Jean TSHIMANGA ILUNGA - Academia.edu (original) (raw)

Papers by Jean TSHIMANGA ILUNGA

Research paper thumbnail of Conjugate-gradients versus multigrid solvers for diffusion-based correlation models in data assimilation

HAL (Le Centre pour la Communication Scientifique Directe), 2014

This paper provides a theoretical and experimental comparison between conjugate-gradients and mul... more This paper provides a theoretical and experimental comparison between conjugate-gradients and multigrid, two iterative schemes for solving linear systems, in the context of applying diffusionbased correlation models in data assimilation. In this context, a large number of such systems has to be (approximately) solved if the implicit mode is chosen for integrating the involved diffusion equation over pseudo-time, thereby making their efficient handling crucial for practical performance. It is shown that the multigrid approach has a significant advantage, especially for larger correlation lengths and/or large problem sizes.

Research paper thumbnail of On a second-order expansion of the truncated singular subspace decomposition

Numerical Linear Algebra With Applications, Jan 25, 2016

We present a second-order expansion for singular subspace decomposition in the context of real ma... more We present a second-order expansion for singular subspace decomposition in the context of real matrices. Furthermore, we show that, when some particular assumptions are considered, the obtained results reduce to existing ones. Some numerical examples are provided to confirm the theoretical developments of this study.

Research paper thumbnail of Parallelisation of MACOPA, a Multi-Physics Asynchronous Solver

HAL (Le Centre pour la Communication Scientifique Directe), 2017

International audienceMacopa is a partial differential equations solver based on a particular loc... more International audienceMacopa is a partial differential equations solver based on a particular local time-stepping technique dedicated to multi-physics and multi-scale problems. Here, some parallelisation strategies – multi-threading, domain decomposition, and hybrid OpenMP/MPI – are introduced for this solver. Their efficiency is evaluated on a few examples

Research paper thumbnail of Comparaison de deux implémentations de la méthode quasi Newton à mémoire limitée utilisées dans un problème d'assimilation de données en océanographie

Research paper thumbnail of Parallelisation of MACOPA, A Multi-physics Asynchronous Solver

Macopa is a partial differential equations solver based on a particular local time-stepping techn... more Macopa is a partial differential equations solver based on a particular local time-stepping technique dedicated to multi-physics and multi-scale problems. Here, some parallelisation strategies – multi-threading, domain decomposition, and hybrid OpenMP/MPI – are introduced for this solver. Their efficiency is evaluated on a few examples.

Research paper thumbnail of Conjugate gradients versus multigrid solvers for diffusion-based correlation models in data assimilation

Quarterly Journal of the Royal Meteorological Society, 2013

This paper provides a theoretical and experimental comparison between conjugate-gradients and mul... more This paper provides a theoretical and experimental comparison between conjugate-gradients and multigrid, two iterative schemes for solving linear systems, in the context of applying diffusionbased correlation models in data assimilation. In this context, a large number of such systems has to be (approximately) solved if the implicit mode is chosen for integrating the involved diffusion equation over pseudo-time, thereby making their efficient handling crucial for practical performance. It is shown that the multigrid approach has a significant advantage, especially for larger correlation lengths and/or large problem sizes.

Research paper thumbnail of On A Class of Limited Memory Preconditioners For Large Scale Linear Systems With Multiple Right-Hand Sides

SIAM Journal on Optimization, 2011

This work is concerned with the development and study of a class of limited memory preconditioner... more This work is concerned with the development and study of a class of limited memory preconditioners for the solution of sequences of linear systems. To this aim, we consider linear systems with the same symmetric positive definite matrix and multiple right-hand sides available in sequence. We first propose a general class of preconditioners, called Limited Memory Preconditioners (LMP), whose construction involves only a small number of linearly independent vectors and their product with the matrix to precondition. After exploring and illustrating the theoretical properties of this new class of preconditioners, we more particularly study three members of the class named spectral-LMP, quasi-Newton-LMP and Ritz-LMP, and show that the two first correspond to two well-known preconditioners (see [8] and [20], respectively), while the third one appears to be a new and quite promising preconditioner, as illustrated by numerical experiments.

Research paper thumbnail of Differentiating the Method of Conjugate Gradients

SIAM Journal on Matrix Analysis and Applications, 2014

The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse ... more The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations = , where ∈ ℜ × is symmetric positive definite. Let denote the-th iterate of CG. This is a nonlinear differentiable function of. In this paper we obtain expressions for , the Jacobian matrix of with respect to. We use these expressions to obtain bounds on ∥ ∥2, the spectral norm condition number of , and discuss algorithms to compute or estimate and for a given vector. * The research presented in this paper was conducted with the support of the "Assimilation de Données pour la Terre, l'Atmosphère et l'Océan (ADTAO)" project, funded by the "Fondation Sciences et Technologies pour l'Aéronautique et l'Espace (STAE)", Toulouse, France, within the "Réseau Thématique de Recherche Avancée (RTRA)".

Research paper thumbnail of A reduced and limited-memory preconditioned approach for the 4D-Var data-assimilation problem

Quarterly Journal of the Royal Meteorological Society, 2011

Data assimilation is a concept involving any method which estimates the initial state of a dynami... more Data assimilation is a concept involving any method which estimates the initial state of a dynamical system by combining both the information from a numerical model and from observations. The computed estimated initial state of the system can then be integrated in time to obtain a forecast. There are two main ways to solve data assimilation problems. The sequential methods are based on statistical estimation theory and regroup the different Kalman filtering approaches.

Research paper thumbnail of Sides

This work is concerned with the development and study of a class of limited memory preconditioner... more This work is concerned with the development and study of a class of limited memory preconditioners for the solution of sequences of linear systems. To this aim, we consider linear systems with the same symmetric positive definite matrix and multiple right-hand sides available in sequence. We first propose a general class of preconditioners, called Limited Memory Preconditioners (LMP), whose construction involves only a small number of linearly independent vectors and their product with the matrix to precondition. After exploring and illustrating the theoretical properties of this new class of preconditioners, we more particularly study three members of the class named spectral-LMP, quasi-Newton-LMP and Ritz-LMP, and show that the two first correspond to two well-known preconditioners (see [8] and [20], respectively), while the third one appears to be a new and quite promising preconditioner, as illustrated by numerical experiments.

Research paper thumbnail of Sensitivity and Conditioning of the Truncated Total Least Squares Solution

SIAM Journal on Matrix Analysis and Applications, 2013

We present an explicit expression for the condition number of the truncated total least squares (... more We present an explicit expression for the condition number of the truncated total least squares (TLS) solution of ≈. This expression is obtained using the notion of the Fréchet derivative. We also give upper bounds on the condition number which are simple to compute and interpret. These results generalize those in the literature for the untruncated TLS problem. Numerical experiments demonstrate that our bounds are often a very good estimate of the condition number, and provide a significant improvement to known bounds.

Research paper thumbnail of On a second-order expansion of the truncated singular subspace decomposition

Numer. Linear Algebra Appl., 2016

Matrix subspace decompositions have numerous applications in science and technology. One of these... more Matrix subspace decompositions have numerous applications in science and technology. One of these matrix decompositions is the singular subspace decomposition. Signal processing, image processing, and discrete rank-deficient problems, to cite only few, constitute classes of practical problems where such decompositions play an important role. To begin with, let the real‡ matrix A 2 <n m have rank r 6 min.n;m/ and have a singular subspace decomposition

Research paper thumbnail of The Exact Condition Number of the Truncated Singular Value Solution of a Linear Ill-Posed Problem

The main result in this paper is the investigation of an explicit expression of the condition num... more The main result in this paper is the investigation of an explicit expression of the condition number of the truncated least squares solution of Ax = b. The result is derived using the notion of the Fréchet derivative together with the product norm [αA, βb] F , with α, β > 0, for the data space and the 2-norm for the solution. We also derive a lower and an upper bounds to estimate the condition number for the general case. Finally, we carry out numerical experiments and compare our results with respect to a finite difference approach.

Research paper thumbnail of Range-Space Variants and Inexact Matrix-Vector Products in Krylov Solvers for Linear Systems Arising from Inverse Problems

SIAM Journal on Matrix Analysis and Applications, 2011

The object of this paper is to introduce range-space variants of standard Krylov iterative solver... more The object of this paper is to introduce range-space variants of standard Krylov iterative solvers for unsymmetric and symmetric linear systems, and to discuss how inexact matrix-vector products may be used in this context. The new range-space variants are characterized by possibly much lower storage and computational costs than their full-space counterparts, which is crucial in data assimilation applications and other inverse problems. However, this gain is achieved without sacrifying the inherent monotonicity properties of the original algorithms, which are of paramount importance in data assimilation applications. The use of inexact matrix-vector products is shown to further reduce computational cost in a controlled manner. Formal error bounds are derived on the size of the residuals obtained under two different accuracy models, and it is shown why a model controlling forward error on the product result is often preferable to one controlling backward error on the operator. Simple numerical examples finally illustrate the developed concepts and methods.

Research paper thumbnail of Conjugate-gradients versus multigrid solvers for diffusion-based correlation models in data assimilation

HAL (Le Centre pour la Communication Scientifique Directe), 2014

This paper provides a theoretical and experimental comparison between conjugate-gradients and mul... more This paper provides a theoretical and experimental comparison between conjugate-gradients and multigrid, two iterative schemes for solving linear systems, in the context of applying diffusionbased correlation models in data assimilation. In this context, a large number of such systems has to be (approximately) solved if the implicit mode is chosen for integrating the involved diffusion equation over pseudo-time, thereby making their efficient handling crucial for practical performance. It is shown that the multigrid approach has a significant advantage, especially for larger correlation lengths and/or large problem sizes.

Research paper thumbnail of On a second-order expansion of the truncated singular subspace decomposition

Numerical Linear Algebra With Applications, Jan 25, 2016

We present a second-order expansion for singular subspace decomposition in the context of real ma... more We present a second-order expansion for singular subspace decomposition in the context of real matrices. Furthermore, we show that, when some particular assumptions are considered, the obtained results reduce to existing ones. Some numerical examples are provided to confirm the theoretical developments of this study.

Research paper thumbnail of Parallelisation of MACOPA, a Multi-Physics Asynchronous Solver

HAL (Le Centre pour la Communication Scientifique Directe), 2017

International audienceMacopa is a partial differential equations solver based on a particular loc... more International audienceMacopa is a partial differential equations solver based on a particular local time-stepping technique dedicated to multi-physics and multi-scale problems. Here, some parallelisation strategies – multi-threading, domain decomposition, and hybrid OpenMP/MPI – are introduced for this solver. Their efficiency is evaluated on a few examples

Research paper thumbnail of Comparaison de deux implémentations de la méthode quasi Newton à mémoire limitée utilisées dans un problème d'assimilation de données en océanographie

Research paper thumbnail of Parallelisation of MACOPA, A Multi-physics Asynchronous Solver

Macopa is a partial differential equations solver based on a particular local time-stepping techn... more Macopa is a partial differential equations solver based on a particular local time-stepping technique dedicated to multi-physics and multi-scale problems. Here, some parallelisation strategies – multi-threading, domain decomposition, and hybrid OpenMP/MPI – are introduced for this solver. Their efficiency is evaluated on a few examples.

Research paper thumbnail of Conjugate gradients versus multigrid solvers for diffusion-based correlation models in data assimilation

Quarterly Journal of the Royal Meteorological Society, 2013

This paper provides a theoretical and experimental comparison between conjugate-gradients and mul... more This paper provides a theoretical and experimental comparison between conjugate-gradients and multigrid, two iterative schemes for solving linear systems, in the context of applying diffusionbased correlation models in data assimilation. In this context, a large number of such systems has to be (approximately) solved if the implicit mode is chosen for integrating the involved diffusion equation over pseudo-time, thereby making their efficient handling crucial for practical performance. It is shown that the multigrid approach has a significant advantage, especially for larger correlation lengths and/or large problem sizes.

Research paper thumbnail of On A Class of Limited Memory Preconditioners For Large Scale Linear Systems With Multiple Right-Hand Sides

SIAM Journal on Optimization, 2011

This work is concerned with the development and study of a class of limited memory preconditioner... more This work is concerned with the development and study of a class of limited memory preconditioners for the solution of sequences of linear systems. To this aim, we consider linear systems with the same symmetric positive definite matrix and multiple right-hand sides available in sequence. We first propose a general class of preconditioners, called Limited Memory Preconditioners (LMP), whose construction involves only a small number of linearly independent vectors and their product with the matrix to precondition. After exploring and illustrating the theoretical properties of this new class of preconditioners, we more particularly study three members of the class named spectral-LMP, quasi-Newton-LMP and Ritz-LMP, and show that the two first correspond to two well-known preconditioners (see [8] and [20], respectively), while the third one appears to be a new and quite promising preconditioner, as illustrated by numerical experiments.

Research paper thumbnail of Differentiating the Method of Conjugate Gradients

SIAM Journal on Matrix Analysis and Applications, 2014

The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse ... more The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations = , where ∈ ℜ × is symmetric positive definite. Let denote the-th iterate of CG. This is a nonlinear differentiable function of. In this paper we obtain expressions for , the Jacobian matrix of with respect to. We use these expressions to obtain bounds on ∥ ∥2, the spectral norm condition number of , and discuss algorithms to compute or estimate and for a given vector. * The research presented in this paper was conducted with the support of the "Assimilation de Données pour la Terre, l'Atmosphère et l'Océan (ADTAO)" project, funded by the "Fondation Sciences et Technologies pour l'Aéronautique et l'Espace (STAE)", Toulouse, France, within the "Réseau Thématique de Recherche Avancée (RTRA)".

Research paper thumbnail of A reduced and limited-memory preconditioned approach for the 4D-Var data-assimilation problem

Quarterly Journal of the Royal Meteorological Society, 2011

Data assimilation is a concept involving any method which estimates the initial state of a dynami... more Data assimilation is a concept involving any method which estimates the initial state of a dynamical system by combining both the information from a numerical model and from observations. The computed estimated initial state of the system can then be integrated in time to obtain a forecast. There are two main ways to solve data assimilation problems. The sequential methods are based on statistical estimation theory and regroup the different Kalman filtering approaches.

Research paper thumbnail of Sides

This work is concerned with the development and study of a class of limited memory preconditioner... more This work is concerned with the development and study of a class of limited memory preconditioners for the solution of sequences of linear systems. To this aim, we consider linear systems with the same symmetric positive definite matrix and multiple right-hand sides available in sequence. We first propose a general class of preconditioners, called Limited Memory Preconditioners (LMP), whose construction involves only a small number of linearly independent vectors and their product with the matrix to precondition. After exploring and illustrating the theoretical properties of this new class of preconditioners, we more particularly study three members of the class named spectral-LMP, quasi-Newton-LMP and Ritz-LMP, and show that the two first correspond to two well-known preconditioners (see [8] and [20], respectively), while the third one appears to be a new and quite promising preconditioner, as illustrated by numerical experiments.

Research paper thumbnail of Sensitivity and Conditioning of the Truncated Total Least Squares Solution

SIAM Journal on Matrix Analysis and Applications, 2013

We present an explicit expression for the condition number of the truncated total least squares (... more We present an explicit expression for the condition number of the truncated total least squares (TLS) solution of ≈. This expression is obtained using the notion of the Fréchet derivative. We also give upper bounds on the condition number which are simple to compute and interpret. These results generalize those in the literature for the untruncated TLS problem. Numerical experiments demonstrate that our bounds are often a very good estimate of the condition number, and provide a significant improvement to known bounds.

Research paper thumbnail of On a second-order expansion of the truncated singular subspace decomposition

Numer. Linear Algebra Appl., 2016

Matrix subspace decompositions have numerous applications in science and technology. One of these... more Matrix subspace decompositions have numerous applications in science and technology. One of these matrix decompositions is the singular subspace decomposition. Signal processing, image processing, and discrete rank-deficient problems, to cite only few, constitute classes of practical problems where such decompositions play an important role. To begin with, let the real‡ matrix A 2 <n m have rank r 6 min.n;m/ and have a singular subspace decomposition

Research paper thumbnail of The Exact Condition Number of the Truncated Singular Value Solution of a Linear Ill-Posed Problem

The main result in this paper is the investigation of an explicit expression of the condition num... more The main result in this paper is the investigation of an explicit expression of the condition number of the truncated least squares solution of Ax = b. The result is derived using the notion of the Fréchet derivative together with the product norm [αA, βb] F , with α, β > 0, for the data space and the 2-norm for the solution. We also derive a lower and an upper bounds to estimate the condition number for the general case. Finally, we carry out numerical experiments and compare our results with respect to a finite difference approach.

Research paper thumbnail of Range-Space Variants and Inexact Matrix-Vector Products in Krylov Solvers for Linear Systems Arising from Inverse Problems

SIAM Journal on Matrix Analysis and Applications, 2011

The object of this paper is to introduce range-space variants of standard Krylov iterative solver... more The object of this paper is to introduce range-space variants of standard Krylov iterative solvers for unsymmetric and symmetric linear systems, and to discuss how inexact matrix-vector products may be used in this context. The new range-space variants are characterized by possibly much lower storage and computational costs than their full-space counterparts, which is crucial in data assimilation applications and other inverse problems. However, this gain is achieved without sacrifying the inherent monotonicity properties of the original algorithms, which are of paramount importance in data assimilation applications. The use of inexact matrix-vector products is shown to further reduce computational cost in a controlled manner. Formal error bounds are derived on the size of the residuals obtained under two different accuracy models, and it is shown why a model controlling forward error on the product result is often preferable to one controlling backward error on the operator. Simple numerical examples finally illustrate the developed concepts and methods.