# Presentation

This ANR project concerns the development of *new numerical
methods *for *wave propagation problems *using
tools from *microlocal analysis*. We plan to focus on
three important and fundamental directions of research

### Microlocal analysis and numerical methods for acoustic and electromagnetic wave scattering (Tasks 1, 2 and 3):

The aim here concerns the improvement of existing numerical simulation techniques for*high frequency acoustic and electromagnetic scattering*problems by hybrid algorithms involving asymptotic microlocal analysis approximations. The model problems are based on the exterior Helmholtz' and Maxwell's equations.### Microlocal analysis and numerical methods for Schrödinger-type equations (Tasks 4 and 5):

This direction concerns the derivation of accurate approximations of the Dirichlet-to-Neumann (DtN) operator for linear and nonlinear Schrödinger equations and their applications in artificial boundary conditions methods and domain decomposition techniques. To achieve this, we will develop approaches based on pseudo- and para-differential operators techniques.### Crossing directions 1 and 2 (Task 6):

The goal of this last task is to bring together people from directions 1 and 2 to apply our new numerical techniques from Helmholtz/Maxwell to Schrödinger equations, and conversely. One example could be constructing microlocal approximation methods for nonlinear optics using our advances in linear Maxwell's equations.

Problems 1. and 2. lead to central and difficult questions in scientific computing.
Indeed, some fundamental numerical problems are related to the special features
of the structure of the underlying partial differential operators governing
such problems (for example highly indefinite operators or/and nonlinearities)
or to the problem itself (like the unbounded character of the computational
domain). The originalities of our project are the development of new numerical
methods and the improvement of existing standard simulation techniques by using
*asymptotic microlocal analysis*. Furthermore,
in addition to directions 1 and 2, we will develop selected crossing points
between the two problems to increase the impact of our work (Task 6). Finally,
our motivations are related to exciting physical applications (e.g. Radar Cross
Section computation of large targets, quantum chemistry computations,
Bose-Einstein condensates) that will be considered as benchmarks for applying
our methods.

The theory of microlocal analysis and pseudo-differential operators has been developed by mathematicians like L. Hörmander in the sixties for studying regularity properties of solutions to PDEs. Many theoretical developments have been pursued during the last four decades and in particular by the french school of PDEs. The application of such techniques has been applied for the first time in the area of numerical simulation by B. Engquist and A. Majda in their famous paper on artificial boundary conditions for linear hyperbolic equations and systems of PDEs (Math. Of Comp. 77). Since then, their approach has given rise to many important developments worldwide in the field of absorbing boundary conditions. This historical step has shown that theoretical tools like pseudodifferential operators and associated microlocal analysis can lead to very important contributions in the numerical simulation of PDEs. Our goal is to show that other new ideas can be developed by using microlocal analysis with some applications in numerical methods.

## State-of-the-art :

### 1. Hybridization of classical numerical methods and microlocal analysis for high frequency scattering.

Acoustic and electromagnetic scattering problems lead to the resolution of a PDE, Helmholtz or Maxwell's equations, in an exterior domain (the complementary set of the scatterer). A boundary condition on the target's surface must be added to the problem and a suitable condition must be imposed at infinity to select the outgoing wave (Sommerfeld's and Silver-Müller's radiation condition). Such a problem is well-posed in a suitable functional framework. Scattering problems find many applications in technology like the computation of Radar Cross Sections of complex objects (radar signature). Moreover, they yield challenging basic questions in scientific computing that may find applications in other scientific fields.

Two difficulties are basically known when one wishes to solve numerically the scattering boundary-value problem

- 1.a) First, the equation is set in an unbounded domain. This requires for a numerical procedure to reformulate the problem as a bounded domain problem.
- 1.b)
Second, when the wavelength of the incident field is small compared to the
characteristic size of the obstacle, the problem is called
*high-frequency*. From the numerical viewpoint, a very fine grid is needed for solving the small wavelength problem.

Concerning problem 1.a), different robust procedures exist for setting the problem in a bounded domain. A first approach, for a volumetric finite element solution, consists in introducing an absorbing boundary condition or a Perfectly Matched Layer (PML) on a fictitious boundary/domain delimiting hence a finite computational domain [1]. This point is no longer a technical problem even if difficulties remain when high frequencies are considered. Recent improvements have been obtained in particular by members of our ANR group in this direction. A second solution is to equivalently reformulate the problem on the finite boundary of the scatterer as an integral equation. This strategy is also known to be robust and is studied by many people in our project. In particular, we have two- and three-dimensional codes for solving such situations. However, these two standard procedures find some limitations when large frequencies are considered, in particular because of point 1.b). This has given rise to many important contributions and efforts from scientists and engineers. We refer to the recent review paper [1] for acoustics which gives a state-of-the-art of finite element strategies in this area. According to this paper, we can roughly consider that two fundamental difficulties remain unsolved :

- i)
The finite element methods suffer from
*pollution effects for large wavenumbers*. This leads to considering very fine 3D grids. Therefore, new approximation methods or alternative formulations must be considered. During the last decade, important contributions have been made like for example by incorporating plane wave solutions into the standard polynomial basis functions. However, the question of reducing the pollution remains an important open question since all the techniques developed until now are limited in frequency. - ii) A preconditioned Krylov subspace iterative solver like the GMRES does not converge if the wavenumber (« the inverse of the wavelength ») is large. Indeed, the matrix becomes then highly indefinite. In particular, purely algebraic methods fails rapidly for 3D problems.

For integral equations, similar problems arise:

- i) It is important to work with a coarse surface grid to reduce the number of degrees of freedom of the Boundary Element Method (BEM). In particular, we would like to work with one (or even less) point per wavelength (we notice here that there is no pollution in the boundary element methods compared to volumetric FEMs). Recent developments (with contributions from our group) have been obtained in this direction by using modified boundary element methods or integral formulations.
- ii) The numerical solution to the discrete integral equations is obtained by coupling a multilevel Fast Multipole Method (introduced by L. GREENGARD and V. ROKHLIN in the 80's) for accelerating matrix-vector products operations in a Krylov iterative solver combined with a preconditioner. Again, it is shown that the convergence fails if the frequency becomes too large. This proves that purely algebraic approaches are limited in such complex situations.

Moreover, for points i) and ii), it is often admitted that scattering by non convex objects is harder to solve since multiple scattering effects arise. In our project, we will focus on points i) and ii) for both FEMs and BEMs, considering possibly non convex scatterers, for acoustics and electromagnetic problems. Some contributions have already been obtained by members of our project on this intensively studied problem.

### 2. Microlocal
approximations of DtN maps for linear and nonlinear Schrödinger equations and
applications in scientific computing.

Our aim here is to develop accurate approximations of the DtN operators for linear and nonlinear time-domain and stationary three-dimensional Schrödinger equations in unbounded domains. Two applications in terms of computational methods are considered here

- 2.a) We want to build some robust and accurate absorbing boundary conditions to truncate the unbounded domain (and propose associated efficient and robust schemes)
- 2.b) We wish to use these boundary conditions as transmission conditions between subdomains when Domain Decomposition Methods (DDMs) are considered.

Fundamental physical applications that will be considered by our numerical methods come from quantum mechanics, underwater acoustics propagation (Standard and Wide-Angle Parabolic Equations), optics (uncoupled and coupled nonlinear Schrödinger equations, Schrödinger equations with stochastic coefficients), quantum chemistry (Kohn-Sham equations with variable integro-differential potentials) and Bose-Einstein condensates (Gross-Pitaevskii equations). These problems represent challenging applications for new numerical methods.

Because of all these applications, there is a growing interest in absorbing and transparent boundary conditions for Schrödinger equations since the beginning of the eighties. Since then, many improvements have been addressed but mainly for linear, constant coefficients and one-dimensional Schrödinger equations. Much more details are given e.g. in the recent review paper [2]. Contributions to higher-dimensional problems, variable coefficients equations and nonlinear problems are much more recent. In particular, many contributions come from people of our group [3]. Most other noticeable developments have been made by A. ARNOLD (Vienna), M. EHRHARDT (Berlin), C. ZHENG and H. HAN (Tsinghua University) for these extensions. We have strong links and papers with these researchers which make our project visible at the highest international level. Concerning Domain Decomposition Methods for Schrödinger equations, in a submitted joint paper with L. HALPERN, J. SZEFTEL was the first one to develop some applications of DtN approximations for one-dimensional linear Schrödinger equations to DDMs. This important first step will be the keystone of other numerical and algorithmic developments for more complex Schrödinger equations. Furthermore, the Schrödinger equation can be seen as the basic example of a dispersive equation, exhibiting most of the crucial difficulties.

*Bibliography.*

[1] L.L. Thompson, *A
Review of Finite-Element Methods for Time-Harmonic Acoustics*, Journal of
the Acoustical Society of America A, 119 (3) (2008), pp. 1315-1330.

[2] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, A.
Schädle, *A Review of Transparent and
Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger
Equations*, Communications in Computational Physics 4 (4) (2008), pp.
729-796.

[3] X. Antoine, C. Besse, J. Szeftel, *Towards Accurate Artificial Boundary Conditions for Nonlinear PDEs*, invited paper in Cubo
A Mathematical Journal (2009) (20 pages).

## Scientific/technical
goals :

Our goal is to develop new numerical methods and to improve existing reference methods by using microlocal analysis techniques.

The first objective is to contribute to the numerical methods for solving acoustics and electromagnetic scattering problems at high frequency. Basically, we will focus on

- Improving variational/integral formulations and FEM/BEMs by using microlocal asymptotic properties of the solution to the corresponding boundary-value problem.
- Building new preconditioners based on approximations at the continuous level of the inverse of the scattering boundary-value or integral equation operators (with possible coupling with standard algebraic approaches).

We will essentially use methods related to the symbolic calculus of pseudodifferential operators.

The second objective is to build accurate and robust approximations of the DtN operators for complex Schrödinger equations. The operators will involve the following features: nonlinearities, variable coefficients, stochastic terms, three-dimensional version, time-domain and stationary problems, coupled equations. We will focus on techniques based on fractional microlocal analysis for pseudodifferential and paradifferential operators calculus.

The third goal, which justifies the presentation of the two
topics in the same ANR project, is *to
investigate crossing techniques developed in the two previous objectives*.
For example, one interesting question is: How can we develop accurate
microlocal finite element approximations of the ground states of a stationnary
nonlinear Schrödinger equation with possibly integral potentials computed by an
adapted Fast Multipole Method ? This is a crucial point in practice for
example in quantum chemistry.

## Originalities/scientific
advances/objectives:

We list the different main originalities of our project

- To
apply microlocal analysis calculus to develop efficient and robust new
scientific computing algorithms for linear and nonlinear PDEs. Until now,
microlocal analysis has essentially found applications in the analysis of
PDEs but not so much to
*numerical methods*. This is a challenging new area to explore and develop. - To develop and validate free-downloadable solvers based on these techniques for real three-dimensional situations related to Helmholtz, Maxwell and Schrödinger equations.
- To apply our new microlocal numerical methods developed in the case of the Helmholtz/Schrödinger equation to design improved numerical solutions for the Schrödinger/Helmholtz equation.

We think that these three points significantly contribute to the development of new ideas and methods in scientific computing.

## Scientific
bottlenecks to solve:

Many important difficulties need to be solved. Mainly, we focus on

- Contributing to the advances of efficient and robust algorithms for solving high frequency scattering problems which find important applications in industry. This problem is a very challenging area for applied mathematicians.
- Developing new approximations of DtN maps for nonlinear equations with contributions to associated algorithms. The ultimate goal is to produce accurate solutions to nonlinear/stochastic, three-dimensional Schrödinger equations. This would contribute to large scale simulation of complex systems with applications in Physics.

## Bibliography of the
project :

[1] L.L. Thompson, *A Review of Finite-Element Methods for Time-Harmonic Acoustics*, Journal of
the Acoustical Society of America A, 119 (3) (2008), pp. 1315-1330.

[2] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, A. Schädle, *A Review of Transparent and
Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger
Equations*, Communications in Computational Physics 4 (4) (2008), pp.
729-796.

[3] X. Antoine, C. Besse, J. Szeftel, *Towards Accurate Artificial Boundary Conditions for Nonlinear PDEs*, Cubo, A
Mathematical Journal 11 (4) (2009), pp. 29-48.

[4] X. Antoine, *Advances in the On-Surface Radiation Condition Method: Theory, Numerics and Applications*,
Book Chapter, Computational Methods for Acoustics Problems, Editor F. Magoulès,
Saxe-Coburg, 2007, pp. 169-194.

[5] Y.Y. Lu, *Some Techniques for Computing Wave Propagation in Optical Waveguides*,
Communications in Computational Physics, 1 (6) (2006), pp.1056-1075.

[6] C. Geuzaine, J. Bedrossian and X. Antoine, *An Amplitude Formulation to Reduce the
Pollution Error in the Finite Element Solution of Time-Harmonic Scattering
Problems*, IEEE Transactions on Magnetics 44 (6) (2008), pp. 782-785.

[7] T. Abboud, J.C. Nédélec, B. Zhou, *Integral Equation Method for High-Frequencies*, CRAS 318 (2) (1994),
pp. 165-170.

[8] E. Darrigrand, *Coupling
of Fast Multipole Method and Microlocal Discretization for the 3D Helmholtz
Equation, *Journal of Computational Physics, 181 (1) (2002), pp 126-154.

[9] E. Darrigrand and P. Monk, *Coupling of the Ultra-Weak Variational Formulation and an Integral
Representation using a Fast Multipole Method in Electromagnetism*, Journal of Computational and Applied
Mathematics, 204 (2) (2007), pp 400-407.

[10] I. Andronov, D. Bouche, F. Molinet, *Asymptotic And Hybrid Methods in
Electromagnetics, *IEEE 2005*.*

[11] X. Antoine, A. Bendali and M. Darbas, *Analytic Preconditioners for the Boundary
Integral Solution of the Scattering of Acoustic Waves by Open Surfaces*,
Journal of Computational Acoustics, 13 (3), (2005), pp. 477-498.

[12] X. Antoine and M. Darbas, *Alternative Integral Equations for the Iterative Solution of Acoustic
Scattering Problems*, Quarterly Journal of Mechanics and Applied Mathematics
58 (1) (2005), pp. 107-128.

[13] M. Darbas, *Generalized
CFIE for the Iterative Solution of 3-D Maxwell Equations*, Applied
Mathematics Letters, 19 (8) (2006), pp. 834-839.

[14] X. Antoine and Y. Boubendir, *An Integral Preconditioner for Solving the Two-Dimensional Scattering
Transmission Problem using Integral Equations*, International Journal of
Computer Mathematics 85 (10) (2008), pp. 1473-1490.

[15] X. Antoine, C. Chniti and K. Ramdani, *On the Numerical Approximation of
High-Frequency Acoustic Multiple Scattering Problems by Circular Cylinders*,
Journal of Computational Physics 227 (3) (2008), pp. 1754-1771.

[16] C. Geuzaine, O. Bruno, F. Reitich, *On the O(1) Solution of Multiple-Scattering
Problems*, IEEE Transactions on Magnetics 41 (5) (2005), pp. 1488-1491.

[17] X. Antoine, C. Geuzaine and K. Ramdani, *Computational Methods for Multiple
Scattering at High Frequency with Applications to Periodic Structures
Calculations*, M. Ehrhardt (ed.), Wave Propagation in Periodic Media -
Analysis, Numerical Techniques and Practical Applications, Progress in
Computational Physics, Vol. 1, Bentham Science Publishers Ltd., to appear: Fall
2009.

[18] C. Sulem and P.L. Sulem, *The Nonlinear Schrödinger Equation, Self-Focusing and Wave *

*Collapse,* Series:
Applied Mathematical Sciences , Vol. 139. 1999.

[19] X. Antoine and C. Besse, *Construction, Structure and Asymptotic Approximations of a
Microdifferential Transparent Boundary Condition for the Linear Schrödinger
Equation*, Journal de Mathématiques Pures et Appliquées 80 (7) (2001), pp.
701-738.

[20] J. Szeftel, *Design
of Absorbing Boundary Conditions for Schrödinger Equations in IR ^{d}*,
SIAM Journal on Numerical Analysis 42 (4) (2004), pp.1527-1551.

[21] J. Szeftel, *A
Nonlinear Approach to Absorbing Boundary Conditions for the Semilinear Wave
Equation*, Mathematics of Computation 75 (2006) pp.565-594.

[22] J. Szeftel, *Absorbing
Boundary Conditions for Nonlinear Scalar Partial Differential Equations*,
Computer Methods in Applied Mechanics and Engineering 195 (2006), pp.3760-3775.

[23] J. Szeftel, *Absorbing
Boundary Conditions for Nonlinear Schrödinger Equations*, Numerische
Mathematik 104 (2006), pp.103-127.

[24] X. Antoine, C. Besse and P. Klein, *Absorbing Boundary Conditions for the One-Dimensional Schrödinger
Equation with an Exterior Repulsive Potential*, to appear in Journal of
Computational Physics (2009).

[25] X. Antoine, C. Besse and S. Descombes, *Artificial Boundary Conditions for
Non-Linear Schrödinger Equations*, SIAM Journal on Numerical Analysis 43 (6) (2006), pp. 2272-2293.

[26] J. M. Bony, *Calcul
Symbolique et Propagation des Singularités pour les Equations aux Dérivées
Partielles non Linéaires*, Annales Scientifiques de l'Ecole Normale
Supérieure 14, (1981), pp.209-246.

[27] J. Garnier and R. Marty, *Effective Pulse Dynamics in Optical Fibers with Polarization Mode
Dispersion*, Wave Motion 43 (2006), pp. 544-560.

[28] R. Marty, *On a
Splitting Scheme for the Nonlinear Schrödinger Equation in a Random Medium*,
Communications in Mathematical Sciences 4 (4) (2006), pp. 679-705.

[29] C. Besse, B. Bidégaray and S. Descombes, *Order Estimates in Time of Splitting Methods
for the Nonlinear Schrödinger Equation*, SIAM Journal on Numerical Analysis
40 (1) (2002), pp.26-40.

[30] A. De Bouard, A. Debussche and L. Di Menza, *Theoretical and Numerical Aspects of
Stochastic Nonlinear Schrödinger Equations*, Monte-Carlo Methods and
Applications 7 (1-2) (2001), pp.55-63.

[31] M. Barton-Smith, A. Debussche, L. Di Menza, *Numerical Study of Two-Dimensional
Stochastic NLS Equations*, Numerical Methods for PDE's, 21 (4) (2005),
pp.810-842.

[32] L. Di Menza, *Transparent
and Artificial Boundary Conditions
for the Linear Schrödinger Equation*, Numerical Functional Analysis and Optimization, 18 (7-8) (1998),
pp.759-776.

[33] C. Besse, *A
Relaxation Scheme for Nonlinear Schrödinger Equation*, SIAM Journal on
Numerical Analysis 42 (3) (2004), pp.934-952.

[34] L. Di Menza and J.C. Saut, *Solitons in Quadratic Media*, submitted.

[35] A. Jungel and I. Violet, *The Quasineutral Limit in the Quantum Drift-Diffusion Equations*,
Asymptotic Analysis 53 (3) (2007), pp.139-157.

[36] X. Antoine, C. Besse and V. Mouysset, *Numerical Schemes for the Simulation of the
Two-Dimensional Schrödinger Equation using Non-Reflecting Boundary Conditions*,
Mathematics of Computation 73 (2004), pp.1779-1799.

[37] X. Antoine and C. Besse, *Unconditionally Stable Discretization Schemes of Non-Reflecting
Boundary Conditions for the One-Dimensional Schrödinger Equation*, Journal
of Computational Physics 181 (1) (2003), pp. 157-175.

[38] L. Halpern and J. Szeftel, *Optimized and Quasi-Optimal Schwarz Waveform Relaxation for the
One-Dimensional Schrödinger Equation*, submitted.

[39] W. Bao, *The
Nonlinear Schrödinger Equation and Applications in Bose-Einstein Condensation
and Plasma Physics*, Chapter 3 in Dynamics in Models of Coarsening,
Coagulation, Condensation and Quantization (IMS Lecture Notes Series Volume 9),
World Scientific (2007), pp. 141-240.

[40] X.G. Gong, L. Shen, D. Zhang, A. Zhou, *Finite Element Approximations for
Schrödinger Equations with Applications to Electronic Structure Computations*,
J. Comput. Math. 26 (3) (2008), pp. 310-323.

[41] D. Xiu, *Fast
Numerical Methods for Stochastic Computations : A Review*,
Communications in Computational Physics 5 (2-4) (2009), pp. 242-272.

[42] F. Drouart, G. Renversez, A. Nicolet, C. Geuzaine,
*Spatial Kerr Solitons in Optical Fibers
for Finite Size Cross Section: Beyond the Townes Soliton*, J. Opt. A :
Pure Appl. Opt. 10 (2008), 125101 (13pp).