*Probabilistic approach for a priori and error estimate of solution and approximation to determinist partial differential equations.*

**1.** Mixed geometrical-probabilisitic interpretation of error estimate derived form Bramble-Hilbert lemma: Application to the relative accuracy between two Lagrange finite elements.

**2.** On generalized binomial laws to evaluate finite element accuracy:On generalized binomial laws to evaluate finite element accuracy:toward applications for adaptive mesh refinement.

**3.** Relative accuracy for high order finite element by a mixed functional-probabilistic approach.

**4.** Explicit k−dependency for P_{k} finite elements in W^{m,p} error estimates: application to probabilistic laws for accuracy analysis.

**5.** Continuous probability distribution to compare the approximation error between two Lagrange finite elements.

**Mathematical modelling and****data mining**applied to Particle Physics.

**1. Analysis of numerical approximations of asymptotic solutions of the Vlasov-Maxwell equations modeling a beam of ultra-relativistic particles**, (*Collaboration with Prof. F. Assous, Ariel University, Israel*).

We propose a new approach that consists in using data mining techniques for scientific computing. Indeed, data mining has proved to be efficient in other contexts which deal with huge data like in biology, medicine, marketing, advertising and communications.

Our aim, here, is to deal with the important problem of the exploitation of the results produced by any numerical method. Indeed, more and more data are created today by numerical simulations. Thus, it seems necessary to look at efficient tools to analyze them. In this work, we focus our presentation to a test case dedicated to an asymptotic paraxial approximation to model ultrarelativistic particles. Our method directly deals with numerical results of simulations and try to understand what each order of the asymptotic expansion brings to the simulation results over what could be obtained by other lower-order or less accurate means. This new heuristic approach offers new potential applications to treat numerical solutions to mathematical models.

**2. Error Analysis in Numerical Approximations of PDE's: A Pilot Study Using Data Mining Techniques Data Mining applied to Engineering Sciences,** (*Collaboration with A. Lombardi, CERN, Geneva, Swisserland and Prof. F. Assous, Ariel University, Israel*).

We introduced in 1. a new methodology based on data mining techniques for numerical approximation analysis. Our aim is to extend this methodology to a posteriori error analysis in numerical approximations of partial differential equations. It concerns the treatment of the sources of errors involved in any process of approximation, which describes a given real system by a mathematical model, solved by numerical approximation methods, and whose exploitation, in ne, will provide the understanding, the control and the forecast of this system.

Let us consider a real system (S) modeled by a set of partial differential equations (E). Generally, the solution of such a system is carried out by numerical approximations methods, as finite elements, finite volumes, finite differences, or any other appropriate numerical method. Regarding the production of these approximations, it is essential to consider the sources of errors which spoil the accuracy of the description and the understanding of the real system (S). In this work, we define four types of errors : the modeling error, the approximation error, the parametrization error and the discretization error. In this work, we focus our attention to a test case devoted to the discretization error defined as follow: For a given family of approximations methods - for instance the Pk finite elements, k = 1,2... - we consider two numerical methods of this family, says (MN1) and (MN2). The discretization error is defined as the error due to the difference of order between (MN1) and (MN2).

As an example, we considered numerical solutions to Vlasov Maxwell equations, computed by an asymptotic model, and numerically discretized by P1 and P2 finite elements. A priori, the Bramble-Hilbert theorem claims that, under certain conditions of regularity of the mesh and of the solution, the results obtained by the finite elements P2 (of order 2) will be more precise than those computed by finite elements P1 (of order 1).

However, the estimations of the approximation error contain multiplicative constants, unknown or difficult to estimate. As a consequence, we looked for circumstances for which the finite element method P1 would be locally ”equivalent” or more ”accurate” than the finite element method P2.

Then, we sought to identify and to qualify such situations by exploratory data mining techniques. Typically, the segmentation by decision trees was effective in this perspective. Indeed, the algorithm of segmentation allowed us to identify and to characterize the rows of a suitable database that correspond to an equivalent precision between P1 and P2. Particularly, we showed that it may sometimes be possible to euristically precise the classical Bramble Hilbert’s theorem, that gives a global error estimate, whereas our approach may give an local error estimate.

**Mathematical modeling and data mining tehcniques applied to Engineering Sciences**.

**1. Characterization by Data Mining methods of solutions to Pareto optimal front: application to the identification of optimal trajectories aerodynamic micro-UAVs **(*Collaboration with M. Hamdaoui and P. Sagaut the IJLRDA-UPMC and S. Doncieux of ISIR-UPMC*).

The aim of this work is to present a method to find and analyze maximum propulsive efficiency kinematics for a birdlike flapping-wing unmanned aerial vehicle using multiobjective evolutionary optimization and data-mining tools.

For the sake of clarity and simplicity, simple geometry (rectangular wings with the same profile along the span) and simple kinematics (symmetrical harmonic dihedral motion) are used. In addition, it is assumed that the birdlike aerial vehicle (for which the span and surface area are, respectively, 1m and 0:15 m2) is in horizontal motion at low cruise speed (6 m=s).

The aerodynamic performances of the flapping-wing vehicle are evaluated with a semiempirical flight physics model and the problem is solved using an efficient multiobjective evolutionary algorithm called -MOEA. Groups of attractive solutions are defined on the Pareto surface, and the most efficient solutions within these groups are characterized. Given the high dimensionality of the Pareto surface in the kinematic parameters space, data-mining techniques are used to conduct the study.

First, it is shown that these groups can be qualified versus the whole Pareto surface by accurate mathematical relations on the kinematic parameters. Second, the inner structure of each group is studied and highly accurate mathematical relations are found on the optimized parameters describing the most efficient solutions.

**Mathematical modelling****Data Mining explorations**for medical applications.

**1. Data Mining method applied to "Sleep Apnea Syndrom": Modelling the interaction of treatment of sleep apnea problems and Insomnia, **(Collaboration with Dr. B. Fleury and XL Nguyen Sleep from the Unit of the Pneumology Service at the Hospital Saint Antoine in Paris (*CETTSSA), Paris*).

Obstructive Sleep Apnoea Syndrome (OSAS) and insomnia are common pathologies sharing a high comorbidity. CPAP is a cumbersome treatment. Yet, CPAP compliance must remain optimal in order to reverse excessive daytime sleepiness and prevent the cardiovascular consequences of OSAS.

But chronic insomnia could negatively affect CPAP compliance. To assess the consequences of insomnia symptoms on long-term CPAP we explore a prospective study which was conducted on 148 OSAS patients (RDI = 39.0 ± 21.3/h), age = 54.8 ± 11.8 years, BMI = 29.1 ± 6.3 kg/m2, Epworth Score = 12.2 ± 5.4, on CPAP.

Using the Insomnia Severity Index (ISI) as an indicator of insomnia (ISI P14 = moderate to severe insomnia) and baseline data (anthropometric data, sleeping medication intakes, CPAP compliance, Epworth, Pittsburgh Sleep Quality and ISI scores, polygraphic recording data), Data Mining techniques (mainly, decision trees) identified the major rules explaining the features ‘‘High” or ‘‘Low ISI” and ‘‘High” or ‘‘Low Use” in the groups defined, according to the median values of the ISI and the 6th month-compliance, respectively.

The main results we found are: Median ISI was 15 and median 6th month-compliance was 4.38 h/night. Moderate to severe insomnia complaint was found in 50% of patients. In the ‘‘High” and ‘‘Low ISI,” the 6th month-compliance was not significantly different (3.7 ± 2.3 vs 4.2 ± 2.3 h/night). In the classification models of compliance, the ISI was not a predictor of CPAP rejection or of long-term use, the predictor for explaining CPAP abandonment being the RDI, and the predictor of the 6th month-compliance being the one month-compliance.

As a conclusion, insomnia symptoms were highly prevalent in OSAS patients, but had no impact on CPAP rejection or on long-term compliance.

**2. Evolution of health-related quality of life two to seven years after retropubic radical prostatectomy: evaluation by UCLA prostate cancer index.** (*Collaboration with Prof. M. Zerbib Head of Urology - Cancerology Department of the Hospital Cochin in Paris*).

To determine changes in health-related quality of life (HRQOL) in patients treated with retropubic radical prostatectomy (RP) between two and seven years after surgery, a questionnaire from the University of California Los Angeles was sent to 142 patients previously treated with retropubic RP as mono-therapy for clinically localized prostate cancer.

Patients were divided into five groups according to time from surgery. Demographics, clinical and pathological characteristics of patients were compared between these groups. Correlation coefficients controlled for age at the time of questionnaire between HRQOL scores and time from RP were assessed. A total of 105 patients (74%) returned the questionnaire. The mean time from surgery was 48 months (range 25-84). Demographics, clinical and pathological characteristics of patients were not statistically different between time groups.

Several recoding items were found to decrease significantly with the time from RP including physical functioning, role limitations due to physical health problem, vitality, and general health. In contrary, urinary, bowel and sexual scores were not significantly correlated to time from RP. Although sexual, urinary and bowel scores seem to remain stable from 2 to 7 years following RP, general health appears to significantly deteriorate with time after RP, independent of the patient's age at the time of the questionnaire.

**3. Mathematical modeling of the immune system during injections combined BCG and interleukin-2 in the treatment of Superficial Bladder Cancer **(*Collaboration with Prof. J. Gluckman, Head of Immunology Department of the Hospital Saint -Louis in Paris and Dr. S. Bunimovich from University of Tel-Aviv*).

We developed a mathematical model that describes the growth of superficial bladder cancer and the effect thereupon of immunotherapy based on the administration of Bacillus Calmette-Guerin (BCG) combined or not with interleukin-2 (IL-2). Intravesical instillations of BCG performed after surgical removal of tumors represents an established treatment with approximately 50% success rate. So far, attempts to improve this efficiency have not led to essential changes. However, convincing clinical results have been reported on the combination of IL-2 to BCG, even though this is still not applied in current practice.

The present model provides insights into the dynamical outcomes arising in the bladder from the interactions of immune cells with tumor cells in the course of BCG therapy associated or not with IL-2. Specifically, from the simulations performed using seven ordinary and non-linear differential

equations we obtained indications on the conditions that would result in successful bladder cancer treatment.

We show that immune cells–effector lymphocytes and antigen-presenting cells–expand and reach a sustainable plateau under BCG treatment, which may account for its beneficial effect, resulting from inflammatory ‘‘side-effects’’ which eliminate residual or eventual newly arising tumor cells, providing thus protection from further cancer development.

We find, however, that IL-2 does not actually potentiate the effect of BCG as regards tumor cell eradication. Hence, associating both under the conditions simulated should not result in more efficient treatment of bladder cancer patients.

**4. Statistical techniques applied to the modification of the breathing pattern due to repetitive trnascanial magnetic stimulation **(Collaboration with Prof. T. Similovski - Head of Pneumology Department of Pitié-Salepétrière Hospital in Paris).

In awake humans, breathing depends on automatic brainstem pattern generators. It is also heavily influenced by cortical networks. For example, functional magnetic resonance imaging and electroencephalographic data show that the supplementary motor area becomes active when breathing is made difficult by inspiratory mechanical loads like resistances or threshold valves. This is associated with perceived respiratory discomfort.

We hypothesized that manipulating the excitability of the supplementary motor area with repetitive transcranial magnetic stimulation would modify the breathing pattern response to an experimental inspiratory load possibly respiratory discomfort. Seven subjects (3 men, age 25±4) were studied. Breathing pattern and respiratory discomfort during inspiratory loading were described before and after conditioning the supplementary motor area with repetitive stimulation, using an excitatory paradigm (5Hz stimulation), an inhibitory paradigm, or sham stimulation.

No significant change in breathing pattern during loading was observed after sham conditioning. Excitatory conditioning shortened inspiratory time (p=0.001), decreased tidal volume (p=0.016), and decreased ventilation (p=0.003), as corroborated by an increased end-tidal expired carbon dioxide (p=0.013). Inhibitory conditioning did not affect ventilation, but lengthened expiratory time (p=0.031).

Respiratory discomfort was mild under baseline conditions, and unchanged after conditioning of the supplementary motor area. This is the first study to show that repetitive transcranial magnetic stimulation conditioning of the cerebral cortex can alter breathing pattern. A 5 Hz conditioning protocol, known to enhance corticophrenic excitability, can reduce the amount of hyperventilation induced by inspiratory threshold loading. Further studies are needed to determine whether and under what circumstances rTMS can have an effect on dyspnoea.