Ilya Loshchilov
email: ilya.loshchilov@epfl.ch

Marc Schoenauer

Michèle Sebag

Nikolaus Hansen

Maximum Likelihood-based Online Adaptation of Hyper-parameters in CMA-ES

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is widely accepted as a robust derivative-free continuous optimization algorithm for non-linear and non-convex optimization problems. CMA-ES is well known to be almost parameterless, meaning that only one hyper-parameter, the population size, is proposed to be tuned by the user. In this paper, we propose a principled approach called self-CMA-ES to achieve the online adaptation of CMA-ES hyper-parameters in order to improve its overall performance. Experimental results show that for larger-than-default population size, the default settings of hyperparameters of CMA-ES are far from being optimal, and that self-CMA-ES allows for dynamically approaching optimal settings.

Introduction

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES [START_REF] Hansen | Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES)[END_REF]) is a continuous optimizer which only exploits the ranking of estimated candidate solutions to approach the optimum of an objective function f : R n → R. CMA-ES is also invariant w.r.t. affine transformations of the decision space, explaining the known robustness of the algorithm. An important practical advantage of CMA-ES is that all hyper-parameters thereof are defined by default with respect to the problem dimension n. Practically, only the population size λ is suggested to be tuned by the user, e.g. when a parallelization of the algorithm is considered or the problem at hand is known to be multi-modal and/or noisy [START_REF] Auger | A Restart CMA Evolution Strategy With Increasing Population Size[END_REF][START_REF] Hansen | Benchmarking a weighted negative covariance matrix update on the BBOB-2010 noisy testbed[END_REF]. Other hyperparameters have been provided robust default settings (depending on n and λ), in the sense that their offline tuning allegedly hardly improves the CMA-ES performance for unimodal functions. In the meanwhile, for multi-modal functions it is suggested that the overall performance can be significantly improved by offline tuning of λ and multiple stopping criteria [START_REF] Smit | Beating the 'world champion' Evolutionary Algorithm via REVAC Tuning[END_REF][START_REF] Liao | Benchmark results for a simple hybrid algorithm on the CEC 2013 benchmark set for real-parameter optimization[END_REF]. Additionally, it is shown that CMA-ES can be improved by a factor up to 5-10 by the use of surrogate models on unimodal ill-conditioned functions [START_REF] Loshchilov | Intensive Surrogate Model Exploitation in Self-adaptive Surrogate-assisted CMA-ES (saACM-ES)[END_REF]. This suggests that the CMA-ES performance can be improved by better exploiting the information in the evaluated samples (x, f (x)). This paper focuses on the automatic online adjustment of the CMA-ES hyper-parameters. The proposed approach, called self-CMA-ES, relies on a second CMA-ES instance operating on the hyper-parameter space of the first CMA-ES, and aimed at increasing the likelihood of generating the most successful samples x in the current generation. The paper is organized as follows. Section 2 describes the original (µ/µ w , λ)-CMA-ES. self-CMA-ES is described in Section 3 and its experimental validation is discussed in Section 4 comparatively to related work. Section 5 concludes the paper.

Covariance Matrix Adaptation Evolution Strategy

The Covariance Matrix Adaptation Evolution Strategy [START_REF] Hansen | Adapting Arbitrary Normal Mutation Distributions in Evolution Strategies: The Covariance Matrix Adaptation[END_REF][START_REF] Hansen | Completely Derandomized Self-Adaptation in Evolution Strategies[END_REF][START_REF] Hansen | Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES)[END_REF] is acknowledgedly the most popular and the most efficient Evolution Strategy algorithm.

The original (µ/µ w , λ)-CMA-ES (Algorithm 1) proceeds as follows. At the t-th iteration, a Gaussian distribution N m t , σ t 2 C t is used to generate λ candidate solution x k ∈ R n , for k = 1 . . . λ (line 5):

x t k = N m t , σ t 2 C t = m t + σ t N 0, C t , (1)
where the mean m t ∈ R n of the distribution can be interpreted as the current estimate of the optimum of function f , C t ∈ R n×n is a (positive definite) covariance matrix and σ t is a mutation step-size. These λ solutions are evaluated according to f (line 6). The new mean m t+1 of the distribution is computed as a weighted sum of the best µ individuals out of the λ ones (line 7). Weights w 1 . . . w µ are used to control the impact of selected individuals, with usually higher weights for top ranked individuals (line 1).

The adaptation of the step-size σ t , inherited from the Cumulative Step-Size Adaptation Evolution Strategy (CSA-ES [START_REF] Hansen | Adapting Arbitrary Normal Mutation Distributions in Evolution Strategies: The Covariance Matrix Adaptation[END_REF]), is controlled by the evolution path p t+1 σ . Successful mutation steps m t+1 -m t σ t (line 8) are tracked in the sampling space, i.e., in the isotropic coordinate system defined by the eigenvectors of the covariance matrix C t . To update the evolution path p t+1 σ , i) a decay/relaxation factor c σ is used to decrease the importance of previous steps; ii) the stepsize is increased if the length of the evolution path p t+1 σ is longer than the expected length of the evolution path under random selection E N (0, I ) ; iii) otherwise it is decreased (line 13). Expectation of N (0, I ) is approximated by √ n(1 -1 4n + 1 21n 2 ). A damping parameter d σ controls the change of the step-size. The covariance matrix update consists of two parts (line 12): a rank-one update [START_REF] Hansen | Completely Derandomized Self-Adaptation in Evolution Strategies[END_REF] and a rank-µ update [START_REF] Hansen | Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES)[END_REF]. The rank-one update computes the evolution path p t+1 c of successful moves of the mean m t+1 -m t σ t of the mutation distribution in the given coordinate system (line 10), along the same lines as the evolution path p t+1 σ of the step-size. To stall the update of p t+1 c when σ increases rapidly, a h σ trigger is used (line 9).

The rank-µ update computes a covariance matrix C µ as a weighted sum of the covariances of successful steps of the best µ individuals (line 11). Covariance Algorithm 1 The (µ/µ w , λ)-CMA-ES

1: given n ∈ N+, λ = 4 + ⌊3ln n⌋, µ = ⌊λ/2⌋, wi = ln(µ+ 1 2 )-ln i µ j=1 (ln(µ+ 1
2 )-ln j) for i = 1 . . . µ, µw =

1 µ i=1 w 2 i , cσ = µw +2 n+µw +3 , dσ = 1 + cσ + 2 max(0, µw -1 n+1 -1), cc = 4 n+4 , c1 = 2 (n+1.3) 2 +µw , cµ = 2 (µw -2+1/µw ) (n+2) 2 +µw 2: initialize m t=0 ∈ R n , σ t=0 > 0, p t=0 σ = 0, p t=0 c = 0, C t=0 = I, t ← 0 3: repeat 4:
for k = 1, . . . , λ do 5:

x k = m t + σ t N 0, C t 6:

f k = f (x k ) 7:
m t+1 = µ i=1 wix i:λ // the symbol i : λ denotes i-th best individual on f 8:

p t+1 σ = (1 -cσ)p t σ + cσ(2 -cσ) √ µwC t -1 2 m t+1 -m t σ t 9: hσ = 1 1 p t+1 σ < √ 1-(1-cσ ) 2(t+1) (1.4+ 2 n+1 ) E N(0,I ) 10: p t+1 c = (1 -cc)p t c + hσ cc(2 -cc) √ µw m t+1 -m t σ t 11: C µ = µ i=1 wi x i:λ -m t σ t × (x i:λ -m t ) T σ t 12: C t+1 = (1 -c1 -cµ)C t + c1 p t+1 c p t+1 c T rank-one update +cµ C µ rank-µ update 13: σ t+1 = σ t exp( cσ dσ ( p t+1 σ E N(0,I ) - 1
)) 14: t = t + 1 15: until stopping criterion is met matrix C itself is replaced by a weighted sum of the rank-one (weight c 1 [START_REF] Hansen | Completely Derandomized Self-Adaptation in Evolution Strategies[END_REF]) and rank-µ (weight c µ [START_REF] Hansen | Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES)[END_REF]) updates, with c 1 and c µ positive and c 1 + c µ ≤ 1.

While the optimal parameterization of CMA-ES remains an open problem, the default parameterization is found quite robust on noiseless unimodal functions [START_REF] Hansen | Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES)[END_REF], which explains the popularity of CMA-ES.

The self-CMA-ES

The proposed self-CMA-ES approach is based on the intuition that the optimal hyper-parameters of CMA-ES at time t should favor the generation of the best individuals at time t, under the (strong) assumption that an optimal parameterization and performance of CMA-ES in each time t will lead to the overall optimal performance.

Formally, this intuition leads to the following procedure. Let θ t f denote the hyper-parameter vector used for the optimization of objective f at time t (CMA-ES stores its state variables and internal parameters of iteration t in θ t and the '.'-notation is used to access them). At time t + 1, the best individuals generated according to θ t f are known to be the top-ranked individuals x t 1:λ . . . x t µ:λ , where x t i:λ stands for the i-th best individual w.r.t. f . Hyper-parameter vector θ t f would thus have been all the better, if it had maximized the probability of generating these top individuals.

Along this line, the optimization of θ t f is conducted using a second CMA-ES algorithm, referred to as auxiliary CMA-ES as opposed to the one concerned with the optimization of f , referred to as primary CMA-ES. The objective of the auxiliary CMA-ES is specified as follows:

Given: hyper-parameter vector θ i f and points (x i 1:λ , f (x i 1:λ )) evaluated by primary CMA-ES at steps i = 1, . . . , t (noted as θ i+1 f .f (x 1:λ ) in Algorithm 2), Find: θ t, * f such that i) backtracking the primary CMA-ES to its state at time t-1; ii) replacing θ t f by θ t, * f , would maximize the likelihood of x t i:λ for i = 1 . . . µ. The auxiliary CMA-ES might thus tackle the maximization of g t (θ) computed as the weighed log-likelihood of the top-ranked µ sel individuals at time t:

g t (θ) = µ sel i=1 w sel,i log P (x t i:λ |θ t f = θ ), (2)
where w sel,i ≥ 0, i = 1 . . . µ sel , µ sel i=1 w sel,i = 1, and by construction

P (x i |m t , C t ) = 1 (2π) n |C t | exp (-0.5(m t -x t i )C t -1 (m t -x t i )), (3)
where C t is the covariance matrix multiplied by σ t 2 and |C t | is its determinant. While the objective function for the auxiliary CMA-ES defined by Eq. ( 2) is mathematically sound, it yields a difficult optimization problem; firstly the probabilities are scale-sensitive; secondly and overall, in a worst case scenario, a single good but unlikely solution may lead the optimization of θ t f astray. Therefore, another optimization objective h t (θ) is defined for the auxiliary CMA-ES, where h t (θ) measures the agreement on x t i:λ for i = 1 . . . µ between i) the order defined from f ; ii) the order defined from their likelihood conditioned by θ t f = θ (Algorithm 3). Procedure ReproduceGenerationCMA in Algorithm 3 updates the strategy parameters described from line 7 to line 14 in Algorithm 1 using already evaluated solutions stored in θ t f .x i:λ . Line 4 computes the Mahalanobis distance, division by step-size is not needed since only ranking will be considered in line 5 (decreasing order of Mahalanobis distances corresponds to increasing order of log-likelihoods). Line 6 computes a weighted sum of ranks of likelihoods of best individuals.

Finally, the overall scheme of self-CMA-ES (Algorithm 2) involves two interdependent CMA-ES optimization algorithms, where the primary CMA-ES is concerned with optimizing objective f , and the auxiliary CMA-ES is concerned with optimizing objective h t , that is, optimizing the hyper-parameters of the primary CMA-ES 4 . Note that self-CMA-ES is not per se a "more parameterless" algorithm than CMA-ES, in the sense that the user is still invited to modify the population size λ. The main purpose of self-CMA-ES is to achieve the online adaptation of the other CMA-ES hyper-parameters.

Algorithm 2

The self-CMA-ES Specifically, while the primary CMA-ES optimizes f (x) (line 8), the auxiliary CMA-ES maximizes h t (θ) (line 9) by sampling and evaluating λ h variants of θ t f . The updated mean of the auxiliary CMA-ES in the hyper-parameter space is used as a local estimate of the optimal hyper-parameter vector for the primary CMA-ES. Note that the auxiliary CMA-ES achieves a single iteration in the hyper-parameter space of the primary CMA-ES, with two motivations: limiting the computational cost of self-CMA-ES (which scales as λ h times the time complexity of the CMA-ES), and preventing θ t f from overfitting the current sample x t i:λ , i = 1 . . . µ.

Algorithm 3

Objective function h t (θ)

1: Input: θ, θ t+1 f ,θ t-1 f , θ t f , µ, w sel,i for i = 1, . . . , µ 2: θ ′ t-1 f ← θ 3: θ ′ t f ←ReproduceGenerationCMA(f, θ ′ t-1 f ) using already evaluated θ t f .x i:λ 4: di ← θ ′ t f . √ C -1 • (θ t+1 f .x t i -θ ′ t
f .m) ; for i = 1, . . . , θ t+1 f .λ 5: pi ← rank of di, i = 1 . . . λ sorted in decreasing order 6: h(θ) ← µ i=1 w sel,i p i:λ { i : λ denotes the rank of θ t+1 .xi } 7: Output: h(θ)

Experimental Validation

The experimental validation of self-CMA-ES investigates the performance of the algorithm comparatively to CMA-ES on the BBOB noiseless problems [START_REF] Hansen | Real-Parameter Black-Box Optimization Benchmarking[END_REF]. Both algorithms are launched in IPOP scenario of restarts when the CMA-ES is restarted with doubled population size once stopping criteria [START_REF] Hansen | Benchmarking a BI-population CMA-ES on the BBOB-2009 function testbed[END_REF] are met 5 .

5 For the sake of reproducibility, the source code is available at https://sites.google.com/site/selfcmappsn/ The population size λ is chosen to be 100 for both CMA-ES and self-CMA-ES. We choose this value (about 10 times larger than the default one, see the default parameters of CMA-ES in Algorithm 1) to investigate how sub-optimal the other CMA-ES hyper-parameters, derived from λ, are in such a case, and whether self-CMA-ES can recover from this sub-optimality. The auxiliary CMA-ES is concerned with optimizing hyper-parameters c 1 , c µ and c c (Algorithm 1), responsible for the adaptation of the covariance matrix of the primary CMA-ES. These parameters range in [0, .9] subject to 0 ≤ c 1 + c µ ≤ 0.9; the constraint is meant to enforce a feasible C update for the primary CMA-ES (the decay factor of C should be in [0, 1]). Infeasible hyper-parameter solutions get a very large penalty, multiplied by the sum of distances of infeasible hyper-parameters to the range of feasibility.

We set w sel,i = 1/µ for i = 1, . . . , µ and µ = ⌊λ/2⌋ to 50. The internal computational complexity of self-CMA-ES thus is λ h = 20 times larger than the one of CMA-ES without lazy update (being reminded that the internal time complexity is usually negligible compared to the cost per objective function evaluation). In self-CMA-ES, the hyper-parameters are uniformly initialized in [0, 0.9] (therefore the medians are close to 0.45) and they gradually converge to values which are estimated to provide the best update of the covariance matrix w.r.t. the ability to generate the current best individuals. It is seen that these values are problem and dimension-dependent. The values of c 1 are always much smaller than c µ but are comparable to the default c 1 . The values of c µ and c c and c 1 are almost always larger than the default ones; this is not a surprise for c 1 and c µ , as their original default values are chosen in a rather conservative way to prevent degeneration of the covariance matrix.

Results

Several interesting observations can be made about the dynamics of the parameter values. The value of c µ is high most of the times on the Rosenbrock functions, but it decreases toward values similar to those of the Sphere functions, when close to the optimum. This effect is observed on most problems; indeed, on most problems fast adaptation of the covariance matrix will improve the performance in the beginning, while the distribution shape should remain stable when the covariance matrix is learned close to the optimum.

The overall performance of self-CMA-ES on the considered problems is comparable to that of CMA-ES, with a speed-up of a factor up to 1.5 on Sharp Ridge functions. The main result is the ability of self-CMA-ES to achieve the online adaptation of the hyper-parameters depending on the problem at hand, side-stepping the use of long calibrated default settings6 .

Discussion

self-CMA-ES offers a proof of concept for the online adaptation of three CMA-ES hyper-parameters in terms of feasibility and usefulness. Previous studies on parameter settings for CMA-ES mostly considered offline tuning (see, e.g., [START_REF] Smit | Beating the 'world champion' Evolutionary Algorithm via REVAC Tuning[END_REF][START_REF] Liao | Benchmark results for a simple hybrid algorithm on the CEC 2013 benchmark set for real-parameter optimization[END_REF]) and theoretical analysis dated back to the first papers on Evolution Strategies. The main limitation of these studies is that the suggested hyper-parameter values are usually specific to the (class of) analyzed problems. Furthermore, the suggested values are fixed, assuming that optimal parameter values remain constant along evolution. However, when optimizing a function whose landscape gradually changes when approaching the optimum, one may expect optimal hyperparameter values to reflect this change as well.

Studies on the online adaptation of hyper-parameters (apart from σ, m and C) usually consider population size in noisy [START_REF] Beyer | Controlling population size and mutation strength by meta-es under fitness noise[END_REF], multi-modal [START_REF] Auger | A Restart CMA Evolution Strategy With Increasing Population Size[END_REF][START_REF] Loshchilov | Alternative Restart Strategies for CMA-ES[END_REF] or expensive [START_REF] Hoffmann | Controlled Model Assisted Evolution Strategy with Adaptive Preselection[END_REF] optimization. A more closely related approach was proposed in [START_REF] Schaul | Comparing natural evolution strategies to bipop-cma-es on noiseless and noisy black-box optimization testbeds[END_REF] where the learning rate for step-size adaptation is adapted in a stochastic way similarly to Rprop-updates [START_REF] Igel | Empirical evaluation of the improved rprop learning algorithms[END_REF].

Conclusion and Perspectives

This paper proposes a principled approach for the self-adaptation of CMA-ES hyper-parameters, tackled as an auxiliary optimization problem: maximizing the likelihood of generating the best sampled solutions. The experimental validation of self-CMA-ES shows that the learning rates involved in the covariance matrix adaptation can be efficiently adapted on-line, with comparable or better results than CMA-ES. It is worth emphasizing that matching the performance of CMA-ES, the default setting of which represent a historical consensus between theoretical analysis and offline tuning, is nothing easy.

The main novelty of the paper is to offer an intrinsic assessment of the algorithm internal state, based on retrospective reasoning (given the best current solutions, how could the generation of these solutions have been made easier) and on one assumption (the optimal hyper-parameter values at time t are "sufficiently good" at time t + 1). Further work will investigate how this intrinsic assessment can support the self-adaptation of other continuous and discrete hyper-parameters used to deal with noisy, multi-modal and constrained optimization problems.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Evolution of learning rates c1, cµ, cc (lines with markers, left y-axis) and log10(objective function) (plain line, right y-axis) of CMA-ES (left column) and self-CMA-ES (right column) on 10-and 20-dimensional Sphere and Rosenbrock functions from [4]. The medians of 15 runs are shown.

Figures 1 and 2

 2 Figures 1 and 2 display the comparative performances of CMA-ES (left) and self-CMA-ES (right) on 10 and 20-dimensional Sphere, Rosenbrock, Rotated Ellipsoid and Sharp ridge functions from the noiseless BBOB testbed [4] (medians out of 15 runs). Each plot shows the value of the hyper-parameters (left y-axis) together with the objective function (in logarithmic scale, right y-axis). Hyperparameters c 1 , c µ and c c are constant and set to their default values for CMA-ES while they are adapted along evolution for self-CMA-ES.In self-CMA-ES, the hyper-parameters are uniformly initialized in [0, 0.9] (therefore the medians are close to 0.45) and they gradually converge to values which are estimated to provide the best update of the covariance matrix w.r.t. the ability to generate the current best individuals. It is seen that these values are problem and dimension-dependent. The values of c 1 are always much smaller than c µ but are comparable to the default c 1 . The values of c µ and c c and c 1 are almost always larger than the default ones; this is not a surprise for c 1 and c µ , as their original default values are chosen in a rather conservative way to prevent degeneration of the covariance matrix.Several interesting observations can be made about the dynamics of the parameter values. The value of c µ is high most of the times on the Rosenbrock functions, but it decreases toward values similar to those of the Sphere functions, when close to the optimum. This effect is observed on most problems; indeed, on most problems fast adaptation of the covariance matrix will improve the performance in the beginning, while the distribution shape should remain stable when the covariance matrix is learned close to the optimum.The overall performance of self-CMA-ES on the considered problems is comparable to that of CMA-ES, with a speed-up of a factor up to 1.5 on Sharp Ridge functions. The main result is the ability of self-CMA-ES to achieve the

This scheme is actually inspired from the one proposed for surrogate-assisted optimization[START_REF] Loshchilov | Self-Adaptive Surrogate-Assisted Covariance Matrix Adaptation Evolution Strategy[END_REF], where the auxiliary CMA-ES was in charge of optimizing the surrogate learning hyper-parameters.

cc = 4 n+4 , c1 = 2 (n+1.3) 2 +µw , cµ = 2 (µw -2+1/µw ) (n+2) 2 +µw.

Acknowledgments We acknowledge anonymous reviewers for their constructive comments. This work was supported by the grant ANR-2010-COSI-002 (SIMINOLE) of the French National Research Agency.


G Toulon

A Bourennane

K Isoird

Impact of a backside Schottky contact on the thyristor characteristics at high temperature

Keywords: Pulsed power, High voltage thyristor, TCAD simulations, high temperature, Schottky contacts

In this paper, a thyristor structure presenting improved electrical characteristics at high temperature is analysed through 2D physical simulations. The replacement of the P emitter of a standard symmetrical thyristor by a judicious association of P diffusions and Schottky contacts at the anode side contributes to the reduction of the leakage current in the forward direction and hence improves the forward blocking voltage at high temperature. A fine-tune of the anode side configuration will improve the forward off-state behaviour with only a negligible on-state voltage drop degradation. Moreover, the comparison with the conventional anode short thyristor shows that the insertion of Schottky contacts leads to the same improvements that the anode short in terms of off-state characteristics, while keeping the reverse blocking capability.

INTRODUCTION

In high voltage and high current applications, such as HVDC transmission [START_REF] Peter | Lips: Technology Trends for HVDC thyristor Valves[END_REF] or pulsed power application [START_REF] Spahn | The use of Thyristors as Main Switches in EML Applications[END_REF], a thyristor is commonly used as a power switch because of its high switching power. However, the thyristor architecture composed of four layers of different doping forming an N-P-N and a P-N-P bipolar transistor limit its maximum operating temperature to 125°C. Under high operating temperature, the leakage current increases amplified by the transistor gains, leading to the parasitic turn-on of the thyristor. Moreover, the temperature induced high leakage current can degrade the functioning of the application in some cases. In pulsed power circuits, such as that shown in figure 1 [START_REF] Scharnholz | Performance study of a novel 13.5 kV multichip thyristor switch[END_REF], where a capacitor is charging while the thyristor is maintained in the off-state and is discharging through an R-L circuit when the thyristor is turned on. However, at high temperature when the thyristor leakage current becomes high, part of the current for the capacitor charging is deviated by the thyristor and flows through the R-L circuit, which results in increasing the time required for charging the capacitor.

Current gain reduction can be achieved by different ways, such as:

-Local electron or proton irradiation at the main junctions in order to reduce the carrier lifetimes [START_REF] Nakagawa | 8kV/3.6kA Light Triggered Thyristor[END_REF] -N-type buffer layer between the N-base and P emitter -Anode short-circuits [START_REF] Kokosa | A High-Voltage, High-Temperature Reverse Conducting Thyristor[END_REF] The two last methods lead to a degradation of the reverse blocking capability, which means that additional components, such as a diode connected in series with the thyristor, must be added.

Silicon carbide, thanks to its physical proprieties, could be considered. However, some technological issues still need to be solved, such as the reduction of basal plane dislocation density during thick epitaxial layers growth, which leads to a degradation of the carrier lifetime and then the increase of the on-state voltage drop, or the development of novel termination techniques for bidirectional blocking capability [START_REF] Zhang | SiC super GTO thyristor for technology development: Present status and future perspective[END_REF].

The thyristor leakage current at high temperature can also come from surface currents at the chip periphery [START_REF] Obreja | The Operation Temperature of Silicon Power Thyristors and the Blocking Leakage Current[END_REF]. Adequate edge termination and passivation techniques are then necessary [START_REF] Obreja | On the High Temperature Operation of High Voltage Power Devices[END_REF] in order to minimize theses currents which can represent a significant part of the total leakage current in the thyristor device.

Consequently, we propose a thyristor structure based on the utilization of an Schottky contacts associated to P diffusions at the backside of a symmetrical silicon thyristor as associated to its termination for symmetric blocking voltage as shown in figure 2. The Schottky contact on the thyristor anode side leads to a reduction of the emitter efficiency of the J 2 junction, similar to conventional anode shorts, while preserving the reverse blocking capability of the device. We study, by means of TCAD simulations, the impact of the insertion of Schottky contacts in the backside of a thyristor on its electrical characteristics.

THYRISTOR LEAKAGE CURRENT

The leakage current in a thyristor can be expressed by the following equation:

NPN PNP C AK I I α α - - = 1 0 ( 1
)
Where I C0 is the leakage current of a PN junction in reverse conducting mode and α PNP et α NPN the gain of the bipolar transistors composing the thyristor shown in figure 2.

The thyristor leakage current reduction can be achieved by minimizing the different elements of equation 1. The leakage current I C0 depends on the N-base doping concentration which govern the breakdown voltage of the device. Cathode shorts, usually used in commercial thyristor, reduce the N-P-N current gain during forward blocking mode through the R Base resistor (figure 2) linking the base and emitter of this transistor. The cofiguration of these cathode shorts [START_REF] Munoz-Yague | Leturcq: Optimum Design of Thyristor Gate-Emitter Geometry[END_REF] has an influence on the N-P-N transistor current gain, the gate triggering current, and the device dV/dt capability. The P-N-P transistor gain can be lowered by the diferent methods presented in the introduction. We present in this paper a solution based on an association of Schottky contact and P diffusions as used in the JBS (Junction Barrier Sschottky) diodes [START_REF] Baliga | The Pinch Rectifier: A Low-Forward-Drop High-Speed Power Diode[END_REF] at the thyristor anode side, replacing the P emitter in the conventional thyrisor.

SIMULATION RESULTS

Studied structures

Three symmetrical thyristor structures are studied: a conventional one, a thyristor with Schottky contacts at the backside (figure 3 (a)) and a conventional anode shorts thyristor (figure 3 (b)). This last structure is identical to the previous one, except that the backside electrode contacts an N+ diffusion instead of the N-base region. In order to achieve a symmetrical breakdown voltage, the N-base doping concentration is uniform and lightly doped in comparison with the P-type regions surrounding it. Cathode shorts are added for the reasons given previously.

Off-state

On figure 4 is represented the forward blocking voltage as a function of the N-base doping concentration and thickness for the three structures at the ambient temperature. For low values of the N-base doping concentration, the breakover voltage is limited by the reach-through of the base, leading to a reduction of the breakover voltage when lowering the doping concentration, while for high values, it is limited by the avalanche breakdown of the reverse biased J 1 junction, which means that the increase of the doping concentration reduces the breakover voltage value. Consequently, for a given N-base layer thickness, there is an optimum doping concentration which gives the highest breakover voltage. From the results on figure 4, at a fixed W N value, the insertion of Schottky contacts or anode short leads to an increase of the maximum breakover voltage. As a result, for an achievable breakover voltage, the N-base layer thickness could be lowered compared to the case of a conventional thyristor, leading to a reduction of the onstate voltage. The leakage current in a Schottky contact can be high because of the barrier lowering effect under the presence of an electric field at the interface [START_REF] Lanois | Caractéristiques électriques simulées et expérimentales de composants Schottky en tranchées[END_REF]. However, this degradation can be limited by a close spacing between two P diffusions, protecting the Schottky contact against excessive electric field values, as used in JBS diodes. This effect is illustrated on figure 5, where one can notice that the Schottky contact must be narrow in order not to degrade the reverse blocking electrical characteristics of the thyristor. On figure 6, the reverse characteristics of the thyristor with Schottky contacts and the conventional one are compared at different temperatures. The anode of the structure with Schottky contacts has been determined by considering the previous results. Both structures present the same breakdown voltage evolution with temperature, but the presence of the Schottky contact leads to an increase of the leakage current and a soft transition near the avalanche breakdown because of the barrier lowering effect. Figure 7 shows the evolution of the forward breakover voltage as a function of the temperature for the three structures. For every structure, the reduction of the N + cathode width (W cathode ) leads to an improvement of the forward breakover voltage at high temperatures. However, when reducing W cathode , the minimum gate current necessary to turn-on the device increases, because of the P-base resistance R base (figure 3) which is proportional to the N + cathode width, meaning that the current flowing through this resistor required to forward bias the P-N junction between P-base and N + will be high if the R base is small. For a fixed value of W cathode , the insertion of Schottky contacts or anode short leads to a reduction of injection efficiency of the J 2 junction, because a part of the carriers are deviated through the Schottky contacts or the anode shorts. Consequently, these thyristor present a degradation of their breakover voltage at a higher temperature compared to a conventional thyristor. The evolution of the breakover voltage with temperature of thyristors with Schottky contacts is identical to that of thyristors with anode shorts, that means that the Schottky contact has the same effect as that the anode short. The forward blocking characteristics of the three structures is presented on figure 8. The lower leakage current in the thyristors with Schottky contacts and anode shorts as compared to the conventional thyristor case explain their improved off-state capability at high temperature. Figure 9, representing the evolution of the breakover voltage as a function of the ratio of the Schottky contact width (L s on figure 2) to the anode cell width (L a on figure 2), shows that the increase of the Schottky contact width leads to a slight improvement in the breakover voltage. At high temperatures, where the improvement is more noticeable, the widening of the Schottky contact results in an increase of some hundred of volts.

On-state

The on-state voltage drop of the three structures is represented on figure 10 for a current density of 200 A.cm -2 . The on-state voltage drop for the thyristors with Schottky contacts and anode shorts is represented as a function of the L s to L a ratio. When the L s to L a ratio increases, the P emitter width decreases, which leads to the reduction of the injection of carriers into the N-base region and consequently to an increase of the on-state voltage drop. This voltage drop increase with the L s /L a ratio is more pronounced in the case of thyristors with anode shorts. On the other hand, for larger P emitter, the on-state voltage drop tends to that of the conventional thyristor value. However, as described previously, the increase of the width of the Schottky contact compared to the P emitter leads to only a low improvement in the breakover voltage. Consequently, the Schottky contact surface should have to be maintained small compared to the P emitter one in order not to increase the conduction power losses.

DEVELOPMENT OF TERMINATIONS

Specific technological bricks for the symmetrical thyristors

Power devices can present premature breakdown if no adequate junction terminations are used. Symmetrical blocking devices present some issues in the reverse blocking state since standard planar terminations are not easy to adapt for the reverse blocking mode, because of the backside common electrode. In high voltage -high current rating thyristors, double bevel termination is usually chosen because a device necessitates a large silicon area, then it can be mechanically processed prior to the thyristor process. Bevel terminations improve the breakdown voltage by reducing the electric field at the device edge. However the P-base region has to be deep and lightly doped in order to have efficient negative bevel terminations. Moreover, it has been experimentally demonstrated that negative bevel shows reduced breakdown voltage because of enhanced electric field in the bulk at the vicinity of the bevel surface [START_REF] Adler | A General Method for Predicting the Avalanche Breakdown Voltage of Negative Beveled Devices[END_REF]. For lower voltage devices (until 1200 V), the termination for reverse blocking can be realized by deep trenches [START_REF] Tokuda | A ultra-small isolation area for 600V class Reverse Blocking IGBT with Deep Trench Isolation process (TI-RB-IGBT)[END_REF] P+ filled and connected to a JTE (Junction Termination Extension) realized at the chip front side [START_REF] Causse | A new peripheral planar structure allowing a symmetrical blocking voltage[END_REF].

Forward blocking edge termination technique

In the forward blocking mode, the J 1 junction of the thyristor is reverse biased. In this case, a standard planar termination can be used. For the thyristor developed in this paper, field limiting ring associated with field plates were used. This termination does not require additional process steps, since the P-base mask can be used to define the P rings and the field plates can be realized with the cathode and gate metallization. Figure 11 shows an example of a termination using two floating rings at the end of the P base associated with their metallization. With 18 floating field rings with their field plate, it is possible to reach 4800V, corresponding to more than 95% of the breakdown voltage of the plane junction. Figure 12 represents the electrostatic potential of this termination at breakdown.

Reverse blocking edge termination technique

A positive bevel termination appears to be the best protection for the reverse blocking mode. Moreover, similar realization of bevel edge by means of a wet etch has been reported [START_REF] Shimoyama | A new isolation technique for reverse blocking IGBT with ion implantation and laser annealing to tapered chip edge sidewalls[END_REF] and it is also reported that it could be realized on several chips at the same time. The bevelled side of the chips is aligned to the crystallographic planes. On a (100) silicon wafer, the taper angle becomes 54.7° to the wafer surface (θ on figure 11) [START_REF] Shimoyama | A new isolation technique for reverse blocking IGBT with ion implantation and laser annealing to tapered chip edge sidewalls[END_REF].

Due to the cut along an angle non perpendicular to the reverse biased PN junction, the missing part of charges in the N-base region due to the bevel must be compensated by an increase of the depletion width as seen on figure 13. The larger extension of the depletion width along the bevel will lead to a reduction of the electric field along the termination, as seen on figure 14, where the electric field along the bevel and perpendicular to the P-emitter/N-base junction is represented. The reduction of the electric field magnitude along the bevel ensure the breakdown in the bulk of the device and not in the termination.

CONCLUSION

In this paper, we studied the impact of the insertion of a Schottky contact on the backside of symmetric thyristor on its electrical characteristics. The simulation results showed that the thyristor with Schottky contacts could, with only a marginal degradation of the on-state voltage drop, highly improve the forward breakover voltage at high temperature thanks to the reduction of the leakage current. The improved performances is made possible with an optimized arrangement of the Schottky contacts. Thanks to the use of Schottky contacts, the proposed thyristor structure exhibits the same performance gain of the anode short thyristor at high temperature, while preserving the reverse blocking capability, as same as in the conventional thyristor structure. Moreover, in this paper, terminations for the symmetrical breakdown voltage has been proposed and some technological elements have been discussed for their realization. All of the simulation results would be validated with experimental realization.

Figure 1 :

 1 Figure 1: Example of a pulsed power circuit highlighting the thyristor role.

Figure 2 :

 2 Figure 2: Overview of the thyristor structure with Schottky contacts on the anode side and both junction terminations for symmetrical blocking voltage.

Figure 3 :

 3 Figure 3: Schematic cross section of thyristor structures with (a) Schottky contact and (b) anode shorts.

Figure 4 :

 4 Figure 4: Influence of the N-base parameters on the forward breakover voltage at ambient temperature.

Figure 5 :

 5 Figure 5: Influence of the Schottky contact width on the leakage current, the value S = 0 corresponding to a plane P-N junction.

Figure 6 :

 6 Figure 6: Influence of temperature on the reverse characteristics of the thyristor with Schottky contacts and the conventional one.

Figure 7 :

 7 Figure 7: Temperature effect on the forward breakdown for different N + cathode widths.

Figure 8 :

 8 Figure 8: Forward blocking characteristics of the three structures for different temperatures.

Figure 9 :

 9 Figure 9: Breakover voltage as a function of the Schottky contact ratio on the anode width.

Figure 10 :

 10 Figure 10: On-state voltage drop as a function of the Schottky contact ratio on the anode width for the three structures.

Figure 11 :

 11 Figure 11: Schematic cross section of the terminations used for the symmetric breakdown voltage.

Figure 12 :

 12 Figure 12: Electrostatic potential line contours at breakdown (V DRM =4800V) of the termination used in the forward blocking direction, each line corresponding to a voltage variation of 200V.

Figure 13 :

 13 Figure 13: Electrostatic potential contour lines at breakdown (V RRM =5400V) of the termination used for the reverse blocking direction, each line corresponding to a voltage variation of 200V.

Figure 14 :

 14 Figure 14: Electric field at breakdown in the reverse blocking direction in the bevel termination


Frédéric Darboux
email: frederic.darboux@orleans.inra.fr

J.-B Paroissien

Alain A Couturier

Benoît B Devillers

Joël J Daroussin

Yves Le Bissonnais

Damien D Raclot

Jean-Baptiste Paroissien

How to assign a value to soil sustainability? An example based on water erosion modeling

Keywords: soil sustainability, water erosion, modeling, life expectancy

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Soil erosion is an example of natural phenomenon that can be greatly accelerated by the global change but that can also be mitigated by better practices. In order to setup environmental and land use management plans, many methods have been developed for assessing soil erosion: soil erosion hazard which refers to the probability of the soil erosion phenomena and the soil erosion risk which requires a combination between the soil vulnerability and the soil erosion hazard [START_REF] Taubenböck | A conceptual vulnerability and risk framework as outline to identify capabilities of remote sensing[END_REF]. Hazard maps are produced at different scales [START_REF] Renschler | Evaluating spatial and temporal variability in soil erosion risk-rainfall erosivity and soil loss ratios in andalusia, spain[END_REF][START_REF] Bissonnais | Mapping erosion risk for cultivated soil in France[END_REF][START_REF] Van Rompaey | Validation of soil erosion risk assessments in Italy[END_REF] and based on empirical and expert model as RUSLE and MESALES [START_REF] Bissonnais | Mapping erosion risk for cultivated soil in France[END_REF]. Most of studies use indicators of soil erosion such as soil erosion rate (expressed T.ha -1 .y -1 ) changes in soil thickness (mm/y). However, more advanced concepts have been proposed, such as soil loss tolerance [START_REF] Stamey | A concervation definition of erosion tolerance[END_REF] and soil life time for an agricultural production [START_REF] Sparovek | Temporal erosion-induced soil degradation and yield loss[END_REF]. These two concepts use the vulnerability of the soil production capacity to assess the soil erosion risk, the soil being considered as an economical resource. Nevertheless, some of basic soil functions can be depleted, or destroyed due to an increase in soil erosion [START_REF] Lal | Soil erosion and the global carbon budget[END_REF] (e.g., soil fertility, soil structure, carbon storage…), limiting future developments of human activities. Thus, it appears important to assess the soil erosion risk by considering the soil as a patrimonial resource for future generation. For such purpose the concept of Safe Minimum Standards and Environmental Minimum Requirements [START_REF] Bastian | Determining environmental minimum requirements for functions provided by agro-ecosystems[END_REF] may be better suited. Soil erosion risk can be taken into account in the assessments of environmental problems [START_REF] Bazzoffi | Soil erosion tolerance and water runoff control: minimum environmental standards[END_REF][START_REF] Neumann | Spatial modelling for the development of agrienvironmental programs[END_REF]. They used the soil regeneration capacity to define the Environmental Minimum Requirements for soil erosion.

The aim of this paper is to develop and apply a method to evaluate soil erosion risk. It is applied in a catchment located in the South-East of France. It uses two models, one to generate land-use change scenarios and one to assess consequences for soil sustainability to soil erosion. It shows the concept of life expectancy of soil is a valuable indicator of soil sustainability, both for soil specialists and decision makers.

II. MATERIAL AND METHODS

Study catchment

The study catchment is located in the South-East of France, close to the Mediterranean sea. The catchment is along the Peyne river. The valley covers an area of about 75 km 2 , encompassing two contrasted zones: The northern part of the Peyne catchment is hilly and consists of Palaeozoic rocks covered by lithosols (0 to 30 cm deep). Land use is mostly shrubs with few arable lands. The southern part is made of small hills consisting of Neogene formations covered by deeper soils (> 80 cm) classified as Fluvisols, Calcisols and Brunisols. Land use is vines (75%) and cereals (20%). The climate is Mediterranean sub-humid with a long dry summer season. The average annual temperature is 14°C and annual rainfall varies between 500 and 1400 mm, showing a bimodal temporal distribution with two major rainy periods, one in spring and another in autumn. Storms with high rainfall intensity may occur in summer.

Materials and Methods

Map of current land use

The land use map was established by using air photo interpretation of IGN (National Institute of Geography) orthophotographies taken in the spring 2005. The land use classes differentiate the vine (vine and orchard), the arable land, the forest (grouping abandonment land, scrubland and pine forest), and urban areas (road network, natural and artificial hydrological network, villages). The cultural practices of the vineyard are added owing to their important impacts on the soil surface characteristics on water erosion. The main cultural practices are (i) chemical weeding (62.5%), (ii) mechanical weeding (27.5%) and (iii) permanent intercrop (10%) [START_REF] Biarnès | Analyzing the determinants of spatial distribution of weed control practices in a languedoc vineyard catchment[END_REF]. These practices are attributed randomly on vine-classed areas.

Scenarios of land use change with CLUE-s

Land use changes are simulated with the CLUE-s ("Conversion of Land-Use and its Effects") model [START_REF] Veldkamp | Clue-cr : An integrated multi-scale model to simulate land use change scenarios in costa rica[END_REF][START_REF] Verburg | Modeling the spatial dynamics of regional land use: The clue-s model[END_REF]. The model is based on an empirical analysis created by the user who specifies the relations between the spatial distribution of land use and a series of factors that drive and constrain land use (e.g. slope, distance to a river). In addition, a set of function is specified by the user to address a hierarchic organization of land use change, a spatial connectivity between the destinations of land use change and stability to the change of certain land use type. It enabled to generate land use map for future scenarios. These scenarios were design to encompass possible changes and were established under the general basis of the work of Kok et al.

Modeling soil erosion with STREAM

STREAM model (Sealing and Transfer by Runoff and Erosion related to Agricultural Management

) is used to model soil erosion by water. It is a non dynamic model that uses a raster-based distributed approach to calculate total runoff volume and soil erosion for a given rainfall event [START_REF] Souchère | Modelling the impact of agrienvironmental scenarios on runoff in a cultivated catchment (normandy, france)[END_REF]. This expert-based model assumes that the main factors of runoff and infiltration at the field scale are the surface characteristics: soil crusting, surface roughness, total cover and moisture content. For each combination of surface characteristics an infiltration rate is attributed. The flow network is obtained from a digital elevation model. The soil erosion is simulated by two dedicated modules: The module for sheet erosion relies on a table assigning a sediment concentration according to the soil surface characteristics. This module takes into account the sediment deposit. The module for rill erosion, based on the slope value and parameters affecting flow velocity and soil resistance (vegetation type, vegetable cover, surface roughness, soil crusting). In this module, the sediment deposit is not taken accounted for.

The main results are maps of soil erosion rate (expressed in T.ha -1 .y -1 ). 9 years of rainfalls are simulated and an average map of soil erosion rate is computed for each scenario.

Results are aggregated at the scale of a several fields to avoid over-interpretation that may result from pixel map. This leads to maps consisting of polygons. Considering that polygon containing large erosion rates should be put forward, the 9 th decile of soil erosion rate is considered.

Map of life expectancy of soils

The soil erosion risk is estimated through the life expectancy of soils and calculated using the soil erosion rate map, the soil depth map and the soil bulk density map:

Ser Sbd Sd SLE =
with Sd the soil depth (m), Ser the soil erosion rate (T.ha -1 .y -1 ), Sb the soil bulk density (kg.m -3 ) and SLE the soil life expectancy (y). Soil production is not accounted for because of too much uncertainty on this variable. One map of soil life expectancy is computed for each scenario. It represents the amount of time remaining before the soil completely disappears. Thus, it is a reasonable proxy for estimating long-term erosion impact [START_REF] Sparovek | Temporal erosion-induced soil degradation and yield loss[END_REF] from a patrimonial point of view. As for maps of soil erosion rates, results are aggregated at the scale of several fields. For the same reasons, the first decile of life expectancy is considered.

Results are aggregated at the scale of a several fields to avoid over-interpretation that may result from pixel map. This leads to a map consisting on polygons. Considering that polygon containing large erosion rates should be put forward, the 9 th decile of life expectancy is plotted.

III. RESULTS AND DISCUSSION

Scenarios of land use change

The four scenarios give contrasted results in term of land use (fig. 1), even if the main features of the watershed are preserved. Specifically, the Northern part remains mostly with forests while the Southern part remains predominantly with agriculture.

Soil erosion rates

In the current situation, the highest soil erosion rates are encountered in the Southern area (fig. 2): the highest soil erosion rates are scattered among soil erosion rates of 3-5 T.ha -1 .y -1 . In this Southern part, some areas with erosion rates lower than 3 T.ha -1 .y -1 exists. An opposite situation is encountered in the northern area, with a few large erosion rate areas scattered among a large area with low erosion rates. The scenario Production not alter much this spatial pattern. On the contrary, the scenario Accentuation increases the surface area with high erosion rates. Both the scenarios Protection and Environmental law cause a decrease in soil erosion rates, but this decline is much sharper for the scenario Environmental law. Using soil erosion rates as a index of soil sustainability, it looks like the scenario Environmental law is able to alleviate all the erosion problems in the watershed. Life expectancy of soils concerned with life expectancy (fig. 3), the current situation looks rather alarming, with large areas having life expectancy lower than 100 years. It must be noticed that the Northern part has low life expectancies. This result contrasts with its low erosion rates (comparison of fig. 2 and3). This is explained by the shallow soils encountered in this area of the watershed. The scenarios Accentuation, Production and Protection do not give results very different from the current situation. Soil sustainability is lower than 100 years in most of the watershed whatever the scenarios. Only the scenario Environmental law appears different, with a noticeable increase in life expectancy in the Southern part of the area. This scenario is the only one enabling a more sustainable soil management.

Soil erosion rate vs. life expectancy

To assert sustainability of soils concerned with water erosion, the use of soil erosion rate can be misleading because it does not account for the soil thickness. Hence, life expectancy is a better index of soil sustainability from a scientific point of view. Moreover, soil erosion rate is difficult to grasp by non-specialists such as decision-makers. Because the concept of life expectancy is part of the regular knowledge, and expressed directly in years, its use should not lead to misunderstanding by decision-makers. The method presented here allows assessing soil erosion risk caused by global change. The coupling between STREAM model and CLUE-s model allows the analysis of the effect of land use change on the soil erosion. The method asserts soil sustainability with a patrimonial point of view by using life expectancy as an index of soil sustainability. For this purpose, life expectancy appears better suited that soil erosion rate. Applied on a watershed located in France as a test-case, the method can be used in other countries.

  [START_REF] Kok | Multi-scale narratives from an ia perspective : Part i. european and Mediterranean scenario development[END_REF]. These general scenarios allow positioning the evolutions of land use in a global context linked to economical, social and climatic context of the region. Four land use change scenarios are considered : (i) Accentuation. This scenario is similar to the scenario Convulsive Changes of Kok et al. It expresses a irreversible climate change acceleration with the formation of permanent desert in the South of Europe. Rainfed agricultures decreases sharply and cultural practices of viticulture are more and more linked to the chemical weeding to compensate the hydric and nitrogenous impact of weeds. (ii) Production. This scenario is based on the scenario Big is Beautiful of Kok et al. It transcribes an increase in market liberalization. This causes an increase competition of agriculture with a severe impact on smaller farmers, wine-growers and forest. (iii) Protection. This scenario is based on the scenario Knowledge is King of Kok et al. It marks a rapid increase in the importance of information and communication technologies and large developments in techniques (e.g., water desalination techniques, high-yielding crop varieties). In this context, cultural practices respectful to the environment develop and forests are protected. (iv) Environmental law. Close to the Protection scenario, it is based on the use of law to enforce the use of permanent intercrop by farmers in vineyards.

FIGURE 1 :

 1 FIGURE 1: CURRENT LANDUSE AND SCENARIOS OF FUTURE LANDUSES

FIGURE 2 :

 2 FIGURE 2: SOIL EROSION RATES FOR CURRENT LANDUSE AND SCENARIOS OF FUTURE LANDUSES

FIGURE 3 :

 3 FIGURE 3: SOIL LIFE EXPECTANCYFOR CURRENT LANDUSE AND SCENARIOS OF FUTURE LANDUSES IV. CONCLUSIONS

ACKNOWLEDGMENT

This study is part of the ANR -MESOEROS21 project, granted by the French National Agency for Research. Authors are grateful to the LAD group of Wageningen University for making CLUE-s available to them.


Jessalyn Alvina

Caroline Appert

Olivier Chapuis

Emmanuel Pietriga

RouteLens: Easy Route Following for Map Applications

Keywords: H.5.2 [[Information Interfaces and Presentation]]: User Interfaces -Graphical user interfaces Design, Human Factors, Experimentation, Performance Focus+Context, Route, Map, Steering law

Millions of people go to the Web to search for geographical itineraries. Inspecting those map itineraries remains tedious because they seldom fit on screen, requiring much panning & zooming to see details. Focus+context techniques address this problem by displaying routes at a scale that allows them to fully fit on screen: users see the entire route at once, and perform magnified steering using a lens to navigate along the path, revealing additional detail. Navigation based on magnified steering has been shown to outperform pan & zoom for large steering tasks. Yet, this task remains challenging, in part because paths have a tendency to "slip off" the side of the lens. RouteLenses automatically adjust their position based on the geometry of the path that users steer through. Route-Lenses make it easier for users to follow a route, yet do not constrain movements too strictly, leaving them free to move the lens away from the path to explore its surroundings.

INTRODUCTION

Web-based mapping service applications have become the tool of choice for exploring geographical areas and locating points of interest. Millions of people, both novice and expert computer users, go on sites such as Google Maps, OpenStreetMap or Bing Maps to search for itineraries, compare them, and eventually select one. This task remains tedious because map itineraries seldom fit on screen, requiring much panning & zooming of the map to see particular details. For instance, a tourist faced with several alternative itineraries to go from one place to another in a city may want to explore each one at street level to pick the one that passes through the most places of interest. To address this problem, Web-based mapping services feature panning & zooming capabilities, augmented with an overview widget to prevent excessive context loss when zooming in. However, the scale difference between the overview and the detailed view remains limited. Furthermore, decoupling the overview from the detailed view is not necessarily the best-suited strategy to effectively support navigation along itineraries.

Focus+context techniques offer an alternative, well-suited to following lengthy routes at varying levels of magnification. The route is displayed at a scale that allows it to fully fit on screen, and the focus+context technique, typically a fisheye lens [START_REF] Sarkar | Graphical fisheye views of graphs[END_REF][START_REF] Pindat | JellyLens: content-aware adaptive lenses[END_REF], provides an in-place magnification of the locally-bounded region of interest around the cursor. Users see the entire route at once, and perform magnified steering [START_REF] Gutwin | Fisheyes are good for large steering tasks[END_REF] to navigate along the path, displaying -incontext and in more detail -the portion that falls below the lens. Navigation based on magnified steering has been shown to outperform pan & zoom for large steering tasks [START_REF] Gutwin | Fisheyes are good for large steering tasks[END_REF]. Yet, this task remains a challenging one for users, in part because paths have a tendency to "slip off" the side of the lens.

We introduce RouteLens, a new content-aware technique that automatically adjusts the lens' position based on the geometry of the path that users steer through, so as to keep the lens on track in case of overshoot (Figure 1). RouteLens makes it easier for users to follow a route, yet do not constrain movements too strictly. The lens is more or less strongly attracted to the path depending on its distance to it, and users remain free to move the lens away from it to explore its surroundings. RouteLenses only affect the motor behavior of lenses, and can easily be combined with any type of graphical magnification lens [START_REF] Appert | High-precision magnification lenses[END_REF][START_REF] Pindat | JellyLens: content-aware adaptive lenses[END_REF]. After a brief overview of related work, we describe the behavior and implementation of RouteLenses. We then report on a laboratory experiment in which RouteLenses improve user performance over conventional magnification lenses on a path following task, as predicted by steering law [START_REF] Accot | Beyond Fitts' law: models for trajectory-based HCI tasks[END_REF].

RELATED WORK

There are two main ways of showing itineraries in mapping applications: representing them as a list of turn-by-turn driving instructions, or as an overlay on top of an interactive 2D map. The two strategies actually complement one another, and serve different purposes. The former, even if enhanced (e.g., LineDrive [START_REF] Agrawala | Rendering effective route maps: improving usability through generalization[END_REF], Detail Lenses [START_REF] Karnick | Route visualization using detail lenses[END_REF]), is designed to match drivers' cognitive representation of a route they are following in real-time [START_REF] Tversky | Cognitive maps, cognitive collages, and spatial mental models[END_REF]. However, they fail, because of their linear nature, to support the initial exploration and planning phase. The latter strategy better supports this phase, letting users freely explore the area surrounding the points of interest that the itinerary will go through, and plan alternative routes.

But interactive maps usually require users to perform a lot of panning and zooming actions to both see the complete itinerary and look at particular details, possibly causing some disorientation. One option to address this issue consists in displaying portions of the itinerary at different scales, for instance using PolyZoom [START_REF] Javed | PolyZoom: multiscale and multifocus exploration in 2D visual spaces[END_REF], a visualization that organizes multiple linked views of the same map at different scales in a hierarchical manner, letting users pan & zoom any of those. The technique has been shown to outperform pan & zoom for some multi-scale search tasks, but can be cognitively demanding as it requires users to manage and mentally relate numerous spatially-disconnected views.

Focus+context techniques address this latter issue by smoothly integrating a zoomed-in representation of the current area of interest into a smaller-scale overview of the entire map [START_REF] Appert | High-precision magnification lenses[END_REF]. The wired fisheye lens [START_REF] Yamamoto | Wired fisheye lens: A motion-based improved fisheye interface for mobile web map services[END_REF] lets mobile users explore their surroundings on a map by tilting their device. But the technique ties the lens to a particular point on the map using a rubber-wire-like mechanism, and does not help steer along paths [START_REF] Accot | Beyond Fitts' law: models for trajectory-based HCI tasks[END_REF]. JellyLenses [START_REF] Pindat | JellyLens: content-aware adaptive lenses[END_REF] adapt their shape to the geometry of the objects they magnify, so as to optimize the visual representation of the focus, context, and transition areas. However, they are only concerned with optimizing the visual representation when performing magnified steering, and do not consider the motor aspects of the task. High-Precision Magnification Lenses [START_REF] Appert | High-precision magnification lenses[END_REF] do look into the motor aspects of lens positioning, but do not provide support for steering.

While fisheyes have been shown to perform well for large steering tasks [START_REF] Gutwin | Fisheyes are good for large steering tasks[END_REF], those tasks remain challenging ones, in part because paths have a tendency to "slip off" the side of the lens. This specific problem could be addressed by combining lenses with LinkSliding [START_REF] Moscovich | Topology-aware navigation in large networks[END_REF], a technique for navigating large networks that strictly constrains movements to the path itself. However, as it translates the viewport (and thus the context view) whenever the cursor moves, it makes exploration of a route's surroundings impractical.

ROUTELENS

A RouteLens facilitates steering along a route by behaving as if it were attracted by it. This behavior is achieved by decoupling the lens' position from the cursor's position, following an approach similar to that of Semantic Pointing [START_REF] Blanch | Semantic pointing: improving target acquisition with control-display ratio adaptation[END_REF], that enlarges targets of interest in motor space by decoupling the latter from its visual counterpart. When using a RouteLens, all route segments whose distance to the system cursor is less than ∆ apply an attraction force to the lens. The lens' position L is computed as a function of the system cursor's position C by using a weighted mean between all attracting route segments:

L = C + dmin • n i=1 wi • Ai n i=1 wi
where Ai is the force vector that route segment i applies at position C to attract the lens (see below) and dmin is the distance between the cursor and the closest route segment. To ensure continuous lens movements when a route segment starts or stops having an influence on the lens, wi is set to ∆ -dc,i, where dc,i is the distance between the cursor and route segment i.

For a given route segment, the attraction vector is computed as:

A = α(dc) • (Rc -C)/dc
where Rc is the point on the route closest to the cursor, and dc the distance between the cursor and the route segment. α is a power function of dc that parameterizes the force vector a route segment applies to the lens:

α(dc) = 1 -dc ∆ p if dc ≤ ∆ 0 otherwise.
Figure 2 illustrates the progressive attraction effect of a straight route on the lens' position. As the cursor gets away from the route, the lens moves away from it more slowly than the cursor does. It then progressively moves faster, so as to match the cursor's position as soon as the latter leaves the attraction area. This area corresponds to a tunnel of diameter ∆ centered on the route. Power parameter p is used to fine-tune the attraction effect, controlling how quickly the lens leaves the route to match the cursor's position. The solid and dashed black lines in Figure 2 illustrate two different p values. Our composition of different attraction forces is in the spirit of techniques that rely on force vectors to adapt control-display gain [START_REF] Ahlström | An evaluation of sticky and force enhanced targets in multi target situations[END_REF][START_REF] Fung | Kinematic templates: end-user tools for content-relative cursor manipulations[END_REF][START_REF] Hurst | Dirty desktops: Using a patina of magnetic mouse dust to make common interactor targets easier to select[END_REF]. Among these, Kinematic Templates [START_REF] Fung | Kinematic templates: end-user tools for content-relative cursor manipulations[END_REF] are the closest to our approach, as they consider fixed field vectors around an arbitrary path to create visual magnetic guides that can be combined together to generate artistic effects in a mouse-operated drawing application. Our approach is also related to that employed in Snap-and-go [START_REF] Baudisch | Snap-and-go: Helping users align objects without the modality of traditional snapping[END_REF]. But while this technique works well for 1D line snapping, it is not straightforward to generalize to arbitrary 2D routes. It also makes users feel like the cursor temporarily stops and potentially causes exaggerated movements compared to the more progressive transitions that RouteLens enables.

When steering along a magnified route, users want to minimize the distance d l between the lens' center and the route. In Accot & Zhai's steering law [START_REF] Accot | Beyond Fitts' law: models for trajectory-based HCI tasks[END_REF], d l represents the movement's variability along the tunnel centered on the route, i.e., the tunnel's width. The law stipulates that the larger the variability, the easier the movement. Figure 2 shows how RouteLens makes steering easier than a regular fisheye lens does, by allowing for a wider variability in usercontrolled cursor movements. To keep a regular lens at a distance d l from the route, users have to keep the cursor at a distance dc = d l . With a RouteLens, this distance can be larger: dc = d l + dc • α(dc).

With a regular fisheye lens, keeping the route visible in the flattop requires that variability d l be less than F M (half of the tunnel width according to the steering law), where F is the radius of the flat-top, and M the magnification factor. With a RouteLens, the tunnel width can be larger if the radius of the attractive area goes beyond F M (the actual tunnel width is 2 p+1 F •∆ p M ). Preliminary testing and pilot studies showed that setting ∆ = 2F

M and p = 2 yield a good balance between the magnitude of the attraction and the smoothness of the transition, resulting in an overall lens behavior that is hardly noticed by users. Figure 2 illustrates how these values make the maximal variability of the cursor movement ∼ 1.6 larger than a regular fisheye lens does.

EVALUATION

RouteLenses' motor behavior can be implemented with any type of lenses, including the shape-shifting JellyLenses [START_REF] Pindat | JellyLens: content-aware adaptive lenses[END_REF] that adapt their geometry to that of objects of interest or the high-precision magnification lenses [START_REF] Appert | High-precision magnification lenses[END_REF] that address problems of quantization at high magnification factors. In this study we consider conventional fisheye lenses as a baseline to both isolate the benefits of Route-Lenses' motor behavior and keep a reasonable experiment length for participants. The experimental task consisted in following a route with a lens, always keeping the route visible in the flat-top (Figure 3). We hypothesized that RouteLens outperforms Regular-Lens on this route following task, since the former facilitates the underlying steering task [START_REF] Accot | Beyond Fitts' law: models for trajectory-based HCI tasks[END_REF], as discussed above.

Participants

12 volunteers (6 female), all right-handed, aged 23 to 39 years-old (average 28.2, median 27), daily mouse users, participated in the experiment. 11 use mapping applications frequently. 7 of them are familiar with magnifying lenses, and 5 with fisheye lenses.

Apparatus

We conducted the experiment on a Mac Pro workstation running Mac OS X, equipped with an ATI Radeon HD 5870 video card driving a 30" LCD monitor (2560×1600, 100 dpi), and a standard optical mouse (800 dpi resolution) set with the default system acceleration. The software was implemented in JavaScript and We-bGL using three.js (http://threejs.org/) and ran in Google Chrome. We used fisheye lenses with a magnification factor of 4, a lens size of 260 pixels and a flat top size of 180 pixels.

Design and Procedure

Our goal was to evaluate the motor aspects of the magnified steering task, focusing on the interaction technique itself. We thus decided not to use a real map to avoid any noise and bias due to the additional information that it would have featured. Instead, we operationalized the task using quantitative description factors. We could better control those, while still ensuring that the task was sufficiently representative of actual route following.

The experiment is a 2×2×2×4 within-subjects design with factors: TECH, ANGLE, DISTRACTOR, and DIR. TECH is the primary factor with two values: RegularLens and RouteLens. ANGLE and DISTRACTOR are secondary factors that define characteristics of the route, illustrated in Figure 3: ANGLE (Acute = π/4 and Obtuse = 3π/4) defines the angle between two route segments. DISTRAC-TOR defines the presence or absence of distractor routes, that also attract the cursor. When DISTRACTOR = With, additional (grey) routes are added at each turn of the (black) target route. DIR defines the direction of steering: left-to-right, right-to-left, top-to-bottom, or bottom-to-top. This factor was introduced for ecological reasons, and we do not consider it further in our analyses.

To start a trial, participants have to click on a black circle at the beginning of the first segment of the route. They then follow the target route and click on a rectangle located at the route's other endpoint. They are instructed to keep the route visible in the lens' flat-top all the way. As soon as the route leaves the flat-top, participants get notified on-screen and have to restart the whole steering task until they succeed. For each trial, we record (i) the (successful) task completion time TCT; (ii) the number of failed attempts, NumError; and (iii) the average distance from the lens to the target route.

We grouped trials into 2 blocks, one per technique, half of the participants starting with RouteLens. A block contains four subblocks of 16 trials corresponding to all ANGLE × DISTRACTOR × DIR conditions, presented in a random order. The first sub-block is for training purposes only. We collect measures for analyses during the three other sub-blocks. The overall experiment lasts about 50 minutes, including final debriefing and questionnaire. TECH has a significant effect on TCT: RouteLens is significantly faster than RegularLens, a difference of ∼ 15%. We also observe significant effects of DISTRACTOR and ANGLE on TCT: tasks are ∼ 5% faster with distractors than without, and ∼ 6% faster with Acute angles than with Obtuse angles.

Results

More importantly, we observe a significant TECH × DISTRAC-TOR interaction (Figure 4-b). Post-hoc t-tests with Holm correction for multiple pair comparisons reveal that RouteLens is faster than RegularLens, both with distractors (p < 0.0001) and without (p = 0.0002). However, distractors have a stronger impact on RouteLens. RouteLens is significantly faster with distractors than without, but this DISTRACTOR effect is not significant for Regular-Lens (p = 0.1420). The presence of distractors actually improves TCT for RouteLens without introducing more errors (p = 0.98). This may be due to the specific route layout we considered: the presence of a distractor route in the middle of the turn may make the turning movement easier. This middle route applies additional force vectors, resulting in a stronger global attraction towards the route at the end of the turn.

Regarding the number of errors NumError, we observe a significant difference for ANGLE only. Participants made more errors (Figure 4-c) with Obtuse than with Acute angles (p = 0.0072). Thus, participants were both slower (see above) and made more errors in the Obtuse condition. It may appear as a surprising result, as movements that involve highly curved portions are generally slower [START_REF] Viviani | Trajectory determines movement dynamics[END_REF] than straighter movements. We attribute this to the more frequent mouse clutching actions the operator noticed for Obtuse angles, the path being lengthier in the direction of steering (see Figure 3).

As expected, RouteLens' attraction effect also makes users steer along the route with a movement that exhibits less variability. The average distance from the lens' center to the route is significantly lower for RouteLens than for RegularLens (F 1,11 = 331, p < 0.0001). The distance is 13.3 ±0.9 pixels for RouteLens and 33.3 ± 1.7 pixels for RegularLens (expressed with respect to the flat-top's coordinate system).

At the end of the experiment, we collected qualitative feedback using a post-hoc questionnaire. Three participants did not even notice the difference between the regular fisheye and RouteLens. Among the nine participants who did notice a difference, three participants said they had no idea what that difference was (they just felt like they could go faster with RouteLens) and seven had a preference for RouteLens (only one had a preference for RegularLens).

CONCLUSION

RouteLenses are designed to make it easier for users to follow map itineraries by dynamically adapting properties of the motor space, based on both cursor position and route geometry. This is achieved without constraining users, who remain free to explore the itinerary's surroundings without having to perform any explicit action to either engage or disengage the modified motor behavior. In our laboratory experiment, RouteLens better helped users stay close to the path they follow than a regular fisheye did, without making the lens stick too strongly to it. Future work will evaluate RouteLens' performance beyond steering, e.g., when disengaging from the route to explore its local surroundings. We would also like to further study the effect of the number and geometrical configuration of distractors routes present at intersections. We also plan to investigate the combination of RouteLenses' motor behavior with techniques designed for high magnification factors [START_REF] Appert | High-precision magnification lenses[END_REF] and dynamic visual adaptation [START_REF] Pindat | JellyLens: content-aware adaptive lenses[END_REF].

  Jessalyn Alvina, Caroline Appert, Olivier Chapuis & Emmanuel Pietriga. RouteLens: Easy Route Following for Map Applications. In AVI '14: Proceedings of the International Working Conference on Advanced Visual Interfaces, 125-128, ACM, may 2014. c ACM, 2014. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version will be published in AVI '14, May 27-29 2014, Como, Italy. http://dx.doi.org/10.1145/2598153.2598200

Figure 1 :

 1 Figure 1: Following an itinerary. (a) Conventional lens: the user overshoots at a right turn in Harrisburg; losing the route that falls in the distorted region. (b) RouteLens: the route's attraction compensates the overshoot; the lens remains closer to the route, which remains in focus.

Figure 2 :

 2 Figure 2: Position of the cursor (grey line), and of the Route-Lens (black line), that is vertically attracted (p = 2) by the route (bold blue line). The dashed black line shows the positions of the RouteLens when p = 6. In this figure, ∆ is equal to the lens' flat-top diameter in motor space, and the black (resp. grey) circles show the part of the context displayed in the RouteLens' (resp. regular fisheye lens') flat-top.

Figure 3 :

 3 Figure 3: The four path configurations used in the experiment. The target route (black) always consists of 5 segments, 300 pixels in length each. Distractor routes are painted gray.

Figure 4

 4 Figure 4 shows trial completion time TCT for each TECH by DIS-TRACTOR, and the results of the ANOVA for the full factorial model TCT ∼ TECH × DISTRACTOR × ANGLE × Rand(PARTICIPANT).TECH has a significant effect on TCT: RouteLens is significantly faster than RegularLens, a difference of ∼ 15%. We also observe significant effects of DISTRACTOR and ANGLE on TCT: tasks are ∼ 5% faster with distractors than without, and ∼ 6% faster with Acute angles than with Obtuse angles.More importantly, we observe a significant TECH × DISTRAC-TOR interaction (Figure4-b). Post-hoc t-tests with Holm correction for multiple pair comparisons reveal that RouteLens is faster than RegularLens, both with distractors (p < 0.0001) and without (p = 0.0002). However, distractors have a stronger impact on RouteLens. RouteLens is significantly faster with distractors than without, but this DISTRACTOR effect is not significant for Regular-Lens (p = 0.1420). The presence of distractors actually improves TCT for RouteLens without introducing more errors (p = 0.98).

Figure 4 :

 4 (a) ANOVA for the full factorial model TCT ∼ TECH × DISTRACTOR × ANGLE × Rand(PARTICIPANT). (b) TCT by TECH×DISTRACTOR condition. (c) Average number of errors by trial, by TECH × ANGLE condition. Error bars show the confidence interval for the mean over all trials.