## Laurent Gosse on Mathematical Computation Applied to Real-World Problems

#### Scientist Interview: December 2010

**Conservation laws lie at the heart of physics
and they provide a rigid framework for mathematical modeling. The
modeling of real-world situations, such as fluid flow over topography,
or atmospheric flow in a nuclear explosion, or shock waves, is
extremely demanding in terms of computing power. Brute force number
crunching takes too long or demands massively parallel supercomputers.
In computational mathematics success often comes from the cunning
development of algorithms that lend themselves to efficient
coding.**

*In*
Essential Science Indicators^{SM} from Thomson
Reuters*, the work of Dr. Laurent Gosse ranks in the
top 1% in the field of Mathematics, with a total of 24 papers cited
211 times between January 1, 2000 and August 31, 2010. His record
over all fields in the database for this same period includes 34
papers cited a total of 376 times.*

*Gosse is a Researcher at the Istituto per le Applicazioni del
Calcolo in Bari, Italy. His work has applied computational mathematics
to a wide range of problems in fluid mechanics, and even to the
analysis of stock market fluctuations.*

**I am interested in how you got started in
this field, and who influenced your early career.**

I studied for higher degrees at French universities, in Lille for my Master's, followed by the Université Paris-Dauphine, which awarded my Ph.D. in 1997. I decided to go for mathematics because I was stronger at math than physics. I chose the field of analysis, which refers to the solution of problems by reducing them to equations.

The main influences on my career date from my time at Paris-Dauphine, where there were outstanding people, such as the Fields Medalist Pierre-Louis Lions. He was the best teacher of non-linear partial differential equations (PDEs) that I ever encountered. Yves Meyer, who won the Gauss Prize this year, taught us the recent theory of wavelets. The atmosphere was very stimulating but also highly competitive.

**After finishing training, where did you find
your first position as a professional in applied mathematics?**

First I was hired by the French Atomic Energy Commission in Saclay. That's where I completed the Ph.D. thesis in the LETR (Laboratoire d'Etudes Thermiques des Réacteurs) thanks to a recommendation by Grégoire Allaire.

**Your papers can mostly be grouped into
discrete areas of research. Your most-cited papers propose well-balanced
schemes for hyperbolic balance laws. What's the science behind this
research?**

"I have a wild idea that macroscopic (I mean a daily time scale) movements of stock indexes can contain trends, fluctuations, and noise to which advanced techniques of signal processing can be applied in order to decompose them, and hopefully extrapolate them."

Many applications exist for this theory, such as simulations of dam breaks, atmospheric flow computations after the explosion of a nuclear weapon at high altitude, or simple fluid flow over topographic obstacles. One had to solve equations of inviscid fluid mechanics in which the effects of external forces, the simplest one being gravity, are rendered with supplementary terms that were creating numerical instabilities.

At Saclay, I further developed this subject of my Ph.D. in an industrial
context. In 2002 and 2004, after my TMR postdocs and with my colleagues
Debora Amadori, Paola Goatin, and Graziano Guerra, I published papers
giving a rather complete theoretical overview of the field. The 2002 paper
(*Arch. Ration. Mech. Anal.* 162[4]: 327-66, May 2002) is ranked at
#4 in my *Essential Science Indicators* record.

**Your two most-cited papers ( Comput. Math.
Appl. 39[9-10]: 135-59, May 2000; Math. Model. Method. Appl.
Sci. 11[2]: 339-65, March 2001) feature well-balanced schemes for
hyperbolic balance laws. What are the novel features of those
computations?**

At Saclay I worked on realistic problems: it was an industrial environment where we simulated accidents in nuclear plants that involved two-phase flows, of liquid and vapor, in a high-pressure environment. Typically this was performed to see how to react to the case of a hole appearing in the primary cooling circuit passing through the reactor. This was modeled by inviscid compressible Euler equations with source terms for heating, vaporization, gravity, and so on.

At the numerical level, the inclusion of source terms turns the situation into a generalized Riemann problem that was too intricate for numerical solution. With the help of A.-Y. LeRoux, we came up with the "well-balanced solver" that was probably the best compromise between feasibility and accuracy. Later these ideas have been used extensively for computing shallow water flows over various types of topography. I think that's an important reason why these papers attracted so much attention.

**Have your later papers extended the line of
research started in those two papers?**

They absolutely have. The work with Drs. Amadori, Goatin, and Guerra (the
previously mentioned 2002 *Arch. Ration. Mech. Anal*. paper; *J.
Differential Equations* 198[2]: 233-74, 10 April 2004; *Proc. Amer.
Math. Soc*. 132[6]: 1627-37, 2004) extend it from a theoretical
perspective.

Five papers jointly authored by me and Giuseppe Toscani (*C. R.
Math*. 334[4]: 337-42, 28 February 2002; *SIAM J. Numer. Anal*.
43[6]: 2590-606, 2006; *Numer. Math*. 98[2]: 223-50, August 2004;
*SIAM J. Sci. Comput*. 28[4]: 1203-27, 2006; and *SIAM J. Numer.
Anal*. 41[2]: 641-58, 2003) pushed toward diffusion approximation of
kinetic models, the "Asymptotic-Preserving" schemes as Shi Jin calls them.
We had also important discussions with A.-E. Tzavaras in order to recast
the non-conservative products in the framework of weak limits in order to
put them on a solid theoretical base.

Very recently the method has been applied to a simple model of cellular
dynamics. The well-balanced approach is also used in the 2007 *SIAM J.
Sci. Comput*. paper (29[1]: 376-96) with Philippe Bechouche, which
models a particular type of semiconductor junction diode.

**Turning to another of your interests, you
have worked on solutions to linear PDEs that approximate physical
conditions in the transition from quantum mechanics to classical
mechanics. This is the territory in which the semiclassical WKB
approximation was used in the past to make calculations of
wavefunctions. This brings me to your 2002 J. Comput. Phys.
paper (180[1]: 155-82, 20 July 2002) on K-branch entropy solutions for
geometrical optics calculations. What's the story with this
one?**

The motivation for so-called K-multivalued solutions comes from high-frequency asymptotics for linear wave equations. Usually the brute force computations fail because the computing power for really high frequency is enormous even for today. The issue with classical WKB equations lies in the fact that they are not time-reversible and they are non-linear, so the waves they represent collide and dissipate rather than superimpose.

I've developed the algorithms and procedures in conjunction with Olof
Runborg (*Commun. Math. Sci.* 3[3]: 373-92, September 2005; *C.R.
Math*. 341[12]: 775-80, 15 December 2005; and *SIAM J. Appl.
Math*. 68[6]: 1618-40, 2007), principally for use in multiphase
geometric optics. But the applications include all fields that require the
computation of high frequency asymptotics of wave propagation: in optics
the light beams behind a focusing lens, in micro-electronics electrons
flowing in periodic crystals (a problem carried out with Peter Markowich
and Norbert Mauser), and even the propagation of elastic waves inside the
Earth for oil prospecting.

**Some of your latest papers demonstrate a
remarkable interest in applying mathematics to biological modeling. Tell
me more about that.**

"...these ideas have been used extensively for computing shallow water flows over various types of topography."

My work on diffusive relaxation and asymptotic-preserving schemes took me rather naturally toward these questions: in the same way that macroscopic fluid mechanics equations can be derived by passing to the limit in Boltzmann kinetic equation, many diffusion models can be obtained by setting to zero a small parameter in a much simpler kinetic model. Giuseppe Toscani and I have developed several numerical schemes based on the well-balanced framework that permit us to visualize this limiting process.

Recently, I used them to study a common model of biological modeling, the Cattaneo model of chemotaxis early studied by Greenberg and Alt, more recently by Natalini and collaborators, which relaxes towards the classical Keller-Segel system. Hopefully, this work can be extended to more elaborate kinetic models.

**Many of your papers have applications to
physics, but more recently you have developed some new
interests—stock market movements, for example. What's the
connection between the gyrations of markets and your research?**

I have a wild idea that macroscopic (I mean a daily time scale) movements of stock indexes can contain trends, fluctuations, and noise to which advanced techniques of signal processing can be applied in order to decompose them, and hopefully extrapolate them.

Trends and fluctuations are thought to result from a relaxation step at the market fixing each day and thus can be smooth enough to be extrapolated by means of existing techniques. But it is a severely ill-posed inverse problem which has to be handled carefully.

Somehow, this is a way of saying that, at every instant, we hear something of the future, but extremely badly. Therefore sophisticated techniques are required to extract correctly the hints. Some progress has been achieved in three years and there's a SciLab code now working rather nicely.

**What are you working on right now?**

I am very into Compressed Sensing, either because it might be used for stock market extrapolation studies, but also because it is actually a beautiful theory with a huge number of applications. Perhaps there's also a possibility to apply the well-balanced framework to several significant models of mathematical biology going beyond the Cattaneo semilinear system. Another direction is to extend the well-balanced ideas to general kinetic models, like e.g. radiative transfer: one possibility I explore nowadays is to take advantage of the classical method of "elementary solutions" in a time-dependent framework.

**Laurent Gosse, Ph.D.
Istituto per le Applicazioni del Calcolo
Bari, Italy**

LAURENT GOSSEE'S MOST CURRENT MOST-CITED PAPER IN ESSENTIAL SCIENCE INDICATORS:

Gosse L, "A well-balanced scheme using non-conservative products designed
for hyperbolic systems of conservation laws with source terms," *Math.
Model. Method. Appl. Sci*. 11(2): 339-65, March 2001 with 50 cites.
Source:
*Essential Science Indicators* from
Clarivate.

ADDITIONAL INFORMATION:

- Read a previous (November 2005) interview with Laurent Gosse.

KEYWORDS: MATHEMATICAL MODELING, REAL-WORLD SITUATIONS, COMPUTATIONAL MATHEMATICS, ALGORITHMS, CODING, FLUID MECHANICS, STOCK MARKET FLUCTUATIONS, PARTIAL DIFFERENTIAL EQUATIONS, HYPERBOLIC BALANCE LAWS, INVISCID COMPRESSIBLE EULER EQUATIONS, SOURCE TERMS, GENERALIZED RIEMANN PROBLEM, DIFFUSION APPROXIMATION, KINETIC MODELS, CELLULAR DYNAMICS, SEMICONDUCTOR JUNCTION DIODE, K-MULTIVALUED SOLUTIONS, WKB EQUATIONS, MULTIPHASE GEOMETRIC OPTICS, CATTANEO MODEL, CHEMOTAXIS, KELLER-SEGEL SYSTEM, COMPRESSED SENSING.