Jean-Bernard Chatelain
email: jean-bernard.chatelain@univ-paris1.fr

Kirsten Ralf
email: kirsten.ralf@esce.fr.

Ramsey Optimal Policy versus Multiple Equilibria with Fiscal and Monetary Interactions

Keywords: Ramsey optimal policy, Interest Rate Rule, Fiscal Rule JEL classi…cation numbers: E63, C61, C62, E43, E44, E47, E52, E58 Frictionless endowment economy, Fiscal theory of the Price Level, Ramsey optimal policy, Interest Rate Rule, Fiscal Rule

. Under this formulation there exists a unique Ramsey equilibrium, with an interest rate peg and a passive …scal policy.

Introduction

Monetary and …scal interactions are often presented using [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and …scal policies[END_REF] model of frictionless endowment economies. This model includes the intertemporal budget constraint of the government and a Fisher relation with a constant real interest rate equal to the representative household's discount factor. In [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and …scal policies[END_REF], for monetary policy, the interest rate responds in proportion to in ‡ation. For …scal policy, a lump-sum tax responds in proportion to the stock of real public debt.

Assuming in ‡ation, interest rate and lump-sum tax are forward-looking variables for seeking equilibria with [START_REF] Blanchard | The solution of linear di¤erence models under rational expectations[END_REF] determinacy condition, [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and …scal policies[END_REF] obtains two equilibria for monetary and …scal interactions. These two equilibria are de…ned by ranges of values of the parameters of ad hoc policy rules. In the …rst equilibrium, the interest rate rule parameter destabilizes in ‡ation ("active monetary policy" according to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and …scal policies[END_REF]) and the …scal rule parameter stabilizes public debt ("passive …scal policy" according to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and …scal policies[END_REF]). In the second equilibrium related to the …scal theory of the price level, the …scal rule parameter destabilizes public debt ("active policy" according to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and …scal policies[END_REF]) and the interest rate rule parameter stabilizes in ‡ation ("passive monetary policy" according to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and …scal policies[END_REF]). A peg of the interest rate with a lack of response of the interest rate to in ‡ation is a particular case of a "passive monetary policy" equilibrium.

This note demonstrates that Ramsey optimal policy is a third equilibrium with an optimal interest rate peg and "passive" …scal policy.

Section two presents the policy transmission mechanism, the two equilibria with ad hoc policy rules and the Ramsey optimal policy equilibrium. Section three concludes. Consider an in…nitely-lived representative consumer who receives a constant endowment of goods each period in the amount y and derives utility only from consumption c t . The government purchases a constant quantity of goods g > 0 from the consumer each period. We impose equilibrium in the goods market, so that consumption: c t = y g.

The consumer makes a consumption-saving decision that produces the Fisher relation where the real rate is here equal to a constant discount rate:

E t 1 t+1 = 1 1 R t (1)
where R t is both the gross one-period nominal interest rate on nominal bonds bought at t and pay o¤ in t + 1 and the monetary policy instrument, and t+1 = P t+1 =P t is the gross rate of in ‡ation between t and t + 1, with P t the aggregate price level. In the steady state, P t = P t+1 ) = 1 where is the in ‡ation target. Then, R = = = 1= , is the nominal interest rate consistent with the in ‡ation target according to the Fisher relation. The equilibrium real interest rate is constant at r = (1= ) 1 where 0 < < 1 is the consumer's discount factor. The Fisher relation in deviation of steady state values is:

E t 1 t+1 1 = 1 1 R t 1 R ) E t 1 t+1 1 = 1 1 R t . ( 2
)
It is linearized around the steady state equilibrium:

E t t+1 = (R t R ) ) E t t+1 1 = R t 1 :
Fiscal policy levies lump-sum taxes of t and sets purchases to be constant, g > 0 with primary surplus s t = t g. Government issues one-period nominal bonds, B t , that satisfy the ‡ow constraint, where P t is the aggregate price level and real debt is de…ned as b

t = B t =P t . b t = B t P t = ( t g) + R t 1 P t 1 P t B t 1 P t 1 = ( t g) + R t 1 t b t 1 :
Using the Fisher relation, we substitute the constant real interest rate in the government intertemporal budget constraint:

b t = ( t g) + 1 b t 1 :
The dynamics of real debt does not depend on in ‡ation or on the nominal rate R t . The steady state level of real government debt has an exogenous value: b t+1 = b t = b . To be consistent with the government intertemporal budget constraint, the steady state level of tax revenue is equal to the steady state interest expense:

g = 1 1 b :
The government intertemporal budget constraint written in deviation of steady state is

b t b = (s t s ) + 1 (b t 1 b ):
The linearized dynamics are (in deviation from steady state):

E t t+1 E t b t+1 b = A = 0 A b = 0 A b = 0 A b = 1 | {z } =A t b t b + B R = B s = 0 B bR = 0 B s = 1 | {z } =B R t R s t+1 s :
The log-linearized dynamics of in ‡ation is:

E t t+1 = R R t R R = R t R R :
The log-linearized dynamics of real debt is:

E t b t+1 b b = 1 b t b b s b s t+1 s s with g b = 1 1:
Log-linearized dynamics are:

Et t+1 Etb t+1 b b = A = 0 A b = 0 A b = 0 A b = 1 | {z } =A t bt b b + 1 0 0 1 1 ! | {z } =B log Rt R R s t+1 s s

Ad Hoc Policy Rules

The …scal authority adjusts lump-sum tax in response to the level of real government debt. Monetary policy follows an interest rate rule that responds to in ‡ation:

s t s = G b (b t 1 b ) + " s t and R t R = F ( t ) + " R t
Shocks " R t , " s t are assumed to be independently and identically distributed, with mean zero and a non-zero variance-covariance matrix. The linearized dynamics are:

E t 1 t+1 1 E t 1 b t+1 b = 1 F 0 0 1 G b t b t b + 0 0 1 " R t " s t with b 0 given.
Public debt is the only predetermined variable. Blanchard and Kahn's (1980) determinacy condition implies that one eigenvalue should be inside the unit circle and one should be outside the unit circle. Either the in ‡ation eigenvalue is outside the unit circle j F j > 1 and the public debt eigenvalue 1 G b < 1 is inside the unit circle (…rst equilibrium) or it is the reverse (second equilibrium): j F j < 1 and 1 G b > 1 (Leeper (1991)).

Ramsey Optimal Policy under Quasi-Commitment

In a monetary policy regime indexed by j, a policy maker may re-optimize on each future period with exogenous probability 1 q strictly below one. Following [START_REF] Schaumburg | An investigation of the gains from commitment in monetary policy[END_REF], we assume that the mandate to minimize the loss function is delegated to a sequence of policy makers with a commitment of random duration. The degree of credibility is modelled as if it is a change of policy-maker with a given probability of reneging commitment and re-optimizing optimal plans. The length of their tenure or "regime" depends on a sequence of exogenous independently and identically distributed Bernoulli signals f t g t 0 with E t [ t ] t 0 = 1 q, with 0 < q 1. If t = 1; a new policy maker takes o¢ ce at the beginning of time t. Otherwise, the incumbent stays on. A higher probability q can be interpreted as a higher credibility. A policy maker with little credibility does not give a large weight on future welfare losses. The policy maker j solves the following problem for regime j, omitting subscript j, before policy maker k starts:

V j 0 = E 0 t=+1 X t=0 ( q) t 1 2 Q ( t ) 2 + Q b (b t b ) 2 + R (R t R ) 2 + s (s t s ) + (1 q) V k t R t R = q (E t t+1 ) + (1 q) E t k t+1 b t b = q(E t b t+1 b ) + (1 q) E t b k t+1 b + q(s t s ) + (1 q) (s k t s ).
In ‡ation and public debt expectations are an average between two terms. The …rst term, with weight q is the in ‡ation and public debt, respectively, that would prevail under the current regime upon which there is commitment. The second term with weight 1 q is the in ‡ation and public debt, respectively, that would be implemented under the alternative regime by policy maker k.

This optimal program is a discounted linear quadratic regulator (LQR) with a "credibility adjusted" discount factor q. The log-linear version of the dynamic system can also be used instead of the linear version. Preferences of the policy maker are …rstly given by positive weights for the two policy targets Q

0, Q 0 (Q = diag(Q ; Q b )).
Secondly, there are at least two non-zero policy maker's preferences for interest rate smoothing and primary surplus and tax smoothing, with strictly positive weights for these two policy instruments in the loss function: R > 0, s > 0 (R = diag( R ; s )). This ensures the strict concavity of the LQR program which implies the determinacy (uniqueness) of the solution of this Stackelberg dynamic game, if the system of the transmission mechanism is controllable.

All the results are valid for nearly negligible cost of changing the policy instruments: a minimal interest rate smoothing parameter ( R 10 7 ), a minimal tax smoothing parameter ( s 10 7 ) and a minimal credibility (a non zero probability of not reneging commitment next period: 10 7 q 1).

If R = 0 or s = 0 or if the costs of changing the policy instruments are not strictly convex or if q = 0, the results are no longer valid. If q = 0, the policy maker knows that he is replaced next period. He does static optimization (( q) 0 = 1) of his current period quadratic loss function subject to a static transmission mechanism where he considers the expectations terms related to the next period policy maker as exogenous intercepts (Chatelain and Ralf (2020)).

Proposition 1 For the transmission mechanism of the Fisher relation with constant real rate and the government intertemporal budget constraint, Ramsey optimal policy has a unique equilibrium. The interest rate is pegged at its long run value R t = R . A feedback Taylor rule is not optimal and the Taylor principle should not be satis…ed: F = 0 < . This is a "passive monetary policy". The optimal auto-correlation of monetary policy shocks is zero: " R = 0. The optimal variance of monetary policy shocks is zero 2 " R = 0. In ‡ation jumps to its steady state value instantaneously following monetary policy shocks 0 = 0, as in a degenerate rational expectations model without predetermined variables. Hence, the price level is constant. Ramsey optimal …scal rule has a negative feedback parameter ("passive …scal policy") with ensures the local stability of public debt dynamics. There are two stable eigenvalues giving the optimal persistence of in ‡ation ( = 0) and the optimal persistence of public debt (0 < b < 1).

Proof. Ramsey optimal policy amounts to …nd the solution of a linear quadratic regulator [START_REF] Chatelain | A Simple Algorithm for Solving Ramsey Optimal Policy with Exogenous Forcing Variables[END_REF]). The autocorrelation parameters of policy maker's shocks are optimally set to zero, else they increase the volatility of in ‡ation and of public debt in the policy maker's loss function. The optimal expected value of the loss function is:

L = 1 2 b 0 b b 0 P b 0 b b 0 T with P = P P b P b P b , (3)
b 0 = 0 and b b 0 = b 0 b . ( 4
)
where P is a positive symmetric square matrix of dimension two which is the solution of the discrete algebraic Riccati equation (DARE) of the LQR. The optimal initial anchor of in ‡ation 0 on public debt b 0 is given by:

@L @ t t=0 = P b 0 + P b b b 0 = 0 ) b 0 = P b P b b 0 if P 6 = 0: (5)
This initial transversality (or natural boundary) condition eliminates the indeterminacy of initial in ‡ation 0 put forward for the ad hoc policy rules solution. Because the system is decoupled (A b = A b = 0) and because the weight on the product t b t is zero in the loss function Q b = 0, this implies a zero weight P b = 0 in the optimal value of the loss function. Therefore, P 0 and P b 0 are solutions of scalar discrete algebraic Riccati equations (DARE). Because the system of the policy transmission mechanism is decoupled, the optimal program is identical if we consolidate the central bank and the treasury as a single policy maker or if we consider that they are distinct policy makers. For in ‡ation, this DARE equation is:

P = Q + A 0 P A 0 A 0 P B R ( R + B 0 R P B R ) 1 B 0 R P A = Q : Because A b = A b = A = 0,
this implies that the optimal loss function parameter P is equal to the in ‡ation weight in the loss function: P = Q . Therefore, if the policy maker has a non-zero weight on in ‡ation volatility in his loss function Q > 0, optimal initial in ‡ation is zero because P b = 0: 0 = 0. Because in ‡ation dynamics are decoupled from the dynamics of predetermined public debt, in ‡ation behaves exactly as in a degenerate rational equilibrium model where there is no predetermined variable. At any date t, if ever there is a monetary policy shock " R t , in ‡ation instantaneously jumps back to equilibrium t = 0. There are no transitory dynamics. The volatility of in ‡ation is zero.

Because the open-loop system is already at the zero lower bound of in ‡ation persistence (A = 0) and because there is a non-zero cost of interest rate volatility ( R > 0) in the loss function, it is optimal to set an interest rate peg R t = R . Therefore, the quadratic term of interest rate volatility R (R t R ) 2 is minimized at its zero lower bound zero in the loss function. This implies a Taylor rule parameter equal to zero: F = 0 < , a zero auto-correlation of monetary policy shocks " R = 0, and a zero variance of monetary policy shocks 2 " R = 0 initiated by the policy maker. If there is a zero cost of in ‡ation in the loss function (Q = 0), the policy maker does not care about in ‡ation. There is an indeterminacy for the optimal choice of the initial value of in ‡ation 0 . However, the implied volatility originated by this indeterminacy does not matter in the policy maker's loss function.

For public debt, the solution is a scalar case of Ramsey optimal policy under quasicommitment (we use the linear dynamics instead of the log-linear dynamics).

E t b t+1 b = 1 p q (b t b ) p q(s t s ) with A b = 1 q or qA = 1 and B bs = 1:
Optimal public debt persistence (or auto-correlation or closed-loop eigenvalue) is the stable root of the characteristic polynomial of the Hamiltonian system:

0 = 2 S + 1 q with 2 = 1 q with S = A + 1 A q + B 2 A Q b s = 1 + 1 q + q Q b s 0 < b ( s ; q) = 1 2 S r S 2 4 q < 1:
The optimal public debt persistence or autocorrelation b increases with the cost s of changing the policy instrument (lump sum taxes). Its boundaries are given by limits when the cost of changing policy instrument s tends to zero and when tends s to in…nity:

lim s!0 b ( s ; q) = lim b !0 1 2 q s q s = 0 and lim s!+1 b ( s ; q) = 1 qA = 1:
The …scal rule parameter G b remains in the range of values so that the persistence of public debt is strictly positive and strictly below one:

1 q 1 < G b = b A B = 1 q b < 1 q .
The optimal loss function parameter P b is:

P b = 1 A Q b q 1 b = Q b 1 b .
It is also the solution of a scalar algebraic Riccati quadratic equation. For a given initial value of predetermined public debt b 0 and for preferences Q b 0 which can be equal to zero, the optimal expected value of the policy maker loss function is:

If Q > 0 : L = 1 2 0 b b 0 Q 0 0 Q b 1 b ! 0 b b 0 T = 1 2 Q b 1 b (b 0 b ) 2 : If Q = 0 : L = 1 2 b 0 b b 0 0 0 0 Q b 1 b ! b 0 b b 0 T = 1 2 Q b 1 b (b 0 b ) 2 :
A rational policy maker with quadratic preferences including a convex cost of changing policy instruments (interest rate smoothing R > 0 and tax smoothing s > 0) taking into account private sector expectations with a minimal credibility (q > 0) eliminates the indeterminacy of initial in ‡ation 0 if the policy maker's preferences includes a non-zero weight (Q > 0) on in ‡ation volatility in his loss function. Else, a rational policy maker neglects the indeterminacy of initial in ‡ation because in ‡ation volatility has zero weight (Q = 0) in his loss function. A SCILAB code is available from the authors to solve Ramsey optimal policy with numerical values.

Conclusion

For the model of frictionless endowment economies, Ramsey optimal policy is an interest rate peg and a "passive" …scal policy. It is a third equilibrium with respect to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and …scal policies[END_REF] two equilibria with simple feedback rules.


domain: spi phys

text:
B Alzaix

L Giraud

B L Michielsen

J.-R Poirier

A Plane Wave Scattering Dedicated Integral Equation

Keywords: Plane Wave Scattering, Boundary Integral Equations, Model Reduction

We present a variant of a boundary integral equation proposed by M. Herberthson in 2008-2010 for high-frequency plane wave scattering by perfectly conducting obstacles. This integral equation, as some of its predecessors, is entirely dedicated to a single direction of incidence. A significant reduction of the system size can then be obtained by solving for a pseudocurrent which is the actual surface current multiplied by the conjugate of the incident wave's phase function. This advantage is mostly lost when multi-incidence problems have to be solved because, in addition to the source terms, also the discretized operator must be recomputed for every new incidence. In this paper, we make an explicit representation of the new operator as the sum of a conventional Electric Field Integral Equation (EFIE) which is independent of the direction of incidence, and an additional operator which takes into account the dependency on the direction of incidence. The latter can be computed very efficiently such that a net performance gain is obtained with respect to the EFIE even for multi-incidence problems.

I. INTRODUCTION

T HE scattering of a plane wave by an impenetrable obstacle can be studied numerically using universal methods like boundary integral equations (Electric Field Integral Equation (EFIE) or Magnetic Field Integral Equation (MFIE)). In this paper, we follow a different strategy in which the computational technique is completely dedicated to the scattering of a given incident plane wave. This has the disadvantage that if the plane wave changes, one needs to reconstruct the model. However, it has the possible advantage that the dedicated model leads to more efficient computations. Such "goal-oriented" approaches to plane wave scattering have been developed before, in particular, with high-frequency scattering problems in mind. The work of Abboud, Nédélec and Zhou (see [START_REF] Zhou | Méthode des équations intégrales pour la résolution des problèmes de diffraction à hautes fréquences[END_REF], [START_REF] Abboud | Méthodes des équations intégrales pour les hautes fréquences[END_REF], [START_REF] Abboud | Improvement of the integral equation method for high frequency problems[END_REF]) published in 1995, is one of the first boundary integral equations where a physical optics approximation is combined with the EFIE. These authors modified the edge element discretization by multiplying the edge elements with the phase function of the incident wave. In a certain way, this makes solving the EFIE to compute a correction with respect to the physical optics approximation. Later work by Darrigrand (see [START_REF] Darrigrand | Couplage Méthodes Multipôles -Discrétisation Microlocale pour les Equations Intégrales de l'Electromagnetisme[END_REF]) extended this approach to an alternative integral equation method proposed by B. Després. In these earlier publications, error estimates, based on asymptotic expansions B. Alzaix is with Naval GROUP, Ollioules, France L. Giraud is with INRIA, 200 Avenue de la vieille tour, 33405 Talence, France.

B.L. Michielsen is with Onera/DEMR, Université de Toulouse, 31055 Toulouse France, Bastiaan.Michielsen@Onera.fr J.-R. Poirier is with LAPLACE, Université de Toulouse, CNRS, Toulouse, France of the solution in the transition to the shadow region, showed that, for convex obstacles, the phase regularization method is warranted to be efficient at high frequencies because the required mesh size decreases in the order of only λ 1/3 . In 2008, Herberthson followed a different track with the same purpose of improving the EFIE performance at high frequencies (see [START_REF] Herberthson | Validation of the Potential Method ; Comparing measurements of a dihedral with calculations[END_REF], [START_REF] Herberthson | EM Scattering Calculations Using Potentials[END_REF], [START_REF] Herberthson | Application of the Potential Method to the Combined Field Integral Equation[END_REF]). In his formulation, the incident wave's phase function was incorporated in the kernel distribution of the integral equation, thereby obtaining an integral equation for a pseudo-current defined as the product of the complex conjugate of the incident plane wave's phase function and the physical current distribution. He observed that multiplying both sides of the integral equation with this conjugate phase function yields an exact gradient in the righthand side. He exploited this further to reduce the conventional EFIE to a system of coupled integral equations for two scalar potentials. Where the previous method was applicable to convex scatterers, Herberthson's version is based on specific topologies (with genus equal to 0) such that the Helmholtz decomposition holds. Roughly speaking, the two points of view lead to increased performance for nearly the same type of geometries.

In this paper, we study a variant of Herberthson's version of the EFIE in order to overcome the restriction to obstacles with special topology. As with the previously mentioned work, the integral equation is specific for plane wave scattering with one chosen direction of incidence. However, representing the operator as an additive perturbation of the conventional one, the perturbation part, which is the only part depending on the phase function of the incident plane wave, can be computed very efficiently.

This splitting allows for the design of a new Galerkin system relaxing the usual constraint of the EFIE with regards to the mesh size and offers the possibility to reduce the number of degrees of freedom required to get an accurate solution of the problem. Considering the conventional equation, and the diffraction of a plane wave with a frequency f 0 , a good approximation of the surface current requires the edges of the mesh to have a maximum size h, smaller than λ 0 /7. Working with the pseudo-current and this perturbation splitting, it will be shown that we can effectively work mesh sizes of the order of λ 0 /4.

When considering several directions of incidence, the recomputation of only the perturbation operator leads to a rather low computational overhead.

II. HERBERTHSON'S WORK AND A NEW FORMULATION

In this section, we present Herberthson's modified version for the EFIE. Most of the development can be found in [START_REF] Herberthson | Validation of the Potential Method ; Comparing measurements of a dihedral with calculations[END_REF], [START_REF] Herberthson | EM Scattering Calculations Using Potentials[END_REF], [START_REF] Herberthson | Application of the Potential Method to the Combined Field Integral Equation[END_REF]. To set the notations, we assume that an incoming plane wave, with radian frequency ω, illuminates a surface Γ.

The plane wave electric field is given in Cartesian coordinates, r ∈ R 3 , by E inc (r) = eψ(r) where ψ(r) = e -jku•r is the phase function of the wave with u the unit vector in the direction of propagation and k = ω c the wavenumber (we write r for the vector representation of the coordinates r). The polarization given by e is transverse to the propagation direction, u • e = 0. The associated magnetic field is given by H

inc (r) = Y 0 u × eψ(r) (Y 0 = 1 µ 0 c
). We let J denote the surface current density. With these notations, the EFIE reads

∀r ∈ Γ tr (ζ I - 1 η ∇∇•)A J (r) = tr(eψ)(r), (1)
where ζ = jω µ 0 , η = jωε 0 and for a vector-valued function X on R 3 , we note tr(X)(r) ≡ (n(r) × X(r)) × n(r) the tangential component of X in r ∈ Γ. The vector potential, A J , is defined by

A J (r) = r ∈Γ G(r, r )J (r )A (r ).
Here G(r, r ) is the Green function of the Helmholtz operator defined by G(r, r ) = e -jk r-r 4π rr , with rr the Euclidian distance function. Finally, in the surface integral, A (r ) denotes the area element on Γ. In the rest of this paper, we shall leave these area elements implicit in the definition of the integral over Γ and use Γ f ≡ Γ f A = r∈Γ f (r)A (r). Now, Herberthson observed that, modulo the phase function ψ, the right-hand side of (1) is the gradient of a scalar function, e = ∇(e•r). By multiplying both sides of (1) with the complex conjugate of the phase function of the incident plane wave, ψ, he obtained the following equation,

tr ψ(ζ I - 1 η ∇∇•) A J (r) = -e| T r Γ , (2)
with

A J (r) = r ∈Γ G(r, r )ψ(r ) J (r )A (r )
for a new unknown, J (r ) = ψ(r )J (r ), which we refer to as the pseudo-current.

The second element introduced by Herberthson, is the Helmholtz decomposition. For the special case that Γ is a closed surface diffeomorphic to a sphere, every vector field J on Γ can be decomposed as the sum of a surface gradient, ∇ s , and a co-gradient,

J = ∇ s Φ + n × ∇ s Ψ, (3)
where Φ and Ψ are two scalar functions. In this way, the integral equation ( 2) can be further transformed into a system of two coupled equations for the scalar potentials Φ and Ψ. In the Galerkin discretization of this system of equations a natural 2 × 2 block structure appears according to the two potentials and in the right-hand side the terms corresponding to Φ have the value zero.

At this point, our development differs from the one chosen by Herberthson. We remain with the pseudo current vector distribution as the fundamental unknown on an arbitrary surface Γ. The resulting integral equation can, of course, also be seen as a system of coupled equations for the two components of the vector distribution. In that formulation, we can use a special case of the Hodge decomposition which splits off ker(div). This also gives zero values in the right-hand side corresponding to the testing with ker(div) functions (with the usual edge finite elements for surface current distributions one can identify a conformal subspace of divergence free functions). Because we have, as yet, not obtained any numerical advantages of using such a decomposition, this aspect will not be further developed here.

III. ALGEBRAIC PROPERTIES A. Algebraic equivalence between conventional and modified equations

The modified integral equation ( 2) and the conventional EFIE [START_REF] Zhou | Méthode des équations intégrales pour la résolution des problèmes de diffraction à hautes fréquences[END_REF], are actually algebraically equivalent. To show this, we give a description of the two operators E and H which define the two equations:

E : H -1/2 div (Γ) -→ H -1/2 curl (Γ) u -→ tr (ζ I - 1 η ∇∇•)A u H : H -1/2 div (Γ) -→ H -1/2 curl (Γ), v -→ tr ψ(ζ I - 1 η ∇∇•) A v ,
where H

-1/2 div (Γ) and H -1/2
curl (Γ) are two spaces of distributions on a surface Γ, which can be a piecewise smooth surface such as a triangle mesh. These are, in fact, the spaces of boundary limits and surface restrictions of locally squareintegrable vector fields, of which also the curl is locally square integrable. This is precisely the kind of fields which we need for numerical computations with frequency domain electromagnetic fields with locally finite energy.

With these notations, the conventional EFIE reads

E u = -tr(E inc ),
whereas the new formulation can be written as:

H v = -tr( ψE inc ). (4)
Then, if we introduce the operator Ψ to denote the multiplication by the Γ-trace of the analytic phase function and Ψ -1 its conjugate and inverse, the system (4) can easily be re-written as:

H v = [Ψ -1 • E • Ψ]v = -Ψ -1 tr(E inc ). ( 5
)
Because ψ is a nowhere vanishing C ∞ function, the operator Ψ is a homeomorphism in H s (Γ) for any s. For the domain of div or the domain of curl, we have the following. For any

u ∈ H -1/2 div (Γ), div(ψu) = ∇ • (ψu) = (∇ψ) • u + ψ(∇ • u) is also in H -1/2 div (Γ), because (∇ψ)
• u is a sum of H -1/2 coefficients multiplied by C ∞ functions, and ψ(∇ • u) is the product of a C ∞ function and a H -1/2 function, by the definition of u. Similarly, for any u ∈ H

-1/2 curl (Γ), curl(ψu) = ∇ • (ψu × n) = (∇ψ) • (u × n) + ψ∇ • (u × n),
where, again, the first term of the right-hand side is a linear combination of components of u × n which are in H -1/2 (Γ) whereas the second term on the right-hand side is in H -1/2 (Γ) by definition of u and the regularity of ψ.

We conclude that the conventional EFIE and the modified EFIE are algebraically equivalent. This means that if the EFIE has a unique solution the modified EFIE also has a unique solution and vice versa. We know that on closed surfaces at discrete frequencies of internal resonance, nonradiating surface currents of undetermined amplitude destroy the uniqueness of the EFIE solution. The algebraic equivalence means that the modified integral EFIE on such closed surfaces has the same non-uniqueness problem at the same set of discrete frequencies. The way these uniqueness problems do actually appear in the discretized integral equation depends on the specific implementation and on the way the resulting linear systems are solved approximately.

IV. GALERKIN DISCRETIZATION OF THE EQUATIONS

Approximate solutions for the integral equations can be obtained by a Galerkin finite element discretization of the variational formulation.

A. Discretization of the conventional equations

The variational formulation for the conventional EFIE is:

Find J such that for all J the following equality is satisfied:

ζ a E (J , J ) + 1 η φ E (J , J ) = -J , e ,
where , is the standard distribution evaluation and the bilinear forms in the left-hand side are defined by

a E (J , J ) = (r,r )∈Γ 2 G(r, r )J (r) • J (r )
and

φ E (J , J ) = (r,r )∈Γ 2 G(r, r )(∇ • J )(r)(∇ • J )(r ),
respectively. A conformal finite element discretization follows by restricting J and J to a finite dimensional subspace of H -1/2 div (Γ), such as the usual edge element spaces on triangulations.

B. Discretization of the modified equations

The weak formulation of the modified EFIE:

Find J , such that for any tangent vector distribution J on Γ,

ζ a H (J , J ) + 1 η φ H (J , J ) = -J , e , (6)
where

a H (J , J ) = (r,r )∈Γ 2 H(r, r )J (r) • J (r ), φ H (J , J ) = (r,r )∈Γ 2 G(r, r )∇ • ( ψJ )(r)∇ • (ψJ )(r )
and where we introduced the kernel

H : R 3 × R 3 (r, r ) → H(r, r ) = ψ(r)G(r, r )ψ(r ).
C. The modified EFIE as a perturbation of the EFIE If we describe the operator H as the sum of the conventional EFIE operator, E , and a perturbation, K , defined by

K = H -E , (7)
we have isolated the part of H which depends on the plane wave's direction of incidence. This will appear to be an essential feature of the modified formulations.

For the corresponding bilinear forms a K = a H -a E and φ K = φ Hφ E of the perturbation, we obtain

a K (J , J ) = (r,r )∈Γ 2 H 1 (r, r )J (r) • J (r ), with H 1 (r, r ) = H(r, r ) -G(r, r ),
and

φ K (J , J ) = (r,r )∈Γ 2 G(r, r )(∇ • ( ψJ ))(r)(∇ • (ψJ ))(r ) -(∇ • J )(r)(∇ • J )(r ),
which we split into four terms,

φ K (J , J ) = (r,r )∈Γ 2 H 1 (r, r )(∇ • J )(r)(∇ • J )(r ) + k 2 (r,r )∈Γ 2 H(r, r )(u • J (r))(u • J (r )) + jk (r,r )∈Γ 2 H(r, r )(u • J (r))(∇ • J )(r ) -jk (r,r )∈Γ 2 H(r, r )(∇ • J )(r)(u • J (r )).
The integrals involved in a K and φ K are regular enough to be evaluated numerically since they present no real singularities. This regularity can be shown regarding the kernel's behavior for r → r . The smoothness of the phase function implies that the kernel H has the same regularity as the Green function of the Helmholtz operator. This means that the numerical evaluation of the bilinear forms in finite element discretizations can be done with the same techniques as used in the implementations of the conventional EFIE. The kernel H 1 is more regular than the Green function of the Helmholtz operator. Indeed, we have

H 1 (r, r ) ≈ -jku • v(r, r ) + O( r -r ) r → r , with v(r, r ) = (r -r ) r -r ,
which is a bounded function (albeit with an undefined value in r = r ). This implies that we can numerically compute the internal integrals of a K and φ K (with Gaussian integration) by applying a specific treatment to the kernel which accounts for this asymptotic behavior when r is close to r .

In the end, the computation of the perturbation only involves numerical integrations (with Gauss' methods, for example) and a few integrals of singular expressions that are also present in the EFIE and for which efficient implementations have already been found (see, for example, [8, Chapter 2, Section 2.2.1] for a summary of such techniques and further references).

We give in Table I an overview of the floating-point operation count of the computation of H regarding this choice of implementation.

V. REDUCTION OF THE NUMBER OF DEGREES OF FREEDOM

A. Ideas and motivations

In the previous section, we have presented, the formulation of the modified integral equation. However, we did not yet show how to take benefit from its potential advantages. With the modified integral equations, we are solving for a pseudocurrent J , which is less oscillating than the physical current J on large parts of the surface (see [START_REF] Zhou | Méthode des équations intégrales pour la résolution des problèmes de diffraction à hautes fréquences[END_REF]). Fig. 1 illustrates this, with the exact current and pseudo-currents, for the case of plane wave scattering by a perfectly conducting unit sphere.

Even though the pseudo current distribution can be accurately discretized on a rather coarse triangulation, the Galerkin coefficients require integration against a kernel function with possibly great oscillations. On the triangles of a coarse mesh this would require integration rules of much higher order than those usually associated with finite elements. As has already been demonstrated in [START_REF] Zhou | Méthode des équations intégrales pour la résolution des problèmes de diffraction à hautes fréquences[END_REF], this makes that the modified EFIE

z x y k ϑ E H 0.5 1 1.5 2 2.5 3 -6 -4 -2 0 2 4 6 •10 -3
Angle (rad)

J ϑ (Am -1 ) J ψJ Figure 1.
Comparing the oscillations in the physical surface current, J, and the pseudo-current, ψJ, induced on a unit sphere by a plane wave at 1GHz (k = ku). The curves represent, in standard spherical coordinates, the real parts of J ϑ and ψJ ϑ as a function of ϑ for ϕ ≡ 0.

suffers from the same constraint as the conventional EFIE. In the following paragraphs, we elaborate a technique to deal with this problem.

Figure 2. On the left, a fine mesh satisfying the constraints for the EFIE at a given frequency f = f 0 . On the right, coarse mesh on which we could expect the modified EFIE to give correct results at the same frequency.

B. Macro elements and reduced system

In order to cope with the computational problem observed above, we consider two meshes: a fine mesh satisfying, h < λ 0 /7 (see Fig. 2 on the left), well adapted to work with the EFIE at given frequency f 0 and allowing proper computations of the Galerkin coefficients, and a coarse mesh, on which we want to discretize the pseudo-currents in the modified equation and which does not necessarily comply with the constraint h < λ 0 /7 (see Fig. 2

on the right).

To avoid ambiguities in the interpretation of the comparisons, we consider the polyhedron of the coarse mesh to be the true geometry. In this way, the finite element spaces in which the modified integral equations are discretized are true subspaces of finite element spaces over a finer mesh. This contrasts to practical applications, where one should use the fine mesh to get a better geometric approximation to the actual boundary surface of the scattering object (For example, by slightly re-distributing the vertices on the actual surface). We shall from now on speak of macro edge elements on macro cells even though the two finite element spaces are more tightly related.

For practical reasons, we start with the coarse mesh and derive the fine mesh from it. For this, we divide each triangle of the coarse mesh into four small triangles by adding a new vertex on the middle of each edge. In this way, starting from a coarse mesh with V c vertices, E c edges and F c faces, we end up with a finer mesh with

V f = V c + E c vertices, E f = 2E c + 3F c
edges and F f = 4F c faces. For a closed surface where 3F = 2E, we also have E f = 4E c , such that the dimension of the edge element space on the fine mesh equals four times the dimension of edge element space on the coarse mesh. The given refinement divides the mesh size by two.

The set of edges of the fine mesh can be partitioned using the macro edges and macro triangles containing them,

E f = Γ∈E c E Γ f ∪ T ∈F c E T f
, where

E Γ f = {Γ -, Γ + }
the pair of micro edges arising from the subdivision of the macro edge Γ and

E T f = {γ T 0 , γ T 1 , γ T 2 }
. the triplet of micro edges in the interior of macro triangle T .

v 0 v 1 v 2 ∂ 0 T ∂ 1 T ∂ 2 T ∂ 0 T ∂ 1 T ∂ 2 T T v 0 v 1 v 2 γ T 0 γ T 1 γ T 2
Figure 3. Relation between edge basis functions defined on the coarse mesh and edge basis functions defined on the fine mesh.

The above relations allow us, to construct the Galerkin coefficients of a reduced system over the coarse mesh, using the computations as for the micro elements on the fine mesh, in an update algorithm. Let H RED be the Galerkin matrix over the coarse mesh and H FIN the Galerkin matrix over the fine mesh. As can be inferred from Fig. 3, each macro edge carries a degree of freedom (DoF) which is a linear combination of 8 micro DoF's. We can therefore express the matrix reduction as

H Γ ,Γ RED = ∑ γ ∈e Γ ∑ γ∈e Γ c Γ γ c Γ γ H γ ,γ FIN .
using e Γ for the set of 8 micro edges carrying the micro DoF's contributing to the macro DoF on Γ. The eight coefficients c Γ γ have the values 1 2 or ± 1 4 (see the appendix for the details). Because the kernel of the perturbation operator is more regular than the original kernel of the H operator, we propose an alternative discretization,

H RED,alt = E RED + K COA ,
where K COA is the Galerkin matrix of the perturbation operator K, computed directly in the macro element space, whereas E RED is the reduced EFIE Galerkin matrix defined in the same way as H RED . Again, the coefficients of H RED and E RED can be updated "on the fly" from the computation of the coefficients of H FIN and E FIN , respectively, which themselves do not need to be stored (except when we want to use the EFIE on the fine mesh for comparison purposes, of course).

VI. NUMERICAL RESULTS

In this section, we evaluate the modified EFIE with respect to the accuracy of the current distribution on a scattering obstacle and the accuracy of radar cross-section coefficients.

A. Accuracy of the current distribution

In this section, we can compare the results of a conventional EFIE solution on the fine mesh to the solution obtained via the modified EFIE on the coarse mesh. For that purpose, the pseudo-current distribution multiplied by the phase function of the incident wave is represented on the fine mesh. Taking the EFIE result as the reference the relative error is computed via L 2 norms on the fine mesh,

Error(J ) = J -J FIN L 2 J FIN L 2 .
1) The unit sphere: The first example concerns a perfectly conducting unit sphere. However, the coarse mesh is again used as the reference geometry and the new vertices of the fine mesh are not displaced to lie on the sphere. In Fig. 4, we show Error(J ) as a function of frequency where 0.2 GHz is the frequency where the mesh size of the coarse mesh equals λ /10. We observe that H RED and H RED,alt are very close with a slowly increasing error up to 0.4 GHz (λ /5). Beyond 0.4 GHz, the error increases more rapidly. The error associated with the regular EFIE based on the coarse mesh is significantly larger and deteriorates rapidly already from 0.05 GHz. From these results, we conclude that H RED,alt offers the best trade-off between accuracy and computational cost. 2) A flat plate with added surface roughness: We further investigate the accuracy of the various methods on the example of a rectangular plate represented by (x, y, z) ∈ (0, L) × (0, L) × (-D, 0) ⊂ R 3 with L = 1 m, D = 0.1 m. In order to test the performance of the method on a non-convex geometry the flat plate was deformed into a non-convex rough plate using a height profile for the z = 0 face, given by

h(x, y) = H ((1 -cos(8πx/L)) (1 -cos(8πy/L)))
where the maximum deformation is H = 0.1 m (see Fig. 5). The relative errors in the surface current distributions on the 6. The observations made for the case of the sphere also apply to these examples. It can be seen that the error jump is significant for the regular EFIE for frequencies larger than 0.5 GHz (corresponding to a mesh size of the order of λ /6). The results obtained for the flat plate and the rough plate lead to observations similar to those for the sphere. Nevertheless we can note here that a stronger increase of the error is observed when the mesh size on the rough plate exceeds the value λ /10 (0.3 GHz). This may be due to the fact that, on this far-from-convex surface, the total variation of the pseudocurrent and true surface current are comparable on a large part of the surface.

3) A torus: The original method proposed by Herberthson consisted of representing the pseudo-current distribution as derived from two scalar potentials using a Helmholtz decomposition. This Helmholtz decomposition relies on the nonexistence of so-called harmonic surface current distributions, i.e., current distributions with zero Laplacian ∆J = 0. Whether harmonic current distributions exist or not depends on the topology of the surface and the torus is an example where harmonic current distributions do exist indeed. We consider a torus generated by a small circle with radius r = 0.5 m with its center sweeped along a large circle with radius R = 1.5 m. We computed the surface current distribution induced by a plane wave with propagation direction and electric polarization in the plane of the large circle. The frequency chosen is 0.2 GHz, this means that the circumference of the small circle equals π m, i.e. about two wave lengths, and the largest distance between points on the torus is 4 m, i.e, a bit more than two and a half wavelengths.

In Fig. 7, the induced current distribution is shown on the surface of a torus. It has been computed on a mesh with 3976 triangles using the conventional EFIE and H RED,alt . The error in the current distribution computed with the modified EFIE (relative to the one obtained with the conventional EFIE) is of Figure 7. Current distribution on a torus induced by a plane wave at frequency 0.2 GHz the same order as observed with the rough plate. Again, this is explained by the fact that non-convexity implies multiple scattering between opposite faces, which locally diminishes the correspondence between the current's phase and the phase of the incident wave.

B. Accuracy of bi-static Radar Cross Section coefficients

We further investigate the accuracy of results obtained with the modified EFIE, but this time by looking at bi-static Radar Cross Section computation, which amounts to evaluating the current distributions on plane waves. Note that such bistatic RCS computations are very cheap compared to the computation of the current distribution and we use the same computation. For the perfectly conducting unit sphere, we compare with the analytic solution computed with the Mie series. For the flat plate and the rough plate example, we will use the solution computed with the regular EFIE on the fine mesh as the reference. Because, it appeared as the most promising alternative in the previous section, we only report on experiments with the H RED,alt version.

1) The unit sphere: Fig. 8 and9 show a bi-static RCS coefficient computed, for various mesh sizes, with the EFIE and the modified EFIE, respectively. As we are no longer comparing different versions of the modified EFIE, we now place the vertices of the fine meshes on the sphere, which is what one would do in applications of the method.

In Fig. 8, it can be observed that a fairly accurate solution is obtained with the EFIE for a mesh size of λ /7. This is in good agreement with the classical rule of thumb λ /10. Similarly, in Fig. 9, we can see that a comparable solution can be obtained with the modified EFIE already for mesh sizes λ /3 or λ /4. These experiments illustrate that the new method allows us to relax the mesh size constraint while preserving the same level of accuracy.

2) A flat plate with added surface roughness: For the RCS of a plate with surface roughness, we come to the same conclusions as for the example of the sphere. We consider a plane wave, at 1 GHz, normally incident on the rough face of the plate and with electric polarization parallel to one of the edges of the plate. This time, we compare the bi-static RCS to results computed with the commercial code FEKO on a mesh Exact mesh λ /2 mesh λ /3 mesh λ /4 mesh λ /5

Figure 9. Representation of the bi-static RCS of the unit sphere obtained from the various meshes with the modified EFIE system using H RED,alt at f = 0.6 GHz.

satisfying the constraint h < λ /10 on the mesh size. It can be seen that the modified EFIE H RED,alt obtains the reference result with a rather coarse mesh when the conventional EFIE is significantly different from the actual values. The efficiency of the modified EFIE is even more pronounced for oblique incidence (see Fig. 11) 45 • off the normal in the same plane of incidence. (We use these results also as a validation of our implementation of the conventional EFIE.) 3) A torus: For the non-convex geometry of a torus, we compute the bi-static RCS graphs in the plane of incidence containing the center of the torus and orthogonal to the electric polarization lying in the plane of the large circle of the torus. Even though the current distribution obtained with the modified EFIE can have relative errors up to 10 percent with respect to the exact current distribution, the bi-static-RCS results are practically indistinguishable (see Fig. 12). This means that even for this type of non-convex objects an important reduction

C. Performance comparison

In this section, we compare the performances of the modified EFIE and the conventional EFIE with respect to memory and CPU time requirements, for a given desired accuracy. The example problems presented in the preceding sections are relatively small and have been solved with direct methods (numpy.linalg.solve). In order to investigate the advantages for larger problems, requiring iterative solution methods, we also solved the EFIE and modified EFIE problems with a GMRES (Generalized Minimal RESidual) iterative solution algorithm. We included the GMRES stopping criterion, ε GMRES , as a parameter in our tests.

The modified EFIE has clear advantages from a memory point of view because, as we have seen, it requires about 4 times less degrees of freedom than the conventional EFIE for a given accuracy. With a given amount of available memory, this allows one to handle objects of larger size relative to the wavelength than is possible with the EFIE. This is illustrated in Fig. 13, where one observes that for a given available memory, the modified EFIE can reach better precision than the conventional EFIE.

In Fig. 14, we show, for the EFIE and the modified EFIE, at a given required accuracy, the total execution time, i.e., the sum of the time needed to construct the linear system and the time to obtain the solution.

In both Fig. 13 and Fig. 14, we observe that the two methods have roughly the same performance. The total solution time with the modified EFIE depends less on the stopping criterion, ε GMRES , than with the conventional EFIE. This is because the system construction time in the examples dominates the total execution time. The fact that for better accuracies the EFIE results are absent is due to the fact that the EFIE memory requirements exceeded the available memory on the computer used.

In the context of multi-incidence problems the advantages of the modified EFIE depend on the cost of recomputation of the K COA matrix for each new incidence compared to the time spent in each iteration with the conventional operator. Because computation times of iterative solution methods strongly depend on the chosen operator representation (Fast Multipole Method (FMM), Adaptive Cross Approximation (ACA), hierarchical (HCA), . . . ), we cannot make universal statements here. We just want to note that for the GMRES methods iterating with the full Galerkin matrices, as used in the tests which led to Figs. 13 and 14, the additional cost of computing K COA for a new incidence is largely compensated by the gain in computation time spent in the iterations with much smaller matrices. The fact that the dimension of the problem is reduced by a factor 4 is clearly the dominant feature.

VII. CONCLUSION AND PERSPECTIVES

We have presented a modified version of the electric field integral dedicated to plane wave scattering problems. This formulation allows one to escape from certain limitations of the conventional EFIE in terms of discretization dimensions at high frequencies. The results of various comparisons shown in this paper lead to the conclusion that the new formulation is clearly advantageous. This is, above all, due to the fact that, for comparable accuracy at identical problems, roughly 4 times less unknowns are needed than with the conventional EFIE. According to previous work, this ratio will be more and more in favor of the modified EFIE when the frequency increases.

Although the numerical results obtained are promising, a more exhaustive study deserves to be undertaken. In particular, comparisons should be performed with numerical implementations accommodating compression techniques such as FMM or hierarchical matrix calculations. It can also be noticed that the new integral equation has an advantage in terms of system solving time, when using a GMRES iterative method. This lets us expect to obtain advantageous performance for multiple right-hand sides (multiple incidence directions) where the gain in solving time outweighs the additional cost needed to construct the incidence direction dependent perturbation matrix (K = H -E) which can be done on a rather coarse mesh.

However, these perspectives need to be confirmed by a numerical study for objects of larger size. This work will include the elaboration of acceleration techniques like FMM, ACA or HCA with H-Matrix calculations. The HCA techniques (see [START_REF] Börm | Hybrid cross approximation of integral operators[END_REF], [START_REF] Daquin | Hybrid Cross Approximation for the Electric Field Integral Equation[END_REF]) will be our most favorable candidates because they allow us to work directly on the optimization of the numerical integration rules.

APPENDIX DETAILS OF THE MODEL REDUCTION

In this appendix, we present the details of the reduction of the edge element Galerkin matrix on the fine mesh to one on the coarse mesh. Fixing the ordering of the added edges makes it easier to handle the relation between the edge element spaces on the coarse and the fine mesh. Here, we choose to order the edges added in the interior of a triangle of the coarse mesh according to the positive orientation of this triangle (see Fig. 3). Each macro edge Γ, corresponds to an ordered pair of two micro edges Γ = (Γ -, Γ + ) oriented consistently with the orientation of Γ.

For the degree of freedom (DoF) associated with the macro edge ∂ p T , we have two micro DoF's associated with the two micro edges contained by the macro edge, and three micro-DoF's on the micro-edges, γ T p , p ∈ {0, 1, 2}, internal to each of the two macro-triangles, T = ∂ * + Γ and T = ∂ * -Γ, adjacent to the macro-edge Γ (only one of these macro-triangles, ∂ * + Γ, is shown in Fig. 3).

We write

e Γ = E Γ f ⊕ E ∂ * -Γ f ⊕ E ∂ * + Γ f
for the ordered octuple of micro-edges associated with Γ.

Let basis functions of an edge finite element space be labeled by the edge fixing the degree of freedom. We write J Γ for a macro basis function associated with a macro edge Γ and j γ for a micro basis function associated with a micro edge γ.

The macro basis function, J Γ , can be written generically as a sum

J Γ = ∑ γ∈e Γ c Γ γ j γ .
The coefficients are defined as follows:

c Γ γ =    1 2 if γ ∈ {Γ -, Γ + ∓ 1 4 if γ = γ T
p and Γ = ±∂ p T ± 1 4 if γ = γ T p and Γ = ±∂ q T with p = q. The value 1 2 , is obtained because of the subdivision of any macro edge in two equal halves. The 4 sub triangles of a triangle T , have equal area, |T | 4 , because they are spanned by edges which are parallel to and exactly half the size of two macro edges. As the edge elements have constant divergence, each flow entering a micro triangle should leave it with a flow 1 4 smaller. The specific distribution of signs follows from the choice of orientation of the micro edges and the orientation of the macro edge relative to the positive orientation of the macro triangle. These must be taken as they come with the coarse mesh (we cannot suppose any specific rule here).

It should be clear that all the mappings and associated data structures can be defined while constructing the refinement by sweeping through the coarse mesh for which the boundary operator and all DoF administration is already available (See [START_REF] Alzaix | Mathematical and numerical analysis of the Herberthson integral equation dedicated to electromagnetic plane wave scattering[END_REF] for more details and [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Efficient Algorithms[END_REF] for alternative methods).

Luc Giraud holds a doctorate in computer science from the Institut National Polytechnique de Toulouse. He joined CERFACS successively as a researcher, ENSEEIHT as a full professor in applied mathematics and finally INRIA as senior researcher where he is currently leading a team addressing numerical techniques for large computing platforms.

Bastiaan Michielsen received the Engineering degree from Delft University of Technology in 1980. After military service at TNO's microwave laboratory he worked from 1982 to 1991, at PHILIPS Research Laboratories in Eindhoven, The Netherlands, and in Redhill, United Kingdom. Since 1991 he works at Onera in France. His main research interests are in electromagnetic theory and boundary integral equations, including the analysis of geometrical uncertainties.

Jean-René

Figure 4 .

 4 Figure 4. Evolution of the relative error in the surface current on the unit sphere for each of the proposed discretizations (E, H RED and H RED,alt ).

Figure 5 .

 5 Figure 5. The geometry of the rough plate test case

7 Figure 8 .

 78 Figure 8. Representation of the bi-static RCS of the unit sphere obtained from the various meshes with the EFIE at f = 0.6 GHz.

m 2 ) 5 Figure 10 .m 2 ) 5 Figure 11 .

 25102511 Figure 10. Comparison of the bi-static RCS for normal incidence on the rough plate at f = 1 GHz obtained with FEKO and with H RED,alt . The FEKO and EFIE at λ /10 curves coïncide.

Figure 12 .

 12 Figure 12. Bi-static RCS result of a torus in the plane of incidence computed with the conventional EFIE on a λ /10 mesh and the modified EFIE on a λ /5 mesh at frequency 0.2 GHz

  Poirier received the Engineer diploma from INSA, Toulouse, France in 1996. He was a Ph.D. student at Onera from 1997 to 2000 and received the Ph.D. degree in applied mathematics from INSA in december 2000. From 2001 to 2003, he joined the Analysis and Scientifc Computing group from EPFL, Switzerland. He is currently working as an associate professor at INPT-ENSEEIHT, Toulouse, and joined the LAPLACE laboratory, in January 2009. His current research interests focus on computational electromagnetics and include rough surface scattering, boundary impedance condition and fast solvers for boundary element method.

Table I .

 I FLOATING-POINT OPERATION COUNT FOR THE COMPUTATION OF E AND ADDITIONAL COST FOR THE COMPUTATION OF THE MATRIX H.

	N G	Operation count for E	Additional cost for K
	3	∼2.600 F 2	∼1.150 N 2	∼1.482 F 2	∼659 N 2
	7	∼13.200 F 2	∼5.850 N 2	∼5.694 F 2	∼2.531 N 2

(F: number of faces, N: number of internal edges, N G : number of Gauss points considered).

  Evolution of the relative error in the surface current distribution as function of the frequency on the flat plate and on the rough plate for each of the proposed discretizations E, H RED and H RED,alt .

			E flat		H RED flat	H RED,alt flat
			E rough	H RED rough	H RED,alt rough
		10 1
	Error(J)	10 0
	10 -1
	10 -2	0.2	0.4	0.6	0.8	1
				Frequency (GHz)
	Figure 6.

  GMRES =0.01 H h ε GMRES =0.001 Figure13. Relative error in the computed current distribution as a function of the available memory for several values of ε GMRES . All methods show the same tendency, but the modified EFIE allows to obtain better precision in when limited memory is available. Relative error in the computed current distribution as a function of the total computation time for several values of ε GMRES . Given the current implementation, if computation time as allowed to increase the relative error goes down in roughly the same way with all methods tested.

			EFIE ε GMRES =0.01	EFIE ε GMRES =0.001
	Error(J) Error(J)	10 -1 10 0 10 -1 10 0	10 2 EFIE ε GMRES =0.01 Memory (MB) 10 3 H h ε 10 2.5 H h ε GMRES =0.01	10 3 10 3.5 EFIE ε GMRES =0.001 H h ε GMRES =0.001
			Time (sec)
	Figure 14.


The paths of creation or How can I help my Dybbouk to get out of me?

For a long time, I have been maintaining some sort of schizophrenia, divided between my artistic work, and my research and teaching theater at the University. As long as I can assault theatrical works or literary works to adapt, sometimes translate, and even play when it's possible, but always direct, my artistic gesture has been and will remain for me an art of living. These is the only way I can assume my human condition. I shaped my identity in this artistic gesture : first as a daughter (my father was an architect and my mother an actress in her youth) , as a woman, as a jew, as a citizen, and also -but it took me a long time to realize it -as a university researcher… My artistic creation accompanied me at all times, at all stages of my life, drawing an evolutive path that the research at the University could help me to inscribe write down.

Would we say, as Pablo Picasso did in nineteen twenty six : "I don't search, I find" 1 or on the contrary in theater : "I don't find, I search". Nobody can ever say in theater that he has found, we are always searching. However, creation is not research and of course research is not creation. Research is feeding creation which is feeding research. So we will try to follow the paths of creation, full of sound and fury, of laugh and work, to try to answer to these essential question s: "Was the chicken before the egg or the egg before the chicken?". This is a fundamentally circular question that gives to research and creation, to both of them, an absolutely wonderful status of uselessness in our neoliberal society -to integrate the subject -while this uselessness is really a beginning of hope: the soul supplement that we need to live and not only to survive. Let's say that we are the cherry on the cake -as art could be perceived -but the cherry gives its flavour to the cake, gives the real taste of it.

The second point that comes after uselessness is the ability in creative practice research to give objectivity to our totally subjective artistic point of view. Let's ask how research, in the area of creative practice research, might be necessarily autobiographic. These objectification of our artistic subjectivity could be what Paul Claudel calls "co-naissance" : the new "born with" of the artist, a new birth to the world will show us how to follow the path through creation to reach knowledge.

1) Theatrical adaptation of novels : a gesture of (re) writing

So what does happen between knowledge as knowhow and knowledge as practice and research? The first step would be to understand what to catch, what to snatch, what to extract on these paths of creation to make it and transmute it into knowledge. Creation means intuition and research means analysis. In creative practice research, we take the risk of not knowing, following his our own intuition. It is, or something is searching during the process of creation more than we are searching voluntarily -a little bit like during a psychoanalytic cure. We are going to a mysterious end, far away from the limited date of the production submitted to the market law of the market : we follow the path of our project.. And because my own material could only be autobiographic, I will take as an example of my creative practice research the intuition that made me lean on the novel for a theatrical adaptation, or more precisely how to transfer great mythological characters of the novel's repertoire into theater, into theatrical shapes or scenic writings.

I began with Scrooge -from Charles Dickens's Christmas Tale. I wrote an adaptation for one actress and fourteen characters. The costume set (because her gigantic dress was the set from where every puppet and object were hidden) was representative of the Book from where the whole characters of the novel magically appeared as in a dream or a nightmare. The main character wasn't Scrooge but the Book itself. The words were becoming shapes, characters, incarnations. To integrate my explanations and because once more our material in creative practice research is our own plays and directions, I would be obliged to refer to video 's links : https://vimeo.com/118144784. Here, the narration was becoming theatrical in using dialogs between the different parts of the story that were told by the actress. I just kept very few sentences of the tale to get into a playing with the metaphorical incarnation of the characters (the employee is only represented by his muffler and a candle).

The second step of my path was the creation of Quichotte. I began once more with the set as an opening Book to create the whole word world ? of the play, but this time the two main characters were Don Quichotte and Sancho Pança as two faces of the same mythological figure. As Franz Kafka said : "Sancho Panca, who has never boasted about it, succeeds over the years, devouring stories of brigand and novels of chivalry during the nights and evenings, to completely divert his demon entirely from himself. He did so well that he -whom he later called Don Quixote -threw himself now and again into the wildest adventures" 2 . In contrast to Scrooge, there was no more storyteller because it was no longer a tale. Here, we entered into the book but everything was seen through Quichotte's eyes first and Pança's eyes in the second part, except for the Countess, who is the repulsive character that judges Don Quichotte as a foolish and distractive jester, that makes fun of him and seems to be the external look of the reality, of the "reasonable" materialism: https://vimeo.com/7556354. The third step was my adaptation of "Alice in Wonderland" and "Alice through the looking glass" called : "Du côté d'Alice", in reference to the famous Proustian book "Du côté de chez Swann". I tried to manufacture the theatrical writing as the story itself seemed to be manufactured. I wanted the spectator to see how this unbelievable and mysterious, mystical story of Alice was built. It is not a tale, it is like a puzzle. So I did as if I was inventing the story with two Lewis Carroll -one was writing with the sounds (a musician) and the other with the drawing (a painter). So that I could write directly on stage, in the space with my words -my spatial, flesh words -the signs of Carroll's writing. https://vimeo.com/75768948

The fourth step in my creative practice research was Un gros grand gras Gargantua. Here my point of view was different because I wanted to rewrite in a modern area the keys that let us understand these mysterious and metaphoric Rabelais'novels, full of symbols and fables (I chased to mix different books as Gargantua, Pantagruel, Le Quart Livre…around the main character of Gargantua). So I began to take the keys of my own vision as a director : what I saw in the narrative story as the way men eat or think, the way we feed ourselves just to fulfill our emptiness and on the contrary how to taste and feel the life nourishing our brain and our body -as humanism can teach us. That was of course my reading of what gigantism could represent in our imaginations and instead of writing this new story only with the actor's bodies and the space, I decided to mix original parts I chased and a new (re)writing by a French playwriter : https://vimeo.com/174277583 My next step will be a play about a more contemporary character, Mangeclous, extracted from Albert Cohen's novels. The main interest will be to work on a literary character speaking about other literary characters (Anna Karenine)…as too much literary! A book into a book, a sort of directing the deconstruction of a writing process… All these theatrical adaptations are working on the "subground" (between the subconscious and the underground) of the writing process : the invisible part we have to make visible if we want to let appear the artistic gesture, this mysterious moment of creation that underlies the final "product".

2) "Full of hints for the workers in theater"

So if I come back to the link between intuition and analysis, research does'nt kill creation as all artists fear about. It seems like the very common fear artists have to go to a shrink because we think that understanding our process of fantasy that is the main background of our creation will stop any ability of creation. On the contrary, knowledge and in this case our own knowledge of ourselves digs the furrow of our creation which stays always mysterious and invisible. Research is drawing this mysterious way of creation to a conscious path -which helps to create and transmit. Grotowski spoke wonderfully about "robbed transmission" in a "robbed film" 3 . And Grotowski tells us that we can't teach or give an artistic personal gesture, but only technical tricks and -more important and more invisible -the way to re-do it, to re-do the visible part of creation becoming at these point research : creative practice research. Edward Gordon Craig did say the same when he asked his pupils to draw, to take notes as the best mean to know how to grab : "You have only a school now. Let no public see this grim "revival" -lovely here and there -full of hints for workers in theatres -no more." 4 . In my point of view, these "full of hints for the workers in theater" are the best definition of creative practice research. Research goes with the creation, step by step. It allows us to know how to re-do what intuition just did.

Research helps us to glimpse what is absolutely invisible in creation. Art is digging the mysterious hole of the world, the matrix hole of human condition, deep as inferno and glowing as paradise 5 . We can't see inside (who could see the hinted face of God?), we can just turn around, hear voices that we can reinterpret, give shapes to these voices so that they could become visible. And as in alchemy where science and mysticism go together to search the philosophical stone, made of material and immaterial, or as in ancient Greek theater, at the origin of western theater, where sacred rituals become the foundations of democracy, research help us to find the way of turning around the black hole of our own human mythology. It allows us to stay, as a tightrope walker, in a delicate balance between intuition and cleverness, emotion and structure, Dionysus and Apollo. And before coming back to my own creative practical research, let's assert that it is the best way, in our hopelessness liberal society, to avoid falling into fascism or fundamentalism which is the religious way not to know about mystery but to fall in fear, trying to think (or precisely not to think!) to the incomprehensible of the world. We want, on the contrary, to know the way to turn, as in antique chorus, around the matrix hole and agree that consciousness and knowledge are part of our creation and of the creation's mystery. We don't fall in the dark deep hole, we are allowed to think the hole, to imagine it which is yet conceptualization, to think about his infinite deepness as Pascal did with the infinitely small and the infinitely big. And when Pascal gives as a philosopher an answer -or a non-answer -about God, I personally think that he gives us the answer as a poet which means : the path of thinking our inability to thin is creation, which is what philosophical rationalized thinking can't afford 6 . So, after a long time, suffering from being between research as a lecturer and creation as an artist, I resolved, as I said at the beginning, these schizophrenic pathology with creative practice research. When I tried to put words on my actions -and acting too -, to follow the steps of my intuition, of my emotion like through "a conscious diary", it was like reversing internal knowledge in external 5 Victor Hugo said that « The sublime is down » -speaking about literary creation… 6 « Mais si notre vue s'arrête là, que l'imagination passe outre ; elle se lassera plutôt de concevoir, que la nature de fournir. Tout ce monde visible n'est qu'un trait imperceptible dans l'ample sein de la nature. Nulle idée n'en approche. Nous avons beau enfler nos conceptions au-delà des espaces imaginables, nous n'enfantons que des atomes, au prix de la réalité des choses. C'est une sphère infinie dont le centre est partout, la circonférence nulle part. Enfin, c'est le plus grand caractère sensible de la toute puissance de Dieu, que notre imagination se perde dans cette pensée » ;Pascal, Blaise (1951). Les Pensées ; Luxembourg : Lafuma : fragment 199. knowledge, transforming an artistic gesture in ideas : an "Idea's Theater" as Antoine Vitez called it7 , putting words on his artistic way, as Peter Brook or many others did. Many of those whom we could call "Masters" are writing on their creation as it is a part of their creation process.

3) How did I let my own Dibbouk get away

So could we agree with Valere Novarina telling us that the thought is made, is manufactured in the mouth? I would say yes, as Marcel Proust does, when he offers us a kind of methodology for creative practice research with his "madeleine". The sensation comes first in the narrator's mouth eating a madeleine, and intuition arrives to open and let you just follow the path of your knowledge -which is exactly path of creation. It makes Marcel Proust as one of the major artist working as a searcher about his sensations, his intuition. Once more, Apollo builds shapes on the drunkness of Dionysus : reason gives frames to spontaneous gushing of our artistic intuition.

Creation would be to let emerge the vision (and not only the hearing) of God's voice8 or, in other words, to dig the mysterious matrix hole of the world, this invisible universe that we make visible in creation. Research would be to interprete as Nietzsche could speak about "The being as being interpreted". Research will be the interpretation of these interpretation of the world -which is creation. Antoine Vitez, one of the greatest director in the 80 th , said that "art has to let us hear what can't be heard, show what can't be showed and tell what we can't tell"9 also said : "I think that everybody will not understand immediately what is in the play. Does'nt matter. When a tale is sufficiently exemplary, if it is sent in the public with enough violence, it stays in the memories and gives us the ability to think, and understand after. I send ideas, like punches"10 .

Research helps to create our Dibbouk and give him the ability to get out of yourselves. I speak about Dibbouk because it is for me the perfect metaphor of theater and what is acting (possession, exteriorization…). The Dibbouk (or Between two worlds ), play written by Shalom An-Ski inn 1917, was the first step of the creation of an Hebrew theater in Russia when it was directed by Vakhtangov. It is also the beginning of all my academic research on jewish theater. I wrote my PHD on the prohibition to make images : what could be a theater born with the absolute prohibition to make images? That is probably what brought me, consciously and unconsciously, to work in my plays on grotesque in theater.

My artistic gesture mixed identity and creation in grotesque -as we can see how it is manufactured in Yiddish theater which influenced so much Kafka, and it allowed me in the same time to understand expressionism from an internal point of view. As a director and as an actress, I had to experiment the difference between caricature and stylization in expressionism. Expressionism is not overacting as it could seem to be. It is the body 's writing in the space as if the actor's bodies were like letters on the white page of the stage. I had to experiment the way to drive an actor to essentialize his gesture, to stylize it as to use his own body as a sign and not as full of sense -as psychology can give to actor's play. So work with the actors on what could be expressionism in Yiddish theater helped me in my knowledge, as a reader at University, to transmit the meanings of expressionism, the operation that gives its aesthetics. This research went from "Kafka's dance" to "The man in the attic", two plays written by Timothy Daly, a famous Australian playwriter. : https://vimeo.com/174277910 My main tracks in this last play were :

-How the grotesque can allow to re-play the "big" History in a small history which would be the transposition in a stylized tale of the philosophical features that are underground -How to tell History from an internal point of view which could be metaphorical : full of signs and not giving sense (interpretation) -How the fable, told through this small world enclosure in the infernal circle between the Neighbor, the Man, the Woman and the Jew, allows us to understand how barbarity happens, where is located the step where man falls in inhumanity. So the play was a sort of long scale going from an historical process to the internal human being.

I did some part of the same work during my last creation "The tango of wandering stars" which is a musical theatrical about Yiddish tango in relation to the trip of the jewish people at the twentieth century from Poland to United States and Argentina11

These is not only a way to do a therapy (letting the others pay for it!) but once more how to transform my own story in an universal tale -which gives us the definition of art.

4) In conclusion : how to open the door…

To come to conclusion, creative practice research allows us to name the doing, to name what is acting. Research in creative practical research would be to think the creation, I mean the process of creation so that we would be able to transmit neither creation which is invisible, neither the technics of creation -which is the way we teach theater in practical classes -but to transmit the process of creation, the visible paths that brings us to creation, as Antoine Vitez, Peter Brook, Jacques Lassalle, Edward Gordon Craig and so many others did in essays on their theater -trying to understand the way they create so they could be "robbed" by their pupils, as Grotowski said.

Research put distance between us and our intuition, our emotion, not to leave them but on the contrary to follow them in a visible, readable and clear way even if we don't know what is behind the door. Kafka tells the story of a man walking all his life to reach a door behind which he sees the truth of the light . At the end of his path, he arrives exhausted to the door where light is more and more brilliant and asks the guard in front of the door to allow him to enter at the end of his life to see the truth behind these door. And the guard refuses, answering that the truth is not behind the door but is the path that the man did all his life to reach the door… 12 This philosophical and poetical tale is also my creative practice research's personal path, working in my creation on the process of creation itself. Working on the great figures of literature's mythology as Scrooge, Don Quichotte, Alice or either Gargantua. All these characters are following an initiatory trip to reach at the end…the understanding of their own path. Scrooge understands that his real life is only the metaphor of a philosophical process about human being. Alice tries to interpret the puzzle of her growth and her femininity, jumping from a mystery to another one just to accept that her growing is puzzling and enigmatic. Gargantua is the tale of an initiatory travel in his own body which is the world and Quichotte, in particular, begins to travel at fifteen 13 , and go on the world's paths to give sense to his life, returning at the end on his death's bed, telling Sancho Pança that all his apparently crazy path was to discover that : "I am who I am"(soy quien soy).

Charles Dullin could be the final reference about what transmission as the fundamental of these creative practice research : "Past gives you examples. It does'nt say : "Imitate to do as perfect as possible what I did". But he says "Do as I do, search as I searched, work"" 14 . In that sense, creative practice research is at the perfect contrary of our society, built on uselessness, consumption, efficiency, far away of the time of research. We eat the cake but we don't have any more the essential flavor of the cherry : it is a stodgy cake, without any color and flavor.

To know is to do in creative practice research : research is the path to get inside our own creation and to question it. Not to answer but to ask question…As a rabbi said to his pupils : "I got the answer! I got the answer!...But who have the question?". Creative practical research is the research of the truth revealed by creation, the horse hidden in the stone that the sculptor lets appear with his work. In the doing is the process of knowing -coming back to these this incredible Grotowski's definition of transmission about the "ignorant master" and the "robbed talks". "Art is path" said Paul Klee and that will sum what creative practice research has to be. 13 Kabbala is studied only after fourteen which seems the beginning of understanding our own mysteries 14 Dullin, Charles (2011). Introduction, choix et textes, Mayenne : Editions Actes Sud : 61.

Biography

Student of the Ecole Normale Supérieure, "agrégée" of modern letters, associate professor directing PHD in theatrical studies at the University of Evry, Dr. Isabelle Starkier is also a director, actress and company director in theater. She works on the link between theory and practice, bringing together her plays (about forty), her work as a company-in-residence and her research on otherness as well as theater in the heart of the city. Last plays : Le Bal de Kafka, L'homme dans le plafond de Timothy Daly, Le bourgeois Gentilhomme, Un gros grand gras Gargantua, Le tango des étoiles errantes…

Picasso, Pablo (2014). Je ne cherche pas, je trouve-Ecrits et pensées ; Paris : Ed. Cherche Midi.

Kafka, Franz (1975). La muraille de Chine ; trad. Jean Carrive et Alexandre Vialatte, Mayenne : NRF Gallimard : 132 (translation is mine)

Grotowski said : "There are two things in your question, there are two things. One thing: can we take ownership of someone else's practices? Obviously, we can, if we have a special guide who ... in truth it's like history: can we rob a Master? The great Master, indeed, he makes his disciple elected all the possibilities of robbing him. But he does not give it, no, he lets the other go rob, that's what's essential. He's a very bad teacher, if I can say teacher who ... (makes the gesture of giving) no, no ... create the conditions when he sees that we put something in the hat, a hidden rabbit, and we do not do not give him that rabbit, and leave the hat on the table. It's like when everyone who has the real job, he has a secret behind. This secret is never disclosed. So, we just rob him. We try the burglary. It is one of the most essential filiations, it is burglary. But yes, but it's not stealing the secret outside, it's like stealing things related to sources, secret processes, not the ones we see." in « La parole de Jerzy Grotowski : légende » is a registration of Jerzy Grotowski's talking during one of the Action of the Académie Expérimentale des Théâtres : « Le laboratoire d'acteur. Des origines au commencement avec Jerzy Grotowski », from 3 to 5 April 1996 at the Workcenter of Jerzy Grotowski and Thomas Richards at Pontedera (Italie). Vidéo realized by Pierre-Henri Magnin. Document from the Académie Expérimentale des Théâtres at the Institut des Mémoires de l'édition contemporaine (Imec). Thanks to Pascale Butel-Skrzyszowski, head of the archive division of the funds at Imec et of the funds of Académie Expérimentale des Théâtres and thanks to Michelle Kokosowski.

Gordon Craig, Edward, "A note on sanity in stage production of Shakespearean plays", quoted by Le Boeuf Patrick, « Attention!(ce n'est qu'une) école », in Faire théâtre sous le signe de la recherche, sous la direction de Losco-Lena Mireille (2017), Presses Universitaires de Rennes.

Vitez, Antoine (1991). Le théâtre des Idées ; Saint-Amand : Ed. Le Messager Gallimard : 271

As it is told in the Bible (translation in french byChouraqui, André (1989),Exode, 20, 20, La Flèche : Ed. Desclée de Brouwer : « Tout le peuple voit les voix » Exode 20,18

Vitez, Antoine (1991). Le théâtre des Idées ; Saint-Amand : Ed. Le Messager Gallimard : 271

https://vimeo.com/233634431

InKafka, Franz (2000). Récits, romans et journaux, trad. Marthe Robert, Classiques modernes, Librairie Générale Française, La Pochothèque : 1054-1056

Bibliography Isabelle Starkier

The Bible (translation in french by Chouraqui, André (1989), La Flèche : Ed. Desclée de Brouwer. Dullin, Charles (2011). Introduction, choix et textes, Mayenne : Editions Actes Sud.

Gordon Craig, Edward, "A note on sanity in stage production of Shakespearean plays", quoted by Le Boeuf Patrick, « Attention!(ce n'est qu'une) école », in Faire théâtre sous le signe de la recherche, sous la direction de Losco-Lena Mireille (2017), Presses Universitaires de Rennes. Kafka, Franz (1975). La muraille de Chine ; trad. Jean Carrive et Alexandre Vialatte, Mayenne : NRF Gallimard. Kafka, Franz (2000). Récits, romans et journaux, trad. Marthe Robert, Classiques modernes, Librairie Générale Française, La Pochothèque. Pascal, Blaise (1951). Les Pensées ; Luxembourg : Lafuma : fragment 199. Picasso, P. (2014). Je ne cherche pas, je trouve-Ecrits et pensées ; Paris : Ed. Cherche Midi.

Vitez, Antoine (1991). Le théâtre des Idées ; Saint-Amand : Ed. Le Messager Gallimard

Abstract

Understand what, in our creations, follow a path that research must bring out, means to start from my deepness intimacy to help my Dybbouk to get out and become an performance for humanity, an artistic gesture that I can share , that I can transmit to others. Starting from my theater director experiment as a knowhow with more than thirty plays and my academic knowledge, I will try to follow this path that leads me to a theoretical research on both theatrical forms and the prohibition of representation , to an artistic gesture that leads once again to its capacity for transmission, pedagogy of the gesture through its theorization. From experiment to practice and from practice to experiment, we will try to put words on our own path in regard to what we call practice research which is at his beginning in France particularly in theater.