JabRef references

Author	Title	Year	Journal/Proceedings	Reftype	DOI/URL
Vande Velde, S., Piron, A., Leo, O. and Loris, I.	SmartFACS: a computational deconvolution method for blood and tumor samples in humans and mice [Abstract] [BibTeX]	2024		article
Abstract: An important confounding factor in the analysis of expression data is the cellular composition of the samples. This is often overlooked and can lead to a loss of information from less abundant cell types. The distribution and abundance of immune cells play a central role in cancer immunology and hematology. For this reason, we developed a computational cell sorter called SmartFACS to estimate the percentages of over 20 immune cell types from bulk RNA-seq data. The algorithm at the heart of this program is used for the first time in the field of deconvolution. The reference profiles contained in SmartFACS for the detection of the cell types enable highly accurate analysis of tumor samples as well as blood samples from healthy patients or those suffering from autoimmune diseases. SmartFACS is also suitable for mouse data, which is very important for research applications. A comparison with current deconvolution methods has been carried out to demonstrate the powerful and unique function of SmartFACS in the estimation of tumor infiltrates and the analysis of blood samples. Finally, two concrete applications are presented: one in immuno-oncology and the other in vaccinology.
BibTeX: @article{Velde2024, author = {Vande Velde, Sylvie and Anthony Piron and Oberdan Leo and Ignace Loris}, title = {SmartFACS: a computational deconvolution method for blood and tumor samples in humans and mice}, year = {2024}, note = {To be submitted.} }
Fest, J.-B., Heikkilä, T., Loris, I., Martin, S., Ratti, L., Rebegoldi, S. and Sarnighausen, G.	On a fixed-point continuation method for a convex optimization problem [Abstract] [BibTeX]	2023	Advanced Techniques in Optimization for Machine learning and Imaging	inproceedings	URL
Abstract: We consider a variation of the classical proximal-gradient algorithm for the iterative minimization of a cost function consisting of a sum of two terms, one smooth and the other prox-simple, and whose relative weight is determined by a penalty parameter. This so-called fixed-point continuation method allows one to approximate the problem's trade-off curve, i.e. to compute the minimizers of the cost function for a whole range of values of the penalty parameter at once. The algorithm is shown to converge, and a rate of convergence of the cost function is also derived. Furthermore, it is shown that this method is related to iterative algorithms constructed on the basis of the 𝜖-subdifferential of the prox-simple term. Some numerical examples are provided.
BibTeX: @inproceedings{Fest2022, author = {Jean-Baptiste Fest and Tommi Heikkilä and Ignace Loris and Ségolène Martin and Luca Ratti and Simone Rebegoldi and Gesa Sarnighausen}, title = {On a fixed-point continuation method for a convex optimization problem}, booktitle = {Advanced Techniques in Optimization for Machine learning and Imaging}, publisher = {Springer}, year = {2023}, note = {Accepted.}, url = {https://arxiv.org/abs/2212.12256} }
Loris, I. and Rebegoldi, S.	Convergence analysis of a primal-dual optimization-by-continuation algorithm [Abstract] [BibTeX]	2023	Journal of Computational and Applied Mathematics, pp. 22	article	URL
Abstract: We present a numerical iterative optimization algorithm for the minimization of a cost function consisting of a linear combination of three convex terms, one of which is differentiable, a second one is prox-simple and the third one is the composition of a linear map and a prox-simple function. The algorithm's special feature lies in its ability to approximate, in a single iteration run, the minimizers of the cost function for many different values of the parameters determining the relative weight of the three terms in the cost function. A proof of convergence of the algorithm, based on an inexact variable metric approach, is also provided. As a special case, one recovers a generalization of the primal-dual algorithm of Chambolle and Pock, and also of the proximal-gradient algorithm. Finally, we show how it is related to a primal-dual iterative algorithm based on inexact proximal evaluations of the non-smooth terms of the cost function.
BibTeX: @article{Loris2023, author = {Ignace Loris and Simone Rebegoldi}, title = {Convergence analysis of a primal-dual optimization-by-continuation algorithm}, journal = {Journal of Computational and Applied Mathematics}, year = {2023}, pages = {22}, note = {Submitted.}, url = {https://arxiv.org/abs/2311.09123} }
Denneulin, L., Pustelnik, N., Langlois, M., Loris, I. and Thiébaut, E.	Primal-dual splitting scheme with backtracking for handling with epigraphic constraint and sparse analysis regularization [Abstract] [BibTeX]	2020	International Traveling Workshop on Interactions between low-complexity data models and Sensing Techniques (iTWIST 2020)	inproceedings	URL
Abstract: The convergence of many proximal algorithms involving a gradient descent relies on its Lipschitz constant. To avoid computing it, backtracking rules can be used. While such a rule has already been designed for the forward-backward algorithm (FBwB), this scheme is not flexible enough when a nondifferentiable penalization with a linear operator is added to a constraint. In this work we propose a backtracking rule for the primal-dual scheme (PDwB), and evaluate its performance for the epigraphical constrained high dynamical reconstruction in high contrast polarimetric imaging, under TV penalization.
BibTeX: @inproceedings{Denneulin2020, author = {Laurence Denneulin and Nelly Pustelnik and Maud Langlois and Ignace Loris and Eric Thiébaut}, title = {Primal-dual splitting scheme with backtracking for handling with epigraphic constraint and sparse analysis regularization}, booktitle = {International Traveling Workshop on Interactions between low-complexity data models and Sensing Techniques (iTWIST 2020)}, year = {2020}, url = {https://arxiv.org/abs/2009.03576} }
Chen, J. and Loris, I.	On starting and stopping criteria for nested primal-dual iterations [Abstract] [BibTeX]	2019	Numerical algorithms Vol. 82(2), pp. 605-621	article	DOI 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.
BibTeX: @article{Chen.Loris2019, author = {Jixin Chen and Ignace Loris}, title = {On starting and stopping criteria for nested primal-dual iterations}, journal = {Numerical algorithms}, year = {2019}, volume = {82}, number = {2}, pages = {605--621}, url = {https://arxiv.org/abs/1806.07677}, doi = {https://doi.org/10.1007/s11075-018-0616-x} }
Bonettini, S., Loris, I., Porta, F., Prato, M. and Rebegoldi, S.	On the convergence of a linesearch based proximal-gradient method for nonconvex optimization [Abstract] [BibTeX]	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.
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} }
Bonettini, S., Loris, I., Porta, F. and Prato, M.	Variable metric inexact line-search based methods for nonsmooth optimization [Abstract] [BibTeX]	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.
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} }
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 [Abstract] [BibTeX]	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.
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} }
Porta, F. and Loris, I.	On some steplength approaches for proximal algorithms [Abstract] [BibTeX]	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.
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} }
Loris, I.	Numerical algorithms for non-smooth optimization applicable to seismic recovery [Abstract] [BibTeX]	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.
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} }
Nassiri, V. and Loris, I.	An efficient algorithm for structured sparse quantile regression [Abstract] [BibTeX]	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.
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} }
Schretter, C., Loris, I., Dooms, A. and Schelkens, P.	Total Variation reconstruction from quasi-random samples [Abstract] [BibTeX]	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.
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} }
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 [Abstract] [BibTeX]	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.
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} }
Loris, I. and Verhoeven, C.	An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography [Abstract] [BibTeX]	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.
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} }
Nassiri, V. and Loris, I.	A generalized quantile regression model [Abstract] [BibTeX]	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.
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} }
Cloquet, C., Loris, I., Verhoeven, C. and Defrise, M.	GISTA reconstructs faster with a restart strategy and even faster with a FISTA-like reconstruction [Abstract] [BibTeX]	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.
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} }
Loris, I.	A generalization of the iterative soft-thresholding algorithm for non-separable penalties [Abstract] [BibTeX]	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.
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} }
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} }
Loris, I. and Verhoeven, C.	Iterative algorithms for total variation-like reconstructions in seismic tomography [Abstract] [BibTeX]	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.
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} }
Nassiri, V. and Loris, I.	On log-concavity of skew-symmetric distributions and their applications in penalized linear models [Abstract] [BibTeX]	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.
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/} }
Loris, I. and Verhoeven, C.	On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty [Abstract] [BibTeX]	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.
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} }
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 [Abstract] [BibTeX]	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.
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} }
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 [Abstract] [BibTeX]	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
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} }
Loris, I., Douma, H., Nolet, G., Daubechies, I. and Regone, C.	Nonlinear regularization techniques for seismic tomography [Abstract] [BibTeX]	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.
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} }
Loris, I. and Verhoeven, C.	Practical error estimates for sparse recovery in linear inverse problems. [Abstract] [BibTeX]	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.
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} }
Brodie, J., Daubechies, I., De Mol, C., Giannone, D. and Loris, I.	Sparse and stable Markowitz portfolios [Abstract] [BibTeX]	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.
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} }
Loris, I.	On the performance of algorithms for the minimization of _1-penalized functionals [Abstract] [BibTeX]	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.
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} }
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 [Abstract] [BibTeX]	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.
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} }
Daubechies, I., Fornasier, M. and Loris, I.	Accelerated projected gradient method for linear inverse problems with sparsity constraints [Abstract] [BibTeX]	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.
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} }
Loris, I.	L1Packv2: A Mathematica package for minimizing an _1-penalized functional [Abstract] [BibTeX]	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.
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} }
Loris, I., Nolet, G., Daubechies, I. and Dahlen, F.A.	Tomographic inversion using _1-norm regularization of wavelet coefficients [Abstract] [BibTeX]	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.
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} }
Khare, A., Loris, I. and Sasaki, R.	Affine Toda-Sutherland systems [Abstract] [BibTeX]	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.
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} }
Loris, I. and Sasaki, R.	Quantum and classical eigenfunctions in Calogero and Sutherland systems [Abstract] [BibTeX]	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
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} }
Loris, I. and Sasaki, R.	Quantum vs classical mechanics: role of elementary excitations [Abstract] [BibTeX]	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.
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} }
Loris, I.	Bilinear representations of integrable equations [Abstract] [BibTeX]	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.
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} }
Lambert, F., Loris, I. and Springael, J.	Classical Darboux transformations and the KP hierarchy [Abstract] [BibTeX]	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).
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} }
Lambert, F., Loris, I., Springael, J. and Willox, R.	On the Hirota representation of soliton equations with one tau-function [Abstract] [BibTeX]	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.
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} }
Loris, I.	Dimensional reductions of BKP and CKP hierarchies [Abstract] [BibTeX]	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.
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} }
Loris, I.	Solutions of coupled Korteweg-de Vries systems [Abstract] [BibTeX]	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.
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} }
Loris, I.	Recursion operator for a constrained BKP system [Abstract] [BibTeX]	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.
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} }
Loris, I.	On reduced CKP equations [Abstract] [BibTeX]	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.
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} }
Loris, I. and Willox, R.	Symmetry reductions of the BKP hierarchy [Abstract] [BibTeX]	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
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} }
Willox, R. and Loris, I.	An algebraic description of generalized k-constraints [Abstract] [BibTeX]	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.
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} }
Willox, R. and Loris, I.	KP constraints from reduced multi-component hierarchies [Abstract] [BibTeX]	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.
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} }
Loris, I.	Symmetry reductions in the tau-function approach to integrability [Abstract] [BibTeX]	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''. 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) … 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. 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. 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. 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. 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.
BibTeX: @phdthesis{Loris1998, author = {Loris, Ignace}, title = {Symmetry reductions in the tau-function approach to integrability}, school = {Vrije Universiteit Brussel}, year = {1998} }
Willox, R., Tokihiro, T., Loris, I. and Satsuma, J.	The fermionic approach to Darboux transformations [Abstract] [BibTeX]	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.
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} }
Loris, I. and Willox, R.	Bilinear form and solutions of the k-constrained Kadomtsev-Petviashvili hierarchy [Abstract] [BibTeX]	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
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} }
Loris, I. and Willox, R.	KP symmetry reductions and a generalized constraint [Abstract] [BibTeX]	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).
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} }
Loris, I. and Willox, R.	On solutions of constrained Kadomtsev-Petviashvili equations: Grammians [Abstract] [BibTeX]	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
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} }
Loris, I. and Willox, R.	On solutions of constrained KP equations [Abstract] [BibTeX]	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,…).
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} }
Loris, I. and Willox, R.	Recent results on dimensional reductions of the Kadomtsev-Petviashvili equation [Abstract] [BibTeX]	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).
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} }
Pelinovsky, D., Springael, J., Lambert, F. and Loris, I.	On modified NLS, Kaup and NLBq equations: differential transformations and bilinearization [Abstract] [BibTeX]	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.
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} }
Willox, R., Loris, I. and Gilson, C.R.	Binary Darboux transformations for constrained KP hierarchies [Abstract] [BibTeX]	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.
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} }
Loris, I., Lambert, F. and Willox, R.	New ways of applying the Hirota method in soliton theory [Abstract] [BibTeX]	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.
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.} }
Loris, I. and Willox, R.	Soliton solutions of Wronskian form to the nonlocal Boussinesq equation [Abstract] [BibTeX]	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.
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} }
Springael, J., Hu, X.B. and Loris, I.	Bilinear characterization of higher order Ito-equations [Abstract] [BibTeX]	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.
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} }
Willox, R. and Loris, I.	Symmetry constraints of the KP hierarchy and a nonlocal Boussinesq equation [Abstract] [BibTeX]	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.
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} }
Willox, R., Loris, I. and Springael, J.	The nlBq-hierarchy as a pq=C reduction of the KP-hierarchy [Abstract] [BibTeX]	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.
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} }
Lambert, F., Loris, I., Springael, J. and Willox, R.	A direct bilinearization scheme based on the use of partition polynomials [Abstract] [BibTeX]	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.
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} }
Willox, R., Loris, I. and Springael, J.	Bilinearization of the nonlocal Boussinesq equation [Abstract] [BibTeX]	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.
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} }
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 [Abstract] [BibTeX]	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.
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} }
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} }