Varvara Isyuk
email: varvara.isyuk@univ-paris1.fr

CPP

Keywords: Bailouts, bank recapitalisation, CPP funds, systemic risk JEL Classification Numbers: E52, E58, G21

The U.S. Federal Reserve responded to liquidity shortage through compulsory

Introduction

Financial crisis of 2007 had a large cost for the economy as a whole. Besides the welfare loses for financial institutions, private firms and households, government interventions all over the world have become a heavy burden for the society. Government interventions included various measures that mostly constituted from loans guarantee schemes on newly issued senior unsecured debt and bank recapitalisations. The U.S. Treasury offered a substantial amount of liquidity to banks through purchases of preferred equity stakes under Capital Purchase Program (further referred to as CPP, for more details see [START_REF] Acharya | The Financial Sector Bailout. Sowing the Seeds of the Next Crisis ?[END_REF][START_REF] Panetta | An assessment of financial sector rescue programmes[END_REF][START_REF] King | Time to buy or just buying time ? The market reaction to bank rescue packages[END_REF][START_REF] Cooley | The Role of the Federal Reserve[END_REF][START_REF] Khatiwada | Stimulus Packages to Counter Global Economic Crisis : A Review[END_REF].

The debate concerning the effectiveness of the U.S. rescue packages offered to commercial banks still continues. On one hand, the regulators were concerned about "moral hazard" problem [START_REF] Dam | Bank bailouts, interventions, and moral hazard[END_REF][START_REF] Gale | Dollarization, Bailouts, And The Stability Of The Banking System[END_REF][START_REF] Stiglitz | Restoring growth and Stability in a World of Crisis and Contagion : Lessons from Economic Theory and History[END_REF], on the other hand, bank recapitalisations were unavoidable to support solvent but illiquid banks and to stabilise the financial markets [START_REF] Fender | Overview : global financial crisis spurs unprecedented policy actions[END_REF].

Comparing to other types of government support, purchase of preferred or common shares is often seen as one of the most efficient types of capital infusion (see [START_REF] Wilson | Common (stock) sense about risk-shifting and bank bailouts[END_REF].

Another pro-argument of the CPP is that the program did not end up costing much to taxpayers. It used 204.6 billion dollars of the 250 billion dollars stipulated earlier (more than a third of the total TARP funding) and approximately half of the commercial banks repurchased their preferred shares from the Treasury by the year 2012.

In this article the focus is on the determinants of liquidity provisions under CPP. The goal of the paper is to define, first of all, factors that determined the final bailout allocation (both the fact of receiving CPP funds and the relative size of CPP funds) and the bailout repayments1 . Based on that, it is possible to assess the effectiveness of CPP according to the goals of the program and the risks for the taxpayers.

The analysis is motivated by several hypotheses. First hypothesis is that the distribution of CPP funds and its repayment are dependent on financial fragility of the commercial banks just before the crisis. These variables are similar to those used in the probability of default models. The regulators are expected to provide liquidity to more financially fragile banks as well as exposed to so-called "tail risk" that materialized after a secular collapse in housing markets banks.

The second hypothesis is that CPP funds were designed to minimise the spillover effect of the crisis to the economy at large. First, there was a risk of reduction in credit availability due to deterioration of the financial industry's intermediary role. Second, there was a significant counterparty risk, mostly from the side of LCFIs2 .

The contribution of the banks to systemic risk is estimated through Marginal Expected Shortfall (MES) [START_REF] Acharya | Regulating Sytemic Risk[END_REF], ∆CoV aR [START_REF] Adrian | CoVaR[END_REF], bank size and beta (last two are individual systemic risk indicators).

The third hypothesis is that political contributions (including lobbying activities) and locational advantages of the commercial banks may induce generous distribution of CPP funds towards some banks. [START_REF] Duchin | The politics of government investment[END_REF] find the evidence of higher likelihood of politically connected firms to be funded.

The last hypothesis is that excessive risk-taking of the bank have triggered the need of the bank for CPP funds in [2008][2009] [START_REF] Kibritcioglu | Excessive Risk-Taking, Banking Sector Fragility, and Banking Crises[END_REF]. The more risk the bank was undertaking prior to crisis of 2007 (proxied by the change in bank's stock returns), the higher should have been its loses during the crisis and, thus, the greater its need in CPP funds is expected to be comparing to other banks.

The article is completing the literature on bailouts and the effectiveness of liquidity provisions. It tends to improve the assessment of bailout programs through additional analysis of bailout repayments during four years following the disbursement of CPP funds. It is potentially an important source of information concerning the realised risks of funding allocation that were tried to be avoided by the Federal Reserve when designing the bailout packages.

Various econometric models are applied to analyse the capital injections under CPP and their reimbursement : logit, OLS, polytomous and duration models.

Results of logit and OLS regressions suggest that the rescue packages in the U.S. were designed in a way to provide liquidity to more financially distressed and more systemically important commercial banks. From this point of view, the CPP program can be assessed as a successful bailout program that helped to recapitalise large financial institutions with temporal liquidity problems and to avoid the collapse of entire banking system.

According to the results of polytomous and duration models banks that were larger contributors to systemic risk (measured through MES, beta and bank size) were also the ones that repaid their loan totally and in the shorter period of time. Thus, from the point of view of taxpayer the risks of non-repayment and the time until the CPP funds repayment were minimised.

On the other hand, bailout packages failed to support the banks that were actually exposed to mortgage and non-performing loans as well as mortgage-backed securities. Banks with higher share of mortgages in their total loans are found to exhibit a smaller probability of receiving CPP funds and if approved for the program to receive a smaller size loan. The same pattern is found for the commercial banks more exposed to MBSs and non-performing loans.

It could be explained by the fact that regulators classified such banks as insolvent with higher risk of bailout non-repayment. The banks with larger share of mortgage and nonperforming loans on their balance sheets are associated with higher probability of CPP

OLS regressions

Allocation of CPP funds among commercial banks in the U.S. is analysed using logistic regression and cross-section OLS. The former one focuses on the probability of the CPP funds disbursement to the particular commercial bank, while the latter one studies the relative size of the distributed CPP funds.

Logit regression allows to estimate the probability of the dependent variable to be equal to 1 (probability that some event occurs). Here BD i represents a binary variable that takes a value of 1 if a particular bank i received CPP funds in 2008-2009 ; 0 otherwise. There is no data available separately on applications for the bailouts and approvals for participation in CPP. The banks from the sample which did not receive CPP funds were, on one hand, banks that were refused the funds due to insolvency and high risk of taxpayerfunded credit non-repayment. On the other hand, these were the banks that did not require that kind of aid or could attract alternative external financing. Banks that received the bailout and repaid it totally during 2009-2012 should have been more financially stable than those that did not repay the bailout, otherwise banks potentially abused the bailout program in order to keep additional liquidity in their financial turnover and to generate extra income.

Cross

The model requires to set one outcome as a base one, thus, setting coefficients for that outcome equal to zero. That is, when setting outcome "bailout and total repayment" (y = 1) as an arbitrary, the coefficients for remaining outcomes measure the change relative to that base group.

Duration analysis

Under the Capital Purchase Program financial institutions have received the funds at more or less the same period (mostly fourth quarter of 2008), while the date of repayment has been defined by each bank individually. The time until the bailout repayment is another measure describing the realised risks of CPP funds allocation.

A central component of the analysis in this section is the hazard rate which is the probability of the CPP refunds at time t i , conditional on not having repaid the bailout before (or having survived to time t i ).

The semiparametric Cox proportional hazards model does not make assumptions about the shape of h 0 (t). However, when a correct form of the h 0 (t) is chosen, the model could fit the data better and could produce better results.

Parametric models can be based, on one hand, on the proportional hazards assumption, on the other hand, on the accelerated failure time assumption.

To capture the monotonically increasing shape of the hazard function (the graph of Kaplan-Meier estimate is available on demand), the Weibull distribution is chosen (see details in appendix).

The flattening of the Kaplan-Meier survival estimate at the end of distribution, however, suggests a possibility of non-monotonic pattern of duration dependence. The log-logistic distribution is chosen among other AFT models.

The choice between the parametric models is made using the Akaike Information Criterion (AIC) and log-likelihood. The AIC scores are compared between the parametric models. The lowest value of AIC is found for the Weibull model of baseline hazard, even though the graph of hazard function estimates (available on demand) suggests a greater resemblance with loglogistic and log-normal models. Log-logistic distribution of hazard function is preferred to log-normal one according to AIC criterion, besides it is commonly used when fitting data with censoring. Thus, three duration models are finally fitted : the Cox proportional hazards model (no specific parametrization), the Weibull proportional hazards model (monotonically increasing hazard function) and the log-logistic model (non-monotonic unimodal hazard).

3 Data and summary statistics

Data description

To construct the sample of firms, U.S. domestically controlled commercial banks were selected in DataStream. These financial firms operated on the U.S. market in U.S. dollars and were still active on December, 2008. After selecting variables needed for estimation, around 650 commercial banks were left in the sample.

The data on bailouts (promised amount, actually disbursed amount, the date of entering the program) and bailout reimbursement (amount repaid, date of repayment) is obtained from Treasury's Office of Financial Stability. The data on political contributions and lobbying expenditures of PACs3 related to banks comes from the website of the Federal Election Commission in the U.S.

The data from three sources is merged (see details in appendix). Bailouts under CPP were provided to domestically controlled banks, bank holding companies, savings associations, and savings and loans holding companies. 707 financial institutions finally received financing under CPP. Commercial banks that appear on the bailout list received the bailout, while others did not. Only actually disbursed amount is considered as a fact of the bank bailout.

After outliers selection procedure, 597 banks are left in the sample.

Dependent variables

CPP funds allocations in 2008-2009

• Bailout dummy

The variable BD i is a dummy that takes on two values, 0 and 1 (see table 1) to distinguish between the banks that did not receive CPP funds and those who did.

Banks that have finally received CPP funds applied for Capital Purchase Program (CPP), have been approved for funding and then accepted the funds. Out of 597 banks in the sample approximately 320 banks did not receive a bailout, while around 280 received the CPP funds.

-INSERT TABLE 1 -

• Bailout continuous variable

Bailout continuous variable represents actually disbursed CPP funds amount normalised by bank total assets. The maximum relative size of the bailout reaches 7% (see table 1). Among the bailed out banks more than 50% obtained capital of the relative size between 2% and 3% of their total assets. The correlation with dependent variables is shown in table 2. Correlation coefficients are higher in its absolute values for the bailout dummy but the most correlated explanatory variables remain the same for the relative size of bailout.

-INSERT TABLE 2 HERE -

CPP refunds between 2009 and 2012

• Bailout repayments

This discrete dependent variable classifies the banks in four groups : banks that did not receive the bailout at all (y = 0), banks that received the bailout and repaid it totally (y = 1), repaid it partly (y = 2) or did not repay the bailout at all (y = 3).

Slightly more than half of represented banks did not receive the CPP funds during 2008-2009. Around 20% of the banks received the bailout and repaid it totally, another 20% of the banks never repaid it by 31 of July, 2012, while small fraction of the banks (less than 5%) repaid it partly (majority of these banks repaid at least 50% of the total amount).

• Time-to-repayment

The time at risk or time until the event occurs (here the CPP funds repayment) is analysed in duration models.

Only bailed out banks are considered for the analysis. Thus, around half of observations are left in the sample, around 280 banks. The analysed period is limited between the distribution of the CPP funds in 2008 and July 31, 2012. In this period approximately half of these banks repaid the bailouts while others did not.

CPP refund is only taken into account if the bank managed to repurchase the total amount of preferred shares from the Treasury by the end of the analysed period (total refund). Time to repayment is counted in days.

The data and the repayment announcements suggest the first repayments to take place in March of 2009, around half a year after the start of CPP program. For the rest the probability of CPP refunds increases with time (see 2.2.2 for details).

Bank balance sheet characteristics

Bank balance sheet characteristics are financial statement variables that define the "financial health" of the bank, or, in other words, determine the probability of the bank's default [START_REF] Duchin | The politics of government investment[END_REF][START_REF] Ratnovski | Why Are Canadian Banks More Resilient ?[END_REF]. Here indicators from the next three models are included : Altman's Z-score, KMV Moody's RiskCalc for U.S. banks, BondScore (Credit Sights) model. Some indicators appear to be highly correlated with each other and need to be excluded from the final estimation.

Altman's Z-score

Z-score bankruptcy model proposes Z-score indicator for each firm as a discriminant score that represents level of distress of that firm. Five financial ratios constitute the score with defined weights for each of them (see details in appendix). Higher Z-score is interpreted as an indicator of a "safer" or, in other words, more financially healthy firm, while low Z-score indicates higher level of distress of the firm.

It is expected that safer financial firms before the crisis should experience less need in bailout funds during the crisis and, thus, probability of the bailout for such banks should be lower. As expected default frequency measures as well as the formula for computing them are not available in public access, the input variables of Moody's model are plugged directly in the regressions (taking into account multicollinearity issues with indicators from other models).

Moody

Each category is represented by at least one variable, descriptive statistics is provided in table 1. Some variables are discussed below.

Asset Concentration group consists of two variables : relative size of real estate mortgage loans (AC 1 in tables) and commercial and industrial loans (AC 2 in tables) in the bank's portfolio.

Real Estate Mortgage Loans (AC 1 ) include commercial and construction mortgages, thus, its relative size could be positively correlated with the size of commercial and industrial loans (AC 2 ). It appears though that two normalised sizes of these groups of loans are highly but negatively correlated with each other (correlation coefficient is -0.89, table 2). It means that if bank loan portfolio is concentrated in real estate mortgage loans, the bank loans less for commercial and industrial purposes4 . That gives an impression of a bank "specialisation". In the recent crisis MBSs became highly risky and illiquid assets. That is why the initial indicator proposed in Moody's RiskCalc model that brings together Treasury securities and mortgage-backed securities (as both representing liquid groups of assets) is replaced by two separate ratios.

Asset Quality group is represented by the share of non-performing loans in total loans. Lower asset quality is expected to increase the probability of default and, consequently, the probability of the bailout. Nevertheless, the correlation coefficient between bailout dependent variables and normalised non-performing loans in 2007 is negative (-0.11 and -0.12 with bailout dummy and relative size of disbursed amount, table 2).

BondScore Model

BondScore Credit Model is another model that calculates credit risks for all U.S. nonfinancial corporations with total assets in excess of $250 millions and publicly traded equity.

Three variable from BondScore Model are analysed (others are similar or even same as variables from Moody's RiskCalc Model) : ratio of EBITDA to bank's net sales (EBITDA margin, EM ), leverage (Lev) and volatility of EBITDA (V ol). It is expected that commercial banks with higher margin, smaller leverage and smaller volatility exhibit a smaller probability of default and, consequently, less shortage in liquidity during the crisis.

However, first two BondScore variables cannot be kept in regressions due to the high risk of multicollinearity.

Systemic Risk variables

The bailout packages were designed in large part to limit systemic risk in the financial system -that is, risk of a spillover effect of a crisis from one big institution to another and from the financial sector to the economy at large. One of the most frequently used proxies of systemic risk is a firm's size (standardised, Size, table 3). It supports "too big to fail" argument : the lender of last resort cannot deny support to large financial institutions whose closure would significantly affect the rest of the market [START_REF] Freixas | The lender of last resort of the 21st century[END_REF]. Correlation coefficients are presented in table 4. Bank's size is indeed highly and positively correlated with bailout dependent variables.

-INSERT TABLE 3 HERE -

The second variable that represents the systemic risk is Beta. It is the correlation between stock returns of financial institution and the overall market. In the period of crisis market in general performed very badly, thus, firms with higher beta should exhibit a higher probability of default and thus, require government intervention.

-INSERT TABLE 4 HERE -∆CoV aR was developed by Adrian and Brunnermeier (2009). ∆CoV aR represents the difference between the Value-at-Risk of financial sector conditional on the institution "i" being under distress and Value-at-Risk in regular times (see details in appendix). All the measures of systemic risk are calculated in a way that the higher value of the variable indicates higher contribution of the commercial bank to the systemic risk. Correlation coefficients from table 4 are positive confirming that higher contribution to systemic risk is associated with the higher probability of the bailout and larger relative size of the disbursed amount.

Marginal Expected Shortfall (M ES

Political Involvement and Location related variables

The financial sector is one of the largest contributors to federal political campaigns.

Political contributions are not done directly by the financial firms but through especially organised political action committees (PACs). The data on PACs contributions only contains information on official contribution of the bank-related PACs so only 3.3% of the financial firms are found to be contributors between 2006 and 2008.

Lobbying expenditures is another form of political support of government agencies. The political involvement dummy is then constructed which takes on value of 1 if in the underlined period the PAC related to the particular bank made a political campaign or lobbying contribution, 0 otherwise. Correlation of political involvement dummy with dependent variables suggests positive influence of the former on the latter one (0.12, 0.08 and 0.08 with bailout dummy, relative size of disbursed amount and repayment categorical variable).

The distribution of bank assets across the states is not the same. Even when the size of bank assets in two states is similar, the distribution of bank assets across organisation size is never identical [START_REF] Berger | The transformation of the U.S. Commercial Banking Industry : What A Long Strange Trip It's Been[END_REF] in these two states. Thus, the state dummy is then included into regression.

Excessive bank's risk-taking related variables

Several attempts to use the past history of financial institution were made in the literature to see if banks with riskier strategies learned in past crises or continued following original business concepts5 .

The representative variable from this group aims to account for individual risk-taking of the bank.

Change in stock prices is calculated as a difference of log stock prices between 2003 and 2006. Firms that take on more risk and follow riskier strategies with higher returns are expected to experience an important increase in their stock prices in that period (due to the development of securitization and other financial instruments). These should also be the firms that are hit the most during the crisis and that require government intervention to remain active on the market.

It is logical that several financial ratios proposed by different probability of default models are similar to each other. Including all of them could be a reason of high multicollinearity in the regressions that would lead to greater standard errors and larger estimated coefficients.

The correlation matrix for all financial indicators is shown in table 2. In order to avoid large tables and concentrate on significant variables, regressions are conducted using stepwise backward selection method with significance level for removal of 0.05. Balance sheet variables, systemic risk and individual excessive risk-taking variables are standardized. It makes the size of parameters (or effects of the explanatory variables) comparable within each column. out, received a smaller amount).

Treasury securities remained one of the most liquid and, thus, high demanded financial assets during the financial crisis. Hence, more liquid banks in terms of higher share of treasury securities (Liq 1 in tables) on their balance sheets should have been financially more stable during the crisis and should not have applied for the financial funds (or applied for a relatively smaller amount). The estimated coefficients from table 5 confirm that argument.

Part of originated before the financial crisis loans remained on the banks' balance sheets.

It could be argued that banks preferred to leave on their balance sheets high quality loans while subprime loans have been mostly securitised and sold to other entities [START_REF] Acharya | Regulating Sytemic Risk[END_REF]. That problem is otherwise referred to as an "adverse selection" problem of lenders.

However, subprime and lower quality loans have had much higher yield than any other type of loans and could be left on the banks' balance sheets (as well ABCP, CDOs and CLOs).

Furthermore, due to the abrupt rise in the interest rates even prime borrowers experienced difficulties with repayment of their debt. Results suggest that banks with higher share of non-performing loans in 2007 (AQ in tables), however, had a smaller probability to receive CPP funds and if received, their size was smaller.

Systemic risk variables beta (Beta i,2007 ), bank size (Size i,2007 ) and MES (M ES i,2000(M ES i, -2007 ) ) have positive significant coefficients which means that regulators concentrated in offering liquidity to larger, more correlated with the market and riskier institutions. Another measure of systemic risk, CoV aR i,2000-2007 is not significant which can be due to not long enough period of estimation and low frequency of used data (no other data is available in free access).

Political involvement variable appears positively significant only in one regression which also can be due to the poor data available in the free access.

State dummy is not significant at all, as well as bank's individual risk-taking measured by the past performance of the bank's stock prices.

Models fit the data well, R-squared (pseudo R-squared for logit) reaches 17%-18%.

Polytomous logistic model

This multinomial model is used to define the factors that determine the probability of bailout repayment/non-repayment under the CPP during 2009-2012. Dependent variable describes if the bank was bailed out or not, and if it was, which share of the bailout did the bank repay : total amount, a part of the disbursed amount or nothing at all (see section 3.2.2 for details).

Results for multinomial regressions are presented in terms of signs of the estimated coefficients in table 6. The full results with exact values of the coefficients are available on demand. The base outcome is a positive decision concerning the bank bailout under CPP and its total repayment by commercial bank during 2009-2012.

-INSERT TABLE 6 HERE -

The third column of table 6 reports results for the group of banks that did not receive CPP funds ("No bailout"). The signs of the multinomial estimates show the change (positive or negative) in logit of outcome "no bailout" (group "0") relative to the group of banks that received CPP funds and repaid them totally (group "1") given the other variables in the model are held constant.

These results confirm those for logit and OLS regressions from the previous section. Safer banks or more financially stable banks (with higher Altman's Z-score in 2007) are less likely to receive CPP funds during the crisis. Positive sign in table 6 indicates that increase in bank's Altman's score (Z) in 2007 is associated with higher probability of a bank not to be bailed out by the government in 2008-2009.

Commercial banks with higher shares of real estate mortgage loans (AC 1 ) and nonperforming loans (AQ) are more likely to receive no CPP funds as it is suggested by the signs of coefficients from the third column "No bailout". Besides, even if banks with higher share of that sort of loans received the bailout, they exhibit a higher probability not to repay it (table 6, column 5 "Bailout and no repayment"). An opposite effect is found for the banks that are more exposed to commercial and industrial loans (AC 2 ) : they are more likely to be bailed out and less likely not to repay it during 2009-2012.

These results should be thought of in two ways, from the point of view of minimising the risks for taxpayers and from the point of view of restoring the financial stability. From the point of view of taxpayer, the risks have been minimised. The CPP funds were provided to the banks with the highest probability of repaying them in short term : banks with smaller exposure to mortgage and non-performing loans and higher exposure to commercial loans.

However, from the point of view of consumers and borrowers, the program had a potentially ineffective side. Commercial banks with higher exposure to mortgage loans and non-performing loans did not get enough external financing from the government in the period of the crisis. Banks more specialised in real estate mortgage loans were the ones affected the most by the crisis and without appropriate help from the government potentially failed to restructure a large part of mortgage loans and continued to perform poorly due to a large amount of foreclosures. Facing liquidity problems such banks needed to raise mortgage interest rates putting even creditworthiness homeowners in difficult financial position.

Banks with stronger positions in treasury securities (Liq 1 ) and MBSs (Liq 2 ) before the crisis are less likely to be bailed out in [2008][2009]. The first relationship is justified by high safety and liquidity of Treasury bills, especially in the time of crisis ("flight to security" argument). The second relationship is less clear as MBSs became highly illiquid during the crisis. There are two potential explanations : first, banks kept on their balance sheets MBSs of a good quality6 ("adverse selection" argument), second, government help was provided to the banks that have had higher chances to repay it, thus, to those banks that were less exposed to MBSs.

If the results for "No bailout" outcome (column 3) are compared with the results for "Bailout and no repayment" (column 5) outcome, it can be noticed that liquidity indicators described above as well as Z-score are not significant in predicting the repayment/nonrepayment of CPP funds by the commercial banks. It means that, first of all, that more financially stable and liquid firms prior to crisis applied less for CPP funds (and potentially were rarely approved for participation in the program). Secondly, if they applied, their level of safety and liquidity before the crisis did not affect the probability of the CPP funds repayment. Besides, higher exposure to MBSs is not associated with later or earlier CPP funds repayment.

All systemic risk variables are significant with negative coefficients when predicting the "No bailout" (column 3) and "Bailout and no repayment" (column 5) outcomes. The signs of the estimates tell us that larger banks with higher potential contribution to systemic risk (higher beta Bets i,2007 and MES M ES i,2000-2007 ) were less likely not to receive CPP funds.

That confirms that CPP was designed to bail out systemically important and "too big to fail" financial institutions.

Besides, a large part of banks with higher contribution to systemic risk repaid the bailout totally during 2008-2009 (or had a lower probability of CPP funds non-repayment). It allows to conclude that the focus of CPP on bailing out systematically important institutions was effective from both sides. From the point of view of taxpayer these banks were the ones that repurchased their shares in the shortest notice. From the point of view of borrowers and restoring of financial stability the failure of large institutions was avoided which could have led to the large externality costs for the other sectors of economy.

Thus, more systematically important, large and interconnceted financial institutions could be finally characterised as "temporarily illiquid but healthy" during the crisis. Nevertheless, it shoud not be forgotten that larger banks have greater capacities of restoring their activities that are represented in diverse forms and of attracting alternative sources of financing partly due to their "too big to fail" position.

It is at some point logical that these institutions could repurchase their shares form the Treasury fast enough and it does not confirm the soundness of "too big to fail". The big collapse of these institutions was avoided, however, more reforms should be introduced (expanding the Dodd-Frank Wall Street Reform and Consumer Protection Act of 2010, see [START_REF] Acharya | Regulating Wall Street : the Dodd-Frank Act and the new architecture of global finance[END_REF] for the discussion) in order to limit the propensity of the financial sector to put the entire system at risk and to benefit from its too-big-to-fail position.

It can be concluded then that in terms of efficiency for taxpayers the CPP funds been disbursed in the correct way, in order to limit the probability of CPP funds nonrepayments. Besides, the collapse of large financial institutions has been avoided that could have been translated into the collapse the entire banking sector. However, banks which could if properly motivated help to avoid the bankruptcy of the large number of homeowners and other borrowers have not been supported.

Time to repayment analysis

This section presents the results for duration analysis where dependent variable is the time until CPP funds repayment. The choice of parametrizations for that analysis is described in section 2.2.2. Each continuous variable that enters the model is checked for correlation with the dependent variable. Besides, the models with single continuous predictors are considered as well as the results of the Chi-squared tests in order to choose predictors for the final model.

Results for three types of regressions (with Cox PH, Weibull and log-logistic parametrizations) are presented in table 7. Besides, the signs of estimated coefficients for log-logistic AFT are also included in table 6 to compare with previous results.

The interpretation of coefficients is different for proportional hazards models (Cox PH and Weibull PH, columns 3, 4, 5 and 6, table 7) and accelerated failure time models (log-logistic AFT, columns 7 and 8, same table). The coefficients from the first pair of models indicate how coviariates affect the hazard rate. Positive coefficients increase the hazard rate and, therefore, reduce the expected duration. The positive coefficients from AFT models indicate how covariates affect the logged survival time and, hence, increase expected duration.

The first view on the signs of the regression coefficients confirms that results of all three models are coherent. The impacts of covariates are similar in terms of the signs of coefficients, and their level of significance does not change from one model to another.

The results are in line with those presented in the previous sections (columns 5 and 6, table 6). More systemically risky banks managed to repurchase their preferred shares faster than the rest, while those with higher cash flows per share, higher concentration of non-performing and mortgage loans had more difficulties with repaying the bailouts. It can be also the case that banks with higher cash flow per share did not wish to repurchase their shares from the Treasury too fast (as this predictor also has a positive impact on the probability of partial repayment, column 4, table 6) as it was a comfortable and relatively cheap source of external financing comparing to the market conditions.

The goodness of fit is checked by using, first of all, plots of estimated hazard functions and, secondly, Cox-Snell residuals. The plot of the Nelson-Aalen cumulative hazard function for Cox-Snell residuals for the log-logistic function has the poorest fit with respect to the 45 • line with the slope of 1. The plots for the models with Weibull and log-logistic parametrizations seem to be the both close to linearity (graphs are available on demand).

Conclusion

In the aftermath of the crisis one of the questions that is still discussed by the policymakers and academics is the effectiveness of the government and Central Bank responses to the financial crisis.

Various econometric techniques are used in this article to study the factors that determined the CPP funds disbursement and their repayment in the period between 2009 and July, 2012. The results from each section are coherent and complement each other.

Bailout packages provided under CPP seem to be efficient in responding to liquidity crisis subject to the large banks that contributed the most to systemic risk. Due to that, the propagation of the financial crisis was slowed down even though there is many criticism concerning the late response (announcement of loan guarantees and recapitalisations) of the Federal Reserve. However, it is logical that these institutions could repurchase their shares from the Treasury fast enough and it does not confirm the soundness of "too big to fail". The big collapse of these institutions was avoided, however, more reforms have to be introduced (expanding the Dodd-Frank Wall Street Reform and Consumer Protection Act of 2010) in order to limit the propensity of the financial sector to put the entire system at risk and to benefit from its too-big-to-fail position.

The allocation of CPP funds was also effective from the point view of taxpayer. Larger firms with smaller shares of mortgage and non-performing loans, higher shares of commercial loans and greater contributions to systemic risk were the ones with the highest probability to receive the CPP funds and to receive larger amounts but also to reimburse them totally in shorter notice.

Thus, larger and more systemically important banks were mostly considered viable and were recapitalised, while other banks that were actually exposed to the mortgage market were denied financial aid (some of them went bankrupt). In this sense, CPP potentially did not succeed in supporting cosumers' loan recapitalisations and restructurings that could become a reason of large welfare loses for the homeowners.

Designing the programs addressing the housing crisis through minimisation of mortgage default losses could incite faster recovery of financial markets and economy as a whole.

Supporting mortgage-specialised banks could reduce the number of delinquent borrowers, foreclosures and other externalities.

∆CoV aR i p = V aR F S|"i"distress p -V aR F S p
Institution "i" is said to be in distress when it exhibits the lowest growth rates of its market-valued total financial assets defined through the p t h percentile (here 5 t h percentile, thus, bottom 5% of the returns). V aR F S p is the mean of growth rates of the financial sector in the 5th percentile of its distribution unconditionally on the state of separate institutions.

The growth rate of market valued total assets (X i t ) is calculated in the next way :

X i t = M E i t • LEV i t -M E i t-1 • LEV i t-1 M E i t-1 • LEV i t-1 = A i t -A i t-1 A i t-1
Knowing that

A i t = M E i t • LEV i t = BA i t • ( M E i t BE i t )
where M E i t is the market value of an intermediary i 's total equity, LEV i t is the ratio of total assets to book equity, A i t are market-valued total assets, BA i t are book-valued total assets,

M E i t BE i t
is market-to-book ratio of the institution "i".

Following [START_REF] Adrian | CoVaR[END_REF] the growth rate of the whole financial sector is calculated as the average market-valued returns of all financial institutions, weighted by the lagged market value of their assets :

X F S t = i (X i t • w i t-1 )
where w i t-1 is the weight of the financial institution i in the financial sector at the period t-1. The (unconditional) Value-at-Risk of the financial sector is then defined as the bottom 5% of the growth rate of the financial sector between 1990 and July, 2008 (quarterly data from Compustat). The Value-at-Risk of the financial system conditional on institution i being in distress is calculated as the mean of growth rates of financial sector in the periods when institution i was found to be in distress. The difference between the two measures is ∆CoV aR i p , the contribution of the institution i to the risks of financial sector in general.

Marginal Expected Shortfall (M ES α ) is expected percentage loss in market value faced by one institution given that a shock drives the market beyond the threshold C (market drop by more than a certain threshold). Expected shortfall is the average of financial market returns on days when the portfolio's loss exceeds its VaR limit. The financial market return R is the sum of each bank's return r i :

R = i w i • r i
where w i is the weight of bank "i" in the overall banking system. The expected shortfall of the financial market can be then represented as a weighted sum of individual banks' expected shortfall :

ES α = - i w i E[r i |R ≤ -V aR α ]
The derivative of the financial system expected shortfall with respect to the weight of bank y i is then bank "i"'s expected shortfall :

∂ES α ∂w i = -E[r i R ≤ -V aR α ] = M ES i α
Here M ES i α is calculated with 100% -α = 95% confidence. Thus, M ES i 5% is computed in the next way :

M ES i 5% = 1 N t:R-in-its-5%-tail r i t (1)
where 1 N t:R-in-its-5%-tail r i t is the average of returns of financial firm i in the 5% worst-case periods of aggregate stock market losses (measured on a daily basis through S&P 500 index).

M ES

  funds non-repayment during 2009-2012 and longer duration until repurchase of their stock from the Treasury. The rest of the paper is structured as follows. Section 2 presents the estimation methodology. Section 3 introduces the data, describes dependent and explanatory variables. Empirical results for cross-section logit, OLS, polytomous and time-to-event regressions analysing the factors that determined CPP funds disbursement and their repayments are presented in 5 Documents de Travail du Centre d'Economie de la Sorbonne -2012.72 Section 4. Section 5 concludes. 2 Model specifications for studying CPP funds allocation and their repayments 2.1 Determinants of CPP funds allocation in 2008-2009 : logit and

  -sectional bailout probability P r(BD i ) is modeled in the next way : P r(BD i ) = α 0 + βBC i,2007 + γSR i,2007 + χP L i,2006-2008 + ηRT i,2003-2006 + i where BC i,2007 represent bank balance sheet characteristics for 2007 (indicators from private bank and company default models) ; SR i,2007 are measures of bank systemic risk (and contribution to systemic risk) in 2007 ; P L i,2007 are variables measuring political influence and locational advantages of the bank ; RT i,2003-2006 include variables associated with bank excessive individual risk-taking during the four years prior to crisis. The detailed description of the variables and its selection is provided in section 3 and in appendix. OLS model includes the same predictive variables while dependent variable is continuous 6 Documents de Travail du Centre d'Economie de la Sorbonne -2012.72 and represents the relative size of disbursed amounts. 2.2 Determinants of CPP refunds during 2009-2012 2.2.1 Multinomial (polytomous) logistic regression This type of regression is used in order to define which factors influenced the probability of bailout repayments by the U.S. banks. The discrete dependent variable in that model represents the bank's progress in CPP refund by July 31, 2012. The set of coefficients for explanatory variables is estimated for each outcome : no bailout y = 0, bailout and total repayment y = 1, bailout and partial repayment y = 2, bailout and no repayment y = 3 (see details in appendix).

  's KMV RiskCalc V3.1 U.S. Banks More recent Moody's KMV RiskCalc V3.1 is the Moody's rating agency model for predicting probability of the bank default. It comprises financial statement variables and equity market information on the bank's prospects and business risk.

  Liquidity-related variables (Liquidity group) should measure the share of liquid assets on the balance sheet of the bank. Moody's RiskCalc v3.1 U.S. Banks model (2006) and Basel II regulation classified mortgage-backed securities (MBS) as safe and liquid holdings. That was indeed the case at the time, MBSs also included government mortgages provided by Government National Mortgage Association or other U.S. agencies.

13

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.72

  α ) is expected percentage loss in market value faced by one institution given that a shock drives the market beyond the threshold C (market drop by more than a certain threshold, see details in appendix). M ES is calculated over three different periods (it could not be done with ∆CoV aR as there are not enough observations) : for the year 2007, for the period of 8 years preceding the crisis (from 2000 to 2007) and for the periods of Bear Stearns and Lehman Brothers collapses (February and March of 2008 ; September and October of 2008).

  for Logit and OLS regressions analysing the bank bailout probability and the relative size of disbursed amount under CPP during 2008-2009 in the U.S. The results are reported in table 5. Columns 3, 5 and 7 present results for logit regressions with binary outcome : bailout or no bailout. Columns 4, 6 and 8 present results for OLS cross-sectional regressions where the dependent variable is the actually disbursed amount normalised by bank's total assets. For each model, the results for three alternative regressions are reported with different measures of systemic risk : beta (Beta i,2007 ), columns 3 and 4 ; bank size (Size i,2007 ), columns 5 and 6 and Marginal Expected Shortfall over 7 years from 2000 to 2007 (M ES i,2000-2007 ), columns 7 and 8.

4. 2

 2 Polytomous and time-to-repayment regressions analysing the bank repayments of CPP funds during 2009-2012 in the U.S.

Table 1 :

 1 i 5% is calculated for the year2007, 2006 and over 8 years between 2000 and 2007. Summary of dependent variables and balance sheet characteristics from Altman's and Moody's models for U.S. commercial banks

	Variable	Name	Obs	Mean	Std. Dev. Min	Max
	Bailout dummy	BD i	644	0.44	0.5	0	1
	Size of the bailout normalised by	B i	644	0.01	0.01	0	0.07
	bank total assets
	Bailout and repayment categorical	R i	644	0.87	1.16	0	3
	variable
	Z-score,	Z	597	0	1.00	-2.92	4.27
	standardized
	Moody's RiskCalc U.S. Banks
	Total equity to total assets,	CS 1	661	0	1	-1.20	3.54
	winsorized at 2% level, standardized
	Total deposits to total assets,	CS 2	642	0	1	-1.67	2.80
	winsorized at 1% level, standardized
	Net revenues to total assets,	P 1	654	0	1	-2.25	3.68
	winsorized at 1% level, standardized
	Cash flow per share,	P 2	640	0	1	-0.84	3.50
	winsorized at 2% level, standardized
	Mortgage Real Estate Loans to to-	AC 1	661	0	1	-3.56	2.02
	tal loans ratio (in percentage),
	standardized
	Consumer and Industrial Loans to	AC 2	653	0	1	-1.22	3.21
	total loans ratio (in percentage),
	winsorized at 2% level, standardized
	Treasury Securities to total assets	Liq 1	607	0	1	-0.56	3.59
	ratio (in percentage),
	winsorized at 2% level, standardized
	Mortgage-Backed Securities to to-	Liq 2	641	0	1	-1.04	3.36
	tal assets ratio (in percentage),
	winsorized at 2% level, standardized
	Non-performing loans to total	AQ	661	0	1	-.91	3.69
	loans ratio (in percentage),
	winsorized at 2% level, standardized

Table 3 :

 3 Summary of BondScore balance sheet characteristics, systemic risk, political involvement and individual risk-taking related variables

	Variable		Name	Obs	Mean	Std. Dev. Min	Max
	BondScore U.S.
	EBIT DA 2007 Sales 2007 ,		EM	632	0	1	-3.24	1.83
	winsorized at 2%level, standardized
	Debt 2007 M arketCap+BookV alueDebt 2007	,	Lev	604	0	1	-3.57	2.52
	winsorized at 1%level, standardized
	standardized V olatility 2007 ,		V ol	502	0	1	-1.88	3.43
	Systemic risk
	Size (logarithm of total as-	Size i,2007	661	0	1	-2.84	3.49
	sets),
	standardized
	Beta, standardized		Beta i,2007	621	0	1	-1.76	2.78
	Marginal expected shortfall	M ES i,2007	626	0	1	-2.41	2.95
	(MES) for 2007,
	standardized
	Marginal expected shortfall	M ES i,2000-2007	632	0	1	-1.87	3.65
	(MES) over 8 years between
	2000 and 2007,
	winsorized at 1%level, standardized
	Marginal expected shortfall	M ES i,BSLB	608	0	1	-1.81	2.51
	(MES) for the Bear Stearns
	and Lehman Brothers near-collapse,
	winsorized at 1%level, standardized
	Conditional Value-at-Risk,		∆CoV aR i,1990-2007	628	0	1	-3.13	1.97
	standardized
	Political influence and lo-
	cation
	Political influence dummy		P D 2006-2008	658	0.03	0.18	0	1
	State		State	661	25.90	14.39	1	51
	Individual risk-taking
	Change in log stock prices du-ring 2003-2006,	ln(q i,2003-2006 )	525	0	1	-3.09	2.64
	winsorized at 1% level, standardized

Table 4 :

 4 Correlation between dependent variables (dummy and relative size of bailout) and explanatory systemic risk, political influence, location and individual risk variables for US banks

	ln(q i,2003-2006 )												1.00
	∆CoV aR P D State									1.00	0.15 1.00	0.01 0.01 1.00	-0.11 -0.02 -0.03
	M ES i,BSLB								1.00	0.17	0.18	0.02	-0.08
	M ES i,2000-2007							1.00	0.74	0.20	0.24	0.02	-0.09
	M ES i,2007						1.00	0.80	0.90	0.19	0.19	0.02	-0.09
	BD B R Size Beta	1.00	0.90 1.00	0.82 0.77 1.00	0.24 0.14 0.18 1.00	0.26 0.26 0.19 0.62 1.00	0.12 0.06 -0.02 0.74 0.62	0.16 0.10 0.00 0.78 0.61	0.15 0.09 0.00 0.68 0.59	0.05 0.02 0.00 0.25 0.14	0.12 0.08 0.08 0.39 0.20	-0.04 -0.04 -0.03 -0.01 -0.06	-0.04 -0.02 -0.04 -0.12 -0.09
	Var	BD	B	R	Size	Beta	M ES i,2007	M ES i,2000-2007	M ES i,BSLB	∆CoV aR	P D	State	ln(q i,2003-2006 )
	Type	Depend	var-s		Systemic	Risk

Table 5 :

 5 Determinants of the bank bailout probability and the relative size of the bank bailout under the Troubled Asset Relief Program in 2008-2009, U.S. commercial banks, stepwise cross-sectional estimation

	OLS	With M ES
	Logit	With M ES
	OLS	With Size
	Logit	With Size
	OLS	With Beta
	Logit	With Beta
	Name
	Type of var	sheet Balance	char-s	Altman's

Table 7 :

 7 Time to the CPP funds repayment analysis, U.S. commercial banks, proportional hazards (PH) and accelerated failure time (AFT) models

	Log-logistic	AFT	With M ES		-0.103	(-1.610)	0.085***	(2.708)	0.177***	(2.659)	0.264***	(3.062)			-0.291***	(-4.384)	7.010***	(73.900)			-0.701***	(-8.339)	537.188	279
	Weibull PH Weibull PH Log-logistic	AFT	With Beta With M ES With Beta		0.212*** 0.143* -0.143**	(2.693) (1.775) (-2.294)	-0.107** -0.114*** 0.073**	(-2.468) (-2.653) (2.260)	-0.303*** -0.289*** 0.206***	(-3.415) (-3.171) (2.862)	-0.381*** -0.434*** 0.250***	(-2.986) (-3.196) (2.910)	0.213** -0.134**	(2.275) (-2.018)	0.392***	(4.611)	-11.796*** -12.108*** 7.040***	(-12.849) (-12.979) (74.831)	0.483*** 0.510***	(5.976) (6.372)	-0.654***	(-7.690)	546.147 536.265 549.771	275 279 275
	Cox PH		With M ES		0.128	(1.569)	-0.113***	(-2.625)	-0.301***	(-3.209)	-0.418***	(-3.131)			0.395***	(4.639)							1406.519	279
	Cox PH		With Beta		0.199**	(2.562)	-0.108**	(-2.468)	-0.315***	(-3.488)	-0.366***	(-2.919)	0.221**	(2.364)									1406.630	275
	Name				Z		P 2		AC 1		AQ		Beta i,2007		M ES i,2000-2007		Constant		Ln(p)		Ln(gamma)		AIC	Obs
	Type of var		Balance sheet	char-s	Altman's	Z-score	Moody's	RiskCalc					Systemic Risk	var-s
												34

Documents de Travail du Centre d'Economie de la Sorbonne -2012.72

The bailout repayments under CPP mean repurchase of the Treasury's equity stake 3 Documents de Travail du Centre d'Economie de la Sorbonne -2012.72

Large Complex Financial Institutions

Documents de Travail du Centre d'Economie de la Sorbonne -2012.72

Political Active Committees

Commercial and industrial loans represent a general amount of loans made to business and industry excluding commercial mortgages and including consumer loans 12 Documents de Travail du Centre d'Economie de la Sorbonne -2012.72

For instance, through the performance of the banks during LTCM crisis in 1998,[START_REF] Fahlenbrach | This Time Is the Same : Using Bank Performance in 1998 to Explain Bank Performance During the Recent Financial Crisis[END_REF]

Agency-backed MBSs of the prime loan type. 20 Documents de Travail du Centre d'Economie de la Sorbonne -2012.72

Documents de Travail du Centre d'Economie de la Sorbonne CPP funds allocation : restoring financial stability or minimising risks of non-repayment to taxpayers ? funds allocation: restoring financial stability or minimising risks of non-repayment to taxpayers? loan guarantee scheme and bank recapitalisations mainly under Capital Purchase Program (CPP) for commercial banks. The bailout packages provided under CPP seem to be

The empirical evidence is in favor of the main argument provided in the beginning of this article : the bailouts have been provided to the more financially distressed firms (illiquid commercial banks). Banks with higher Z-score in 2007 (thus, "safer" or more financially stable banks according to Altman's Z-score) exhibit a smaller probability of bailout in 2008-2009 ( see columns 3, 5 and7, table 5) and if bailed out, they receive CPP funds of a smaller size (see columns 4, 6 and 8, table 5).

-INSERT TABLE 5 HERE -Besides, OLS regression with beta (column 4, table 5) suggests that an increase in Z-score by one (variable is standardised, thus its standard deviation is equal to one) is associated with reduction in the relative bailout size by 0.3%. Among Moody's RiskCalc model indicators these are asset concentration, liquidity and asset quality variables that are significant and that remain robust in each regression.

The empirical results suggest that the banks more specialised in commercial and industrial loans (AC 2 in tables) than in mortgage lending (AC 1 in tables) were more likely to be bailed out (and to receive larger amounts of financial aid) in 2008-2009. On one hand, the crisis of 2007 was originated in the subprime loan market and one could expect regulators to start saving banks with higher share of mortgage loans. On the other hand, under both Basel I and Basel II the weight of mortgage loans in risk-weighted assets was smaller than that of corporate loans. Thus, banks exposed to corporate loans could be considered as more affected during the crisis.

In many cases banks that have originated large amounts of mortgage loans securitised them, repacked and then sold to third-party investors in form of mortgage-backed securities.

These assets (including AAA-rated mortgage-backed tranches) became illiquid following the collapse in housing markets. The one could expect monetary regulators to focus on saving the banks with higher exposure to mortgage-backed securities in order to raise the liquidity in financial market. Results show that it was not the case : firms with higher exposure to mortgage-backed securities (Liq 2 in tables) were less likely to be bailed out (and if bailed

Appendices A Construction of variables

A.1 Altman's Z-score

Altman's Bankruptcy model suggests an index based on the five main financial ratios where weight of each variables defined using discriminant analysis :

where X 1 is the ratio of difference between current assets and current liabilities to total assets ; X 2 is the ratio of retained earnings to total assets ; X 3 is the ratio of earnings before interest and taxes (EBIT) to total assets ; X 4 is the ratio of market value of equity to total liabilities ; X 5 is the ratio of sales to total assets.

A.2 Systemic Risk related indicators

Bank size (Size i,2007 ) is calculated as the logarithm of total assets of the bank.

Beta (Beta i,2007 ) is taken from Datastream and represents the measure of the asset's risk with relation to the market (correlation with the market) over the past 5 years. Thus, beta for 2007 is actually calculated for the period from 2002 to 2007.

∆CoV aR p is a systemic risk indicator that measures the marginal contribution of a separate financial firm to the risk of the whole financial sector. As the name of the variable shows it, the prefix "co" stands for conditional, contagion and comovement [START_REF] Adrian | CoVaR[END_REF]. This measure of COntribution of the institution i to the systemic risk of the financial system is measured as the difference between Value-at-Risk of the financial sector conditional on institution i being in distress (V aR F S|"i"distress p

) and the Value-at-Risk of the financial sector in normal times (V aR F S p ) : 0.02 0.04 0.07 -0.15 -0.12 0.12 0.22 -0.02 0.10 -0.01 -0.11 -0.06 0.18 -0.14 0.02 1.00

Z


Xavier Antoine
email: xavier.antoine@iecn.u-nancy.fr

Christophe Besse
email: christophe.besse@math.univ-lille1.fr

Jérémie Szeftel
email: jszeftel@math.princeton.edu.

Towards accurate artificial boundary conditions for nonlinear PDEs through examples

The aim of this paper is to give a comprehensive review of current developments related to the derivation of artificial boundary conditions for nonlinear partial differential equations. The essential tools to build such boundary conditions are based on pseudodifferential and paradifferential calculus. We present various derivations and compare them. Some numerical results illustrate their respective accuracy and analyze the potential of each technique.

Introduction

The subject of designing artificial boundary conditions for linear scalar and systems of Partial Differential Equations (PDEs) has been studied since more than thirty years now. Essentially, the goal of these boundary conditions is to truncate an infinite domain into a finite one for computational considerations. To this end, a fictitious boundary Γ is introduced delimiting therefore a finite domain Ω. All the difficulty is then to build an accurate, robust and easy-to-implement approximate boundary condition on this fictitious boundary. These boundary conditions can be found in the literature under different names (which in fact have different subtle meanings) like artificial boundary conditions, absorbing boundary conditions, non-reflecting or transparent boundary conditions and sometimes Dirichlet-to-Neumann operators. Among the major contributions written on the topic and without being exhaustive, let us quote e.g. the papers by Engquist and Majda [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF][START_REF] Engquist | Radiation boundary conditions for acoustic and elastic wave calculations[END_REF], Bayliss and Turkel [START_REF] Bayliss | Radiation boundary-conditions for wave-like equations[END_REF], Mur [15] or also Bérenger [START_REF] Bérenger | A perfectly matched layer for the absorption of electromagnetic-waves[END_REF]. A few review papers have also been published to establish the current state-of-the-art on the subject (see e.g. [START_REF] Antoine | A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations[END_REF][START_REF] Givoli | Nonreflecting boundary conditions[END_REF][START_REF] Hagstrom | Radiation boundary conditions for the numerical simulation of waves[END_REF][START_REF] Tsynkov | Numerical solution of problems on unbounded domains. A review[END_REF]).

While many improvements have been achieved over the past years, most of them are developed for linear equations. Practically, most available methods for linear equations do not apply to nonlinear equations since they often rely on the explicit computation of the transparent boundary condition by using the Fourier or Laplace transforms (note however that in the particular case of integrable equations, one may use the inverse scattering transform to explicitly compute the transparent boundary condition, see e.g. [START_REF] Zheng | Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations[END_REF]). Now, Nonlinear problems have recently received some increasing special care because of their importance in applications like e.g. in wave propagation, quantum mechanics, fluid mechanics,... The aim of this paper is to give a comprehensive introduction to possible solutions to handle such nonlinear situations. They are mainly based on pseudodifferential [START_REF] Taylor | Pseudodifferential Operators[END_REF] and paradifferential operators techniques [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF].

The paper is organized as follows. In Section 2, we analyze in detail the way of constructing approximate artificial boundary conditions for a general one-dimensional wave equation with variable coefficients using pseudodifferential calculus. Then, we test numerically these absorbing boundary conditions on a model problem showing that they yield accurate computations at least for small times. In a third Section, we consider a general one-dimensional nonlinear Schrödinger equation. We present several ways to extend the method of Section 2 to this nonlinear equation depending on the kind of nonlinearity involved in the equation. The various types of absorbing boundary conditions are obtained using either the pseudodifferential or the paradifferential calculus. In a fourth Section, some numerical comparisons are developed to test the accuracy of the various absorbing boundary conditions. The last Section draws a conclusion and suggests some future directions of research.

2 Artificial boundary conditions for linear variable coefficients equations: the case of the wave equation

The case of the constant coefficients wave equation

Before directly going to the case of the wave equation with variable coefficients, let us first consider the simple wave equation

(∂ 2 t -∂ 2 x )u = 0, (1)
with initial data u(0, x) = u 0 (x) and ∂ t u(0, x) = u 1 (x), where the field u = u(t, x) is defined on the whole space (t, x) ∈ [0; +∞[×R. For simulation purposes, it is standard to introduce a bounded spatial computational domain setting now (t, x) ∈ [0; +∞[×[x ℓ ; x r ], where x ℓ (respectively x r ) is a left (respectively right) fictitious endpoint introduced to get a bounded domain Ω = [x ℓ ; x r ]. Let us assume that the initial data of our problem, that is u 0 and u 1 , are compactly supported in Ω.

Then, we can define the extension of u (which is still denoted by u) for negative times by fixing its value to zero so that u is a solution to (1) for all times t as long as x ∈ Ω c , where

Ω c = R -Ω.
Let us denote by ût (τ, x) = F t (u)(τ, x) the partial time-Fourier transform of u, where τ ∈ R.

Applying F t to (1) for (t, x) ∈ R × Ω c leads to the Helmholtz-type constant coefficients equation

(∂ 2 x + τ 2 )û t (τ, x) = 0, (2)
where the wavenumber is τ . The solution of this equation can be written as the superposition of two waves

ût (τ, x) = A + e iτ x + A -e -iτ x , (3)
where A ± are two smooth functions depending on τ . Computing the derivative ∂ x ût , we obtain

(∂ x -iτ )û t = -2iτ e -iτ x A -, (4)
and

(∂ x + iτ )û t = 2iτ e iτ x A + , (5)
and we obviously check that: (∂ x + iτ )(∂ x -iτ )û t = 0. We also have the following operator factorization

∂ 2 t -∂ 2 x = -(∂ x + ∂ t )(∂ x -∂ t ). (6)
Equation (4) (respectively ( 5)) gives a characterization of the right (respectively left) traveling solution to (1) by setting: A -= 0 (respectively A + = 0). Therefore, the following boundary condition (4)

(∂ n -iτ )û t = 0, at Γ, (7)
acts as a filter in the time-Fourier domain and translates the property that there is no reflection back into the computational domain Ω, where n is the unit normal vector to Γ = {x ℓ ; x r }, outwardly directed to Ω. This is a constraint which forces the wave to be outgoing to Ω. In the time-space domain, the corresponding boundary condition writes down

(∂ n -∂ t )u = 0, at [0; +∞[×Γ. ( 8
)
Since there is no reflection, this boundary condition is usually called Transparent or Non-Reflecting Boundary Condition (TBC). Let us remark at this point that another interpretation of writing a transparent boundary condition is that we require

u ∈ Ker(∂ n -b), setting b(x, t, ∂ t ) = ∂ t .

The case of the variable coefficients wave equation

Let us consider that α, β and γ are three C ∞ functions. Writing a TBC for a variable coefficient model wave equation

(∂ 2 t + β(t, x)∂ t -∂ 2 x + γ(t, x)∂ x + α(t, x))u = 0, (9)
is much more complicate than in the constant coefficients case. Indeed, in such a situation i) directly applying a time-Fourier transform to the equation ( 9) leads to a convolution equation which is extremely difficult to write down explicitly, ii) and even if it is possible to write an inhomogeneous Helmholtz-type equation, solving this equation for general functions associated with α, β and γ cannot be expected.

Building an accurate boundary condition which approximates the TBC can however be expected since the pioneering work of Engquist & Majda [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF][START_REF] Engquist | Radiation boundary conditions for acoustic and elastic wave calculations[END_REF] in the middle of the seventies using the theory of reflection of singularities at the boundary [START_REF] Majda | Reflection of singularities at the boundary[END_REF] and pseudodifferential calculus (see for example [START_REF] Taylor | Pseudodifferential Operators[END_REF]).

Let us develop the main ideas. Like in the previous situation with constant coefficients, we assume that u as been extended by zero for negative times t and that the initial data u 0 and u 1 are compactly supported in Ω. Then, Engquist and Majda prove that there exist two classical pseudodifferential operator a and b of OPS 1 such that we get the following Nirenberg-like factorization

[16]

-∂ 2 x + ∂ 2 t + β(t, x)∂ t + γ(t, x)∂ x + α(t, x) = -(∂ x -a(x, t, D t ))(∂ x -b(x, t, D t )) + R, ( 10
)
where R is a smoothing operator of OPS -∞ . This approximate factorization can be considered as the extension to the variable coefficients case of the exact form [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF]. Actually, the smoothing operator R accounts for the fact that the factorization is now true only at high frequencies.

The two pseudodifferential operators a(x, t, D t ) and b(x, t, D t ), with D t = -i∂ t , have respective associated symbols a(x, t, τ ) and b(x, t, τ ) of S 1 admitting the following asymptotic expansions in homogeneous symbols

a(x, t, τ ) ∼ j≥0 a 1-j (x, t, τ ) and b(x, t, τ ) ∼ j≥0 b 1-j (x, t, τ ), (11)
with classical homogeneous symbols a 1-j and b 1-j of order 1 -j. This means that we have e.g. a 1-j (x, t, λτ ) = λ 1-j a 1-j (x, t, τ ), ∀λ > 0. Developing the factorization (10), we get

-∂ 2 x + γ(t, x)∂ x + ∂ 2 t + β(t, x)∂ t + α(t, x) = -∂ 2 x + (a + b)∂ x -ab + Op(∂ x b) + R (12) since ∂ x (bu) = Op(∂ x b)u + b∂ x u.
In the above equation, we designate by Op(σ) the pseudodifferential operator with symbol σ. If it is possible to compute a and b then, it can be proved that the TBC for equation ( 9) is given by

(∂ n -b(x, t, D t ))u = 0, at [0; +∞[×Γ. ( 13
)
Indeed, the results in [START_REF] Majda | Reflection of singularities at the boundary[END_REF] imply that [START_REF] Hagstrom | On the accurate long-time solution of the wave equation in exterior domains: asymptotic expansions and corrected boundary conditions[END_REF] annihilates the wave reflected back in the computational domain. Generally speaking, this TBC, which extends [START_REF] Engquist | Long-time behavior of absorbing boundary conditions[END_REF], cannot be directly implemented since b is given by an infinite expansion, but it can however be approximated by a k-th order artificial boundary condition by truncating the series [START_REF] Givoli | Nonreflecting boundary conditions[END_REF] to the first k symbolic terms and considering

(∂ n - k-1 j=0 b 1-j (x, t, D t ))u = 0, at [0; +∞[×Γ. ( 14
)
The computation of the terms {b 1-j } k-1 j=0 is therefore needed. To this end, we identify the operators on both sides of equality [START_REF] Hagstrom | Radiation boundary conditions for the numerical simulation of waves[END_REF] and we obtain at the symbolic level the system

a + b = γ -a#b + ∂ x b = -τ 2 + iβτ + α, (15)
which can also be rewritten as

b#b -γb + ∂ x b = -τ 2 + iβτ + α, (16)
by eliminating a. In the above notations, a#b designates the symbol of the composition operator ab which admits the following expansion (see for example [START_REF] Taylor | Pseudodifferential Operators[END_REF]):

b#b ∼ ∞ m=0 (-i) m m! ∂ m τ b ∂ m t b. ( 17
)
Since b is developable in terms of homogeneous symbols using relation [START_REF] Givoli | Nonreflecting boundary conditions[END_REF], we can extract each term b 1-j from ( 16) by identifying the decaying order terms starting from 2 to 2 -j using [START_REF] Szeftel | Calcul pseudodifférentiel et paradifférentiel pour l'étude de conditions aux limites absorbantes et de propriétés qualitatives d'équations aux dérivées partielles non linéaires[END_REF].

Beginning with b 1 , we get that

b 1 (x, t, τ ) = iτ, (18)
fixing also the uniqueness of the expansion [START_REF] Givoli | Nonreflecting boundary conditions[END_REF]. Computing the two next terms gives

     b 0 = β + γ 2 , b -1 = (γ 2 -β 2 ) + 4α -2(∂ x + ∂ t )(γ + β) 8iτ , (19)
at the right point x r . It directly gives the following proposition.

Proposition 1. Let u be the solution to the generalized wave equation ( 9) with C ∞ variable coefficients α, β and γ. Then, the artificial boundary conditions of order k, for k = 1, 2, 3, are respectively given at the right endpoint x r by

∂ x u -∂ t u = 0, at [0; +∞[×{x r }, ∂ x u -∂ t u - β + γ 2 u = 0, at [0; +∞[×{x r }, ∂ x u -∂ t u - β + γ 2 u - (γ 2 -β 2 ) + 4α -2(∂ x + ∂ t )(γ + β) 8 I t u = 0, at [0; +∞[×{x r }, (20)
where I t is defined by I t u(t) = t 0 u(s)ds. Similar formulas can be derived at x ℓ .

Short-time vs long-time behavior

Let us now consider that we wish to compute a numerical solution to the problem

             (∂ 2 t + β(t, x)∂ t -∂ 2 x + γ(t, x)∂ x + α(t, x))u = 0, (t, x) ∈]0; T [×Ω, u(0, x) = u 0 (x), x ∈ Ω, ∂ t u(0, x) = u 1 (x), x ∈ Ω, (∂ n - k-1 j=0 b p 1-j (x, t, D t ))u = 0, at [0; T ] × {x p }, (21)
for a maximal time of computation T and where the k-th artificial boundary condition is defined by operators {b p 1-j } k-1 j=0 , for p = ℓ, r, at the left or right fictitious point x p (see e.g. Proposition 1). Introducing N intervals of discretization in time, we denote by ∆t the time step defined by ∆t = T /N . We next seek to compute an approximate solution u n (x) ≈ u(t n , x) to system [START_REF] Taylor | Pseudodifferential Operators[END_REF], with t n = n∆t, for n ∈ {1, ..., N }. We have seen before that the derivation of the artificial boundary conditions at the continuous level is made under the high frequency assumption |τ | ≫ 1.

Since we consider discrete times t n , for n = 1, ..., N , discrete time frequencies τ n = π/t n are then associated and lie in the interval [π/T ; π/∆t]. Hence, the artificial boundary conditions which work well for high frequencies will be accurate if T ≪ 1. This means that an artificial boundary condition is accurate as long as it is used for small times of computation. They may fail for large computational times which is a known problem of artificial boundary conditions techniques (see e.g. [8] [13]).

As an illustration, we compare in Figure 1 the performances of the artificial boundary conditions of order 1, 2 and 3 in the case of the wave equation (∂

2 t + ∂ t -∂ 2
x )u = 0. We give the relative error in the L 2 (Ω)-norm for times between 0 and 10. As predicted by the theory, we notice indeed an improvement for small times by increasing the order. The second order condition is more efficient than the first order condition for all computed times but the third order condition is more efficient than the second order condition only for t ≤ 5.2.

i∂ t u + ∂ 2 x u + α(u)∂ x u + β(u)u = 0, (t, x) ∈ [0; +∞[×R, u(0, x) = u 0 (x), x ∈ R, (22)
where we assume again that the initial condition u 0 has compact support in Ω and that α and β are two C ∞ functions.

Let us now consider the following associated variable coefficients linear Schrödinger equation

i∂ t u + ∂ 2 x u + A(t, x)∂ x u + B(t, x)u = 0, (t, x) ∈ [0; +∞[×R, u(0, x) = u 0 (x), x ∈ R. ( 23
)
Extending the previous strategy presented for the variable coefficients wave equation in section 2.2 to equation [START_REF] Zheng | Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations[END_REF], one can prove that there exist two pseudodifferential operators a(x, t, D t ) and b(x, t, D t ) such that we have

∂ 2 x + i∂ t + A∂ x + B = (∂ x -a(x, t, D t ))(∂ x -b(x, t, D t )) + R, (24)
where again R ∈ OPS -∞ . The operators a and b are elements of OPS 1/2 admitting the following expansion in homogeneous symbols

a(x, t, τ ) ∼ ∞ j=0 a (1-j)/2 (x, t, τ ) and b(x, t, τ ) ∼ ∞ j=0 b (1-j)/2 (x, t, τ ), ( 25
)
where a (1-j)/2 and b (1-j)/2 are homogeneous symbols of order (1 -j)/2. This means that ∀λ > 0, we have:

a (1-j)/2 (x, t, λτ ) = λ (1-j)/2 a (1-j)/2 (x, t, τ ), b (1-j)/2 (x, t, λτ ) = λ (1-j)/2 b (1-j)/2 (x, t, τ ). ( 26
)
If one considers e.g. the right fictitious point x r , then by fixing b 1/2 (x, t, τ ) = -√ τ , where √ τ is defined by:

√ τ = √ τ if τ ≥ 0, -i √ -τ if τ < 0, (27)
it can be shown (see [START_REF] Szeftel | Calcul pseudodifférentiel et paradifférentiel pour l'étude de conditions aux limites absorbantes et de propriétés qualitatives d'équations aux dérivées partielles non linéaires[END_REF][START_REF] Szeftel | Absorbing boundary conditions for nonlinear Schrödinger equations[END_REF]) that the TBC is given by

(∂ n -b(x, t, D t ))u = 0, at [0; +∞[×{x r }, (28)
and that an approximate artificial boundary condition of order k is

(∂ n - k-1 j=0 b (1-j)/2 (x, t, D t ))u = 0, at [0; +∞[×{x r }, (29)
with the convention that the artificial boundary condition of order zero corresponds to the Neumann boundary condition. The required inhomogeneous symbols can be obtained by adapting relation ( 16)

b#b + Ab + ∂ x b = τ -B, (30)
using the suitable substitutions. Then, using the Leibniz symbolic rule we get the four first symbols

b 1/2 = - √ τ , b 0 = - A 2 , b -1/2 = - 1 2 √ τ ( A 2 4 -B + ∂ x A 2 ), b -1 = - 1 8τ (A∂ x A + ∂ 2 x A -2∂ x B + i∂ t A). (31)
To explain the different strategies which can be considered, we propose now to investigate first the case α = 0 and next to detail the situation when β = 0, where α and β are the functions in [START_REF] Tsynkov | Numerical solution of problems on unbounded domains. A review[END_REF].

3.2 Case I: α = 0

Potential strategy

The point of view adopted in this strategy considers that α(u) and β(u) act as potential functions independent of u. More specifically, they have respectively corresponding functions A and B in Equation ( 23). If we assume that A = 0, then, the symbols in (31) simplify as

b 1/2 = - √ τ , b 0 = 0, b -1/2 = B 2 √ τ , b -1 = ∂ x B 4τ . ( 32
)
Using the definition (29), we obtain the following artificial boundary conditions of order k at

[0; +∞[×{x r }    ∂ n u + e -i π 4 ∂ 1/2 t u = 0, for k = 1, 2, ∂ n u + e -i π 4 ∂ 1/2 t u -e i π 4 B 2 I 1/2 t u = 0, for k = 3, (33)
and finally

∂ n u + e -i π 4 ∂ 1/2 t u -e i π 4 B 2 I 1/2 t u -i ∂ n B 4 I t u = 0, for k = 4. ( 34
)
In the above equations, the fractional half-order derivative operator ∂

1/2 t , with symbol √ -iτ , is
given by

∂ 1/2 t f (t) = 1 √ π ∂ t t 0 f (s) √ t -s ds, (35)
and the half-order integration operator I 1/2 t (with symbol (-iτ ) -1/2 ) is defined by

I 1/2 t f (t) = 1 √ π t 0 f (s) √ t -s ds. ( 36
)
Following our strategy, we replace B by the nonlinearity β(u) which gives the three following artificial boundary conditions of order k

           ∂ n u + e -i π 4 ∂ 1/2 t u = 0, for k = 1, 2, ∂ n u + e -i π 4 ∂ 1/2 t u -e i π 4 β(u) 2 I 1/2 t u = 0, for k = 3, ∂ n u + e -i π 4 ∂ 1/2 t u -e i π 4 β(u) 2 I 1/2 t u -i ∂ n β(u) 4 I t u = 0, for k = 4. ( 37
)
These conditions will be denoted by ABC β 1,k in the sequel of the paper.

Gauge change strategy

Let us remark that the artificial boundary condition (34) is not a transparent boundary condition even when B is a constant potential. Now, in the case of a time-dependent potential B(x, t) = B(t), one can get the transparent boundary condition by using the gauge change

v(x, t) = e -iB(t) u(x, t), (38)
where B(t) = I t B(t), and noticing that v is now solution to the free Schrödinger equation

i∂ t v + ∂ 2 x v = 0. ( 39
)
Then, the transparent boundary condition

∂ n v + e -i π 4 ∂ 1/2 t v = 0 ( 40
)
holds for v and, coming back to the initial unknown u, we obtain the transparent boundary condition for u

∂ n u + e -i π 4 e iB(t) ∂ 1/2 t (e -iB(t) u(x, t)) = 0. (41)
This boundary condition is clearly not exact if B also depends on x. Nevertheless, we can use a similar change of gauge, that is v(x, t) = e -iB(t,x) u(x, t),

with B(t, x) = I t B(t, x). Then, v is sought as the solution to the variable coefficients Schrödinger equation

i∂ t v + ∂ 2 x v + (2i∂ x B)∂ x v + (i∂ 2 x B -(∂ x B) 2 )v = 0, ( 43
)
which is of the general form [START_REF] Zheng | Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations[END_REF] with initial condition v(x, 0) = u 0 (x). We can therefore apply the previous general derivation of artificial boundary conditions of Section 3.1 to this equation of unknown v for suitably defined variable coefficients. As a consequence, if b (1-j)/2 j≥0 designates the symbolic asymptotic expansion of the transparent boundary condition associated with v solution to (43), then an artificial boundary condition of order k is given for u as

∂ n u - k-1 j=0 e iB b (1-j)/2 (x, t, D t )(e -iB u) -i∂ n Bu = 0, at [0; +∞[×{x r }. (44)
More precisely, using (31), the computation of the first four symbols gives

b 1 = - √ τ , b 0 = -i∂ n B, b -1/2 = 0, b -1 = ∂ n B 4τ . (45)
Finally, we obtain [START_REF] Antoine | Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations[END_REF] the second-and fourth-order artificial boundary conditions given respectively by

∂ n u + e -i π 4 e iB ∂ 1/2 t (e -iB u) = 0, at [0; +∞[×{x r }. (46)
and

∂ n u + e -i π 4 e iB ∂ 1/2 t (e -iB u) -i ∂ n B 4 e iB I t (e -iB u) = 0, at [0; +∞[×{x r }. (47)
Replacing B by β(u), the associated nonlinear artificial boundary conditions are then given by

∂ n u + e -i π 4 e iB(u) ∂ 1/2 t (e -iB(u) u) = 0, at [0; +∞[×{x r }. (48)
and

∂ n u + e -i π 4 e iB(u) ∂ 1/2 t (e -iB(u) u) -i ∂ n β(u) 4 e iB(u) I t (e -iB(u) u) = 0, at [0; +∞[×{x r }, (49)
setting B(u)(x, t) = I t (β(u))(x, t). These conditions will be referred to as ABC β 2,j in the sequel, for j = 2, 4.

Let us develop the connection existing between the artificial boundary conditions ABC β m,j , for m = 1, 2 and j = 2, 4. To this end, let us recall the following Leibniz formula for computing the fractional derivative of the product of two functions

∂ p t (f g) = ∞ k=0 Γ s (p + 1) k!Γ s (p -k + 1) ∂ k t f ∂ p-k t g, (50)
for p > 0. The real-valued function f is supposed to be C ∞ and g is a continuous function. The notation Γ s designates the Gamma special function. For p = 1/2, we obtain

∂ 1/2 t (f g) = f ∂ 1/2 t g + 1 2 ∂ t f I 1/2 t g + R, ( 51
)
where R is an error operator in OPS -3/2 . Using a similar formula for the integral operator gives

I t (f g) = f I t g + S (with S ∈ OPS -2
). Using these two relations to approximate the half-order operator in (47) by setting f = e -iB and g = u, we see that (47) exactly corresponds to (34) up to an operator in OPS -3/2 . This error may be not negligible since is involves time derivatives of the potential, and, in the nonlinear case, of β(u). This difference can therefore be significant between the two kinds of artificial boundary conditions. This will be more deeply investigated during the numerical simulations.

3.3 Case II: β = 0

Potential strategy

Let us now consider the second case where β = 0. Then, the potential strategy consists in replacing

B = 0 in (31) leading to b 1/2 = - √ τ , b 0 = - A 2 , b -1/2 = - 1 2 √ τ ( A 2 4 + ∂ x A 2 ), b -1 = - 1 8τ (A∂ x A + ∂ 2 x A + i∂ t A). ( 52
)
This gives the following artificial boundary conditions

         ∂ x u + e -i π 4 ∂ 1/2 t u = 0, ∂ x u + e -i π 4 ∂ 1/2 t u + A 2 u = 0, ∂ x u + e -i π 4 ∂ 1/2 t u + A 2 u + e i π 4 2 ( A 2 4 + ∂ x A 2 )I 1/2 t u = 0, (53)
and

∂ x u + e -i π 4 ∂ 1/2 t u + A 2 u + e i π 4 2 ( A 2 4 + ∂ x A 2 )I 1/2 t u + i 8 (A∂ x A + ∂ 2 x A + i∂ t A)I t u = 0, ( 54
                         ∂ x u + e -i π 4 ∂ 1/2 t u = 0, for k = 1, ∂ x u + e -i π 4 ∂ 1/2 t u + α(u) 2 u = 0, for k = 2, ∂ x u + e -i π 4 ∂ 1/2 t u + α(u) 2 u + e i π 4 2 ( α(u) 2 4 + ∂ x α(u) 2 )I 1/2 t u = 0, for k = 3, ∂ x u + e -i π 4 ∂ 1/2 t u + α(u) 2 u + e i π 4 2 ( α(u) 2 4 + ∂ x α(u) 2 )I 1/2 t u + i 8 (α(u)∂ x α(u) + ∂ 2 x α(u) + i∂ t α(u))I t u = 0, for k = 4. ( 55
)
The set of above j-th order artificial boundary conditions will be called ABC α 1,j is the sequel, for j = 1, ..., 4.

Linearization strategy

We linearize equation ( 22) with β = 0 around a mean state u and call v its linearization. We obtain

i∂ t v + ∂ 2 x v + α(u)∂ x v + α ′ (u)∂ x uv = 0, ( 56
)
which is of the form [START_REF] Zheng | Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations[END_REF] with A = α(u) and B = α ′ (u)∂ x u. Equations (31) give

b 1/2 = - √ τ , b 0 = - α(u) 2 , b -1/2 = - 1 2 √ τ α(u) 2 4 - α ′ (u)∂ x u 2 , b -1 = - 1 8τ (α(u)∂ x α(u) -∂ 2 x α(u) + i∂ t α(u)). (57)
This yields absorbing boundary conditions for the linearized problem. To go back to the original problem, we now need to "unlinearize" these boundary conditions.

The first order absorbing boundary condition for v does not involve u and we immediately obtain for u

∂ x u + e -i π 4 ∂ 1/2 t u = 0, for k = 1. ( 58
)
The second order absorbing boundary condition for v reads

∂ x v + e -i π 4 ∂ 1/2 t v + 1 2 α(u)v = 0, ( 59
)
where α(u)v is the linearization of γ(u), where γ is the primitive of α vanishing at 0. Thus, the unlinearization of (59) is:

∂ x u + e -i π 4 ∂ 1/2 t u + 1 2 γ(u) = 0, for k = 2. ( 60
)
The unlinearization of the third and fourth order absorbing boundary conditions of v are far more challenging. We have to unlinearize:

∂ x v + e -i π 4 ∂ 1/2 t v + 1 2 α(u)v + e i π 4 α(u) 2 8 I 1/2 t (v) - e i π 4 α ′ (u)∂ x u 4 I 1/2 t (v) = 0, (61)
and

∂ x v + e -i π 4 ∂ 1/2 t v + 1 2 α(u)v + e i π 4 α(u) 2 8 I 1/2 t (v) - e i π 4 α ′ (u)∂ x u 4 I 1/2 t (v) + i(α(u)∂ x α(u) -∂ 2 x α(u) + i∂ t α(u)) 8 I t (62)
To achieve this goal, we rely on the paradifferential operators of J. M. Bony [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] which are generalization of pseudodifferential operators well-suited to nonlinear problems. We refer to [START_REF] Szeftel | Calcul pseudodifférentiel et paradifférentiel pour l'étude de conditions aux limites absorbantes et de propriétés qualitatives d'équations aux dérivées partielles non linéaires[END_REF] [20] for details about these operators and about the rigorous unlinearization of ( 61) and (62). We finally obtain the following nonlinear artificial boundary conditions of order k for u at [0; +∞[×{x r }:

                         ∂ x u + e -i π 4 ∂ 1/2 t u = 0, for k = 1, ∂ x u + e -i π 4 ∂ 1/2 t u + γ(u) 2 = 0, for k = 2, ∂ x u + e -i π 4 ∂ 1/2 t u + γ(u) 4 - e i π 4 4 I 1/2 t (α(u)∂ x u) = 0, for k = 3, ∂ x u + e -i π 4 ∂ 1/2 t u - e i π 4 2 I 1/2 t (α(u)∂ x u) + i 8 I t -α ′ (u)(∂ x u) 2 + α(u) 2 ∂ x u = 0, for k = 4, ( 63
)
where γ is the primitive of α vanishing at 0. From now, these j-th order artificial boundary conditions will be referred to as ABC α 2,j , for j = 1, ..., 4.

Remark 1. The unlinearization step of the linearization strategy has been successful not only in the case of ( 22) with β = 0, but also in the case of the semilinear wave equation with various nonlinearities (see [START_REF] Szeftel | A nonlinear approach to absorbing boundary conditions for the semilinear wave equation[END_REF]). However, the unlinearization step of the linearization strategy is not always successful, as shown by the case of the cubic nonlinear Schrödinger equation (see [START_REF] Szeftel | Absorbing boundary conditions for nonlinear Schrödinger equations[END_REF]).

Numerical examples

We consider the model Schrödinger equation [START_REF] Tsynkov | Numerical solution of problems on unbounded domains. A review[END_REF] for the two specific cases I (α = 0) and II (β = 0) of section 3. More specifically, we choose to present results in Case I for the nonlinearity β(u) = |u| 2 , which corresponds to the well-known one-dimensional cubic nonlinear Schrödinger equation, and in Case II for α(u) = u. We therefore will focus on systems (64) and (65) respectively given by

i∂ t u + ∂ 2 x u + |u| 2 u = 0, (t, x) ∈ [0; +∞[×R, u(0, x) = u 0 (x), x ∈ R, (64)
and

i∂ t u + ∂ 2 x u + u∂ x u = 0, (t, x) ∈ [0; +∞[×R, u(0, x) = u 0 (x), x ∈ R. (65)
In both cases, we have to our disposal explicit solutions. Concerning system (64), we consider the so-called soliton solution computed by using the inverse scattering theory and given by

u ex,α=0 (x, t) = √ 2a sech( √ a(x -ct)) exp(i c 2 (x -ct)) exp(i(a + c 2 4 )t). (66)
Concerning system (65), adapting the Cole-Hopf transform [START_REF] Szeftel | Absorbing boundary conditions for nonlinear Schrödinger equations[END_REF] , we have the explicit solution

u ex,β=0 (x, t) = 2 0 exp i (x -y) 2 4t u 0 (y) exp( y 0 u 0 (s) 2 ds)dy × √ iπt - x 0 exp(i y 2 4t )dy + 2 0 exp(i (x -y) 2 4t ) exp( y 0 u 0 (s) 2 ds)dy + exp( 2 0 u 0 (s) 2 ds)( √ iπt - 2-x 0 exp(i y 2 4t )dy) -1 , (67)
which has a compact support in [0, 2] at time t = 0. Durán and Sanz-Serna [START_REF] Durán | The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation[END_REF] and are given by

i u n+1 -u n δt + ∂ 2 x u n+1 + u n 2 + u n+1 + u n 2 2 u n+1 + u n 2 = 0 (68) and i u n+1 -u n δt + ∂ 2 x u n+1 + u n 2 + u n+1 + u n 2 ∂ x u n+1 + u n 2 = 0 (69)
respectively for Cases I and II. We denote here by u n the approximate value of u at time t n . In order to reduce the computational time, we set 2v n+1 = u n+1 + u n , and the schemes read for n ≥ 0

2i v n+1 -u n δt + ∂ 2 x v n+1 + |v n+1 | 2 v n+1 = 0, (70)
and

2i v n+1 -u n δt + ∂ 2 x v n+1 + v n+1 ∂ x v n+1 = 0, (71)
imposing in both cases that v 0 = 0. Clearly, a general form of the previous schemes is

2i v n+1 -u n δt + ∂ 2 x v n+1 + v n+1 f (v n+1 ) = 0, (72)
                         ∂ x u -e -i π 4 ∂ 1/2 t u = 0, for k = 1, ∂ x u -e -i π 4 ∂ 1/2 t u + α(u) 2 u = 0, for k = 2, ∂ x u -e -i π 4 ∂ 1/2 t u + α(u) 2 u - e i π 4 2 ( α(u) 2 4 + ∂ x α(u) 2 )I 1/2 t u = 0, for k = 3, ∂ x u -e -i π 4 ∂ 1/2 t u + α(u) 2 u - e i π 4 2 ( α(u) 2 4 + ∂ x α(u) 2 )I 1/2 t u + i 8 (α(u)∂ x α(u) + ∂ 2 x α(u) + i∂ t α(u))I t u = 0, for k = 4. ( 73
)
Concerning the linearization strategy, we get on [0; +∞[×{x l }:

                         ∂ x u -e -i π 4 ∂ 1/2 t u = 0, for k = 1, ∂ x u -e -i π 4 ∂ 1/2 t u + γ(u) 2 = 0, for k = 2, ∂ x u -e -i π 4 ∂ 1/2 t u + γ(u) 4 + e i π 4 4 I 1/2 t (α(u)∂ x u) = 0, for k = 3, ∂ x u -e -i π 4 ∂ 1/2 t u + e i π 4 2 I 1/2 t (α(u)∂ x u) + i 8 I t -α ′ (u)(∂ x u) 2 + α(u) 2 ∂ x u = 0, for k = 4, ( 74
)
where γ is the primitive of α vanishing at 0.

The boundary conditions are of memory-type and involve half-order fractional derivatives and integrals. To preserve the second-order approximation and the unconditional stability of the Crank-Nicolson schemes, the operators ∂ 1/2 t and I

1/2 t are approximated through quadrature rules which are well suited to the Crank-Nicolson schemes. Namely, we choose the quadrature formulas derived in [START_REF] Antoine | Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation[END_REF], which read for the sequence of complex values {f n } n∈N approximating {f (t n )} n∈N ,

I 1/2 t f (t n ) ≈ √ 2δt 2 n k=0 α k f n-k and ∂ 1/2 t f (t n ) ≈ 2 √ 2δt n k=0 β k f n-k , (75)
where (α k ) k∈N and (β k ) k∈N designate the sequences defined by

   (α 0 , α 1 , α 2 , α 3 , α 4 , α 5 , • • • ) = 1, 1, 1 2 , 1 2 , 1 • 3 2 • 4 , 1 • 3 2 • 4 , • • • , β k = (-1) k α k , ∀k ≥ 0.
The composition of the approximation of I 1/2 t with itself gives the approximation of I t by the trapezoidal rule, which is coherent with the underlying Crank Nicolson scheme. Using these quadratures formulas, the numerical versions of the boundary conditions (37), ( 48), ( 49), ( 55) and

(63) are discrete convolutions which may be represented by the following formulation

∂ n v n+1 + e -i π 4 2 δt v n+1 + g(v n+1 , v n , v n-1 , • • • , v 0 ) = 0,
where g is a function giving information on the construction of the approximation. For example, the approximation of ABC β 1,3 is given by

∂ n v n+1 + e -i π 4 2 δt n+1 k=0 β k v n+1-k - e i π 4 2 |v n+1 | √ 2δt 2 n+1 k=0 α k |v n+1-k |v n+1-k = 0.
The other approximations of ABC α,β j,k , j = 1, 2, k = 1, 2, 3 can be found in [START_REF] Antoine | Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations[END_REF] and [START_REF] Szeftel | Absorbing boundary conditions for nonlinear Schrödinger equations[END_REF]. The complete Crank Nicolson scheme with fixed point procedure therefore takes the form given in Table 1.

let w 0 = u n s = 0 while w s+1 -w s L 2 (Ω i ) > ε do solve the linear boundary-value problem Ω 2i δt w s+1 ψdx - Ω ∂ x w s+1 ∂ x ψdx - Γ e -i π 4 2 δt w s+1 ψdΓ = - Ω f (w s )w s ψdx + Ω 2i δt u n ψdx + Γ g s ψdΓ, setting g s = g(w s , v n , v n-1 , • • • , v 0 )
and ψ is one of the basis functions of the P1 finite element set. end while v n+1 = w s+1 u n+1 = 2v n+1 -u n Table 1: Fixed-point algorithm for solving the nonlinear Schrödinger equation with nonlinear ABC. We present below some numerical experiments to show the effectiveness of the different boundary conditions. Since we have an exact solution in Cases I and II, we choose to evaluate the schemes on the solutions with initial data (66) and (67) respectively. Concerning Case I, the computational domain is limited to the open set (-10, 10) discretized with 4000 points. The time step is fixed to δt = 10 -3 . In Case II, the finite domain is (0, 2) discretized with 2000 points. The time step δt is equal to 2 • 10 -3 . To analyze the accuracy of the different boundary conditions, we compute the relative error for the L 2 (Ω)-norm u ex -u num 0,Ω (t) u ex 0,Ω (t = 0) , where u num denotes the numerical solution.

For case I, we compare in Figure 3 the ABCs obtained with the potential strategy and the gauge change strategy for various orders. We can see that increasing the order improves the accuracy.

Moreover, the gauge change strategy (ABC β 2,k ) provides better accuracy compared to the potential strategy (ABC β 1,k ). Finally, let us also notice that the long-time behaviour of the various ABCs seems correct. To analyze the accuracy behaviour of the computed solution with respect to the velocity parameter, we plot on Fig. 4 the evolution of the relative error with respect to c for the most accurate ABC: ABC β 2,2 . We see that the relative error increases with lower velocities. This is in agreement with the theory developed in the paper since the boundary conditions are constructed under a high-frequency assumption. In our last experiment, we focus on case II. We compare in Figure 5 the ABCs obtained with the potential strategy and the linearization strategy for various orders. Generally, the linearization strategy leads to the most accurate solutions. Increasing the order of the ABC improves the accuracy for small times (t ≤ 2.5 in the experiments). After this time, the relative errors cross and the best results are obtained for ABC α 2,2 . The potential strategy is accurate and competitive for ABC α 1,3 but only for sufficiently large computational times. This shows that each strategy has his own strengths and weaknesses. Finally, let us mention that ABC α 2,2 has been shown to give optimal results within a large class of ABCs (see [START_REF] Szeftel | Absorbing boundary conditions for nonlinear scalar partial differential equations[END_REF]).

Conclusion

We presented an analysis of the construction and some numerical validations of ABCs for nonlinear PDEs considering the example of the nonlinear Schrödinger equation. The methods are mainly based on pseudo-and paradifferential operator techniques. We show that each strategy can lead to powerful solutions. However, much work remains to be done. In particular, developing rigorous extensions for higher dimensions, coupled problems and systems is a complete open problem.

Figure 1 :

 1 Figure 1: (∂ 2 t + ∂ t -∂ 2 x )u = 0. Relative error in L 2 norm in function of time. abc --order 1, -order 2 and order 3 • -•.

  ) at [0; +∞[×{x r }. Again, replacing A by α(u) yields the nonlinear artificial boundary conditions of order k

  (a) Case I, a = 2, c = 15, θ = 0 (b) Case II

Figure 2 :

 2 Figure 2: Exact solutions representations.

  where f designates the map | • | 2 or ∂ x according to the equation. The Crank-Nicolson approximation must be coupled to the boundary conditions (37), (48), (49), (55) and (63). Since the Jacobian of the maps associated to these nonlinear problems is difficult to obtain, we choose to use a classical fixed-point method based on the semi-discrete Crank-Nicolson schemes. The choice of a variational approximation method is thus obvious. Here, we specifically use a P 1 linear Lagrange finite element approximation. The bounded computational domain is the open set Ω =]x l , x r [. The fictitious boundary is limited to the two endpoints Γ = {x l , x r }. At this point, let us note that the boundary conditions for the case β = 0 have been given explicitly only at the right endpoint.The left boundary conditions differ. Concerning the potential strategy, the system of equations (55) is transformed on [0; +∞[×{x l } as:

Figure 3 :

 3 Figure 3: Evolution of the relative error for Case I, a = 2, c = 15, θ = 0.

Figure 4 :

 4 Figure 4: Evolution of the relative error for the simulation of the soliton solution with respect to the velocity c for ABC β 2,2 .

Figure 5 :

 5 Figure 5: Relative error for Case II.

3

  Different approaches for nonlinear equations: the case of nonlinear Schrödinger equations3.1 Nonlinear and linear Schrödinger equationsWe have seen in Section 2 that it is possible to build accurate artificial boundary conditions using techniques based on pseudodifferential calculus in the model case of the linear wave equation with variable coefficients. The aim of the present Section is to develop some applications of the pseudodifferential calculus method and its nonlinear version, the paradifferential calculus strategy, to obtain accurate artificial boundary conditions for nonlinear equations. As a model equation, we consider the time dependent nonlinear Schrödinger equation

The last author is partially supported by NSF Grant DMS-0504720.