Recent Decisions

Many inverse problems can be described by a PDE model with unknown parameters that need to be calibrated based on measurements related to its solution. This can be seen as a constrained minimization problem where one wishes to minimize the mismatch between the observed data and the model predictions, including an extra regularization term, and use the PDE as a constraint. Often, a suitable regularization parameter is determined by solving the problem for a whole range of parameters -- e.g. using the L-curve -- which is computationally very expensive. In this paper we derive two methods that simultaneously solve the inverse problem and determine a suitable value for the regularization parameter. The first one is a direct generalization of the Generalized Arnoldi Tikhonov method for linear inverse problems. The second method is a novel method based on similar ideas, but with a number of advantages for nonlinear problems

Projected Newton Method for noise constrained Tikhonov regularization

Tikhonov regularization is a popular approach to obtain a meaningful solution for ill-conditioned linear least squares problems. A relatively simple way of choosing a good regularization parameter is given by Morozov's discrepancy principle. However, most approaches require the solution of the Tikhonov problem for many different values of the regularization parameter, which is computationally demanding for large scale problems. We propose a new and efficient algorithm which simultaneously solves the Tikhonov problem and finds the corresponding regularization parameter such that the discrepancy principle is satisfied. We achieve this by formulating the problem as a nonlinear system of equations and solving this system using a line search method. We obtain a good search direction by projecting the problem onto a low dimensional Krylov subspace and computing the Newton direction for the projected problem. This projected Newton direction, which is significantly less computationally expensive to calculate than the true Newton direction, is then combined with a backtracking line search to obtain a globally convergent algorithm, which we refer to as the Projected Newton method. We prove convergence of the algorithm and illustrate the improved performance over current state-of-the-art solvers with some numerical experiments

Exploring variable accuracy storage through lossy compression techniques in numerical linear algebra: a first application to flexible GMRES

Large scale applications running on HPC systems often require a substantial amount of memory and can have a large computational overhead. Lossy data compression techniques can reduce the size of the data and associated communication cost, but the effect of the loss ofaccuracy on the numerical algorithm can be hard to predict. In this paper we examine the FGMRES algorithm, which requires the storage of a basis for the Krylov subspace and for the search space spanned by the solutions of the preconditioning systems. We show that the vectors spanning this search space can be compressed by looking at the combination of FGMRES and compression in the context of inexact Krylov subspace methods. This allows us to derive a bound on the normwise relative compression error in each iteration. We use this bound to formulate a number of different practical compression strategies, and validate and compare them through numerical experiments.Les applications à grande échelle fonctionnant sur des systèmes HPC nécessitent souvent une quantité importante de mémoire et peuvent avoir une charge de calcul importante.Les techniques de compression de données avec perte peuvent réduire la taille des données et les coûts de communication associés, mais l’effet de la perte de précision sur l’algorithme numérique peut être difficile à prévoir. Dans cet article, nous examinons l’algorithme FGMRES, qui nécessite le stockage d’une base pour le sous-espace de Krylov et pour l’espace de recherche couvert parles solutions des systèmes de préconditionnement. Nous montrons que les vecteurs couvrant cet espace de recherche peuvent être comprimés en examinant la combinaison de FGMRES et de la compression dans le contexte des méthodes inexactes du sous-espace de Krylov. Cela nous permet de dériver une borne sur l’erreur de compression relative normale dans chaque itération. Nous utilisons cette limite pour formuler un certain nombre de stratégies de compression pratiques différentes, et les valider et les comparer par des expériences numériques

A complementary note on soft errors in the Conjugate Gradient method: the persistent error case

This note is a follow up study to [1], where we studied the resilience of the preconditioned conjugate gradient method (PCG). We complement the original work by performinga similar series of numerical experiments, but using what we called persistent instead of transient bit-flips.Cette note est une étude qui fait suite à [1], où nous avons étudié la résilience de la méthode du gradient conjugué préconditionné (PCG). Nous complétons le travail initial en effectuant une série similaire d’expériences numériques, mais en utilisant ce que nous avons appelé des bit-flips persistants au lieu de transitoires

Les variantes rétro-stables de GMRES en précision variable

In the context where the representation of the data is decoupled from the arithmetic used to process them, we investigate the backward stability of two backward-stable implementations of the GMRES method, namely the so-calledModified Gram-Schmidt (MGS) and the Householder variants. Considering data may be compressed to alleviate the memory footprint, we are interested in the situation where the leading part of the rounding error is related to the datarepresentation. When the data representation of vectors introduces componentwise perturbations, we show that the existing backward stability analyses of MGS-GMRES and Householder-GMRES still apply. We illustrate this backward stability property in a practical context where an agnostic lossy compressor is employed and enables the reduction of the memory requirement to store the orthonormal Arnoldi basis or the Householder reflectors. Although technical arguments of the theoretical backward stability proofs do not readily apply to the situation where only the normwise relative perturbations of the vector storage can be controlled, we show experimentally that the backward stability is maintained; that is, the attainable normwise backward error is of the same order as the normwise perturbations induced by the data storage. We illustrate it with numerical experiments in two practical different contexts. The first one corresponds to the use of an agnostic compressor where vector compression is controlled normwise. The second one arises in the solution of tensor linear systems, where low-rank tensor approximations based on Tensor-Train is considered to tackle the curse of dimensionality.Dans le contexte où la représentation des données est découplée de l’arithmétique utilisée pour les traiter, nous étudions la stabilité inverse des deux implémentations stables de la méthode GMRES, à savoir la variante dite Modified Gram-Schmidt (MGS) et la variante Householder. Considérant que les données peuvent être compressées pour réduire l’empreinte mémoire, nous nous intéressons à la situation où la partie principale de l’erreur d’arrondi est liée à la représentation des données. Lorsque la représentation des données des vecteurs introduit des perturbations par composantes, les analyses de stabilité inverse existantes de MGS-GMRES [27] et Householder-GMRES [15] restent applicables. Nous illustrons cette propriété de stabilité dans un contexte pratique pratique où un compresseur agnostique à perte est utilisé et permet de réduire la mémoire nécessaire pour stocker la base orthonormale d’Arnoldi ou les réflecteurs de Householder. Bien que les arguments techniques des preuves théoriques de de stabilité inversene s’appliquent pas facilement à la situation où seules les perturbations relatives en norme sont utilisées, nous montrons expérimentalement que la stabilité inverse est maintenue ; c’est-à-dire que l’erreur inverse atteignable est du même ordre que les perturbations normalisées induites par le stockage des données. Nous rapportons des expériences numériques dans deux contextes pratiques différents. Le premier correspond à l’utilisation d’un compresseur agnostique. Le deuxième se présente dans la résolution de systèmes linéaires tensoriels, définis sur un produit tensoriel d’espaces linéaires, où les approximations tensorielles à faible rang basées sur Tensor-Train [26] est envisagée pour lutter contre la malédiction de la dimensionnalité

On soft errors in the conjugate gradient method: sensitivity and robust numerical detection

International audienceThe conjugate gradient (CG) method is the most widely used iterative scheme for the solution of large sparse systems of linear equations when the matrix is symmetric positive definite. Although more than 60 years old, it is still a serious candidate for extreme-scale computations on large computing platforms. On the technological side, the continuous shrinking of transistor geometry and the increasing complexity of these devices affect dramatically their sensitivity to natural radiation and thus diminish their reliability. One of the most common effects produced by natural radiation is the single event upset which consists in a bit-flip in a memory cell producing unexpected results at the application level. Consequently, future extreme-scale computing facilities will be more prone to errors of any kind, including bit-flips, during their calculations. These numerical and technological observations are the main motivations for this work, where we first investigate through extensive numerical experiments the sensitivity of CG to bit-flips in its main computationally intensive kernels, namely the matrix-vector product and the preconditioner application. We further propose numerical criteria to detect the occurrence of such soft errors and assess their robustness through extensive numerical experiments

On soft errors in the Conjugate Gradient method: sensitivity and robust numerical detection -revised

The conjugate gradient (CG) method is the most widely used iterative scheme forthe solution of large sparse systems of linear equations when the matrix is symmetric positivedefinite. Although more than sixty year old, it is still a serious candidate for extreme-scalecomputation on large computing platforms. On the technological side, the continuous shrinkingof transistor geometry and the increasing complexity of these devices affect dramatically theirsensitivity to natural radiation, and thus diminish their reliability. One of the most common effectsproduced by natural radiation is the single event upset which consists in a bit-flip in a memory cellproducing unexpected results at application level. Consequently, the future computing facilitiesat extreme scale might be more prone to errors of any kind including bit-flip during calculation.These numerical and technological observations are the main motivations for this work, where wefirst investigate through extensive numerical experiments the sensitivity of CG to bit-flips in itsmain computationally intensive kernels, namely the matrix-vector product and the preconditionerapplication. We further propose numerical criteria to detect the occurrence of such soft errors; weassess their robustness through extensive numerical experiments.La méthode du gradient conjugue (CG) est la méthode itérative la plus utilisée pour résoudre des systèmes linéaires creux de grande taille lorsque la matrice est symétrique définie positive. Bien que vieille de de soixante ans, cette méthode reste une candidate sérieuse pour être mise en œuvre pour la résolution de très grands systèmes linéaires sur des plateformes de calcul de très grande taille. Sur le plan technologique, la réduction permanente de la taille et la complexité croissante des composantes électroniques de ces calculateurs affecte dramatiquement leur sensibilité aux radiations cosmiques ce qui réduit leur fiabilité. L’un des effets les plus courants des rayonnements naturels est la perturbation due à un événement unique qui consiste en un retournement de bit dans une cellule mémoire produisant des résultats inattendus au niveau de l’application. Par conséquent, les futures installations informatiques à très grande échelle pourraient être plus sujettes à des erreurs de toute sorte. y compris le basculement de bit pendant le calcul. Ces observations numériques et technologiques sont les suivantes les principales motivations de ce travail, pour lequel nous étudions d’abord par le biais d’études approfondies et approfondies la sensibilité de la CG aux sauts de bits dans ses principaux domaines d’application.à forte intensité de calcul, à savoir le produit matrice-vecteur et le produit application du préconditionneur. Nous proposons en outre des critères numériques pour détecter l’apparition de tels défauts ; nous évaluons leur robustesse à travers des expériences numériques approfondies

Investigating the validity of the DN4 in a consecutive population of patients with chronic pain

Neuropathic pain is clinically described as pain caused by a lesion or disease of the somatosensory nervous system. The aim of this study was to assess the validity of the Dutch version of the DN4, in a cross-sectional multicentre design, as a screening tool for detecting a neuropathic pain component in a large consecutive, not pre-stratified on basis of the target outcome, population of patients with chronic pain. Patients’ pain was classified by two independent (pain-)physicians as the gold standard. The analysis was initially performed on the outcomes of those patients (n = 228 out of 291) in whom both physicians agreed in their pain classification. Compared to the gold standard the DN4 had a sensitivity of 75% and specificity of 76%. The DN4-symptoms (seven interview items) solely resulted in a sensitivity of 70% and a specificity of 67%. For the DN4-signs (three examination items) it was respectively 75% and 75%. In conclusion, because it seems that the DN4 helps to identify a neuropathic pain component in a consecutive population of patients with chronic pain in a moderate way, a comprehensive (physical-) examination by the physician is still obligate

Avoiding Catch-22:Validating the PainDETECT in a in a population of patients with chronic pain

BACKGROUND: Neuropathic pain is defined as pain caused by a lesion or disease of the somatosensory nervous system and is a major therapeutic challenge. Several screening tools have been developed to help physicians detect patients with neuropathic pain. These have typically been validated in populations pre-stratified for neuropathic pain, leading to a so called "Catch-22 situation:" "a problematic situation for which the only solution is denied by a circumstance inherent in the problem or by a rule". The validity of screening tools needs to be proven in patients with pain who were not pre-stratified on basis of the target outcome: neuropathic pain or non-neuropathic pain. This study aims to assess the validity of the Dutch PainDETECT (PainDETECT-Dlv) in a large population of patients with chronic pain. METHODS: A cross-sectional multicentre design was used to assess PainDETECT-Dlv validity. Included where patients with low back pain radiating into the leg(s), patients with neck-shoulder-arm pain and patients with pain due to a suspected peripheral nerve damage. Patients' pain was classified as having a neuropathic pain component (yes/no) by two experienced physicians ("gold standard"). Physician opinion based on the Grading System was a secondary comparison. RESULTS: In total, 291 patients were included. Primary analysis was done on patients where both physicians agreed upon the pain classification (n = 228). Compared to the physician's classification, PainDETECT-Dlv had a sensitivity of 80% and specificity of 55%, versus the Grading System it achieved 74 and 46%. CONCLUSION: Despite its internal consistency and test-retest reliability the PainDETECT-Dlv is not an effective screening tool for a neuropathic pain component in a population of patients with chronic pain because of its moderate sensitivity and low specificity. Moreover, the indiscriminate use of the PainDETECT-Dlv as a surrogate for clinical assessment should be avoided in daily clinical practice as well as in (clinical-) research. Catch-22 situations in the validation of screening tools can be prevented by not pre-stratifying the patients on basis of the target outcome before inclusion in a validation study for screening instruments. TRIAL REGISTRATION: The protocol was registered prospectively in the Dutch National Trial Register: NTR 3030