FNRS Chercheur Qualifié/Research Associate (01/10/2010-present)

Maître d'enseignement (01/10/2010-present)

President of the Mathematics Department (01/10/2014-02/10/2017)

E-mail: igloris at ulb.ac.be

Phone: ++32-2-650 58 54

Office: Campus Plaine, Building NO, 7th floor, room 2.O.7.107

Mailing address:

Mécanique et mathématique appliquée, CP 217

Département de Mathématique

Faculté des Sciences

Université libre de Bruxelles

Boulevard du Triomphe

B-1050 Bruxelles, Belgium

MATH-F431: Optimisation, algorithmes et applications, 5ECTS, 24h théorie, 12h exercices; fiche de cours: français/English.

MATH-F400: Compléments de mathématiques : Analyse appliquée, 5ECTS, 36h théorie, 12h exercices; fiche de cours: français/English.

Author | Title | Year | Journal/Proceedings | Reftype | DOI/URL |
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Chen, J. and Loris, I. | On starting and stopping criteria for nested primal-dual iterations | 2018 | Numerical algorithms | article | URL |

Abstract: The importance of an adequate inner loop starting point (as opposed to a sufficient inner loop stopping rule) is discussed in the context of a numerical optimization algorithm consisting of nested primal-dual proximal-gradient iterations. While the number of inner iterations is fixed in advance, convergence of the whole algorithm is still guaranteed by virtue of a warm-start strategy for the inner loop, showing that inner loop ``starting rules" can be just as effective as ``stopping rules'' for guaranteeing convergence. The algorithm itself is applicable to the numerical solution of convex optimization problems defined by the sum of a differentiable term and two possibly non-differentiable terms. One of the latter terms should take the form of the composition of a linear map and a proximable function, while the differentiable term needs an accessible gradient. The algorithm reduces to the classical proximal gradient algorithm in certain special cases and it also generalizes other existing algorithms. In addition, under some conditions of strong convexity, we show a linear rate of convergence. |
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BibTeX:
@article{Chen.Loris2018, author = {Jixin Chen and Ignace Loris}, title = {On starting and stopping criteria for nested primal-dual iterations}, journal = {Numerical algorithms}, year = {2018}, note = {Accepted.}, url = {https://arxiv.org/abs/1806.07677} } |
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Bonettini, S., Loris, I., Porta, F., Prato, M. and Rebegoldi, S. | On the convergence of a linesearch based proximal-gradient method for nonconvex optimization | 2017 | Inverse Problems Vol. 33(5), pp. 055005 |
article | DOI URL |

Abstract: We consider a variable metric line-search based proximal gradient
method for the minimization of the sum of a smooth, possibly nonconvex function plus a convex, possibly nonsmooth term. We prove convergence of this iterative algorithm to a minimum point if the objective function satisfies the Kurdyka-Łojasiewicz property at each point of its domain, under the assumption that a limit point exists. The proposed method is applied to a wide collection of image processing problems and our numerical tests show that our algorithm results to be flexible, robust and competitive if compared to recently proposed approaches able to address the optimization problems arising in the considered applications. |
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BibTeX:
@article{Bonettini2016, author = {S. Bonettini and I. Loris and F. Porta and M. Prato and S. Rebegoldi}, title = {On the convergence of a linesearch based proximal-gradient method for nonconvex optimization}, journal = {Inverse Problems}, year = {2017}, volume = {33}, number = {5}, pages = {055005}, url = {https://arxiv.org/abs/1605.03791}, doi = {https://doi.org/10.1088/1361-6420/aa5bfd} } |
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Bonettini, S., Loris, I., Porta, F. and Prato, M. | Variable metric inexact line-search based methods for nonsmooth optimization | 2016 | Siam Journal on Optimization Vol. 26(2), pp. 891-921 |
article | DOI URL |

Abstract: We develop a new proximal-gradient method for minimizing the sum of
a differentiable, possibly nonconvex, function plus a convex, possibly non differentiable, function. The key features of the proposed method are the definition of a suitable descent direction, based on the proximal operator associated to the convex part of the objective function, and an Armijo-like rule to determine the step size along this direction ensuring the sufficient decrease of the objective function. In this frame, we especially address the possibility of adopting a metric which may change at each iteration and an inexact computation of the proximal point defining the descent direction. For the more general nonconvex case, we prove that all limit points of the iterates sequence are stationary, while for convex objective functions we prove the convergence of the whole sequence to a minimizer, under the assumption that a minimizer exists. In the latter case, assuming also that the gradient of the smooth part of the objective function is Lipschitz, we also give a convergence rate estimate, showing the 𝒪(1k) complexity with respect to the function values. We also discuss verifiable sufficient conditions for the inexact proximal point and we present the results of a numerical experience on a convex total variation based image restoration problem, showing that the proposed approach is competitive with another state-of-the-art method. |
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BibTeX:
@article{Bonettini2015, author = {S. Bonettini and I. Loris and F. Porta and M. Prato}, title = {Variable metric inexact line-search based methods for nonsmooth optimization}, journal = {Siam Journal on Optimization}, year = {2016}, volume = {26}, number = {2}, pages = {891--921}, url = {http://arxiv.org/abs/1506.00385}, doi = {https://doi.org/10.1137/15M1019325} } |
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Prato, M., Bonettini, S., Loris, I., Porta, F. and Rebegoldi, S. | On the constrained minimization of smooth Kurdyka-Łojasiewicz functions with the scaled gradient projection method | 2016 | Journal of Physics: Conference Series Vol. 756(1), pp. 012001 |
article | DOI |

Abstract: The scaled gradient projection (SGP) method is a first-order optimization
method applicable to the constrained minimization of smooth functions and exploiting a scaling matrix multiplying the gradient and a variable steplength parameter to improve the convergence of the scheme. For a general nonconvex function, the limit points of the sequence generated by SGP have been proved to be stationary, while in the convex case and with some restrictions on the choice of the scaling matrix the sequence itself converges to a constrained minimum point. In this paper we extend these convergence results by showing that the SGP sequence converges to a limit point provided that the objective function satisfies the Kurdyka-Łojasiewicz property at each point of its domain and its gradient is Lipschitz continuous. |
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BibTeX:
@article{Prato.ea2016, author = {Marco Prato and Silvia Bonettini and Ignace Loris and Federica Porta and Simone Rebegoldi}, title = {On the constrained minimization of smooth Kurdyka-Łojasiewicz functions with the scaled gradient projection method}, journal = {Journal of Physics: Conference Series}, year = {2016}, volume = {756}, number = {1}, pages = {012001}, note = {Proceedings of 6th International Workshop on New Computational Methods for Inverse Problems (Cachan, 20/05/2016).}, doi = {https://doi.org/10.1088/1742-6596/756/1/012001} } |
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Porta, F. and Loris, I. | On some steplength approaches for proximal algorithms | 2015 | Applied Mathematics and Computation Vol. 253, pp. 345-362 |
article | DOI |

Abstract: We discuss a number of novel steplength selection schemes for proximal-based
convex optimization algorithms. In particular, we consider the problem where the Lipschitz constant of the gradient of the smooth part of the objective function is unknown. We generalize two optimization algorithms of Khobotov type and prove convergence. We also take into account possible inaccurate computation of the proximal operator of the non-smooth part of the objective function. Secondly, we show convergence of an iterative algorithm with Armijo-type steplength rule, and discuss its use with an approximate computation of the proximal operator. Numerical experiments show the efficiency of the methods in comparison to some existing schemes. |
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BibTeX:
@article{Porta2015, author = {Federica Porta and Ignace Loris}, title = {On some steplength approaches for proximal algorithms}, journal = {Applied Mathematics and Computation}, year = {2015}, volume = {253}, pages = {345--362}, doi = {https://doi.org/10.1016/j.amc.2014.12.079} } |
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Loris, I. | Numerical algorithms for non-smooth optimization applicable to seismic recovery | 2014 | Handbook of Geomathematics, pp. 1-33 | incollection | DOI |

Abstract: Inverse problems in seismic tomography are often cast in the form
of an optimization problem involving a cost function composed of a data misfit term and regularizing constraint or penalty. Depending on the noise model that is assumed to underlie the data acquisition, these optimization problems may be non-smooth. Another source of lack of smoothness (differentiability) of the cost function may arise from the regularization method chosen to handle the ill-posed nature of the inverse problem. A numerical algorithm that is well suited to handle minimization problems involving two non-smooth convex functions and two linear operators is studied. The emphasis lies on the use of some simple proximity operators that allow for the iterative solution of non-smooth convex optimization problems. Explicit formulas for several of these proximity operators are given and their application to seismic tomography is demonstrated. |
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BibTeX:
@incollection{Loris2013, author = {Ignace Loris}, title = {Numerical algorithms for non-smooth optimization applicable to seismic recovery}, booktitle = {Handbook of Geomathematics}, publisher = {Springer}, year = {2014}, pages = {1--33}, edition = {Second edition.}, doi = {https://doi.org/10.1007/978-3-642-27793-1_65-3} } |
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Nassiri, V. and Loris, I. | An efficient algorithm for structured sparse quantile regression | 2014 | Computational Statistics Vol. 29(5), pp. 1321-1343 |
article | DOI URL |

Abstract: An efficient algorithm is derived for solving the quantile regression
problem combined with a group sparsity promoting penalty. The group sparsity of the regression parameters is achieved by using a 1,∞-norm penalty (or constraint) on the regression parameters. The algorithm is efficient in the sense that it obtains the regression parameters for a wide range of penalty parameters, thus enabling easy application of a model selection criteria afterwards. A Matlab implementation of the proposed algorithm is provided and some applications of the methods are studied. |
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BibTeX:
@article{Nassiri2014, author = {Vahid Nassiri and Ignace Loris}, title = {An efficient algorithm for structured sparse quantile regression}, journal = {Computational Statistics}, year = {2014}, volume = {29}, number = {5}, pages = {1321-1343}, url = {http://arxiv.org/abs/1302.6088}, doi = {https://doi.org/10.1007/s00180-014-0494-1} } |
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Schretter, C., Loris, I., Dooms, A. and Schelkens, P. | Total Variation reconstruction from quasi-random samples | 2014 | Proceedings of the second ``international Traveling Workshop on Interactions between Sparse models and Technology'' (iTWIST 2014), pp. 57-58 | inproceedings | URL |

Abstract: Pseudo-random numbers are often used for generating incoherent uniformly
distributed sample distributions. However randomness is a sufficient -- not necessary -- condition to ensure incoherence. If one wants to reconstruct an image from few samples, choosing a globally optimized set of evenly distributed points could capture the visual content more efficiently. This work compares classical random sampling with a simple construction based on properties of the fractional Golden ratio sequence and the Hilbert space filling curve. Images are then reconstructed using a total variation prior. Results show improvements in terms of peak signal to noise ratio over pseudo-random sampling. |
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BibTeX:
@inproceedings{Schretter2014, author = {Colas Schretter and Ignace Loris and Ann Dooms and Peter Schelkens}, title = {Total Variation reconstruction from quasi-random samples}, booktitle = {Proceedings of the second ``international Traveling Workshop on Interactions between Sparse models and Technology'' (iTWIST 2014)}, year = {2014}, pages = {57--58}, url = {http://arxiv.org/abs/1410.0719} } |
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Charléty, J., Voronin, S., Nolet, G., Loris, I., Simons, F.J., Sigloch, K. and Daubechies, I.C. | Global seismic tomography with sparsity constraints: Comparison with smoothing and damping regularization | 2013 | Journal of Geophysical Research - Solid Earth Vol. 118, pp. 4887-4899 |
article | DOI |

Abstract: We present a realistic application of an inversion scheme for global
seismic tomography that uses as prior information the sparsity of a solution, defined as having few nonzero coefficients under the action of a linear transformation. In this paper, the sparsifying transform is a wavelet transform. We use an accelerated iterative soft-thresholding algorithm for a regularization strategy, which produces sparse models in the wavelet domain. The approach and scheme we present may be of use for preserving sharp edges in a tomographic reconstruction and minimizing the number of features in the solution warranted by the data. The method is tested on a data set of time delays for finite-frequency tomography using the USArray network, the first application in global seismic tomography to real data. The approach presented should also be suitable for other imaging problems. From a comparison with a more traditional inversion using damping and smoothing constraints, we show that (1) we generally retrieve similar features, (2) fewer nonzero coefficients under a properly chosen representation (such as wavelets) are needed to explain the data at the same level of root-mean-square misfit, (3) the model is sparse or compressible in the wavelet domain, and (4) we do not need to construct a heterogeneous mesh to capture the available resolution. |
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BibTeX:
@article{Charlety2013, author = {Jean Charléty and Sergey Voronin and Guust Nolet and Ignace Loris and Frederik J. Simons and Karin Sigloch and Ingrid C. Daubechies}, title = {Global seismic tomography with sparsity constraints: Comparison with smoothing and damping regularization}, journal = {Journal of Geophysical Research - Solid Earth}, year = {2013}, volume = {118}, pages = {4887--4899}, doi = {https://doi.org/10.1002/jgrb.50326} } |
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Loris, I. and Verhoeven, C. | An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography | 2013 | Computational Optimization and Applications Vol. 54(2), pp. 399-416 |
article | DOI URL |

Abstract: We propose an iterative algorithm for the minimization of a 1-norm
penalized least squares functional, under additional linear constraints. The algorithm is fully explicit: it uses only matrix multiplications with the three matrices present in the problem (in the linear constraint, in the data misfit part and in penalty term of the functional). None of the three matrices must be invertible. Convergence is proven in a finite-dimensional setting. We apply the algorithm to a synthetic problem in magneto-encephalography where it is used for the reconstruction of divergence-free current densities subject to a sparsity promoting penalty on the wavelet coefficients of the current densities. We discuss the effects of imposing zero divergence and of imposing joint sparsity (of the vector components of the current density) on the current density reconstruction. |
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BibTeX:
@article{Loris2011, author = {Ignace Loris and Caroline Verhoeven}, title = {An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography}, journal = {Computational Optimization and Applications}, year = {2013}, volume = {54}, number = {2}, pages = {399--416}, note = {Special issue on optimization methods for inverse problems in imaging.}, url = {http://arxiv.org/abs/1202.3362}, doi = {https://doi.org/10.1007/s10589-012-9482-y} } |
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Nassiri, V. and Loris, I. | A generalized quantile regression model | 2013 | Journal of Applied Statistics Vol. 40(5), pp. 1090-1105 |
article | DOI |

Abstract: A new class of probability distributions, the so-called connected
double truncated gamma distribution, is introduced. We show that using this class as the error distribution of a linear model leads to a generalized quantile regression model that combines desirable properties of both least squares and quantile regression methods: robustness to outliers and differentiable loss function. |
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BibTeX:
@article{Nassiri2013, author = {Vahid Nassiri and Ignace Loris}, title = {A generalized quantile regression model}, journal = {Journal of Applied Statistics}, year = {2013}, volume = {40}, number = {5}, pages = {1090--1105}, doi = {https://doi.org/10.1080/02664763.2013.780158} } |
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Cloquet, C., Loris, I., Verhoeven, C. and Defrise, M. | GISTA reconstructs faster with a restart strategy and even faster with a FISTA-like reconstruction | 2012 | 2012 IEEE Nuclear Science Symposium and Medical Imaging Conference, pp. 2334-2338 | inproceedings | DOI |

Abstract: In X-Ray CT, the cone-beam geometry leads to specific artifacts. Moreover,
as the concern about dose-related health effects rises, the struggle to reduce the dose leads to increased image noise. To overcome these image quality issues, a popular recent trend models CT images by flat regions separated by sharp edges, and incorporate this knowledge into a reconstruction algorithm using a total variation (TV) penalty term. Recently, Loris and Verhoeven (2011) designed the Generalization of the Iterative Soft Thresholding Algorithm (GISTA), that has proven convergence and can handle conveniently a non smooth penalty term, like the TV penalty term. To our knowledge, GISTA has not yet been used to reconstruct medical image data. Therefore, we would like to introduce it to the reconstruction community, and present first results of phantom data acquired on a scanner consisting of a cone beam X-ray source and a flat panel detector. We also propose to accelerate the convergence at the initial iterations, using an innovative restart-strategy. |
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BibTeX:
@inproceedings{Cloquet2012, author = {Christophe Cloquet and Ignace Loris and Caroline Verhoeven and Michel Defrise}, title = {GISTA reconstructs faster with a restart strategy and even faster with a FISTA-like reconstruction}, booktitle = {2012 IEEE Nuclear Science Symposium and Medical Imaging Conference}, year = {2012}, pages = {2334--2338}, doi = {https://doi.org/10.1109/NSSMIC.2012.6551530} } |
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Loris, I. and Verhoeven, C. | Iterative algorithms for total variation-like reconstructions in seismic tomography | 2012 | International Journal on Geomathematics Vol. 3(2), pp. 179-208 |
article | DOI URL |

Abstract: A qualitative comparison of total variation like penalties (total
variation, Huber variant of total variation, total generalized variation, is made in the context of global seismic tomography. Both penalized and constrained formulations of seismic recovery problems are treated. A number of simple iterative recovery algorithms applicable to these problems are described. The convergence speed of these algorithms is compared numerically in this setting. For the constrained formulation a new algorithm is proposed and its convergence is proven. |
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BibTeX:
@article{Loris2012, author = {Ignace Loris and Caroline Verhoeven}, title = {Iterative algorithms for total variation-like reconstructions in seismic tomography}, journal = {International Journal on Geomathematics}, year = {2012}, volume = {3}, number = {2}, pages = {179--208}, url = {http://arxiv.org/abs/1203.4451}, doi = {https://doi.org/10.1007/s13137-012-0036-3} } |
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Loris, I. | Wavelets: A Concise Guide. Amir-Homayoon Najmi. 270 pp. The Johns Hopkins University Press, Baltimore, 2012. Price: $45.00 (paper) ISBN 978-1-4214-0496-6 [BibTeX] |
2012 | American Journal of Physics Vol. 80(12), pp. 1113-1113 |
article | DOI |

BibTeX:
@article{Loris2012a, author = {Ignace Loris}, title = {Wavelets: A Concise Guide. Amir-Homayoon Najmi. 270 pp. The Johns Hopkins University Press, Baltimore, 2012. Price: $45.00 (paper) ISBN 978-1-4214-0496-6}, journal = {American Journal of Physics}, year = {2012}, volume = {80}, number = {12}, pages = {1113--1113}, doi = {https://doi.org/10.1119/1.4742757} } |
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Loris, I. | A generalization of the iterative soft-thresholding algorithm for non-separable penalties | 2012 | Vol. 9(2)Oberwolfach Reports, pp. 1811-1814 |
inproceedings | DOI |

Abstract: The present research report focuses on the efficient use of sparse
representations for the regularization and solution of ill-posed problems. In particular, we discuss simple iterative algorithms for the minimization of certain convex functionals encountered in this area. |
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BibTeX:
@inproceedings{Loris2012b, author = {Ignace Loris}, title = {A generalization of the iterative soft-thresholding algorithm for non-separable penalties}, booktitle = {Oberwolfach Reports}, publisher = {European Mathematical Society}, year = {2012}, volume = {9}, number = {2}, pages = {1811--1814}, doi = {https://doi.org/10.4171/OWR/2012/29} } |
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Nassiri, V. and Loris, I. | On log-concavity of skew-symmetric distributions and their applications in penalized linear models | 2012 | Proceedings of the 43rd Annual Iranian Mathematics Conference, pp. 1037-1039 | inproceedings | URL |

Abstract: Log-concavity of the skew-symmetric class of distributions is studied.
Also the possibility of using them as error distribution in a sparse linear model is investigated. A procedure to estimate the penalized model is discussed. |
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BibTeX:
@inproceedings{Nassiri2012, author = {Vahid Nassiri and Ignace Loris}, title = {On log-concavity of skew-symmetric distributions and their applications in penalized linear models}, booktitle = {Proceedings of the 43rd Annual Iranian Mathematics Conference}, year = {2012}, pages = {1037--1039}, note = {The 43rd Annual Iranian Mathematics Conference (27--30/8/2012, Tabriz, Iran).}, url = {http://imc43.tabrizu.ac.ir/en/} } |
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Loris, I. and Verhoeven, C. | On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty | 2011 | Inverse Problems Vol. 27(12), pp. 125007 |
article | DOI URL |

Abstract: An explicit algorithm for the minimization of an 1 penalized least squares functional, with non-separable 1 term, is proposed. Each step in the iterative algorithm requires four matrix vector multiplications and a single simple projection on a convex set (or equivalently thresholding). Convergence is proven and a 1/N convergence rate is derived for the functional. In the special case where the matrix in the 1 term is the identity (or orthogonal), the algorithm reduces to the traditional iterative soft-thresholding algorithm. In the special case where the matrix in the quadratic term is the identity (or orthogonal), the algorithm reduces to a gradient projection algorithm for the dual problem. By replacing the projection with a simple proximity operator, other convex non-separable penalties than those based on an 1-norm can be handled as well. |
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BibTeX:
@article{Loris.Verhoeven2011, author = {Ignace Loris and Caroline Verhoeven}, title = {On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty}, journal = {Inverse Problems}, year = {2011}, volume = {27}, number = {12}, pages = {125007}, url = {http://arxiv.org/abs/1104.1087}, doi = {https://doi.org/10.1088/0266-5611/27/12/125007} } |
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Simons, F.J., Loris, I., Nolet, G., Daubechies, I.C., Voronin, S., Judd, J.S., Vetter, P.A., Charléty, J. and Vonesch, C. | Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity | 2011 | Geophysical Journal International Vol. 187(2), pp. 969-988 |
article | DOI URL |

Abstract: We propose a class of spherical wavelet bases for the analysis of
geophysical models and for the tomographic inversion of global seismic data. Its multiresolution character allows for modeling with an effective spatial resolution that varies with position within the Earth. Our procedure is numerically efficient and can be implemented with parallel computing.We discuss two possible types of discrete wavelet transforms in the angular dimension of the cubed sphere. We discuss benefits and drawbacks of these constructions and apply them to analyze the information present in two published seismic wavespeed models of the mantle, for the statistics and power of wavelet coefficients across scales. The localization and sparsity properties of wavelet bases allow finding a sparse solution to inverse problems by iterative minimization of a combination of the 2 norm of data fit and the 1 norm on the wavelet coefficients. By validation with realistic synthetic experiments we illustrate the likely gains of our new approach in future inversions of finite-frequency seismic data and show its readiness for global seismic tomography |
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BibTeX:
@article{Simons.Loris.ea2011, author = {Frederik J. Simons and Ignace Loris and Guust Nolet and Ingrid C. Daubechies and S. Voronin and J. S. Judd and P. A. Vetter and J. Charléty and C. Vonesch}, title = {Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity}, journal = {Geophysical Journal International}, year = {2011}, volume = {187}, number = {2}, pages = {969--988}, url = {http://arxiv.org/abs/1104.3151}, doi = {https://doi.org/10.1111/j.1365-246X.2011.05190.x} } |
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Simons, F.J., Loris, I., Brevdo, E. and Daubechies, I.C. | Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion | 2011 | Vol. 8138Wavelets and Sparsity XIV, pp. X1-X15 |
inproceedings | DOI URL |

Abstract: Many flexible parameterizations exist to represent data on the sphere.
In addition to the venerable spherical harmonics, we have the Slepian basis, harmonic splines, wavelets and wavelet-like Slepian frames. In this paper we focus on the latter two: spherical wavelets developed for geophysical applications on the cubed sphere, and the Slepian ``tree'', a new construction that combines a quadratic concentration measure with wavelet-like multiresolution. We discuss the basic features of these mathematical tools, and illustrate their applicability in parameterizing large-scale global geophysical (inverse) problems. |
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BibTeX:
@inproceedings{Simons2011, author = {Frederik J. Simons and Ignace Loris and Eugene Brevdo and Ingrid C. Daubechies}, title = {Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion}, booktitle = {Wavelets and Sparsity XIV}, publisher = {SPIE}, year = {2011}, volume = {8138}, pages = {X1--X15}, url = {http://arxiv.org/abs/1109.1718}, doi = {https://doi.org/10.1117/12.892285} } |
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Loris, I., Douma, H., Nolet, G., Daubechies, I. and Regone, C. | Nonlinear regularization techniques for seismic tomography | 2010 | Journal of Computational Physics Vol. 229(3), pp. 890-905 |
article | DOI URL |

Abstract: The effects of several nonlinear regularization techniques are discussed
in the framework of 3D seismic tomography. Traditional, linear, 2 penalties are compared to so-called sparsity promoting 1 and 0 penalties, and a total variation penalty. Which of these algorithms is judged optimal depends on the specific requirements of the scientific experiment. If the correct reproduction of model amplitudes is important, classical damping towards a smooth model using an 2 norm works almost as well as minimizing the total variation but is much more efficient. If gradients (edges of anomalies) should be resolved with a minimum of distortion, we prefer 1 damping of Daubechies-4 wavelet coefficients. It has the additional advantage of yielding a noiseless reconstruction, contrary to simple 2 minimization (`Tikhonov regularization') which should be avoided. In some of our examples, the 0 method produced notable artifacts. In addition we show how nonlinear 1 methods for finding sparse models can be competitive in speed with the widely used 2 methods, certainly under noisy conditions, so that there is no need to shun 1 penalizations. |
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BibTeX:
@article{Loris.Douma.ea2009, author = {Loris, I. and Douma, H. and Nolet, G. and Daubechies, I. and Regone, C.}, title = {Nonlinear regularization techniques for seismic tomography}, journal = {Journal of Computational Physics}, year = {2010}, volume = {229}, number = {3}, pages = {890--905}, url = {http://arxiv.org/abs/0808.3472}, doi = {https://doi.org/10.1016/j.jcp.2009.10.020} } |
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Loris, I. and Verhoeven, C. | Practical error estimates for sparse recovery in linear inverse problems. | 2010 | School: Vrije Universiteit Brussel |
techreport | URL |

Abstract: The effectiveness of using model sparsity as a priori information
when solving linear inverse problems is studied. We investigate the reconstruction quality of such a method in the non-idealized case and compute some typical recovery errors (depending on the sparsity of the desired solution, the number of data, the noise level on the data, and various properties of the measurement matrix); they are compared to known theoretical bounds and illustrated on a magnetic tomography example. |
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BibTeX:
@techreport{Loris.Verhoeven2010, author = {Ignace Loris and Caroline Verhoeven}, title = {Practical error estimates for sparse recovery in linear inverse problems.}, school = {Vrije Universiteit Brussel}, year = {2010}, url = {http://arxiv.org/abs/0908.3636} } |
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Brodie, J., Daubechies, I., De Mol, C., Giannone, D. and Loris, I. | Sparse and stable Markowitz portfolios | 2009 | Proceedings of the National Academy of Sciences of the USA Vol. 106(30), pp. 12267-12272 |
article | DOI URL |

Abstract: We consider the problem of portfolio selection within the classical
Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the absolute values of the portfolio weights. This penalty regularizes (stabilizes) the optimization problem, encourages sparse portfolios (i.e. portfolios with only few active positions), and allows to account for transaction costs. Our approach recovers as special cases the no-short-positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets constructed by Fama and French. Using only a modest amount of training data, we construct portfolios whose out-of-sample performance, as measured by Sharpe ratio, is consistently and significantly better than that of the naive evenly-weighted portfolio which constitutes, as shown in recent literature, a very tough benchmark. |
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BibTeX:
@article{Brodie.Daubechies.ea2009, author = {Brodie, Joshua and Daubechies, Ingrid and De Mol, Christine and Giannone, Domenico and Loris, Ignace}, title = {Sparse and stable Markowitz portfolios}, journal = {Proceedings of the National Academy of Sciences of the USA}, year = {2009}, volume = {106}, number = {30}, pages = {12267--12272}, url = {http://arxiv.org/abs/0708.0046}, doi = {https://doi.org/10.1073/pnas.0904287106} } |
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Loris, I., Bertero, M., De Mol, C., Zanella, R. and Zanni, L. | Accelerating gradient projection methods for 1- constrained signal recovery by steplength selection rules | 2009 | Applied and Computational Harmonic Analysis Vol. 27(2), pp. 247-254 |
article | DOI URL |

Abstract: We propose a new gradient projection algorithm that compares favorably
with the fastest algorithms available to date for 1-constrained sparse recovery from noisy data, both in the compressed sensing and inverse problem frameworks. The method exploits a line-search along the feasible direction and an adaptive steplength selection based on recent strategies for the alternation of the well-known Barzilai-Borwein rules. The convergence of the proposed approach is discussed and a computational study on both well-conditioned and ill-conditioned problems is carried out for performance evaluations in comparison with five other algorithms proposed in the literature. |
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BibTeX:
@article{Loris.Bertero.ea2009, author = {Loris, I. and Bertero, M. and De Mol, C. and Zanella, R. and Zanni, L.}, title = {Accelerating gradient projection methods for 1- constrained signal recovery by steplength selection rules}, journal = {Applied and Computational Harmonic Analysis}, year = {2009}, volume = {27}, number = {2}, pages = {247--254}, url = {http://arxiv.org/abs/0902.4424}, doi = {https://doi.org/10.1016/j.acha.2009.02.003} } |
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Loris, I. | On the performance of algorithms for the minimization of 1-penalized functionals | 2009 | Inverse Problems Vol. 25(3), pp. 035008 (16pp) |
article | DOI URL |

Abstract: The problem of assessing the performance of algorithms used for the
minimization of an 1-penalized least-squares functional, for a range of penalty parameters, is investigated. A criterion that uses the idea of `approximation isochrones' is introduced. Five different iterative minimization algorithms are tested and compared, as well as two warm-start strategies. Both well-conditioned and ill-conditioned problems are used in the comparison, and the contrast between these two categories is highlighted. |
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BibTeX:
@article{Loris2009, author = {Loris, Ignace}, title = {On the performance of algorithms for the minimization of 1-penalized functionals}, journal = {Inverse Problems}, year = {2009}, volume = {25}, number = {3}, pages = {035008 (16pp)}, url = {http://arxiv.org/abs/0710.4082}, doi = {https://doi.org/10.1088/0266-5611/25/3/035008} } |
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Daubechies, I., Fornasier, M. and Loris, I. | Accelerated projected gradient method for linear inverse problems with sparsity constraints | 2008 | Journal of Fourier Analysis and Applications Vol. 14(5-6), pp. 764-792 |
article | DOI URL |

Abstract: Regularization of ill-posed linear inverse problems via 1 penalization
has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an 1 penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to 1-constraints, using a gradient method, with projection on 1-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration. |
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BibTeX:
@article{Daubechies.Fornasier.ea2008, author = {Ingrid Daubechies and Massimo Fornasier and Ignace Loris}, title = {Accelerated projected gradient method for linear inverse problems with sparsity constraints}, journal = {Journal of Fourier Analysis and Applications}, year = {2008}, volume = {14}, number = {5--6}, pages = {764--792}, url = {http://arxiv.org/abs/0706.4297}, doi = {https://doi.org/10.1007/s00041-008-9039-8} } |
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Loris, I. | L1Packv2: A Mathematica package for minimizing an 1-penalized functional | 2008 | Computer Physics Communications Vol. 179(12), pp. 895-902 |
article | DOI URL |

Abstract: L1Packv2 is a Mathematica package that contains a number of algorithms
that can be used for the minimization of an 1-penalized least squares functional. The algorithms can handle a mix of penalized and unpenalized variables. Several instructive examples are given. Also, an implementation that yields an exact output whenever exact data are given is provided. |
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BibTeX:
@article{Loris2008, author = {Loris, Ignace}, title = {L1Packv2: A Mathematica package for minimizing an 1-penalized functional}, journal = {Computer Physics Communications}, year = {2008}, volume = {179}, number = {12}, pages = {895--902}, url = {http://arxiv.org/abs/0710.3728}, doi = {https://doi.org/10.1016/j.cpc.2008.07.010} } |
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Loris, I., Nolet, G., Daubechies, I. and Dahlen, F.A. | Tomographic inversion using 1-norm regularization of wavelet coefficients | 2007 | Geophysical Journal International Vol. 170(1), pp. 359-370 |
article | DOI URL |

Abstract: We propose the use of 1 regularization in a wavelet basis for
the solution of linearized seismic tomography problems Am=d, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used to find a sparse solution m that contains no more fine-scale structure than is necessary to fit the data d to within its assigned errors. |
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BibTeX:
@article{Loris.Nolet.ea2007, author = {Loris, Ignace and Nolet,Guust and Daubechies, Ingrid and Dahlen, F. A.}, title = {Tomographic inversion using 1-norm regularization of wavelet coefficients}, journal = {Geophysical Journal International}, year = {2007}, volume = {170}, number = {1}, pages = {359--370}, url = {http://arxiv.org/abs/physics/0608094}, doi = {https://doi.org/10.1111/j.1365-246X.2007.03409.x} } |
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Khare, A., Loris, I. and Sasaki, R. | Affine Toda-Sutherland systems | 2004 | Journal of Physics A-Mathematical and General Vol. 37(5), pp. 1665-1679 |
article | DOI URL |

Abstract: A cross between two well-known integrable multi-particle dynamics,
an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi-exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorizable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda-Calogero system, with a corresponding rational potential, is also discussed. |
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BibTeX:
@article{Khare.Loris.ea2004, author = {Khare, A. and Loris, I. and Sasaki, R.}, title = {Affine Toda-Sutherland systems}, journal = {Journal of Physics A-Mathematical and General}, year = {2004}, volume = {37}, number = {5}, pages = {1665--1679}, url = {http://arxiv.org/abs/hep-th/0309077}, doi = {https://doi.org/10.1088/0305-4470/37/5/013} } |
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Loris, I. and Sasaki, R. | Quantum and classical eigenfunctions in Calogero and Sutherland systems | 2004 | Journal of Physics A-Mathematical and General Vol. 37(1), pp. 211-237 |
article | DOI URL |

Abstract: An interesting observation was reported by Corrigan-Sasaki that all
the frequencies of small oscillations around equilibrium are 'quantized' for Calogero and Sutherland (CS) systems, typical integrable multi-particle dynamics. We present an analytic proof by applying recent results of Loris- Sasaki. Explicit forms of 'classical' and quantum eigenfunctions are presented for CS systems based on any root system |
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BibTeX:
@article{Loris.Sasaki2004, author = {Loris, I. and Sasaki, R.}, title = {Quantum and classical eigenfunctions in Calogero and Sutherland systems}, journal = {Journal of Physics A-Mathematical and General}, year = {2004}, volume = {37}, number = {1}, pages = {211--237}, url = {http://arxiv.org/abs/hep-th/0308052}, doi = {https://doi.org/10.1088/0305-4470/37/1/015} } |
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Loris, I. and Sasaki, R. | Quantum vs classical mechanics: role of elementary excitations | 2004 | Physics Letters A Vol. 327(2-3), pp. 152-157 |
article | DOI URL |

Abstract: Simple theorems relating a quantum mechanical system to the corresponding
classical one at equilibrium and connecting the quantum eigenvalues to the frequencies of normal modes oscillations are presented. Corresponding to each quantum eigenfunction, a 'classical eigenfunction' is associated. Those belonging to 'elementary excitations' play an important role. |
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BibTeX:
@article{Loris.Sasaki2004a, author = {Loris, I. and Sasaki, R.}, title = {Quantum vs classical mechanics: role of elementary excitations}, journal = {Physics Letters A}, year = {2004}, volume = {327}, number = {2-3}, pages = {152--157}, url = {http://arxiv.org/abs/quant-ph/0308040}, doi = {https://doi.org/10.1016/j.physleta.2004.05.015} } |
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Loris, I. | Bilinear representations of integrable equations | 2002 | Theoretical and Mathematical Physics Vol. 133(2), pp. 1549-1556 |
article | DOI |

Abstract: We present a method for deriving recursion operators and canonical
Lax pairs directly from bilinear identities of the KP type. Examples include the KdV equation, the Boussinesq equation, and a real equivalent of the nonlinear Schrödinger equation. |
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BibTeX:
@article{Loris2002, author = {Loris, I.}, title = {Bilinear representations of integrable equations}, journal = {Theoretical and Mathematical Physics}, year = {2002}, volume = {133}, number = {2}, pages = {1549--1556}, note = {Proceedings of the NEEDS'01 Conference.}, doi = {https://doi.org/10.1023/A:1021103012057} } |
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Lambert, F., Loris, I. and Springael, J. | Classical Darboux transformations and the KP hierarchy | 2001 | Inverse Problems Vol. 17(4), pp. 1067-1074 |
article | DOI |

Abstract: Classical Darboux transformations together with partitional polynomials
are used as elementary tools for the construction of members of a basic hierarchy of integrable nonlinear partial differential equations (the KP hierarchy). |
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BibTeX:
@article{Lambert.Loris.ea2001, author = {Lambert, F. and Loris, I. and Springael, J.}, title = {Classical Darboux transformations and the KP hierarchy}, journal = {Inverse Problems}, year = {2001}, volume = {17}, number = {4}, pages = {1067--1074}, doi = {https://doi.org/10.1088/0266-5611/17/4/333} } |
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Lambert, F., Loris, I., Springael, J. and Willox, R. | On the Hirota representation of soliton equations with one tau-function | 2001 | Journal of the Physical Society of Japan Vol. 70(3), pp. 605-608 |
article | DOI |

Abstract: Alternative Hirota representations in terms of a single tau-function
are derived for a variaty of solition equations, including the sine-Gordon and Tzitzeica equations. The relevance of these representations with respect to known bilinear representations of integrable hierarchies is briefly discussed. The essentials of the derivations method are presented. |
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BibTeX:
@article{Lambert.Loris.ea2001a, author = {Lambert, F. and Loris, I. and Springael, J. and Willox, R.}, title = {On the Hirota representation of soliton equations with one tau-function}, journal = {Journal of the Physical Society of Japan}, year = {2001}, volume = {70}, number = {3}, pages = {605--608}, doi = {https://doi.org/10.1143/JPSJ.70.605} } |
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Loris, I. | Dimensional reductions of BKP and CKP hierarchies | 2001 | Journal of Physics A-Mathematical and General Vol. 34(16), pp. 3447-3459 |
article | DOI |

Abstract: A discussion of dimensional reductions, which are not classical symmetry
reductions, is made for the BKP and CKP hierarchies of integrable evolution equations. A novel direct method for testing Pfaffian solutions to bilinearidentities is presented and applied to these reductions. |
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BibTeX:
@article{Loris2001, author = {Loris, I.}, title = {Dimensional reductions of BKP and CKP hierarchies}, journal = {Journal of Physics A-Mathematical and General}, year = {2001}, volume = {34}, number = {16}, pages = {3447--3459}, doi = {https://doi.org/10.1088/0305-4470/34/16/313} } |
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Loris, I. | Solutions of coupled Korteweg-de Vries systems | 2001 | Journal of the Physical Society of Japan Vol. 70(3), pp. 662-665 |
article | DOI |

Abstract: A class of solutions in determinant form to a set of coupled KdV equations
is derived. The bilinear (Hirota) form of the coupled equations and links with the CKP system serve as a guide. |
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BibTeX:
@article{Loris2001a, author = {Loris, I.}, title = {Solutions of coupled Korteweg-de Vries systems}, journal = {Journal of the Physical Society of Japan}, year = {2001}, volume = {70}, number = {3}, pages = {662--665}, doi = {https://doi.org/10.1143/JPSJ.70.662} } |
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Loris, I. | Recursion operator for a constrained BKP system | 2000 | Proceedings of the workshop on Nonlinearity, integrability, and all that: 20 years after NEEDS '79, pp. 325-330 | inproceedings | |

Abstract: Symmetry reductions of the KP system of B-type are discussed. A recursion
operator (mapping symmetries into symmetries) is derived for the main example. |
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BibTeX:
@inproceedings{Loris2000, author = {Loris, Ignace}, title = {Recursion operator for a constrained BKP system}, booktitle = {Proceedings of the workshop on Nonlinearity, integrability, and all that: 20 years after NEEDS '79}, publisher = {World Scientific, Singapore}, year = {2000}, pages = {325--330} } |
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Loris, I. and Willox, R. | Symmetry reductions of the BKP hierarchy | 1999 | Journal of Mathematical Physics Vol. 40(3), pp. 1420-1431 |
article | DOI |

Abstract: A general symmetry of the bilinear BKP hierarchy is studied in terms
of tau functions. We use this symmetry to define reductions of the BKP hierarchy, among which new integrable systems can be found. The reductions are connected to constraints on the Lax operator as well as on the bilinear formulation. A class of solutions for the reduced equations is derived |
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BibTeX:
@article{Loris.Willox1999, author = {Loris, I. and Willox, R.}, title = {Symmetry reductions of the BKP hierarchy}, journal = {Journal of Mathematical Physics}, year = {1999}, volume = {40}, number = {3}, pages = {1420--1431}, doi = {https://doi.org/10.1063/1.532812} } |
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Loris, I. | On reduced CKP equations | 1999 | Inverse Problems Vol. 15(4), pp. 1099-1109 |
article | DOI |

Abstract: Symmetry reductions of the CKP hierarchy are discussed in a pseudodifferential
and tau-function context. Solutions of the resulting nonlinear partial differential equations are obtained via the methods of gauge transformations and of tau functions. Reductions of other 2+1-dimensional hierarchies related to KP are also briefly investigated. |
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BibTeX:
@article{Loris1999, author = {Loris, I.}, title = {On reduced CKP equations}, journal = {Inverse Problems}, year = {1999}, volume = {15}, number = {4}, pages = {1099--1109}, doi = {https://doi.org/10.1088/0266-5611/15/4/317} } |
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Willox, R. and Loris, I. | KP constraints from reduced multi-component hierarchies | 1999 | Journal of Mathematical Physics Vol. 40(12), pp. 6501-6525 |
article | DOI |

Abstract: The m-vector k-constrained Kadomtsev�Petviashvili KP hierarchy
is shown to be a ''pseudo''-reduction of the (m+1)-component KP hierarchy. To facilitate the implementation of this reduction on the level of the solutions, the typical multicomponent KP solutions are mapped onto solutions of a Toda molecule-type equation from which Wronskian and Grammian solutions for the constrained KP hierarchy follow. The reduction of the associated linear systems is discussed and its importance for the choice of bilinear representation of the reduced systems is explained. |
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BibTeX:
@article{Willox.Loris1999, author = {Willox, R. and Loris, I.}, title = {KP constraints from reduced multi-component hierarchies}, journal = {Journal of Mathematical Physics}, year = {1999}, volume = {40}, number = {12}, pages = {6501--6525}, doi = {https://doi.org/10.1063/1.533104} } |
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Willox, R. and Loris, I. | An algebraic description of generalized k-constraints | 1999 | Journal of Physics A-Mathematical and General Vol. 32(10), pp. 2027-2036 |
article | DOI |

Abstract: The generalized k-constrained KP hierarchy is shown to correspond
to a so-called pseudo-reduction of the two-dimensional Toda lattice hierarchy, described in a free-fermion approach which is adapted to the case of two singularities in the spectral parameter range. Wronskian solutions are discussed and, in particular, soliton solutions are recovered through a p^k + c/p = q^k + c/q reduction of the Toda solitons. |
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BibTeX:
@article{Willox.Loris1999a, author = {Willox, R. and Loris, I.}, title = {An algebraic description of generalized k-constraints}, journal = {Journal of Physics A-Mathematical and General}, year = {1999}, volume = {32}, number = {10}, pages = {2027--2036}, doi = {https://doi.org/10.1088/0305-4470/32/10/018} } |
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Loris, I. | Symmetry reductions in the tau-function approach to integrability | 1998 | School: Vrije Universiteit Brussel |
phdthesis | |

Abstract: It is the aim of this thesis to investigate how different 1+1 dimensional
``soliton'' systems and the various properties characterizing their ``integrability'', may be derived from a single universal equation through reduction procedures. These reductions are implemented by imposing judiciously chosen constraints on the field in which these equations take their most fundamental form: the ``tau-function''. par Soliton systems, such as the non-linear Schrödinger equation or the Korteweg-de Vries equation, feature among the set of integrable non-linear partial differential equations. The term `soliton' refers to the existence of particular solutions which describe the interaction of a (unlimited) number of localized pulses. The soliton phenomenon was intensively studied in the years following its identification. However, over the past decade, attention and research efforts have shifted towards the concept of ``integrability'' in systems with an infinite number of degrees of freedom. For these systems, various criteria and tests of integrability exist. Different criteria require the existence of an infinite number of conservation laws (or an infinite number of symmetries), or the absence of certain movable singularities in the general solution (Painlevé property), or the existence of an underlying linear Lax system and a corresponding inverse scattering problem (IST) … par It is a generally accepted view that, despite the existence of partial results, the problem of understanding the notion of integrability is a difficult one. It is not our purpose to settle this question here. Yet, just as the different integrability criteria are inter-related, so do a number of different 1+1 dimensional soliton systems exhibit astoundingly similar features. It is our intention to show that these features spring from the characteristics of an underlying 2+1 dimensional integrable hierarchy: the Kadomtsev-Petviashvili (KP) hierarchy. The important point to be understood is how 1+1 dimensional soliton systems can be regarded as reductions of a higher dimensional
integrable equation. More precisely, using the symmetries of this KP hierarchy, we shall present a new and unified approach to reductions of the KP hierarchy to 1+1 dimensional integrable systems. par The appropriate technique for the implementation of theses reductions is the method of symmetry reductions. As the set of symmetries (of a given equation) exhibits a linear structure (i.e. the defining equations for symmetries are linear partial differential equations) one can impose the condition that a certain symmetry be identically equal to zero. This condition will not be in contradiction with the integrable structure of the hierarchy, and hence represents a sound reduction technique. In order to apply these reductions successfully, one must express them in the appropriate field variable. par The fundamental field-variable in the KP hierarchy is the `tau-function'. This tau-function (t_1,t_2,) is the solution of Hirota-type partial differential equations, i.e. equation which are quadratically homogeneous in the field variable. The first topic in our investigation is the determination of a general symmetry for the bilinear equations. This symmetry can be written in terms of an eigenfunction potential (eigenfunctions are the solutions of the underlying linear problem). An important result is the fact that this potential can in its turn also be expressed entirely in terms of tau-functions. par The `k-constrained' KP hierarchy is defined by imposing a symmetry constraint between the elementary symmetry t_kτ and a eigenfunction potential symmetry. We show that this reduction can be expressed entirely in terms of tau-functions. A characterization of the corresponding Lax operator is given. Additional bilinear identities for the KP tau-function, which give a bilinear description of the constraint, are derived. Our description of the reduced hierarchies gives access to the determination of general classes of determinant-type solutions. The proposed technique can also be used to determine solutions to the unreduced KP hierarchy and sets itself apart from known determinant techniques by the fact that solutions are shown to exist for all the equations in the hierarchy at once. par Finally, a generalized k-constraint is introduced which is especially adapted to the description of systems with non-zero boundary conditions. This extension is physically relevant in view of the fact that the non-linear Schrödinger equation (modeling light waves in non-linear media) also exhibits solutions with non-vanishing boundary conditions. A bilinear description of this reduction is given and soliton solutions for the reduced systems are derived. Furthermore a Bäcklund transformation and a Lax pair are derived as well. |
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BibTeX:
@phdthesis{Loris1998, author = {Loris, Ignace}, title = {Symmetry reductions in the tau-function approach to integrability}, school = {Vrije Universiteit Brussel}, year = {1998} } |
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Willox, R., Tokihiro, T., Loris, I. and Satsuma, J. | The fermionic approach to Darboux transformations | 1998 | Inverse Problems Vol. 14(3), pp. 745-762 |
article | DOI |

Abstract: Starting from the free fermion description of the one-component KP
hierarchy, we establish a connection between this approach and the theory of Darboux and binary Darboux transformations. Certain difference identities---allowing for the treatment of both continuous as well as discrete evolution equations---turn out to be crucial: first to show that any solution of the associated (adjoint) linear problems can always be expressed as a superposition of KP (adjoint) wavefunctions and then to interpret Darboux (and binary Darboux) transformations as Bäcklund transformations in the fermion language. |
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BibTeX:
@article{Willox.Tokihiro.ea1998, author = {Willox, R. and Tokihiro, T. and Loris, I. and Satsuma, J.}, title = {The fermionic approach to Darboux transformations}, journal = {Inverse Problems}, year = {1998}, volume = {14}, number = {3}, pages = {745--762}, doi = {https://doi.org/10.1088/0266-5611/14/3/022} } |
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Loris, I. and Willox, R. | On solutions of constrained Kadomtsev-Petviashvili equations: Grammians | 1997 | Journal of Mathematical Physics Vol. 38(10), pp. 5190-5197 |
article | DOI |

Abstract: We show the existence of Grammian-type solutions for the (vector)
k-constrained Kadomtsev-Petviashvili (KP) equations. To introduce the method we give a novel proof for the presence of Grammian solutions for the bilinear l-modified KP hierarchies |
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BibTeX:
@article{Loris.Willox1997, author = {Loris, I. and Willox, R.}, title = {On solutions of constrained Kadomtsev-Petviashvili equations: Grammians}, journal = {Journal of Mathematical Physics}, year = {1997}, volume = {38}, number = {10}, pages = {5190--5197}, doi = {https://doi.org/10.1063/1.531937} } |
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Loris, I. and Willox, R. | KP symmetry reductions and a generalized constraint | 1997 | Journal of Physics A-Mathematical and General Vol. 30(19), pp. 6925-6938 |
article | DOI |

Abstract: We study the link between symmetry reductions and constraints of the
Kadomtsev-Petviashvili equations in terms of the tau function. We propose a generalization---adapted to non-zero boundary conditions---of the standard constraints, and show a particular class of solutions (solitons). |
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BibTeX:
@article{Loris.Willox1997a, author = {Loris, I. and Willox, R.}, title = {KP symmetry reductions and a generalized constraint}, journal = {Journal of Physics A-Mathematical and General}, year = {1997}, volume = {30}, number = {19}, pages = {6925--6938}, doi = {https://doi.org/10.1088/0305-4470/30/19/027} } |
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Loris, I. and Willox, R. | On solutions of constrained KP equations | 1997 | Journal of Mathematical Physics Vol. 38(1), pp. 283-291 |
article | DOI |

Abstract: We derive solutions of general Wronskian form for the (vector) constrained
KP hierarchies. As one explicit example we discuss rational solutions. In order to introduce our method, we give a direct, elementary proof of the existence of Wronskian solutions for the l-modified KP hierarchies (l:0,1,…). |
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BibTeX:
@article{Loris.Willox1997b, author = {Loris, I. and Willox, R.}, title = {On solutions of constrained KP equations}, journal = {Journal of Mathematical Physics}, year = {1997}, volume = {38}, number = {1}, pages = {283--291}, doi = {https://doi.org/10.1063/1.531843} } |
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Loris, I. and Willox, R. | Bilinear form and solutions of the k-constrained Kadomtsev-Petviashvili hierarchy | 1997 | Inverse Problems Vol. 13(2), pp. 411-420 |
article | DOI |

Abstract: We show how to derive an alternative bilinear formulation for the
k-constrained Kadomtsev�Petviashvili hierarchy. This Hirota form allows for the easy identification of a broad class of solutions to these equations |
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BibTeX:
@article{Loris.Willox1997c, author = {Loris, I. and Willox, R.}, title = {Bilinear form and solutions of the k-constrained Kadomtsev-Petviashvili hierarchy}, journal = {Inverse Problems}, year = {1997}, volume = {13}, number = {2}, pages = {411--420}, doi = {https://doi.org/10.1088/0266-5611/13/2/014} } |
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Loris, I. and Willox, R. | Recent results on dimensional reductions of the Kadomtsev-Petviashvili equation | 1997 | Proceedings of the Fourth National Conference on Theoretical and Applied Mechanics, pp. 27-30 | inproceedings | |

Abstract: In the present paper, we study a dimensional reductions of the Kadomtsev-Petviashvili
(KP) equations, amonst which one finds the non-linear Schrödinger equation (with non-zero boundary conditions). In particular we use the Darboux transformation to obtain the N soliton solution (∀ N). |
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BibTeX:
@inproceedings{Loris.Willox1997d, author = {Loris, Ignace and Willox, Ralph}, title = {Recent results on dimensional reductions of the Kadomtsev-Petviashvili equation}, booktitle = {Proceedings of the Fourth National Conference on Theoretical and Applied Mechanics}, year = {1997}, pages = {27--30} } |
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Pelinovsky, D., Springael, J., Lambert, F. and Loris, I. | On modified NLS, Kaup and NLBq equations: differential transformations and bilinearization | 1997 | Journal of Physics A-Mathematical and General Vol. 30(24), pp. 8705-8717 |
article | DOI |

Abstract: New transformations between the nonlinear Schrödinger, Kaup and
non-local Boussinesq equations as well as their modified counterparts are found and analysed. The bilinear representations of these equations, including an alternative bilinear form of the Chen-Lee-Liu equation, are obtained by a direct method based on the Bell's exponential polynomials. Explicit Wronskian solutions to these equations are also presented. |
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BibTeX:
@article{Pelinovsky.Springael.ea1997, author = {Pelinovsky, D. and Springael, J. and Lambert, F. and Loris, I.}, title = {On modified NLS, Kaup and NLBq equations: differential transformations and bilinearization}, journal = {Journal of Physics A-Mathematical and General}, year = {1997}, volume = {30}, number = {24}, pages = {8705--8717}, doi = {https://doi.org/10.1088/0305-4470/30/24/029} } |
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Willox, R., Loris, I. and Gilson, C.R. | Binary Darboux transformations for constrained KP hierarchies | 1997 | Inverse Problems Vol. 13(3), pp. 849-865 |
article | DOI |

Abstract: We describe how Darboux transformations and binary Darboux transformations
can be constructed for (vector-) constrained KP hierarchies. These transformations are then used to obtain explicit classes of Wronskian and Grammian solutions for these hierarchies. The relationship between these two types of solutions is also discussed. |
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BibTeX:
@article{Willox.Loris.ea1997, author = {Willox, R. and Loris, I. and Gilson, C. R.}, title = {Binary Darboux transformations for constrained KP hierarchies}, journal = {Inverse Problems}, year = {1997}, volume = {13}, number = {3}, pages = {849--865}, doi = {https://doi.org/10.1088/0266-5611/13/3/019} } |
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Loris, I., Lambert, F. and Willox, R. | New ways of applying the Hirota method in soliton theory | 1996 | Journal of Technical Physics Vol. 37, pp. 519-522 |
article | |

Abstract: It is shown how a sech-squared soliton hierarchy may be obtained from
the classical Boussinesq hierarchy. A collective approach to the problem of its bilinearization is discussed. Solutions and relations to other integrable systems are investigated from a bilinear point of view. |
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BibTeX:
@article{Loris.Lambert.ea1996, author = {Loris, I. and Lambert, F. and Willox, R.}, title = {New ways of applying the Hirota method in soliton theory}, journal = {Journal of Technical Physics}, year = {1996}, volume = {37}, pages = {519--522}, note = {Proceedings of the Conference on Nonlinear Dynamics, Chaotic and Complex Systems (NDCCS'95), Zakopane, Poland.} } |
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Loris, I. and Willox, R. | Soliton solutions of Wronskian form to the nonlocal Boussinesq equation | 1996 | Journal of the Physical Society of Japan Vol. 65(2), pp. 383-388 |
article | DOI |

Abstract: We investigate how one may write the soliton solutions of a nonlocal
Boussinesq equation in Wronskian form, and subsequently prove the existence of N-soliton solutions making use of the bilinear form of this equation. This technique also allows us to construct a bilinear Bäcklund transformation for this equation, mapping N-soliton solutions on (N+1)-soliton solutions. Our analysis extends the results previously obtained by Hirota for the Classical Boussinesq system to actual (c≠ 0) ''pq=c''-reductions performed on Wronskians. |
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BibTeX:
@article{Loris.Willox1996, author = {Loris, I. and Willox, R.}, title = {Soliton solutions of Wronskian form to the nonlocal Boussinesq equation}, journal = {Journal of the Physical Society of Japan}, year = {1996}, volume = {65}, number = {2}, pages = {383--388}, doi = {https://doi.org/10.1143/JPSJ.65.383} } |
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Springael, J., Hu, X.B. and Loris, I. | Bilinear characterization of higher order Ito-equations | 1996 | Journal of the Physical Society of Japan Vol. 65(5), pp. 1222-1226 |
article | DOI |

Abstract: We construct an infinite family of soliton equations, the lowest member
of which corresponds to the Ito-equation: using the bilinear formalism, we obtain a generic bilinear form leading to a recursion operator for this family. A generic Lax-pair and bilinear Bäcklund-transformation are also reported. |
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BibTeX:
@article{Springael.Hu.ea1996, author = {Springael, J. and Hu, X. B. and Loris, I.}, title = {Bilinear characterization of higher order Ito-equations}, journal = {Journal of the Physical Society of Japan}, year = {1996}, volume = {65}, number = {5}, pages = {1222--1226}, doi = {https://doi.org/10.1143/JPSJ.65.1222} } |
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Willox, R., Loris, I. and Springael, J. | The nlBq-hierarchy as a pq=C reduction of the KP-hierarchy | 1996 | Proceedings of the Workshop``Non-linear Physics, Theory and Experiment", pp. 321-329 | inproceedings | |

Abstract: In this paper it is shown that the so-called nonlocal Boussinesq equation
can be regarded as pq=c reduction of the KP hierarchy. The reduction procedure can be carried out explicitly on the N-soliton solutions of the KP hierarchy once a suitable bilinear form of the nonlocal Boussinesq equation is obtained. It is however necessary to impose an extra constraint on the KP evolution in order to implement the reduction on the bilinear forms themselves. This special constraint allows for yet another class of solutions. |
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BibTeX:
@inproceedings{Willox.Loris.ea1996, author = {Willox, R. and Loris, I. and Springael, J.}, title = {The nlBq-hierarchy as a pq=C reduction of the KP-hierarchy}, booktitle = {Proceedings of the Workshop``Non-linear Physics, Theory and Experiment"}, publisher = {World Scientific, Singapore}, year = {1996}, pages = {321--329} } |
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Willox, R. and Loris, I. | Symmetry constraints of the KP hierarchy and a nonlocal Boussinesq equation | 1996 | Vol. 2VII International Conference Symmetry Methods in Physics, pp. 603-609 |
inproceedings | |

Abstract: In this paper we use the Lax pair for the nonlocal Boussinesq equation
in order to show that this equation can be interpreted as a 1-constraint (or corresponding symmetry constraint) on the KP linear problem. A link with density constraints on the modified-KP hierarchy, leading to the classical Boussinesq eaution is also disclosed. |
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BibTeX:
@inproceedings{Willox.Loris1996, author = {Willox, R. and Loris, I.}, title = {Symmetry constraints of the KP hierarchy and a nonlocal Boussinesq equation}, booktitle = {VII International Conference Symmetry Methods in Physics}, publisher = {Joint Institute for Nuclear Research}, year = {1996}, volume = {2}, pages = {603--609} } |
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Lambert, F., Loris, I., Springael, J. and Willox, R. | A direct bilinearization scheme based on the use of partition polynomials | 1995 | Proceedings of the NEEDS'94 workshop at Los Alamos NL, pp. 102-111 | inproceedings | |

Abstract: A systematic procedure for the bilinerization of classes of soliton
equations (and related ordinary differential equations) is presented. The method, based on the use of generalized Bell polynomials, is illustrated with several examples. |
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BibTeX:
@inproceedings{Lambert.Loris.ea1995, author = {Lambert, F. and Loris, I. and Springael, J. and Willox, R.}, title = {A direct bilinearization scheme based on the use of partition polynomials}, booktitle = {Proceedings of the NEEDS'94 workshop at Los Alamos NL}, publisher = {World Scientific, Singapore}, year = {1995}, pages = {102--111} } |
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Willox, R., Loris, I. and Springael, J. | Bilinearization of the nonlocal Boussinesq equation | 1995 | Journal of Physics A-Mathematical and General Vol. 28(20), pp. 5963-5972 |
article | DOI |

Abstract: A single-field bilinear system generating the so-called non-local
Boussinesq equation is constructed. From the bilinearization procedure it can be seen that the associated hierarchy of soliton systems which we construct shares part of the solution set of a hierarchy related to the Kadomtsev-Petviashvili equation. A bilinear representation of the recursion operator for the Kaup hierarchy is essential in the construction and a systematic way of obtaining such a representation from just two-soliton considerations is presented. |
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BibTeX:
@article{Willox.Loris.ea1995, author = {Willox, R. and Loris, I. and Springael, J.}, title = {Bilinearization of the nonlocal Boussinesq equation}, journal = {Journal of Physics A-Mathematical and General}, year = {1995}, volume = {28}, number = {20}, pages = {5963--5972}, doi = {https://doi.org/10.1088/0305-4470/28/20/024} } |
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Lambert, F., Loris, I., Springael, J. and Willox, R. | On a direct bilinearization method - Kaup's higher-order water-wave equation as a modified nonlocal Boussinesq equation | 1994 | Journal of Physics A-Mathematical and General Vol. 27(15), pp. 5325-5334 |
article | DOI |

Abstract: A systematic procedure for the bilinearization of classes of soliton
equations is developed with the help of a generalization of Bell's exponential polynomials. Application of this procedure to Kaup's higher-order wave equation discloses several links with other soliton systems. In particular, it is found that the Kaup equation is the modified version of a sech square soliton system which constitutes an alternative to the good Boussinesq equation. |
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BibTeX:
@article{Lambert.Loris.ea1994, author = {Lambert, F. and Loris, I. and Springael, J. and Willox, R.}, title = {On a direct bilinearization method - Kaup's higher-order water-wave equation as a modified nonlocal Boussinesq equation}, journal = {Journal of Physics A-Mathematical and General}, year = {1994}, volume = {27}, number = {15}, pages = {5325--5334}, doi = {https://doi.org/10.1088/0305-4470/27/15/028} } |
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Loris, I. | Rechtstreekse bilinearisatie van solitonsystemen: Een systematische aanpak met veralgemeende Bell-polynomen [BibTeX] |
1994 | School: Vrije Universiteit Brussel |
mastersthesis | |

BibTeX:
@mastersthesis{Loris1994, author = {Loris, Ignace}, title = {Rechtstreekse bilinearisatie van solitonsystemen: Een systematische aanpak met veralgemeende Bell-polynomen}, school = {Vrije Universiteit Brussel}, year = {1994} } |

Here is my list of publications in the ULB institutional repository: DI-fusion (also includes talks, posters, ...).

Group quantile regression Matlab code: groupQRcode.zip (updated August 6th, 2013).

L1Packv2: A Mathematica package for ...: L1Packv2.zip (updated 2/06/08). A guide is available in the list of publications.

Minisymposium "Advanced optimization methods for image processing'' at SIAM Conference on Imaging Science, 05--08/06/2018: link .

Workshop on "Sparsity in Applied Mathematics and Statistics", 01-02/06/2017: poster.

Minisymposium "Optimization methods for signal and image processing'' at Applied Inverse Problems Conference 2015, 28--29/05/2015: progamme.

Interdisciplinary workshop "Sparsity and Modern Mathematical Methods for High Dimensional Data'', 6--10/04/2010: poster.

International Francqui Professor Prof. Ingrid Daubechies, 01/01--30/06/2010: poster.

Solvay Workshop on "Sparsity, learning and computation", 12--14/02/2009: poster.

Research:
MathSciNet,
ScienceDirect,
Scopus,
arXiv

TeX and friends:
CTAN TeX archive,
MikTeX,
TeXStudio,
JabRef,
pstricks

Computer algebra etc:
Wolfram Mathematica,
Mathworks Matlab,
GNU Octave

(h264 video with AAC audio in an .mp4 container; needs HTML5 compatible browser)

Updated October 12, 2018.

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