye concentration values homogeneously increase from 0, reaching a maximal value close to 0.82 for the 100%-order quantile. At the end of the simulation time, from time step 54 to time step 94, the quantile functions are regularly displaced to the right. The values close to zero disappear and the concentration of strong values becomes more and more important. As a conclusion, thanks to the quantile functions, this graph allows to finely and quantitatively analyze the temporal evolution of this dye concentration phenomena. We remark that, because we have calculated the ubiquitous percentiles, it is possible to obtain Figure 6 and Figure 7 for any location on the simulation domain.

Sobol' Indices

In this section we interpret the Sobol' indices computed during the experiments. Figure 8 presents six spatial first order Sobol' maps extracted from the ubiquitous indices. Looking at these figures an analyst can deduce several things: 1. The width of the injections influence locations far up or down of each injector (Figure 8c and8d). This is because the injectors are located in the center of the inlet segments, as depicted by the arrows in Figure 4. The parameter controlling the aperture of an injector makes a thinner or wider aperture around a central position. When the aperture's width is small, all dye is injected on the center while very few will go up and down. On the opposite, when the injector is at its maximal aperture, dye can easily flow to extreme vertical locations. In any case, the dye always flows in the center of the injectors which makes this parameter not influential around these areas. 2. The Sobol' maps for the duration of the injection (Figure 8e and8f) can be understood by thinking about the temporal evolution of the simulation. When the simulations are just started, they all inject dye and the duration represents the time at which this injection is stopped. Figure 8 represents a state near the end of the simulation. Thus, it is not surprising that this parameter does not influence the right part of the domain (where all simulations were injecting dye) while it strongly influences the left side (where some simulations already stopped injecting while others continued).

3. Finally, the dye concentration mostly influences the areas were the other parameters have less influence, that is to say the center of the top and bottom channels, and the right side of the flow (Figure 8a and8b).

We recommend co-visualizing Sobol' indices with the variance of the whole model output; Figure 8g shows this variance map for the Sobol' indices presented in Figure 8. One of the reasons of this co-visualization is that V ar(Y ) appears as a denominator in Eq. ( 3), consequently when V ar(Y ) is very small or zero, the Sobol' indices have no sense due to numerical errors or can even produce a zero division. Furthermore, it is not conceptually interesting to try to understand which input parameters influence low variance areas of the simulation once we know that 'not much happens' in these areas.

Combining Sobol' Indices

The sum of Sobol' indices should be 1 and they partition the total variance of the outputs. These two characteristics allow Sobol' indices to be grouped (added) or subtracted. If the indices are added, the resulting "Sobol group index" keeps normalised between zero and one. We show two examples of the use of such groups:

1. Figure 9a corresponds to the sum of the Sobol' indices for the upper injector while Figure 9b corresponds to the sum for the bottom injector. We clearly observe that the three parameters that define the behavior of the upper injector have no influence in the lowest half part of the simulation domain. This phenomenon can also be observed by simultaneously looking at the Sobol' maps of the three parameters for the upper injector (Figure 8a, 8c and 8e). However, the grouped Sobol map is clearer and gives a straightforward answer to the question "What is the influence of the upper injector?". Respectively, the parameters of the bottom injector do not influence the upper part of the simulation as directly observed on Figure 9b (or looking at Figure 8b, 8d and 8f). 2. We also calculate S 1 + ... + S n , with n = 6, shown in Figure 9c, which gives us an indication of the importance of the interactions between parameters. Red areas of Figure 9c indicate that the first order Sobol indices are responsible of the behaviour of the simulation. On the contrary, when this sum is near zero (blue in the map) this means that the kind of relationship between input parameters and model outputs is complex and involves cross-effects among several parameters. We observe that the areas where interactions among Sobol' indices are important concentrates spatially around the injectors.

Conclusion

Dealing with uncertainty quantification may require executing from thousands to millions of runs of the same simulation, making it an extremely compute-intensive process that will fully benefit from Exascale machines. However, the large amount of data generated is a strong I/O bottleneck if the intermediate data is saved to disk. The proposed approach, implemented in the Melissa framework, demonstrates that combining one-pass statistics algorithms with an elastic and fault-tolerant in transit data processing architecture drastically reduces the amount of intermediate data stored, making it possible to compute ubiquitous statistics. Melissa currently allows the in transit execution of large scale uncertainty quantification studies. This open-source software2 currently implements the computation of the statistical tools necessary for this purpose: moment-based statistics (mean, standard deviation, higher orders), quantiles, Sobol' indices, and threshold exceedance. Future work will include the integration of Melissa with the OpenTurns uncertainty quantification software to consolidate and broaden their respective capabilities.

Figure 3

 3 Figure 3 Melissa three tier architecture. The launcher oversees the execution in tight link with the batch scheduler. The job scheduler regulates the number of simulation jobs to run according to the machine availability, leading to an elastic resource usage. The parallel server, first, process incoming data as soon as received from the connected simulations. A fault tolerance mechanism automatically restarts the failing simulation runs or a failing parallel server.

Figure 5

 5 Figure 5 Percentiles and inter-percentile ranges maps on a slice of the mesh at timestep 80. The top line corresponds to the percentiles while the bottom line corresponds to inter-percentile ranges. All maps share the same scale.

Figure 6 A

 6 Figure 6 A probe in a cell of the mesh allows an extraction of the temporal evolution of percentiles at a specific spatial location.

Figure 7

 7 Figure 7 Percentile functions of the dye concentration at different time steps of the simulation. Vertical axis represents dye concentration. Horizontal axis represents percentiles. Each curve corresponds to different time steps of the simulation. All curves have been extracted from the probe position shown in Figure 6a.

Figure 8

 8 Figure 8 First order Sobol' index maps on a slice of the mesh at timestep 80. The left column corresponds to the Sobol' indices for the upper injector while right corresponds to the bottom injector. All maps are scaled between zero (blue) and one (red).

Figure 9

 9 Figure 9 Addition of first order Sobol' index maps on a slice of the mesh at timestep 80. The left panel corresponds to the addition of the Sobol' indices for the upper injector while right corresponds to addition for the bottom injector. All maps are scaled between zero (blue) and one (red).


Lina Masood

Keywords: Comportement au travail créatif, Covid-19, État d'esprit créatif, Leadership Encouragement à la créativité, Théorie sociale cognitive, Théorie implicite de la personne Covid-19, Creative Work Behavior; Creative Mindset, Leadership Encouragement of Creativity, Cognitive Social Theory, Implicit Person Theory

. Par conséquent, notre cadre théorique offre des informations précieuses sur la compréhension du comportement de travail créatif des employés dans l'organisation pendant et au-delà de la pandémie de Covid-19.

Introduction

The Covid-19 pandemic has caused tremendous novelty and uncertainty impacting many in the world [START_REF] Barzilay | Resilience, COVID-19-related stress, anxiety and depression during the pandemic in a large population enriched for healthcare providers[END_REF]WHO, 2020bWHO, , 2020a)). In the time of this pandemic, innovation has become an important resource for the success of organizations in responding to the challenges created by the crisis. To be innovative, they have to put more emphasis on the creativity of their employees [START_REF] Tang | Creativity as a Means to Well-Being in Times of COVID-19 Pandemic: Results of a Cross-Cultural Study[END_REF]. Studies covering creativity put more emphasis on the creative work behavior of employees [START_REF] Saunila | Innovation capability for SME success: Perspectives of financial and operational performance[END_REF]. Creative employees can develop new ideas and come up with useful answers to the problems that organizations face during this challenging time. Hence, one important option for organizations to become more creative is to encourage their employees to be creative [START_REF] Alikaj | Proactive personality and creative behavior: Examining the role of thriving at work and high-involvement HR practices[END_REF]. This increasing importance of creative behavior underscores the need to investigate the fundamental factors that foster it (Afsar et al., 2020;Yuan & Woodman, 2010). Several scholars have attempted to determine these factors [START_REF] Alikaj | Proactive personality and creative behavior: Examining the role of thriving at work and high-involvement HR practices[END_REF][START_REF] Aslam | Psychological empowerment on creativity among employees of IT sector: The mediating role of creative process engagement and intrinsic motivation[END_REF][START_REF] Caniëls | The antecedents of creativity revisited: A process perspective[END_REF][START_REF] Dong | Enhancing employee creativity via individual skill development and team knowledge sharing: Influences of dual-focused transformational leadership[END_REF][START_REF] Paek | The indirect effect of teachers' creative mindsets on teaching creativity[END_REF]. One of the factors that seems to have an impact on innovation is the employee's creative mindset [START_REF] Batra | Exploring the linkages of cognitive style and individual innovativeness[END_REF][START_REF] Carnabuci | Social networks, cognitive style, and innovative performance: A contingency perspective[END_REF][START_REF] Liu | Thriving at work: How a paradox mindset influences innovative work behavior[END_REF][START_REF] Newman | The effects of employees' creative self-efficacy on innovative behavior: The role of entrepreneurial leadership[END_REF]. According to the implicit person theory [START_REF] Dweck | Mindset: The new psychology of success[END_REF][START_REF] Dweck | A social-cognitive approach to motivation and personality[END_REF], psychologically trained individuals who believe that their creative abilities and skills can be change, can be described as having a growth creative mindset [START_REF] Dweck | Motivational processes affecting learning[END_REF][START_REF] Dweck | Mindset: The new psychology of success[END_REF][START_REF] Dweck | Mindsets and human nature: Promoting change in the Middle East, the schoolyard, the racial divide, and willpower[END_REF]. Work in the field of developmental psychology has found that the belief about the malleability of personality and traits in general affects how people think and what they do. This is also corroborated by the social cognitive theory [START_REF] Bandura | Self-efficacy mechanism in human agency[END_REF][START_REF] Banduran | Self-efficacy: The Exercise of Control[END_REF], which states that people's beliefs influence their cognition processes, efforts and interactions [START_REF] Blackwell | Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention[END_REF][START_REF] Levy | Lay theories and intergroup relations[END_REF], which in turn, may enable them to share learning and come up with new ideas.

Despite the concept of creative mindset is relatively ubiquitous and well-studied in education literatures (Choi, 2019a), a very limited number of studies have examined the impact of creativity mindset, on creative behavior for employees within the organizational settings [START_REF] Karwowski | Creative mindsets: Prospects and challenges[END_REF][START_REF] O'connor | Consequences of beliefs about the malleability of creativity[END_REF]. Hence, we contribute to social cognitive theory and implicit person theory by expanding the theoretical understanding on how creative mindset may influence employee behaviors like creative behavior. This influence is little explored and developed in the literature (e.g., Jeong et al., 2016). Do employees who feel that creativity is a fixed trait do equally well as those who feel that creativity is learnable?

On top of the employee beliefs, the role of leadership is key in encouraging or discouraging a creative climate [START_REF] Sarros | Building a climate for innovation through transformational leadership and organizational culture[END_REF]. A recent concept has emerged in management literature to describe a specific leadership which is supportive of creativity in the organization. This concept is labelled as "leadership encouragement of creativity" [START_REF] Zhang | Linking empowering leadership and employee creativity: The influence of psychological empowerment, intrinsic motivation, and creative process engagement[END_REF]. To our knowledge, an understanding of the boundary conditions on the creative mindset-creative behavior relationship remains underdeveloped. There is a dearth of knowledge of whether contextual factors at work, such as leadership encouragement of creativity, may accentuate the relationship between employees' creative mindset and their creative behavior. We highlight the significant role that leader encouragement of creativity might play on this relationship. In particular, we argue that this relationship is more likely to be influenced by the extent to which the leader is encouraging or not. In accordance to the above, we developed a theoretical framework based on both social-cognitive theory and implicit person theory to mitigate the pandemic concerns at hands by improving the creative work behavior of employees in organizations.

Literature Review

In this section, we are going to dig deep into the main concepts of our study, i.e., the concept of employee creative behavior and the concept of creative mindset as a potential factor of creative behavior. We will also address the concepts of leadership encouragement of creativity.

Creative Work Behavior

Researches still have to establish an agreement on the definition of creativity; other than 'useful' and'new' properties. In 1953, Stein (1953) described creativity as a process leading to a new output which is desirable to a group or community. [START_REF] Torrance | Adventuring in creativity[END_REF] described creativity as the potential people have to fix the issues using critical thinking and through applying different ways. Creativity was conceptualized by [START_REF] Guilford | Creativity: Yesterday, today and tomorrow[END_REF] as a process by which individuals may develop new ideas in any setting and usefully implement these ideas. As defined by [START_REF] Csikszentmihalyi | Flow and the psychology of discovery and invention[END_REF], creativity is any action, idea or product which alters an existing domain into a new domain. In general, the idea of creativity revolves around the presentation of ideas or products by employees working in the organization that should be a) novel and original and b) helpful and relevant. They considered new if they are novel to the organization and helpful if they add value to it [START_REF] Joo | Creativity and human resource development: An integrative literature review and a conceptual framework for future research[END_REF][START_REF] Shalley | The effects of personal and contextual characteristics on creativity: Where should we go from here[END_REF].

Literature on creativity has been studied in several scholarly fields such as sociology, psychology, management, education and science [START_REF] Runco | Critical creative processes[END_REF]. However, in our study we focus on the literature of creativity in the domain of management and organizational context. Numerous definitions of creativity focus on the work setting. For example, individual creativity is defined by [START_REF] Woodman | Toward a theory of organizational creativity[END_REF] as an individual who is working in a complicated social structure and having the primary production of novel ideas, products, or process which add a societal benefit. Similarly, [START_REF] Zhou | When job dissatisfaction leads to creativity: Encouraging the expression of voice[END_REF] conceptualized creativity as a production of new and valuable ideas, products, services, or processes by individuals working together in a complex social system. However, [START_REF] Pope | Creativity: Theory, history, practice[END_REF] claimed that creativity is something novel when it meets the criteria of being useful and valuable to different shareholders in the organization. [START_REF] Kim | Proactive personality and employee creativity: The effects of job creativity requirement and supervisor support for creativity[END_REF] conceptualized creativity in the workplace as an act by employees to develop new ideas or try new ways regularly in order to reach their objectives.

Since creativity has been conceptualized by several researchers with different ways, the definition that is introduced by [START_REF] Zhou | When job dissatisfaction leads to creativity: Encouraging the expression of voice[END_REF] illustrate the meaning of employee creativity within the fields of the corporate environment. Hence, the researcher decided to adopt this definition that best matches the objective of this study.

Researchers disagree about whether creativity is a talent or skill. Several studies that have been conducted in the period between 1950s and 1960s have suggested that creativity to be a skill rather than a talent and that creative individuals are often creative not because that have a certain talent, but for their approach toward life and work [START_REF] Amabile | Perspectives on the social psychology of creativity[END_REF]. Rather than allowing themselves to react in traditional ways they usually train themselves, react to new challenges and take risk to try new things [START_REF] Sternberg | The assessment of creativity: An investment-based approach[END_REF].

However, other scholars perceive creativity as a talent and that it is more likely to be something you are born with. According to them, a creative person can become a pure genius if you give him the right conditions and support. Furthermore, the talent approach continued to lead the creativity literature for years without looking at environmental or situational factor that may potentially offer understanding to individual creativity behavior [START_REF] Amabile | The social psychology of creativity: A componential conceptualization[END_REF].

Creative Mindset

According to [START_REF] Dweck | Mindset: The new psychology of success[END_REF], mindsets are the beliefs or assumptions that people hold about the plasticity of their abilities. Mindset as a concept is based in the individual implicit theory, which includes incremental implicit theory and entity implicit theory. Incremental implicit theory is known as a "growth mindset", "acquired skill", or "malleable traits". It reflects the underlying belief of the individuals in the malleability of basic personal abilities and characteristics. Growth mindset is exemplified by statements including "Everyone, no matter who they are, can significantly change their basic characteristics" and "No matter what kind of person someone is, they can always change very much" [START_REF] Dweck | Mindset: The new psychology of success[END_REF].

As opposed to incremental implicit theory, entity implicit theory which is known as a "fixed mindset" or "inherent capacity" [START_REF] Dweck | Mindset: The new psychology of success[END_REF], embodies the belief that human basic traits such as abilities and intelligence are static and unchangeable [START_REF] Yeager | Mindsets that promote resilience: When students believe that personal characteristics can be developed[END_REF]. Fixed mindset is demonstrated by statements which stress the limits and weaknesses on the scope for individuals to grow. Examples of those statements include "You can't teach an old dog new trick" and "Everyone is a certain kind of person, and there is not much that can be done to really change that" [START_REF] Dweck | Mindset: The new psychology of success[END_REF].

Although mindsets occur on a continuum between the fixed and growth prototypes, [START_REF] Claro | Growth mindset tempers the effects of poverty on academic achievement[END_REF] have proven that that mindset is a state of mind and not a fixed trait. Everyone is a hybrid of fixed and growth mindset, and this mix evolves constantly with practice [START_REF] Claro | Growth mindset tempers the effects of poverty on academic achievement[END_REF]. In other words, people may hold a fixed mindset or growth mindset about their abilities in some specific areas [START_REF] Burnette | Mind-sets matter: A meta-analytic review of implicit theories and self-regulation[END_REF]. There is no "pure" growth mindset that we need to consider to achieve the advantages we are looking for. For example, an individual may have a fixed mindset about working with difficult clients and a growth mindset about his quantitative abilities. In addition to that, research on social cognitive theory also reveals that some people could have evolving mindsets [START_REF] Lyons | The intersection of mindsets and self-regulated learning. Development and Learning in Organizations[END_REF].

Based on the implicit person theory developed by Dweck and her colleagues [START_REF] Dweck | Mindset: The new psychology of success[END_REF][START_REF] Hong | Implicit theories and evaluative processes in person cognition[END_REF], a new concept of mindset has been introduced, which called creative mindset. Creative mindsets are defined as people's implicit assumptions about the perceived source and nature of creativity [START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF][START_REF] Karwowski | Creative mindsets: Prospects and challenges[END_REF]. Like the concept of mindset, there are two types of creative mindset; namely growth creative mindset and fixed creative mindset. Growth creative mindset is related to growth mindset and it is characterized by the belief that creative abilities and qualities can be developed through efforts and hard work, while fixed creative mindset is related with fixed mindset and it is based on the belief that creativity and creative skills are fixed based on one's inherent nature, and there is no much one can do to improve [START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF].

The importance of creative mindset as a distinct construct different from mindset for two main reasons. First, considering the distinction between intelligence and creativity, perceptions hold by people about their creativity differ from those of their intelligence [START_REF] Sternberg | Creativity and intelligence[END_REF], and hence, the assessment of both constructs have to be distinguished in order to obtain an accurate evaluation. Some studies (e.g., [START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF][START_REF] O'connor | Consequences of beliefs about the malleability of creativity[END_REF][START_REF] Puente-Díaz | The influence of creative mindsets on achievement goals, enjoyment, creative self-efficacy and performance among business students[END_REF] have supported the validity of creative mindset and its relation to the theory of mindset. It has been demonstrated that intelligence did not predict creative problem solving, a growth creative mindset did. Second, scholars have discovered that the belief that individual are born creative is negative to their ability to develop their creative outcomes (e.g., [START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF][START_REF] O'connor | Consequences of beliefs about the malleability of creativity[END_REF].

In sum, with this review, we propose that creative mindset may be central to an employee creativity in the work settings. Thus, in the next section we provide theoretical arguments for how creative behaviors of an employee may be affected by his/her creative mindset. Further to that, the role of leader encouragement of creativity has been demonstrated as a critical variable to encourage people to exchange their knowledge [START_REF] Shamim | Impact of knowledge oriented leadership on knowledge management behaviour through employee work attitudes[END_REF][START_REF] Zia | Knowledge-oriented leadership, knowledge management behaviour and innovation performance in project-based SMEs. The moderating role of goal orientations[END_REF], with will in turn, encourage them to create [START_REF] Yu | Knowledge sharing, organizational climate, and innovative behavior: A cross-level analysis of effects[END_REF]. Hence, we will test if the role of leader encouragement for creativity moderates this relationship.

Theory and Hypotheses

In this section, and as stated earlier, we are describing the theoretical backgrounds for our research, i.e., social cognitive theory and individual implicit theory. These theories act as the theoretical basement of the subsequent hypotheses of our study.

Social Cognitive Theory

As previously stated, social cognitive theory is one of the theoretical basements of our study. It acts as a metatheory which links behavior to personal factors and environmental factors and is a widely accepted model for validating individual behavior [START_REF] Compeau | Computer self-efficacy: Development of a measure and initial test[END_REF]. Creative mindset (personal factor) can be related to creative work behavior (behavior) within the context of Covid-19 pandemic and its effects on employees (environment). Prior to establishing social cognitive theory, [START_REF] Bandura | Social learning theory[END_REF] formed the social learning theory that explained the critical role of social context in a learning environment [START_REF] Bandura | Social learning theory[END_REF]. Later on, [START_REF] Bandura | Social foundations of thought and action[END_REF] incorporate motivational and cognitive mechanisms to the theory.

In an attempt to understand different human phenomena, social cognitive theory influenced a vast and wide range of research practices in the areas of social psychology, education and organizational management. This includes academic success, career decisions, setting goals, selecting a healthier lifestyle, nutrition behavior, motivation, organizational citizenship behavior, employee engagement and job performance [START_REF] Anderson | Self-regulation, self-efficacy, outcome expectations, and social support: Social cognitive theory and nutrition behavior[END_REF][START_REF] Hmieleski | Entrepreneurs' optimism and new venture performance: A social cognitive perspective[END_REF][START_REF] Lent | Toward a unifying social cognitive theory of career and academic interest, choice, and performance[END_REF][START_REF] Lorente | How personal resources predict work engagement and self-rated performance among construction workers: A social cognitive perspective[END_REF][START_REF] Mccormick | Identifying leader social cognitions: Integrating the causal reasoning perspective into social cognitive theory[END_REF][START_REF] Zhao | Socially responsible human resource management and hotel employee organizational citizenship behavior for the environment: A social cognitive perspective[END_REF]. Three elements of social cognitive theory have been focused on in the area of organisation, including the improvement of people's cognitive, social and behavioral skills by creating people's confidence in their capacity to use their strengths and increase their motivation [START_REF] Bandura | Organisational applications of social cognitive theory[END_REF][START_REF] Wood | Social cognitive theory of organizational management[END_REF]).

In the social cognitive theory model, personal factors/cognitions, personal behaviors and environmental influences act as interacting determinants impacting each other (see Figure 1) [START_REF] Bandura | Social foundations of thought and action[END_REF][START_REF] Wood | Social cognitive theory of organizational management[END_REF]. Because of this triadic dynamic causation, human behaviors are affected and guided by variations of individual personal factors as well as the influence of related environmental context [START_REF] Bandura | Social foundations of thought and action[END_REF][START_REF] Bandura | Social cognitive theory: An agentic perspective[END_REF][START_REF] Banduran | Self-efficacy: The Exercise of Control[END_REF].

Social cognitive theory has introduced some human competencies like symbolizing, forethought, vicarious learning, self-efficacy, self-regulatory, and self-reflective competencies, positing that these competencies play a crucial role in human development and functioning [START_REF] Bandura | Social foundations of thought and action[END_REF]. In accordance with this theory, the concept of mindset [START_REF] Dweck | Motivational processes affecting learning[END_REF] plays the role of a significant personal factor to impact employee behavior. Creative mindset relates to a type of self-efficacy and personal beliefs, and beliefs influence behaviors, and both beliefs and behaviors are shaped by environmental impacts.

The aforementioned literature review indicates that an explanation of the employee's motivation of fostering a creative work behavior is to be provided, with the exploration of what nature of environmental facilitators may influence this behavior. The key concept of this theory is known as mindset [START_REF] Dweck | Mindset: The new psychology of success[END_REF]. According to [START_REF] Dweck | Mindset: The new psychology of success[END_REF], mindset is among the most crucial factor of learning that facilitate personal and professional development. Mindset refers to beliefs and assumptions individuals hold about their plasticity of traits (e.g., personality, abilities, talent, intelligence). Such beliefs and assumptions significantly affect their behaviors, motivation and responses to their everyday affairs [START_REF] Blackwell | Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention[END_REF][START_REF] Dweck | Mindset: The new psychology of success[END_REF].

Mindset theory is a mental framework that often applied in many areas. Its interpersonal implications have been revealed in field of learning and education (e.g., [START_REF] Claro | Growth mindset tempers the effects of poverty on academic achievement[END_REF]David S. Yeager et al., 2019), social psychology [START_REF] Schroder | Growth mindset of anxiety buffers the link between stressful life events and psychological distress and coping strategies[END_REF], and organizational context [START_REF] Canning | Cultures of genius at work: Organizational mindsets predict cultural norms, trust, and commitment[END_REF][START_REF] Keating | The potential role of mindsets in unleashing employee engagement[END_REF][START_REF] Zeng | Teachers' growth mindset and work engagement in the Chinese educational context: Well-being and perseverance of effort as mediators[END_REF]. According to this theory, people may have a different sense of their universe as a result of their mindset [START_REF] Daniel | Finding" meaning" in psychology: A lay theories approach to selfregulation, social perception, and social development[END_REF]. Individuals acting with a fixed mindset will perceive their own intellect, skill and abilities as a constant indicator. They want to be intelligent and smart all the time and avoid feeling dumb at all costs, and that drives their attitudes and actions [START_REF] Yeager | Mindsets that promote resilience: When students believe that personal characteristics can be developed[END_REF]. As a result, it is anticipated that people with a fixed mindset would keep away from activities that might indicate an area of deficiency during difficult times, particularly after setbacks [START_REF] Yeager | Mindsets that promote resilience: When students believe that personal characteristics can be developed[END_REF].

The assumption that abilities are unchangeable also prompts those with a fixed to react helplessly to major challenges. During the pandemic crisis, which is characterized by a high level of uncertainty and the need to learn new ways of creative working and communication, one can assume that fixed creative mindset employees will have a hard time to adapt and to accept the situation creatively. Moreover, fixed creative mindset people tend to treat effort as futile [START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF][START_REF] Mueller | Praise for intelligence can undermine children's motivation and performance[END_REF], and ignore feedback that may be beneficial [START_REF] Heslin | Self-regulation derailed: Implicit person theories and feedback-seeking[END_REF]. They also judge people easily for their wrongdoings, and their connections with others can be strained.

On the other hand, individuals with a growth mindset view life as a journey where becoming better every day through learning and practice is the goal [START_REF] Dweck | A social-cognitive approach to motivation and personality[END_REF]. Those people enjoy learning and are prepared to work hard. They even battle to improve and grow to new levels [START_REF] Yeager | Mindsets that promote resilience: When students believe that personal characteristics can be developed[END_REF]. Subsequently, when people have a growth creative mindset, they welcome setbacks and failures. They experience them as chances to progress and tend to embrace them as crucial for mastering creative tasks. Essentially because they assign failure to a lack of efforts rather than a lack of creative skill [START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF]. They appreciate corrective feedback that may help them to improve their creative abilities. They believe people can get more creative if they work at it. One can assume that growth creative mindset individuals face the pandemic crisis with an increased positive attitude than their fixed creative mindset counterparts. Puente-Díaz & Cavazos-Arroyo (2017) showed that a growth creative mindset increased the likelihood of creative achievement and generating creative solutions. Thus, at the professional level, growth creative mindset employees might increase their likelihood to disseminating creative work behavior despite the crisis context.

More recent studies have broadened the concept of mindset as a situational state [START_REF] Burnette | Mind-sets matter: A meta-analytic review of implicit theories and self-regulation[END_REF][START_REF] Claro | Growth mindset tempers the effects of poverty on academic achievement[END_REF][START_REF] Lyons | The intersection of mindsets and self-regulated learning. Development and Learning in Organizations[END_REF]. This implies that creative mindset can be learned and that people could have evolving mindsets [START_REF] Karwowski | Creative mindsets: Prospects and challenges[END_REF][START_REF] Lyons | The intersection of mindsets and self-regulated learning. Development and Learning in Organizations[END_REF]. All these findings suggest that external interventions, like leadership, can potentially impact the relationship between creative mindset. The study by [START_REF] Owusu-Manu | Exploring the linkages between project managers' mindset behaviour and project leadership style in the Ghanaian construction industry[END_REF] showed consistent results with this hypothesis.

To summarize, research has shown that a) employees can learn to move their creative mindset from fixed to growth [START_REF] Paek | The indirect effect of teachers' creative mindsets on teaching creativity[END_REF][START_REF] Puente-Díaz | The influence of creative mindsets on achievement goals, enjoyment, creative self-efficacy and performance among business students[END_REF], and b) leadership can impact the likelihood of this learning to happen (Kouzes & Posner, 2019). Therefore, although this study focuses specifically at the employee level, the role of leadership seems to be a significant variable in the human mechanism between creative mindset and creative work behavior. Specifically, leadership encouragement of creativity that appears as an appropriate leadership style for our study purpose.

Our study applies social cognitive theory [START_REF] Bandura | Social foundations of thought and action[END_REF] and implicit person theory [START_REF] Dweck | Motivational processes affecting learning[END_REF][START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF] to build a theoretical model aiming at understanding how creative mindset could impact creative work behaviors both in the midst of a crisis and beyond. The proposed research model is presented in Figure 2.

The Impact of Mindset on Creative Work Behavior

Based on social cognitive theory [START_REF] Bandura | Social foundations of thought and action[END_REF][START_REF] Banduran | Self-efficacy: The Exercise of Control[END_REF], an individual behavior is affected by his personal perceptions as well as the related environmental influences. In this sense, growth creative mindset is supported to be empowering the creative behavior of the employees. According to mindset theory [START_REF] Dweck | Mindset: The new psychology of success[END_REF][START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF], the assumptions held by individuals about the natural existence of creative abilities and the changeable-fixed nature of them can form their self-beliefs towards creativity, resulting in a fixed or growth creative mindset. These mindsets, in turn, may function as indicators by which people can interpret the world around them [START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF]. Implicit understanding of these abilities could lead people to achieve creatively their tasks [START_REF] Beghetto | Toward untangling creative self-beliefs[END_REF]. However, if they have a wrong understanding of that, particularly if they believe that creative thinking is difficult to reach, then the person's ability to acquire new creative thinking abilities is likely to suffer [START_REF] Hass | Are implicit theories of creativity domain specific? Evidence and implications[END_REF]. Hence, it is critical to pay more attention on people's state of mind in developing their creative abilities as it is a crucial factor in fostering their creative behavior [START_REF] Choi | How Creative Mindset Is Involved in Positive Emotions and Attitude that Affects Creative Design Process[END_REF].

Employees endorse a growth creative mindset believe that creativity is trainable and can be taught in the same way that other psychological traits may be taught. Previous research dedicated to intelligence revealed that people with a growth mindset have a great possibility to perceive setbacks and failures as an opportunity to grow [START_REF] Burnette | Mind-sets matter: A meta-analytic review of implicit theories and self-regulation[END_REF][START_REF] Dweck | Motivational processes affecting learning[END_REF][START_REF] Dweck | Mindset: The new psychology of success[END_REF]. Even if they are troubled, they keep working on problems until they come up with useful ideas and solutions [START_REF] Dweck | Mindset: The new psychology of success[END_REF]. As a result of their insistence on making efforts, they are able to increase their creative skills and competences [START_REF] Dweck | Motivational processes affecting learning[END_REF][START_REF] Intasao | Beliefs about creativity influence creative performance: The mediation effects of flexibility and positive affect[END_REF]. In addition, employees with a growth creative mindset are more likely to participate in creativity training, activities, or hobbies, leading to support the organizational creativity and innovation, thus supporting the organization during crisis periods and high uncertainty contexts as the Covid-19 pandemic one.

Compared to those with a growth creative mindset, an employee having a fixed creative mindset believes that trait is fixed and a person has a certain number of basic creative qualities [START_REF] Dweck | Motivational processes affecting learning[END_REF][START_REF] Karwowski | Creative mindsets: Measurement, correlates, consequences[END_REF]. Previous research on intelligence revealed that entity theorists may not even attempt a difficult activity due to a fear of failure [START_REF] Haimovitz | Dangerous mindsets: How beliefs about intelligence predict motivational change[END_REF]. Due to that, it could be argued that employees with entity beliefs toward creativity view failure as a risk, and this discourages them from participating in risky activities [START_REF] Haimovitz | Dangerous mindsets: How beliefs about intelligence predict motivational change[END_REF]. Creative mindset researchers expect that people with fixed creative mindsets will avoid tasks that are perceived as novel, complex, or creative. Therefore, fixed creative mindset may reduce the possibility of participation in creative solutions for problem-solving tasks [START_REF] Martin | Implicit theories about intelligence and growth (personal best) goals: Exploring reciprocal relationships[END_REF]. In line with that, we hypothesized the following: Hypothesis 1. (a) Growth creative mindset will be positively related to creative work behavior.

Hypothesis 1. (b) Fixed creative mindset will be negatively related to creative work behavior.

The Moderating Impact of Leadership Encouragement of Creativity

There is a need for leaders who are able to create a creative environment in the workplace, especially during times of crisis such as the Covid-19 pandemic. According to social cognitive theory, proxy agents can affect the beliefs and behaviors of people [START_REF] Bandura | Social cognitive theory: An agentic perspective[END_REF]). Leadership is an example of proxy agents and thus it may influence the relationship between an employee's mindset and creative behavior. From the implicit person theory standpoint, many scholars have shown that mindset is more likely to be seen as a state rather than a trait with the potential to evolve in response to contextual environmental stimuli [START_REF] Burnette | Mind-sets matter: A meta-analytic review of implicit theories and self-regulation[END_REF][START_REF] Claro | Growth mindset tempers the effects of poverty on academic achievement[END_REF][START_REF] Lyons | The intersection of mindsets and self-regulated learning. Development and Learning in Organizations[END_REF].

The cultural component of a leader encouragement of creativity entails that manager should understand how to develop an organizational culture that promotes employees' creativity in order to meet the goals of the organization. This culture of support aims to help employees focus on new approaches and strategies that sustain their creativity. All members of an organization can benefit when leaders successfully devote their time and energy to encourage creativity. For instance, they can encourage them to try different solution and take risks to try new techniques in their job (Cheng & Yang, 2019;[START_REF] Zhang | Linking empowering leadership and employee creativity: The influence of psychological empowerment, intrinsic motivation, and creative process engagement[END_REF]. They should also recognize the ones who come up with creative ideas, products or services.

In contrast, leaders can build barriers and obstacles to employees by adapting inappropriate behaviors that discourage them to create (Tu et al., 2019). Leaders can prevent employees from bringing unique perspectives into creative processes. Further, they may diminish perseverance, motivation, risk taking and learning as main keys to creativity. Because of this, employees will be less likely to overcome their fear of failure when taking risk (Politis, 2005). Instead, they will avoid new experiments to avoid the unfavourable feedback and criticism received from their leader. This, in turn, can reduce their enthusiasm for generating new ideas (C.-R. Li et al., 2018). As a result, they fall back on their efforts in the face of setbacks and hard times. Such employees and leaders are less expected to meet the demands and challenges of today's organizations. Employees with a fixed creative mindset, tend to avoid challenging situations that they might not be able to handle. However, they might be expected to react positively to such situations when they have leaders who encourage them and place them in contexts where they can gain skills. With a more encouraging leader, they may share ideas about their ability to create new things beyond their own expectations. Hence, the link between a fixed creative mindset and creative behavior may weaken when leaders support creative work. Hence, the following are hypothesized: Hypothesis 2. (a) Leadership encouragement of creativity moderates the positive relationship between growth creative mindset and creative work behavior; such that the relationship is stronger when leaders encourage employees.

Hypothesis 2. (b) Leadership encouragement of creativity moderates the negative relationship between fixed creative mindset and creative work behavior; such that the relationship is weaker when leaders discourage employees.

Theoretical Contributions

Our study contributes to the literature in several ways. First, most of the studies which examined the relationship between creative mindset and individual outcomes were carried out in psychology settings. Creative mindset is still under-researched in the organizational literature (S. J. [START_REF] Han | Growth mindset for human resource development: A scoping review of the literature with recommended interventions[END_REF]. Puente-Díaz & Cavazos-Arroyo (2017) suggested that there is a need to enrich the organizational literature with the concept of creative mindset. Second, we contribute to social cognitive theory and implicit person theory by expanding the theoretical understanding on how creative mindset may influence employee behaviors like creative behavior. This influence is little explored and developed in the literature (e.g., Jeong et al., 2016). Third, an understanding of the boundary conditions on the creative mindset-creative behavior relationship remains underdeveloped. There is a dearth of knowledge of whether contextual factors at work, such as leadership encouragement of creativity, may accentuate the relationship between employees' creative mindset and their creative behavior. We highlight the significant role that leader encouragement of creativity might play on this relationship. In particular, we argue that this relationship is more likely to be influenced by the extent to which the leader is encouraging or not.

Given this discussion and responding to the scholarly calls to advance the literature of creativity, our study aims at investigating the impact of growth creative mindset and fixed creative mindset of employees on their creative behaviors via the moderating role of leader encouragement of creativity.

Implications for Managerial Practice

Even though this study focuses on the exploration of how employees' creative mindset may influence their creative behavior, we recognize that creative mindset is an important determinant of many organizational performance outcomes (S. J. [START_REF] Han | Growth mindset for human resource development: A scoping review of the literature with recommended interventions[END_REF] and accordingly, should be seriously considered by the organizations. As such, we propose the following implications to provide practical insights for organizations and practitioners looking to compete and sustain their competitive advantage through enhancing the creative behaviors of their employees.

First, one significant implication is that short and easy-to-use psychometric assessments of creative mindset are helpful to companies to recognize the degree of growth and fixed creative mindset of jobseekers during the recruiting process. Thus, managers and supervisors should consciously decide how to deploy these employees within the organization. They may for example, delegate knowledge-intensive and creativity dependent positions to people with a growth creative mindset. Employees with a growth creative mindset are able to generate new ideas and solutions for the challenges and issues they are facing during this era of technological change.

Second, mindset can be viewed as a personal psychological state that can be developed through human resources management and leadership practices (see e.g., Schaufeli & Bakker, 2004). Management and leadership could undertake actions and conduct interventions to develop the mindset of employees into a growth creative mindset direction that could inspire their behavior towards creative practices [START_REF] Heslin | Self-regulation derailed: Implicit person theories and feedback-seeking[END_REF][START_REF] Keating | The potential role of mindsets in unleashing employee engagement[END_REF][START_REF] Puente-Díaz | The influence of creative mindsets on achievement goals, enjoyment, creative self-efficacy and performance among business students[END_REF]. These interventions may include training employees on how they can produce specific outcomes instead of concentrating on their inner creative skills [START_REF] Keating | The potential role of mindsets in unleashing employee engagement[END_REF]. In her book: "Self-Theories", Dweck (2000) suggested processes and methods to help people building a growth mindset. Examples include conducting specific training and learning actions or practicing feedback and criticism to increase self-knowledge. She proposed the using of feedback in a constructive way so that people would not feel dumb but rather work on finding different approaches to succeed (Dweck, 2000).

Third, our study suggest that leader encouragement of creativity strengthens the relationship between creative growth mindset and creative behavior. Moreover, it suggests that leaders can create the conditions to encourage employees with a fixed creative mindset to successfully engage in creative activities, which will motivate them to boost their creative performance. In essence, organizations may consider their managerial leadership styles and match them with employees in such a way as to reap the greatest benefits from them to the company.

Hence, the study may suggest that organizations should employ and value leaders who are best qualified to promote creative behavior. Organizations with encouraging leaders of creativity are better trained for contributing to the active discovery of new ideas. In this perspective, these ideas can be converted and integrated successfully to create new creative results that contribute to firm innovativeness and competitiveness (Bryant, 2003;Donate & de Pablo, 2015;[START_REF] Zia | Knowledge-oriented leadership, knowledge management behaviour and innovation performance in project-based SMEs. The moderating role of goal orientations[END_REF].

Figure

  Figure 1: Social Cognitive Theory Model

Figure 2 :

 2 Figure 2: The Research Model