Publications

2017

• Rare event simulation related to financial risks: efficient estimation and sensitivity analysis
  
  • Agarwal Ankush
  • de Marco Stefano
  • Gobet Emmanuel
  • Liu Gang
  , 2017. In this paper, we develop the reversible shaking transformation methods on path space of Gobet and Liu [GL15] to estimate the rare event statistics arising in different financial risk settings which are embedded within a unified framework of isonormal Gaussian process. Namely, we combine splitting methods with both Interacting Particle System (IPS) technique and ergodic transformations using Parallel-One-Path (POP) estimators. We also propose an adaptive version for the POP method and prove its convergence. We demonstrate the application of our methods in various examples which cover usual semi-martingale stochastic models (not necessarily Markovian) driven by Brownian motion and, also, models driven by fractional Brownian motion (non semi-martingale) to address various financial risks. Interestingly, owing to the Gaussian process framework, our methods are also able to efficiently handle the important problem of sensitivities of rare event statistics with respect to the model parameters.
• Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio
  
  • Agarwal Ankush
  • Sircar Ronnie
  , 2017. We consider an investor who seeks to maximize her expected utility derived from her terminal wealth relative to the maximum performance achieved over a fixed time horizon, and under a portfolio drawdown constraint, in a market with local stochastic volatility (LSV). In the absence of closed-form formulas for the value function and optimal portfolio strategy, we obtain approximations for these quantities through the use of a coefficient expansion technique and nonlinear transformations. We utilize regularity properties of the risk tolerance function to numerically compute the estimates for our approximations. In order to achieve similar value functions, we illustrate that, compared to a constant volatility model, the investor must deploy a quite different portfolio strategy which depends on the current level of volatility in the stochastic volatility model.
• Dynamics of a two-level system with priorities and application to an emergency call center
  
  • Boeuf Vianney
  , 2017. In this thesis, we analyze the dynamics of discrete event systems with synchronization and priorities, by the means of Petri nets and queueing networks.We apply this to the performance evaluation of an emergency call center.Our original motivation is practical. During the period of this work, a new emergency call center became operative in Paris area, handling emergency calls to police and firemen.The new organization includes a two-level call treatment. A first level of operators answers calls, identifies urgent calls and handles (numerous) non-urgent calls.Second level operators are specialists (policemen or firemen) and handle emergency demands.When a call is identified at level 1 as extremely urgent, the operator stays in line with the call until a level 2 operator answers. The call has priority for level 2 operators.A consequence of this procedure is that, when level 2 operators are busy, level 1 operators wait with extremely urgent calls, and the capacity of level 1 diminishes.We are interested in the performance evaluation of various systems corresponding to this general description, in stressed situations.We propose three different models addressing this kind of systems.The first two are timed Petri net models.We enrich the classical free choice Petri nets by allowing conflict situations in which the routing is solved by priorities.The main difficulty in this situation is that the operator of the dynamics becomes non monotone.In a first model, we consider discrete dynamics for this class of Petri nets, with constant holding times on places.We prove that the counter variables of an execution of the Petri net are solutions of a piecewise linear system with delays.As far as we know, this proof is new, even for the class of free choice nets, which is a subclass of ours.We investigate the stationary regimes of the dynamics, and characterize the affine ones as solutions of a piecewise linear system, which can be seen as a system over a tropical (min-plus) semifield of germs.Numerical experiments show that, however, convergence does not always holds towards these affine stationary regimes.The second model is a ``continuization'' of the previous one. For the same class of Petri nets, we propose dynamics expressed by differential equations, so that the tokens and time events become continue.For this differential system with discontinuous righthandside, we establish the existence and uniqueness of the solution.By using differential equations, we aim at obtaining a simpler model in which discrete time pathologies disappear. We show that the stationary regimes are the same as the stationary regimes of the discrete time dynamics.Numerical experiments tend to show that, in this setting, convergence effectively holds.We also model the emergency call center described above as a queueing system, taking into account the randomness of the different call center variables.For this system, we prove that, under an appropriate scaling, the dynamics converges to a fluid limit which corresponds to the differential equations of our Petri net model.This provides support for the second model.Stochastic calculus for Poisson processes, generalized Skorokhod formulations and coupling arguments are the main tools used to establish this convergence.Hence, our three models of an identical emergency call center yield the same schematic asymptotic behavior, expressed as a piecewise linear system of the parameters, and describing the different congestion phases of the system.In a second part of this thesis, simulations are carried out and analyzed, taking into account the many subtleties of our case study (for example, we construct probability distributions based on real data analysis).The simulations confirm the schematic behavior described by our mathematical models.We also address the complex interactions coming from the heterogeneous nature of level 2.
• MODULAR INEQUALITIES FOR THE MAXIMAL OPERATOR IN VARIABLE LEBESGUE SPACES
  
  • Cruz-Uribe David
  • Di Fratta Giovanni
  • Fiorenza Alberto
  , 2017. A now classical result in the theory of variable Lebesgue spaces due to Lerner [A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543] is that a modular inequality for the Hardy-Littlewood maximal function in $L^{p(\cdot)}(\mathbb{R}^n)$ holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality \[ \int_\Omega Mf(x)^{p(x)}\,dx \leq c_1 \int_\Omega |f(x)|^{q(x)}\,dx + c_2, \] where $c_1,\,c_2$ are non-negative constants and $\Omega$ is any measurable subset of $\mathbb{R}^n$. As a corollary we get sufficient conditions for the modular inequality \[ \int_\Omega |Tf(x)|^{p(x)}\,dx \ \leq c_1 \int_\Omega |f(x)|^{q(x)}\,dx + c_2, \] where $T$ is any operator that is bounded on $L^p(\Omega)$, $1<p<\infty$.
• Statistical inference of Ornstein-Uhlenbeck processes : generation of stochastic graphs, sparsity, applications in finance
  
  • Matulewicz Gustaw
  , 2017. The subject if this thesis is the statistical inference of multi-dimensional Ornstein-Uhlenbeck processes. In a first part, we introduce a model of stochastic graphs, defined as binary observations of a trajectory. We show then that it is possible to retrieve the dynamic of the underlying trajectory from the binary observations. For this, we build statistics of the stochastic graph and prove new results on their convergence in the long-time, high-frequency setting. We also analyse the properties of the stochastic graph from the point of view of evolving networks. In a second part, we work in the setting of complete information and continuous time. We add then a sparsity assumption applied to the drift matrix coefficient of the Ornstein-Uhlenbeck process. We prove sharp oracle inequalities for the Lasso estimator, construct a lower bound on the estimation error for sparse estimators and show optimality properties of the Adaptive Lasso estimator. Then, we apply the methods to estimate mean-return properties of real-world financial datasets: daily returns of SP500 components and EURO STOXX 50 Dividend Future prices.
• Structural optimization under overhang constraints imposed by additive manufacturing technologies
  
  • Allaire Grégoire
  • Dapogny Charles
  • Estevez Rafael
  • Faure Alexis
  • Michailidis Georgios
  Journal of Computational Physics, Elsevier, 2017, 351, pp.295-328. This article addresses one of the major constraints imposed by additive manufacturing processes on shape optimization problems-that of overhangs, i.e. large regions hanging over void without sufficient support from the lower structure. After revisiting the 'classical' geometric criteria used in the literature, based on the angle between the structural boundary and the build direction, we propose a new mechanical constraint functional, which mimics the layer by layer construction process featured by additive manufacturing technologies, and thereby appeals to the physical origin of the difficulties caused by overhangs. This constraint, as well as some variants, are precisely defined; their shape derivatives are computed in the sense of Hadamard's method and numerical strategies are extensively discussed, in two and three space dimensions, to efficiently deal with the appearance of overhang features in the course of shape optimization processes. (10.1016/j.jcp.2017.09.041)
  DOI : 10.1016/j.jcp.2017.09.041
• Tropical Kraus maps for optimal control of switched systems
  
  • Gaubert Stéphane
  • Stott Nikolas
  , 2017, pp.1-15. Kraus maps (completely positive trace preserving maps) arise classically in quantum information, as they describe the evolution of noncommutative probability measures. We introduce tropical analogues of Kraus maps, obtained by replacing the addition of positive semidefinite matrices by a multivalued supremum with respect to the L\"owner order. We show that non-linear eigenvectors of tropical Kraus maps determine piecewise quadratic approximations of the value functions of switched optimal control problems. This leads to a new approximation method, which we illustrate by two applications: 1) approximating the joint spectral radius, 2) computing approximate solutions of Hamilton-Jacobi PDE arising from a class of switched linear quadratic problems studied previously by McEneaney. We report numerical experiments, indicating a major improvement in terms of scalability by comparison with earlier numerical schemes, owing to the "LMI-free" nature of our method.
• Exact output stabilization at unobservable points: Analysis via an example
  
  • Lagache Marc-Aurèle
  • Serres Ulysse
  • Gauthier Jean-Paul
  , 2017, pp.6744-6749. (10.1109/CDC.2017.8264676)
  DOI : 10.1109/CDC.2017.8264676
• Analysis of order book flows using a non-parametric estimation of the branching ratio matrix
  
  • Achab M.
  • Bacry Emmanuel
  • Muzy J.
  • Rambaldi M.
  Quantitative Finance, Taylor & Francis (Routledge), 2017, 18 (2), pp.199-212. (10.1080/14697688.2017.1403132)
  DOI : 10.1080/14697688.2017.1403132
• Error estimates for the Euler discretization of an optimal control problem with first-order state constraints
  
  • Bonnans Joseph Frederic
  • Festa Adriano
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2017, 55 (2), pp.445--471. We study the error introduced in the solution of an optimal control problem with first order state constraints, for which the trajectories are approximated with a classical Euler scheme. We obtain order one approximation results in the L ∞ norm (as opposed to the order 2/3 obtained in the literature). We assume either a strong second order optimality condition, or a weaker one in the case where the state constraint is scalar, satisfies some hypotheses for junction points, and the time step is constant. Our technique is based on some homotopy path of discrete optimal control problems that we study using perturbation analysis of nonlinear programming problems.
• Generalization of the Fourier-spectral Eyre scheme for the phase-field equations: Application to self-assembly dynamics in materials
  
  • Demange Gilles
  • Chamaillard M.
  • Zapolsky Helena
  • Lavrskyi M.
  • Vaugeois Antoine
  • Lunéville L.
  • Simeone D.
  • Patte Renaud
  Computational Materials Science, Elsevier, 2017, 144, pp.11-22. (10.1016/j.commatsci.2017.11.044)
  DOI : 10.1016/j.commatsci.2017.11.044
• A Linear-Time Kernel Goodness-of-Fit Test
  
  • Jitkrittum Wittawat
  • Xu Wenkai
  • Szabó Zoltán
  • Fukumizu Kenji
  • Gretton Arthur
  , 2017. We propose a novel adaptive test of goodness-of-fit, with computational cost linear in the number of samples. We learn the test features that best indicate the differences between observed samples and a reference model, by minimizing the false negative rate. These features are constructed via Stein’s method, meaning that it is not necessary to compute the normalising constant of the model. We analyse the asymptotic Bahadur efficiency of the new test, and prove that under a mean-shift alternative, our test always has greater relative efficiency than a previous linear-time kernel test, regardless of the choice of parameters for that test. In experiments, the performance of our method exceeds that of the earlier linear-time test, and matches or exceeds the power of a quadratic-time kernel test. In high dimensions and where model structure may be exploited, our goodness of fit test performs far better than a quadratic-time two-sample test based on the Maximum Mean Discrepancy, with samples drawn from the model.
• Learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data
  
  • Chevallier Juliette
  • Oudard Stéphane
  • Allassonnière Stéphanie
  , 2017. We introduce a hierarchical model which allows to estimate a group-average piecewise-geodesic trajectory in the Riemannian space of measurements and individual variability. This model falls into the well defined mixed-effect models. The subject-specific trajectories are defined through spatial and temporal transformations of the group-average piecewise-geodesic path, component by component. Thus we can apply our model to a wide variety of situations. Due to the non-linearity of the model, we use the Stochastic Approximation Expectation-Maximization algorithm to estimate the model parameters. Experiments on synthetic data validate this choice. The model is then applied to the metastatic renal cancer chemotherapy monitoring: we run estimations on RECIST scores of treated patients and estimate the time they escape from the treatment. Experiments highlight the role of the different parameters on the response to treatment.
• Learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data
  
  • Chevallier Juliette
  • Oudard Stéphane
  • Allassonnière Stéphanie
  , 2017. We introduce a hierarchical model which allows to estimate both a group-representative piecewise-geodesic trajectory in the Riemannian space of shape and inter-individual variability. Following the approach of Schiratti et al. (NIPS, 2015), we estimate a representative piecewise-geodesic trajectory of the global progression and together with spacial and temporal inter-individual variabilities. We first introduce our model in its most generic formulation and then make it explicit for RECIST (Therasse et al., JNCI, 2000) score monitoring, i.e. for one-dimension manifolds and piecewise-logistically distributed data.
• Proceedings of the international conference on computational mathematics and inverse problems honoring Peter Monk
  
  • Haddar Houssem
  • Cakoni Fioralba
  • Sun Jiguang
  • Bacuta Constantin
  , 2017. This special issue is dedicated to Professor Peter Monk in honor of his contribution and leadership in the areas of direct and inverse scattering theory and the numerical analysis of partial differential equations and integral equations, for more than 25 years. The papers in this special issue were solicited from the invited speakers at the International Conference on Computational Mathematics and Inverse Problems honoring Peter Monk, held at Michigan Tech University, Houghton, Michigan, August 15–19, 2016. As organizers of this conference and close collaborators of Peter Monk, we are very honored to have had the opportunity to facilitate this special scientific and social event. It was a special occasion that gathered together long term colleagues, collaborators, former students and friends of Professor Monk. It gave us great pleasure to be guest editors of this special issue of the CAMWA that is a collection of original research papers on topics on numerical methods for PDEs and computational inverse problems. Much of the work presented here has been directly or indirectly influenced by the work of Peter Monk, offering the reader a glimpse of his significant impact in these research areas. We would like to thank all authors who have contributed a paper for this special issue. Special thanks goes to the Editor in Chief of Computers and Mathematics with Applications, Leszek Demkowicz, for supporting and facilitating this publication. We would also like to thank all the participants of the International Conference on Computational Mathematics and Inverse Problems who made such a successful, stimulating and pleasant event possible. Last, but not least, we would also like to thank the sponsors of the conference: Kliakhandler Fellowship, Michigan Tech, University of Delaware, and the National Science Foundation. (10.1016/j.camwa.2017.09.001)
  DOI : 10.1016/j.camwa.2017.09.001
• Tightness and duality of martingale transport on the Skorokhod space *
  
  • Guo Gaoyue
  • Tan Xiaolu
  • Touzi Nizar
  Stochastic Processes and their Applications, Elsevier, 2017. The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of c`adì ag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S−topology and the dynamic programming principle 1 .
• Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers
  
  • Haddar Houssem
  • Nguyen Thi-Phong
  Computers & Mathematics with Applications, Elsevier, 2017, 74 (11), pp.2831-2855. This paper is dedicated to the design and analysis of sampling methods to reconstruct the shape of a local perturbation in a periodic layer from measurements of scattered waves at a fixed frequency. We first introduce the model problem that corresponds with the semi-discretized version of the continous model with respect to the Floquet-Bloch variable. We then present the inverse problem setting where (propagative and evanescent) plane waves are used to illuminate the structure and measurements of the scattered wave at a parallel plane to the periodicity directions are performed. We introduce the near field operator and analyze two possible factorizations of this operator. We then establish sampling methods to identify the defect and the periodic background geometry from this operator measurement. We also show how one can recover the geometry of the background independently from the defect. We then introduce and analyze the single Floquet-Bloch mode measurement operators and show how one can exploit them to built an indicator function of the defect independently from the background geometry. Numerical validating results are provided for simple and complex backgrounds. (10.1016/j.camwa.2017.07.015)
  DOI : 10.1016/j.camwa.2017.07.015
• Blocking strategies and stability of particle Gibbs samplers
  
  • Singh S
  • Lindsten F
  • Moulines Eric
  Biometrika, Oxford University Press (OUP), 2017, 104 (4), pp.953-969. Sampling from the posterior probability distribution of the latent states of a hidden Markov model is nontrivial even in the context of Markov chain Monte Carlo. To address this, Andrieu et al. (2010) proposed a way of using a particle filter to construct a Markov kernel that leaves the posterior distribution invariant. Recent theoretical results have established the uniform ergodicity of this Markov kernel and shown that the mixing rate does not deteriorate provided the number of particles grows at least linearly with the number of latent states. However, this gives rise to a cost per application of the kernel that is quadratic in the number of latent states, which can be prohibitive for long observation sequences. Using blocking strategies, we devise samplers that have a stable mixing rate for a cost per iteration that is linear in the number of latent states and which are easily parallelizable.
• New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data
  
  • Audibert Lorenzo
  • Cakoni Fioralba
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2017, 33 (12), pp.1-30. In this paper we develop a general mathematical framework to determine interior eigenvalues from a knowledge of the modified far field operator associated with an unknown (anisotropic) inhomogeneity. The modified far field operator is obtained by subtracting from the measured far field operator the computed far field operator corresponding to a well-posed scattering problem depending on one (possibly complex) parameter. Injectivity of this modified far field operator is related to an appropriate eigenvalue problem whose eigenvalues can be determined from the scattering data, and thus can be used to obtain information about material properties of the unknown inhomogeneity. We discuss here two examples of such modification leading to a Steklov eigenvalue problem, and a new type of the transmission eigenvalue problem. We present some numerical examples demonstrating the viability of our method for determining the interior eigenvalues form far field data. (10.1088/1361-6420/aa982f)
  DOI : 10.1088/1361-6420/aa982f
• A Bayesian mixed-effects model to learn trajectories of changes from repeated manifold-valued observations
  
  • Schiratti Jean-Baptiste
  • Allassonniere Stéphanie
  • Colliot Olivier
  • Durrleman Stanley
  Journal of Machine Learning Research, Microtome Publishing, 2017, 18, pp.1-33. We propose a generic Bayesian mixed-effects model to estimate the temporal progression of a biological phenomenon from observations obtained at multiple time points for a group of individuals. The progression is modeled by continuous trajectories in the space of measurements. Individual trajectories of progression result from spatiotemporal transformations of an average trajectory. These transformations allow to quantify the changes in direction and pace at which the trajectories are followed. The framework of Rieman-nian geometry allows the model to be used with any kind of measurements with smooth constraints. A stochastic version of the Expectation-Maximization algorithm is used to produce produce maximum a posteriori estimates of the parameters. We evaluate our method using series of neuropsychological test scores from patients with mild cognitive impairments later diagnosed with Alzheimer's disease, and simulated evolutions of symmetric positive definite matrices. The data-driven model of the impairment of cognitive functions shows the variability in the ordering and timing of the decline of these functions in the population. We show also that the estimated spatiotemporal transformations effectively put into correspondence significant events in the progression of individuals.
• Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding
  
  • Gautier Eric
  • Le Pennec Erwan
  , 2017. In the random coefficients binary choice model, a binary variable equals 1 iff an index $X^\top\beta$ is positive. The vectors $X$ and $\beta$ are independent and belong to the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$. We prove lower bounds on the minimax risk for estimation of the density $f_{\beta}$ over Besov bodies where the loss is a power of the $L^p(\mathbb{S}^{d-1})$ norm for $1\le p\le \infty$. We show that a hard thresholding estimator based on a needlet expansion with data-driven thresholds achieves these lower bounds up to logarithmic factors.
• Branching processes for structured populations and estimators for cell division
  
  • Marguet Aline
  , 2017. We study structured populations without interactions from a probabilistic and a statistical point of view. The underlying motivation of this work is the understanding of cell division mechanisms and of cell aging. We use the formalism of branching measure-valued Markov processes. In our model, each individual is characterized by a trait (age, size, etc...) which moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on the trait of the mother and on the number of descendants. First, we study the trait of a uniformly sampled individual in the population. We explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait of a "typical" individual by giving its associated infinitesimal generator. Then, we study the asymptotic behavior of the empirical measure associated with the branching process. Under assumptions assuring the ergodicity of the auxiliary process, we prove that the auxiliary process asymptotically corresponds to the trait along its ancestral lineage of a uniformly sampled individual in the population. Finally, we address the problem of parameter estimation in the case of a branching process structured by a diffusion. We consider data composed of the trait at birth of all individuals in the population until a given generation. We give kernel estimators for the transition density and the invariant measure of the chain corresponding to the trait of an individual along a lineage. Moreover, in the case of a reflected diffusion on a compact set, we use maximum likelihood estimation to reconstruct the division rate. We prove consistency and asymptotic normality for this estimator. We also carry out the numerical implementation of the estimator.
• The spatial structure of genetic diversity under natural selection and in heterogeneous environments
  
  • Forien Raphael
  , 2017. This thesis deals with the spatial structure of genetic diversity. We first study a measure-valued process describing the evolution of the genetic composition of a population subject to natural selection. We show that this process satisfies a central limit theorem and that its fluctuations are given by the solution to a stochastic partial differential equation. We then use this result to obtain an estimate of the drift load in spatially structured populations.Next we investigate the genetic composition of a populations whose individuals move more freely in one part of space than in the other (a situation called dispersal heterogeneity). We show in this case the convergence of allele frequencies via the convergence of ancestral lineages to a system of skew Brownian motions.We then detail the effect of a barrier to gene flow dividing the habitat of a population. We show that ancestral lineages follow partially reflected Brownian motions, of whom we give several constructions.To apply these results, we adapt a method for demographic inference to the setting of dispersal heterogeneity. This method makes use of long blocks of genome along which pairs of individuals share a common ancestry, and allows to estimate several demographic parameters when they vary accross space. To conclude, we demonstrate the accuracy of our method on simulated datasets.
• On the long-time behaviour of an age and trait structured population dynamics
  
  • Roget Tristan
  , 2017. We study the long-time behaviour of a population structured by age and a phenotypic trait under a selection-mutation dynamics. By analysing spectral properties of a family of positive operators on measure spaces, we show the existence of eventually singular stationary solutions. When the stationary measures are absolutely continuous with a continuous density, we show the convergence of the dynamics to the unique equilibrium. 1 Preliminaries and Main Results 1.1 Introduction Our ultimate goal is the understanding of the long-time behaviour of a population where the individuals differ by their physical age a È R and some hereditary variable x È S R d called trait. The population evolves as follows. An individual with trait x È S and age a È R has a death rate DÔx, aÕ cN where D is the intrinsic death rate, N is the population size and c 0 the competition rate. This individual gives birth at rate BÔx, aÕ. At every birth, a mutation occurs with probability p È ×0, 1Ö and the trait of the newborn y È S is choosen according a distribution kÔx, a, yÕdy. Otherwise, the descendant inherits of the trait x È S. In his thesis [19], Tran introduced an individual-based stochastic model to describe such a discrete population. The population is described by a random point measure Z K t 1 K N K t ô i1 δ Ôx i ÔtÕ,a i ÔtÕÕ (1) which evolves as a càdlàg Markov process with values in the set M ÔS ¢ R Õ of positive finite measures on S ¢ R and each jump corresponds to birth or death of individuals. When the order K of the size of the population goes to infinity such that Z K 0 approximates a deterministic measure n 0 È M ÔS ¢ R Õ, it is shown (see [19],[20]) that the process approximates the unique weak solution (in the sense given by (10)) Ôn t Õ t0 È CÔR , M ÔS ¢ R ÕÕ of the partial differential equation t n t Ôx, aÕ a n t Ôx, aÕ ¡ ¡ DÔx, aÕ c S ¢R n t Ôy, αÕdydα © n t Ôx, aÕ, n t Ôx, 0Õ F Ön t × ÔxÕ, Ôt, x, aÕ È R ¢ S ¢ R , (2) 1
• Fast discrete convolution in R2 using Sparse Bessel Decomposition
  
  • Averseng Martin
  , 2017. We describe an efficient algorithm for computing the matrix vector products that appear in the numerical resolution of boundary integral equations in 2 space dimension. This work is an extension of the so-called Sparse Cardinal Sine Decomposition algorithm by Alouges et al., which is restricted to three-dimensional setups. Although the approach is similar, significant differences appear throughout the analysis of the method. Bessel decomposition, in particular, yield longer series for the same accuracy. We propose a careful study of the method that leads to a precise estimation of the complexity in terms of the number of points and chosen accuracy. We also provide numerical tests to demonstrate the efficiency of this approach. We give the compression performance for a N × N linear system for several values N up to 10^7 and report the computation time for the off-line and on-line parts of our algorithm. We also include a toy application to sound canceling to further illustrate the efficiency of our method.