Michel Haïssaguerre
email: michel.haissaguerre@chu-bordeaux.fr

Koonlawee Nademanee

Mélèze Hocini

MD Ghassen Cheniti

MD Josselin Duchateau

MD Antonio Frontera

MD Frédéric Sacher

MD Nicolas Derval

MD Arnaud Denis

MD Thomas Pambrun

MD Rémi Dubois

PhD Pierre Jaïs

MD David Benoist

PhD Richard D Walton

PhD Akihiko Nogami

MD Ruben Coronel

PhD Mark Potse

PhD Olivier Bernus

Prof Michel Haissaguerre

Depolarization versus repolarization abnormality underlying inferolateral J wave syndromes -new concepts in sudden cardiac death with apparently normal hearts

Keywords: Sudden death, ventricular fibrillation, J-wave syndromes, Early repolarization

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Early repolarization indicates a distinct electrocardiographic phenotype affecting the junction (J-point or J-wave) between the QRS complex and the ST segment in inferolateral leads. It was initially described as a benign ECG finding or found in association with hypothermia [START_REF] Shilpey | The four lead electrogram in 200 normal men and women[END_REF][START_REF] Tomaszewski | Changements electrocardiographiques observes chez un homme mort de froid[END_REF][START_REF] Osborn | Experimental hypothermia: respiratory and blood pH changes in relation to cardiac function[END_REF][START_REF] Kambara | Long-term evaluation of early repolarization syndrome (normal variant RS-T-segment elevation)[END_REF] . Subsequently, many other conditions producing this phenotype have been described such as hypercalcemia, acute ischemia, brain injury, and others.

The link with an increased risk of arrhythmic death was later demonstrated in sporadic cases 6- 7 and in case-control studies of unexplained sudden cardiac death (SCD), and finally in association with various types of structural heart disease (SHD) [START_REF] Haissaguerre | Sudden cardiac arrest associated with early repolarization[END_REF][START_REF] Antzelevitch | J-wave syndromes: Brugada and early repolarization syndromes[END_REF][START_REF] Tikkanen | Long-term outcome associated with early repolarization on electrocardiography[END_REF][START_REF] Nam | Mode of onset of ventricular fibrillation in patients with early repolarization pattern vs. Brugada syndrome[END_REF][START_REF] Cheng | Early repolarization is associated with a significantly increased risk of ventricular arrhythmias and sudden cardiac death in patients with structural heart diseases[END_REF] . This article will focus on J-wave syndromes affecting inferolateral leads and review the current knowledge and the limitations in risk stratification. In addition we report new clinical mapping data using highdensity electrode mapping, which provide evidence that two distinct substrates, delayed depolarization and early repolarization abnormality, underlie inferolateral J-wave syndromes.

Diagnosis of Early Repolarization/inferolateral J-wave Syndrome

Expert consensus recommendations [START_REF] Macfarlane | The early repolarization pattern: a consensus paper[END_REF][START_REF] Antzelevitch | J-Wave syndromes expert consensus conference report: Emerging concepts and gaps in knowledge[END_REF] distinguish an early-repolarization ECG pattern (ER) and an early-repolarization syndrome (ERS). The pattern is defined as the presence of a Jpoint elevation≥1 mm in ≥2 contiguous inferior and/or lateral leads of a standard 12-lead ECG. The syndrome is said to be present when the pattern is found in 1) a patient resuscitated from otherwise unexplained ventricular fibrillation (VF) or; 2) an SCD victim with a negative autopsy and medical chart review with a previous ECG. The J-point elevation may be a notch or a slur, with or without elevation of the ST segment.

The definition of J-wave is sometimes ambiguous owing to the small amplitude and spontaneous changes of the signal. The J/ST elevation in inferior leads may be more easily missed as the pattern is less stereotypical than the Brugada syndrome (BrS). Bipolar limb leads (I,III,aVF) may be less sensitive than unipolar precordial leads in identifying discrete J/ST-segment abnormalities, and in assessing the spatial extent of the J-wave. Although unemphasized previously, a J-wave may also be negative and less apparent when it follows a negative QRS complex. Figure 1A illustrates these electrocardiographic variations.

Clinical Significance

The ECG pattern of early repolarization was first reported in 1936 as a normal variant [START_REF] Shilpey | The four lead electrogram in 200 normal men and women[END_REF] . In 1938, Tomaszewski described a slow deflection between the QRS complex and the ST segment in an accidentally hypothermic man [START_REF] Tomaszewski | Changements electrocardiographiques observes chez un homme mort de froid[END_REF] . In 1953, Osborn described a "current of injury" in experimental acidosis and hypothermia in dogs and which was often associated with the development of VF [START_REF] Osborn | Experimental hypothermia: respiratory and blood pH changes in relation to cardiac function[END_REF] .

The ER was known for decades to be an innocent ECG feature that was more common in young men and in athletes, with a prevalence varying between 1 to 24 % in the general population. However, case reports of unexpected SCD associated with an inferolateral J-wave abnormality were published as early as 1984, most of them concerning victims from Japan and Southeast Asia [START_REF] Otto | Ventricular fibrillation causes sudden death in southeast Asian immigrants[END_REF][START_REF] Aizawa | Idiopathic ventricular fibrillation and bradycardia-dependent intraventricular block[END_REF][START_REF] Haissaguerre | Sudden cardiac arrest associated with early repolarization[END_REF][START_REF] Antzelevitch | J-wave syndromes: Brugada and early repolarization syndromes[END_REF] . Then, the potential arrhythmogenicity of ER was shown in wedge preparations by Yan and Antzelevitch [START_REF] Yan | Cellular basis for the electrocardiographic J wave[END_REF] . Finally, a definitive association between J-waves and the pathogenesis of idiopathic VF was described in case-control studies [START_REF] Haissaguerre | Sudden cardiac arrest associated with early repolarization[END_REF][START_REF] Antzelevitch | J-wave syndromes: Brugada and early repolarization syndromes[END_REF][START_REF] Tikkanen | Long-term outcome associated with early repolarization on electrocardiography[END_REF][START_REF] Nam | Mode of onset of ventricular fibrillation in patients with early repolarization pattern vs. Brugada syndrome[END_REF][START_REF] Cheng | Early repolarization is associated with a significantly increased risk of ventricular arrhythmias and sudden cardiac death in patients with structural heart diseases[END_REF][START_REF] Rosso | J-point elevation in survivors of primary ventricular fibrillation and matched control subjects: incidence and clinical significance[END_REF][START_REF] Sinner | Association of early repolarization pattern on ECG with risk of cardiac and all-cause mortality: a population-based prospective cohort study[END_REF][START_REF] Derval | Prevalence and characteristics of early repolarization in the CASPER registry: cardiac arrest survivors with preserved ejection fraction registry[END_REF][START_REF] Junttila | Early repolarization as a predictor of arrhythmic and non-arrhythmic cardiac events in middle-aged subjects[END_REF][START_REF] Seong | Clinical significance of fragmented QRS complexes or J waves in patients with idiopathic ventricular arrhythmias[END_REF] . Although these results have raised some concerns for the affected patients, the high prevalence of Jwaves in the general population causes them to portend a low absolute arrhythmia risk [START_REF] Viskin | Making sense of early repolarization[END_REF][START_REF] Obeyesekere | Early Repolarization -What Should the Clinician Do?[END_REF] .

Risk in asymptomatic subjects

The prevalence of J-waves in athletes is reported to be as high as 44% [START_REF] Tikkanen | Early repolarization: electrocardiographic phenotypes associated with favorable long-term outcome[END_REF][START_REF] Biasco | Clinical, electrocardiographic, echocardiographic characteristics and long term follow up of elite soccer players with J-point elevation[END_REF][START_REF] Quattrini | Benign clinical significance of J-wave pattern (early repolarization) in highly trained athletes[END_REF][START_REF] Ilkhanoff | Clinical characteristics and outcomes associated with the natural history of early repolarization in a young, biracial cohort followed to middle age: the coronary artery risk development in young adults (CARDIA) study[END_REF] . In a study involving 704 athletes (14% harboring a J-wave), with a follow-up of 6 years there was no observation of arrhythmic events [START_REF] Quattrini | Benign clinical significance of J-wave pattern (early repolarization) in highly trained athletes[END_REF] . Similarly, no association with higher risk of death was observed during long-term follow-up in young adults with J-waves [START_REF] Ilkhanoff | Clinical characteristics and outcomes associated with the natural history of early repolarization in a young, biracial cohort followed to middle age: the coronary artery risk development in young adults (CARDIA) study[END_REF] . Interestingly a correlation was found between J-point elevation and septal hypertrophy, as well as a possible link with exercise-induced hypertrophy [START_REF] Biasco | Clinical, electrocardiographic, echocardiographic characteristics and long term follow up of elite soccer players with J-point elevation[END_REF][START_REF] Ilkhanoff | Clinical characteristics and outcomes associated with the natural history of early repolarization in a young, biracial cohort followed to middle age: the coronary artery risk development in young adults (CARDIA) study[END_REF][START_REF] Guillem | Echocardiographic study of the early repolarization syndrome. Demonstration of dynamic obstruction with isoprenaline[END_REF][START_REF] Noseworthy | Early repolarization pattern in competitive athletes: clinical correlates and the effects of exercise training[END_REF] . False tendons have also been related to the genesis of J-waves, potentially through altered depolarization in the Purkinje system due to localized stretching [START_REF] Nakagawa | Electrocardiographic characteristics of patients with false tendon: possible association of false tendon with J waves[END_REF] . Although J-waves are common in African Americans, it is estimated that there is no increased risk in this subgroup whereas a higher risk may be present in Asian populations [START_REF] Olson | Long-term prognosis associated with J-point elevation in a large middle-aged biracial cohort: the ARIC study[END_REF][START_REF] Perez | The prognostic value of early repolarization with ST-segment elevation in African Americans[END_REF][START_REF] Haruta | Incidence and prognostic value of early repolarization pattern in the 12-lead electrocardiogram[END_REF] . Noteworthy, the definition of ER in the prior literature varies in terms of Jwave or ST-elevation pattern; it may include individuals with only ST-elevation and no Jwave, which impacts on the reported prognostic value.

Risk in association with SHD

A large number of studies have suggested that J-waves associated with SHD increase the vulnerability to ventricular arrhythmias but not to non-arrhythmic cardiac events [START_REF] Tikkanen | Association of early repolarization and sudden cardiac death during an acute coronary event[END_REF][START_REF] Ghosh | Early repolarization associated with sudden death: insights from noninvasive electrocardiographic imaging[END_REF][START_REF] Demidova | Transient and rapid QRS-widening associated with a J-wave pattern predicts impending ventricular fibrillation in experimental myocardial infarction[END_REF][START_REF] Patel | Early repolarization associated with ventricular arrhythmias in patients with chronic coronary artery disease[END_REF][START_REF] Pei | The Jwave and fragmented QRS complexes in inferior leads associated with sudden cardiac death in patients with chronic heart failure[END_REF][START_REF] Tsuda | J Waves for Predicting Cardiac Events in Hypertrophic Cardiomyopathy[END_REF] . In a meta-analysis of 19 studies including 7268 SHD patients, a higher risk was associated with J waves specifically in inferior leads, with a notching configuration, and a horizontal or descending ST segment [START_REF] Cheng | Early repolarization is associated with a significantly increased risk of ventricular arrhythmias and sudden cardiac death in patients with structural heart diseases[END_REF] . An increased arrhythmic risk has also been shown in the event of myocardial ischemia [START_REF] Tikkanen | Association of early repolarization and sudden cardiac death during an acute coronary event[END_REF] . In an animal model, J-waves appearing during left anterior descending artery occlusion were associated with 53% risk of VF [START_REF] Demidova | Transient and rapid QRS-widening associated with a J-wave pattern predicts impending ventricular fibrillation in experimental myocardial infarction[END_REF] . J-waves, particularly in the inferior leads, were also associated with increased risk in patients with chronic coronary artery disease, or with dilated or hypertrophic cardiomyopathy [START_REF] Patel | Early repolarization associated with ventricular arrhythmias in patients with chronic coronary artery disease[END_REF][START_REF] Pei | The Jwave and fragmented QRS complexes in inferior leads associated with sudden cardiac death in patients with chronic heart failure[END_REF][START_REF] Tsuda | J Waves for Predicting Cardiac Events in Hypertrophic Cardiomyopathy[END_REF] . Therefore the presence of inferior J-waves associated with SHD increases the incidence of arrhythmic events. J-waves its association with other ECG abnormalities (Brugada, Long QT, short QT) also increase the risk of arrhythmias [START_REF] Laksman | Early repolarization is associated with symptoms in patients with type 1 and type 2 long QT syndrome[END_REF][START_REF] Peters | Early repolarization phenomenon in arrhythmogenic right ventricular dysplasia-cardiomyopathy and sudden cardiac arrest due to ventricular fibrillation[END_REF][START_REF] Letsas | Prevalence of early repolarization pattern in inferolateral leads in patients with Brugada syndrome[END_REF][START_REF] Kamakura | Significance of non-type 1 anterior early repolarization in patients with inferolateral early repolarization syndrome[END_REF] .

J waves and Idiopathic VF-risk stratification

Among patients with a history of idiopathic VF, several reports have provided clinical evidence of an increased prevalence of J-wave patterns. In the most recent study from Korea, J-waves were observed in 35 of 81 patients (43%) with idiopathic VF, and were associated with a higher risk of recurrences [START_REF] Seong | Clinical significance of fragmented QRS complexes or J waves in patients with idiopathic ventricular arrhythmias[END_REF] . The incidence of idiopathic VF associated with inferolateral J-waves is estimated to be 3:100,000. This represents a low absolute risk by comparison with a prevalence of 1 % to 24 % in the population [START_REF] Haissaguerre | Sudden cardiac arrest associated with early repolarization[END_REF][START_REF] Tikkanen | Long-term outcome associated with early repolarization on electrocardiography[END_REF][START_REF] Rosso | J-point elevation in survivors of primary ventricular fibrillation and matched control subjects: incidence and clinical significance[END_REF][START_REF] Sinner | Association of early repolarization pattern on ECG with risk of cardiac and all-cause mortality: a population-based prospective cohort study[END_REF][START_REF] Derval | Prevalence and characteristics of early repolarization in the CASPER registry: cardiac arrest survivors with preserved ejection fraction registry[END_REF][START_REF] Junttila | Early repolarization as a predictor of arrhythmic and non-arrhythmic cardiac events in middle-aged subjects[END_REF][START_REF] Seong | Clinical significance of fragmented QRS complexes or J waves in patients with idiopathic ventricular arrhythmias[END_REF] . Therefore, the majority of individuals with ER are at minimal risk for arrhythmic events so that asymptomatic patients with no family history of SCD should be reassured [START_REF] Viskin | Making sense of early repolarization[END_REF][START_REF] Obeyesekere | Early Repolarization -What Should the Clinician Do?[END_REF] .

In symptomatic patients several ECG markers have been assessed in relation with outcome.

Though, the spontaneous variability of the J-ST pattern is a limiting factor. Clinical variables such as gender or ethnicity have provided conflicting results in terms of prognostic value. A family history of sudden death has been reported to be more common in association with SCD [START_REF] Nunn | Prevalence of J-point elevation in sudden arrhythmic death syndrome families[END_REF] and it likely increases the individual risk. The prognostic value of QRS slurring versus notching, often coexisting or changing in time, is still unclear. The following variables have been shown to be associated with an increased risk of malignant arrhythmias: i) a horizontal or descending type of ST segment in the inferior leads, as opposed to an upward ST segment [START_REF] Rosso | J-point elevation in survivors of primary ventricular fibrillation and matched control subjects: incidence and clinical significance[END_REF][START_REF] Sinner | Association of early repolarization pattern on ECG with risk of cardiac and all-cause mortality: a population-based prospective cohort study[END_REF][START_REF] Derval | Prevalence and characteristics of early repolarization in the CASPER registry: cardiac arrest survivors with preserved ejection fraction registry[END_REF][START_REF] Ilkhanoff | Clinical characteristics and outcomes associated with the natural history of early repolarization in a young, biracial cohort followed to middle age: the coronary artery risk development in young adults (CARDIA) study[END_REF][START_REF] Haruta | Incidence and prognostic value of early repolarization pattern in the 12-lead electrocardiogram[END_REF] ; however the prevalence of this more risky ST pattern in controls may be up to 3%, reducing its specific value; ii) a higher magnitude of J-waves (>2.0 mm) in the inferior leads [START_REF] Tikkanen | Long-term outcome associated with early repolarization on electrocardiography[END_REF] ; iii) an extensive ECG pattern involving anterior (leads V1-V3) and inferior leads [START_REF] Haissaguerre | Sudden cardiac arrest associated with early repolarization[END_REF][START_REF] Junttila | Early repolarization as a predictor of arrhythmic and non-arrhythmic cardiac events in middle-aged subjects[END_REF][START_REF] Kamakura | Significance of non-type 1 anterior early repolarization in patients with inferolateral early repolarization syndrome[END_REF] .

A consistent observation at the time of malignant arrhythmias is the amplification of inferolateral J-waves which then recedes after spontaneous or pharmacological arrhythmia termination [START_REF] Aizawa | Idiopathic ventricular fibrillation and bradycardia-dependent intraventricular block[END_REF][START_REF] Haissaguerre | Sudden cardiac arrest associated with early repolarization[END_REF][START_REF] Aizawa | Dynamicity of the J-wave in idiopathic ventricular fibrillation with a special reference to pause-dependent augmentation of the Jwave[END_REF][START_REF] Mizumaki | Vagal activity modulates spontaneous augmentation of J-wave elevation in patients with idiopathic ventricular fibrillation[END_REF] . A higher J-wave amplitude recorded soon after a syncope (compared with prior or later ECGs) suggest that a malignant arrhythmia may have occurred. The variability of J-waves after longer cycle lengths (post pause) is in our opinion an essential marker of electrical vulnerability. Unfortunately there are no specific means to challenge the dynamicity of J-wave patterns in the vulnerable patients, like e.g. sodium blocker provocation in BrS. A Valsalva maneuver or Holter monitoring are however useful to appreciate the J-wave dynamics on the 12-lead ECG during cycle-length variations. Finally, VF induction maneuvers during an electrophysiology study are of little utility. In a multi-center study involving 81 patients followed up by ICD interrogations, VF inducibility could not predict the incidence of subsequent arrhythmias [START_REF] Mahida | Role of electrophysiological studies in predicting risk of ventricular arrhythmia in early repolarization syndrome[END_REF] .

In summary, from the different published reviews it appears that no clinically strong riskstratification can be performed to identify the small subset of patients at high risk and that may facilitate primary prevention. Genetic variant analysis or specific pharmacological testing may hopefully become of prognostic importance in the future [START_REF] Antzelevitch | J-Wave syndromes expert consensus conference report: Emerging concepts and gaps in knowledge[END_REF] . Currently the decision of therapeutic prevention in high-risk patients (by ICD or sometimes by Quinidine) or loop recorder implantation in intermediate-risk patients, is based on the severity of clinical variables, and electrocardiographic patterns: T-wave negativity, J-wave amplitude, the spatial extent of the J-wave pattern, and dynamic changes in the J-wave [START_REF] Macfarlane | The early repolarization pattern: a consensus paper[END_REF][START_REF] Viskin | Making sense of early repolarization[END_REF][START_REF] Obeyesekere | Early Repolarization -What Should the Clinician Do?[END_REF] . The presence of shortcoupled premature beats is not a risk factor as it indicates imminent threat of VF. Figure 1B summarizes our hierarchical view of risk factors.

The evidence for repolarization and depolarization abnormalities as distinct mechanisms

There is an ongoing controversy on whether the "J-wave syndromes" (Brugada and inferolateral J-waves) are due to repolarization or depolarization abnormalities. The repolarization mechanism is founded on studies in right or left ventricular wedge preparations, demonstrating that J-waves can be a consequence of a transmural repolarization gradient due to a differential distribution and function of the transient outward current [START_REF] Antzelevitch | J-wave syndromes: Brugada and early repolarization syndromes[END_REF] . The depolarization mechanism is founded on studies demonstrating that structural discontinuities can cause conduction disturbances by current-to-load mismatch and display the phenotype of J-waves [START_REF] Hoogendijk | Mechanism of right precordial STsegment elevation in structural heart disease: excitation failure by current-to-load mismatch[END_REF] . In humans, however, the current data indicate heterogeneous mechanisms.

Brugada syndrome

In BrS the best evidence in support of a depolarization abnormality in humans came from Nademanee et al and other groups [START_REF] Nademanee | Prevention of ventricular fibrillation episodes in Brugada syndrome by catheter ablation over the anterior right ventricular outflow tract epicardium[END_REF][START_REF] Nademanee | Fibrosis, Connexin-43, and Conduction abnormalities in the Brugada Syndrome[END_REF][START_REF] Coronel | Right ventricular fibrosis and conduction delay in a patient with clinical signs of Brugada syndrome: a combined electrophysiological, genetic, histopathologic, and computational study[END_REF][START_REF] Sacher | Insight into the mechanism of Brugada syndrome: epicardial substrate and modification during ajmaline testing[END_REF][START_REF] Brugada | Brugada Syndrome Phenotype Elimination by Epicardial Substrate Ablation[END_REF][START_REF] Chung | A novel method to enhance phenotype, epicardial functional substrates, and ventricular tachyarrhythmias in Brugada syndrome[END_REF][START_REF] Pappone | Electrical Substrate Elimination in 135 Consecutive Patients With Brugada Syndrome[END_REF] showing late fractionated electrograms on the epicardial side of the right ventricular outflow tract. These electrograms were then correlated with the presence of interstitial fibrosis and reduced gap-junction expression in biopsies or autopsy studies from affected patients [START_REF] Hoogendijk | Mechanism of right precordial STsegment elevation in structural heart disease: excitation failure by current-to-load mismatch[END_REF][START_REF] Nademanee | Fibrosis, Connexin-43, and Conduction abnormalities in the Brugada Syndrome[END_REF][START_REF] Coronel | Right ventricular fibrosis and conduction delay in a patient with clinical signs of Brugada syndrome: a combined electrophysiological, genetic, histopathologic, and computational study[END_REF] . Such microstructural abnormalities were also reported in prior anatomical studies by Martini et al and Corrado et al [START_REF] Martini | Ventricular fibrillation without apparent structural heart disease[END_REF][START_REF] Corrado | Right bundle branch block, right precordial st-segment elevation, and sudden death in young people[END_REF] . Noteworthy, Antzelevitch's group has demonstrated that fractionated electrograms in animal wedge preparations can also be caused by repolarization disparities and concealed phase 2 reentry [START_REF] Patocskai | Mechanisms Underlying Epicardial Radiofrequency Ablation to Suppress Arrhythmogenesis in Experimental Models of Brugada Syndrome[END_REF] .

Our current clinical experience with high-resolution epi-endocardial mapping techniques (at the full organ level) demonstrates that the late epicardial fragmented potentials are continuous to the main depolarization front. This provides a strong argument in favor of a depolarization abnormality as a primary substrate of BrS in humans.

Inferolateral J wave syndrome

In inferolateral J-waves, repolarization abnormalities have been well established as the dominant substrate. However, a slurred end of the QRS complex is also a well-known marker of delayed activation in SHD (termed as 'peri-infarction block' or 'epsilon wave') [START_REF] Liu | J Wave Syndromes: History and Current Controversies Korean[END_REF] . The Jwave here indicates "post-excited" myocardium in the same way as the delta wave in WPW syndrome indicates "pre-excited" myocardium.

High density invasive mapping data have been collected from 38 patients with inferolateral Jwave syndrome in 3 centers [START_REF] Nademanee | Mapping And Ablation Of Ventricular Fibrillation Associated With Early Repolarization Syndrome[END_REF] . These patients had no demonstrated SHD and most were referred for VF recurring despite antiarrhythmic drugs including Quinidine. Electroanatomical mapping was performed during sinus rhythm to obtain endocardial and epicardial electrograms (2000-6000 recorded points) in unipolar and bipolar mode (unipolar filters of 0.05-250/500 Hz-, and bipolar 30-250/500Hz). A 2mm interelectrode spacing was used to minimize the recording of far-field potentials and specific attention was paid to the electrograms coincident with the J-wave. Abnormal electrograms were defined as prolonged fragmented electrograms with more than 3 components and a local duration superior to 70 ms [START_REF] Haissaguerre | Localized structural alterations underlying a subset of unexplained sudden cardiac death[END_REF] ; these criteria are similar to those defining structural alteration and fibrotic tissue in SHD. The electrograms occurring within (and possibly prolonging) the J-wave were considered as belonging to depolarization if they were sharp and in temporal and spatial continuity of the depolarization field mapped at the end of the QRS complex . They were considered as indicating ventricular repolarization if either they were not in continuity with the surrounding depolarization (presence of an electrical gap > 50ms), or displayed a slow pattern (hump) in unipolar mode that was not related to a near-field potential. Such slow potentials have been reported previously during direct or indirect epicardial recordings, in patients with J-wave or hypothermia conditions. [START_REF] Ghosh | Early repolarization associated with sudden death: insights from noninvasive electrocardiographic imaging[END_REF][START_REF] Emslie-Smith | The significance of changes in the electrocardiogram in hypothermia[END_REF][START_REF] Boineau | The early repolarization variant-an electrocardiographic enigma with both QRS and J-STT anomalies[END_REF][START_REF] Zhang | The Electrophysiological Substrate of Early Repolarization Syndrome: Noninvasive Mapping in Patients[END_REF][START_REF] Nakagawa | Left ventricular epicardial electrogram recordings in idiopathic ventricular fibrillation with inferior and lateral early repolarization[END_REF] .

Based on these definitions, the results [START_REF] Nademanee | Mapping And Ablation Of Ventricular Fibrillation Associated With Early Repolarization Syndrome[END_REF] indicate that the electrocardiographic J-wave can be the phenotypic expression of either delayed depolarization or early repolarization, in the inferior part of ventricles. In patients who had a concomitant BrS (spontaneous or provoked by a sodium channel blocker) inferolateral J-waves were consistently caused by delayed depolarization of inferior myocardium. In patients without BrS, inferolateral J-waves were caused by delayed depolarization in 24% of patients, whereas an ER was the cause of J-waves in 76%. Note that the true prevalence of depolarization versus repolarization abnormalities may be different, as the patients were often only referred after quinidine failure, which is likely to favor a higher proportion of delayed repolarization patients. Ajmaline testing was performed in all patients and resulted in J-wave amplification or ST elevation in the inferior leads in a few patients. The latter had a delayed depolarization whereas no patient with early repolarization had J/ST wave amplification on Ajmaline.

It is noteworthy that in a prior non-invasive mapping study, Zhang et al did not observe conduction abnormalities in inferolateral J-wave syndrome [START_REF] Zhang | The Electrophysiological Substrate of Early Repolarization Syndrome: Noninvasive Mapping in Patients[END_REF] during sinus rhythm. A likely explanation is that the small depolarization fronts that are responsible for the late fractionated electrograms cannot be recognized as such by a mathematical inverse solution from body surface potentials.

Examples of J-waves caused by late depolarization are shown in Figures 2 and3. The fragmented electrograms occurred timely during the inscription of J-waves and were in continuity with the depolarization field. The great majority of abnormal electrograms were found on the epicardium (right ventricle and two cases in left ventricle) while two patients had late electrograms recorded endocardially and epicardially. They were recorded predominantly in the inferior right ventricle, at the sites of terminal activation as predicted by a modeling study [START_REF] Meijborg | Reduced Sodium Current in the Lateral Ventricular Wall Induces Inferolateral J-Waves[END_REF] . In the patients with BrS and inferolateral J-wave syndromes the electrograms recorded in the inferior right ventricle were similar to those in the RVOT. The pathogenesis of inferolateral J-waves here is dominantly due to abnormal delayed conduction; either limited to the inferior myocardium (right or left -endo or epicardial), or combined with other locations (such as the anterior right ventricle). The affection responsible for electrogram fractionation (altered myocardial cells or their connections, fibrotic or fatty tissue infiltrations) is undetermined. In addition not only a heterogeneous delay of activation but also a homogeneous delayed activation may lead to a J-wave [START_REF] Meijborg | Reduced Sodium Current in the Lateral Ventricular Wall Induces Inferolateral J-Waves[END_REF] . Finally, there is probably a potential contribution of a repolarization disparity (secondary to the depolarization alterations) to explain a part of J/ST wave fluctuations or arrhythmogenesis, which requires additional studies [START_REF] Hoogendijk | Mechanism of right precordial STsegment elevation in structural heart disease: excitation failure by current-to-load mismatch[END_REF] .

Examples of J-waves due to (true) early repolarization are illustrated in Figures 4,5. In these patients we could not find electrograms indicative of delayed depolarization coincident with J-waves, but low frequency (hump) potentials were present at the beginning of the ST segment in unipolar recordings. The spatial location and extent of potentials were epicardially dominant in the inferior septal projection and adjacent left ventricle [START_REF] Emslie-Smith | The significance of changes in the electrocardiogram in hypothermia[END_REF][START_REF] Boineau | The early repolarization variant-an electrocardiographic enigma with both QRS and J-STT anomalies[END_REF][START_REF] Zhang | The Electrophysiological Substrate of Early Repolarization Syndrome: Noninvasive Mapping in Patients[END_REF][START_REF] Nakagawa | Left ventricular epicardial electrogram recordings in idiopathic ventricular fibrillation with inferior and lateral early repolarization[END_REF] . Although other hypotheses cannot be totally excluded [START_REF] Hoogendijk | Mechanism of right precordial STsegment elevation in structural heart disease: excitation failure by current-to-load mismatch[END_REF] , the J-wave mechanism may represent a lower epicardial voltage across the ventricular wall as shown by Yan and Antzelevitch [START_REF] Yan | Cellular basis for the electrocardiographic J wave[END_REF][START_REF] Antzelevitch | J-wave syndromes: Brugada and early repolarization syndromes[END_REF][START_REF] Liu | J Wave Syndromes: History and Current Controversies Korean[END_REF] . These authors have emphasized that a short-coupled ectopic beat was "a strong piece of evidence supporting an endo-epicardial myocardial gradient of repolarization leading to phase 2 reentry.". Such arrhythmogenesis due to abbreviated action potentials (loss of the dome) is technically difficult to demonstrate in clinical conditions. However we observed that VF is commonly initiated from Purkinje triggers (also with short coupling interval) whereas initiation from myocardial triggers (potentially due to phase 2 reentry) is more particularly observed in patients with multifocal ectopy and widespread early repolarization [START_REF] Haissaguerre | Sudden cardiac arrest associated with early repolarization[END_REF] .

VF drivers in the 2 forms of inferolateral J wave syndromes

VF drivers were mapped using a non-invasive method [START_REF] Haissaguerre | Localized structural alterations underlying a subset of unexplained sudden cardiac death[END_REF] and showed dominant drivers in the inferior part of myocardium during the initial stages of VF ( Fig 6). VF in patients with Jwaves indicating delayed depolarization was dominantly associated with drivers in the inferior and anterior right ventricle. VF in early repolarization was rather associated with drivers located in the inferior septum and adjacent regions. Note that this epicardial region overlying the inferior septum may be the breakthrough site of activity originating from the Purkinje posterior fascicle. Importantly, the cycle length of VF (measured at the 10 th second after initiation) was significantly shorter in early repolarization than in delayed depolarization (n=16, 148±5ms vs 175±4ms, p=0.013) consistent with shorter ventricular refractory periods.

Table S1 summarizes the differences between J-waves due to early repolarization versus late depolarization.

Implications of repolarization vs depolarization origins of inferolateral J wave -

The classification of inferolateral J-waves in two distinct substrates has implications in terms of terminology, pathogeny, genetics, and therapy.

While the terminology of early repolarization appears adequate in the cases of repolarization abnormality, it is erroneous when the J-wave is due to a late depolarization. The term inferolateral J-wave may thus be more adequate and generic. Further studies are needed to provide additional phenotypic features that may help distinguishing between depolarization and repolarization abnormality. It is likely that a stable J-wave pattern not influenced by long cycle lengths is caused by a depolarization abnormality, potentially exacerbated at short cycle lengths; and vice versa for early repolarization. In a prior study by Roten et al, the response of inferior J-wave patterns to isoproterenol varied individually. Inferior J-waves were persistent in 35% (rather suggestive of delayed depolarization) and decreasing or normalizing in 65% of patients (rather suggestive of early repolarization). Baseline QRS width was significantly larger in patients with persistent J-waves [START_REF] Roten | Heterogeneous response of J-wave syndromes to beta-adrenergic stimulation[END_REF] .

In terms of genetic predisposition, mutations in genes coding for subunits of the IK-ATP channel or cardiac L-type calcium channel have been described, however without validation by functional expression studies in most. Loss-of-function mutations in Na-channel genes have also been reported, including a significant proportion of them associated with an ST elevation in inferior leads under Na channel blocker. We speculate that a late-depolarization J-wave will be more likely associated with gene variants in the Na channel, connexins, and structural proteins, while mutations in the ion channels carrying Ito, IK-ATP or ICa will be more associated with early repolarization.

The immediate benefit of distinguishing J-wave subtypes is its therapeutic potential. In patients with late-depolarization J-waves, substrate ablation targeting the delayed electrograms is feasible like in BrS or other structural heart diseases. In patients with ER Jwaves, we do not know whether ablation of abnormal repolarizing tissue is applicable and safe, but trigger ablation is an effective option when anti-arrhythmic drugs and particularly quinidine have failed.

Mechanistic classification of VF associated with apparently normal hearts

The present review shows that similar ECG phenotypes may be caused by fundamentally different substrates. Inferolateral J-waves can be the expression of voltage gradients at the initial phase of repolarization ('early-repolarization') or the expression of delayed depolarized areas. Delayed depolarization is associated with electrogram fractionation indicating local structural alteration, whereas in the early repolarization group, there is likely no structural abnormality although VF triggers originating from the Purkinje system are often found.

Similar findings have been reported recently in idiopathic VF, using high density electrogram characterization [START_REF] Haissaguerre | Localized structural alterations underlying a subset of unexplained sudden cardiac death[END_REF] . The study involved 24 patients with no electrocardiographic phenotype: Jwaves, long or short QT syndromes were excluded; imaging and ajmaline testing were strictly negative. Localized areas of abnormal depolarization were identified in 62% of them. Most abnormal areas were epicardial, affecting dominantly the right ventricle, and of limited size (5% of the total surface). In the remaining subset of IVF devoid of myocardial alteration (38% of patients), Purkinje triggers were evidenced and appeared as the dominant mechanism.

Therefore, the spectrum of arrhythmogenic diseases leading to SCD in apparently normal hearts, including the J-wave syndromes, appears to comprise an important emerging subgroup in which the dominant substrate are localized depolarization abnormalities that may or may not have an electrocardiographic expression. A simplified mechanistic classification based on the primary pathogenesis is proposed in figure 7.

Conclusion

Inferolateral J-waves are subtle ECG phenotypes that may be responsible of SCD in patients with no apparent structural heart disease. Their occurrence at the QRST junction can be the expression of distinct substrates, early repolarization or delayed activation abnormality or mixed forms. Distinguishing between these substrates could significantly improve genetic interpretation, risk stratification and the therapeutic approach. Further studies are also needed for quantitative assessment of the recorded signals and understanding of the pathogenesis of inferolateral J-waves. Lower maps shows the latest activated regions (blue) in the inferobasal and laterobasal right ventricle. There are no late depolarization electrograms coincident to the J-wave but slow early-repolarization potentials (arrows); which are present in the apical region (white-dotted area). Note that J-wave is small on lead II (right) and underestimates the extent of early repolarization recorded by epicardial mapping.

BFigure 1 :

 1 Figure 1: (A) ECG variations in inferolateral J-Waves. Left: Valsalva or strong inspiration maneuver producing J-wave amplification. Middle: Cyclelength prolongation associated with either unchanged pattern or amplification. Right: negative J-waves in inferior leads and positive J-waves in lateral leads; note the majoration post pause. (B) Hierarchical view of electrocardiographic and clinical risk factors.

Figure 2 :

 2 Figure 2: Inferolateral J-wave syndrome due to abnormal depolarization. Upper central maps shows bipolar electrogram voltage (purple indicating voltage>1 mV) with low voltage in inferior right ventricle. Lower maps show the activation mapping with blue indicating the latest activated regions, here in the inferobasal right ventricle. Right panel shows fragmented electrograms preceding and coincident with the J-wave (white contour) in bipolar and unipolar (arrows) mode.

Figure 3 :

 3 Figure 3: Inferolateral J-wave syndrome due to abnormal depolarization. The maps (center) show activation mapping with blue indicating the latest activated regions; herein the inferobasal right and left ventricles. Low voltage fragmented electrograms coincident with J-wave are only present in inferior right ventricle (white contour) compared with electrograms in inferior left ventricle.

Figure 4 :

 4 Figure 4: Inferolateral J-wave syndrome due to early repolarization. Upper maps show bipolar electrogram voltage without evidence of low voltage area.Lower maps shows the latest activated regions (blue) in the inferobasal and laterobasal right ventricle. There are no late depolarization electrograms coincident to the J-wave but slow early-repolarization potentials (arrows); which are present in the apical region (white-dotted area). Note that J-wave is small on lead II (right) and underestimates the extent of early repolarization recorded by epicardial mapping.

Figure 5 :

 5 Figure 5: Another case of early repolarization. 12-lead ECGs (left) shows a global J-wave pattern. There are no late depolarization bipolar electrograms coincident with the J-wave; but early repolarization potentials are recorded diffusely in the inferior left ventricle.

Figure 6 :

 6 Figure 6: Typical location of VF driver regions. The locations of reentries are shown in red. They are predominantly located in the right ventricle in early repolarization (right) versus the inferior septum in late depolarization J-waves (left).

Figure 7 :

 7 Figure 7: Spectrum of arrhythmogenic diseases leading to SCD in apparently normal hearts, and a mechanistic classification based on the primary pathogenesis.

New concepts in inferolateral J wave Syndromes

This work was supported by the National Research Agency (ANR-10-IAHU04-LIRYC), the European Research Council (FP7/2007-2013 grant agreement number 322886-SYMPHONY) and the Leducq-Foundation (RHYTHM network). The authors acknowledge a research grant from Medtronic.


M Azaïez

F Ben Belgacem

J Casado-Díaz

T Chacón

F Murat

A new Algorithm of Proper Generalized Decomposition for parametric symmetric elliptic problems

We introduce in this paper a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation -in mean parametric norm associated to the elliptic operator-of the error between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the Proper Orthogonal Decomposition (POD) subspaces, excepting that in our case the norm is parameter-depending.

We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the on-line step, and we prove that the partial sums converge to the continuous solution in mean parametric elliptic norm.

We show that the standard PGD for the considered parametric problem is strongly related to the deflation algorithm introduced in this paper. This opens the possibility of computing the PGD expansion by directly solving the optimization problems that yield the optimal sub-spaces.

Introduction

The Karhunen-Loève's expansion (KLE) is a widely used tool, that provides a reliable procedure for a low dimensional representation of spatiotemporal signals (see [START_REF] Ghanem | Stochastic finite elements: A spectral approach[END_REF][START_REF] Loève | Probability Theory[END_REF]). It is referred to as the principal components analysis (PCA) in statistics (see [START_REF] Hotelling | Analysis of a complex of statistical variables into principal components[END_REF][START_REF] Jollife | Principal component analysis[END_REF][START_REF] Pearson | On lines and planes of closest fit system of points in space[END_REF]), or called singular value decomposition (SVD) in linear algebra (see [START_REF] Golub | Matrix Computations[END_REF]). It is named the proper orthogonal decomposition (POD) in mechanical computation, where it is also widely used (see [START_REF] Berkoz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF]). Its use allows large savings of computational costs, and make affordable the solution of problems that need a large amount of solutions of parameter-depending Partial Differential Equations (see [START_REF] Epureanu | Coherent structures and their influence on the dynamics of aeroelastic panels[END_REF][START_REF] Cordier | and 2003-04 on post-processing of experimental and numerical data[END_REF][START_REF] Holmes | Coherent structures, dynamical systems and symmetry[END_REF][START_REF] Lass | Adaptive POD basis computation for parametrized nonlinear systems using optimal snapshot location[END_REF][START_REF] Pearson | On lines and planes of closest fit system of points in space[END_REF][START_REF] Quarteroni | Reduced order methods for modeling and computational reduction[END_REF][START_REF] Schilders | Model order reduction: Theory, research aspects and applications[END_REF][START_REF] Volkwein | Proper orthogonal decomposition: Theory and reduced-order Modelling[END_REF]).

However the computation of the POD expansion requires to know the function to be expanded, or at least its values at the nodes of a fine enough net. This makes it rather expensive to solve parametric elliptic Partial Differential Equations (PDEs), as it requires the previous solution of the PDE for a large enough number of values of the parameter ("snapshots") (see [START_REF] Kahlbacher | Galerkin proper orthogonal decomposition methods for parameter depending elliptic systems[END_REF]), even if these can be located at optimal positions (see [START_REF] Kunisch | Optimal snapshot location for computing POD basis functions[END_REF]). Galerkin-POD strategies are well suited to solve parabolic problems, where the POD basis is obtained from the previous solution of the underlying elliptic operator (see [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF][START_REF] Muller | On the POD method. An abstract investigation with applications to reduced-order modeling and suboptimal control[END_REF]).

An alternative approach is the Proper Generalized Decomposition that iteratively computes a tensorized representation of the parameterized PDE, that separates the parameter and the independent variables, introduced in [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF]. It has been interpreted as a Power type Generalized Spectral Decomposition (see [START_REF] Nouy | A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations[END_REF][START_REF] Nouy | Generalized spectral decomposition method for stochastic non linear problems[END_REF]). It has experienced a fast development, being applied to the low-dimensional tensorized solution of many applied problems. The mathematical analysis of the PGD has experienced a relevant development in the last years. The convergence of a version of the PGD for symmetric elliptic PDEs via minimization of the associated energy has been proved in [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF]. Also, in [START_REF] Falcó | A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach[END_REF] the convergence of a recursive approximation of the solution of a linear elliptic PDE is proved, based on the existence of optimal subspaces of rank 1 that minimize the elliptic norm of the current residual.

The present paper is aimed at the direct determination of a variety of reduced dimension for the solution of parameterized symmetric elliptic PDEs. We intend to on-line determine an optimal subspace of given dimension that yields the best approximation in mean (with respect to the parameter) of the error (in the parametric norm associated to the elliptic operator) between the exact solution and the Galerkin solution calculated on the subspace. The optimal POD sub-spaced can no longer be characterized by means of a spectral problem for a compact self-adjoint operator (the standard POD operator) and thus the spectral theory for compact self-adjoint operators does no apply. We build recursive approximations on finite-dimensional optimal subspaces by minimizing the mean parametric error of the current residual, similar to the one introduced in [START_REF] Falcó | A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach[END_REF], that we prove to be strongly convergent in the "intrinsic"mean parametric elliptic norm. For this reason we call "intrinsic"PGD the method introduced.

In addition, we prove that the method introduced is a genuine extension of both POD and PGD methods, when applied to the solution of parametric elliptic equations. In particular it is strongly related to the PGD method in the sense that the standard formulation of the PGD method actually provides the optimality conditions of the minimization problem satisfied by the optimal 1D sub-spaces. As a consequence of the analysis developed in the paper, the PGD expansion is strongly convergent to the targeted solution in parametric elliptic norm, whenever it is implemented in such a way that all modes are optimal. Furthermore, the characterization of the modes by means of optimization problems opens the door to their computation by using optimization techniques, in addition to the usual Power Iteration algorithm.

The abstract framework considered includes several kind of problems of practical interest, to which the PGD has been and continues being applied. This is the case of the design analysis in computational mechanics. For instance in the design of energy efficient devices (HVACs) or buildings, it is mandatory to address the heat equation with several structural parameters, for instance the thermal diffusivity or transmittance, and the geometric shape of the device, among others. Also, the optimal design of heterogeneous materials with linear behavior law fits into the framework considered, as the parameters model the structural configuration of the various materials (cf. [START_REF] Chen | Optimization of Continuous HeterogeneousModels[END_REF][START_REF] Torquato | Optimal Design of Heterogeneous Materials[END_REF]). Moreover, in practice the structural configuration that optimizes a certain predefined criterion (e.g. construction costs, benefits, etc.) needs to take into account the unavoidable uncertainties in the structural performance. This leads to elliptic problems including modeling of the targeted uncertainty that, when the PDE model is linear, also fits into the abstract framework considered. In addition classical homogenization problems governed by linear symmetric elliptic PDEs formally also fit into this general framework, although the kind of approximation of the solution that is proposed in this work is different than the usual one, that looks for a limit averaged solution. Here we rather approximate the whole family of parameter-depending solutions by a function series. The method, however, does not apply, for instance, to non-symmetric elliptic forms, neither to non-linear problems.

The present paper focuses on theoretical aspects: We study the existence of the intrinsic POD, and give a convergence result for the deflation algorithm. We keep the quantitative analysis of the convergence as well as numerical investigations for future works.

The paper is structured as follows: In Section 2 we state the general problem of finding optimal subspaces of a given dimension. We prove in Section 3 that there exists a solution for 1D optimal subspaces, characterized as a maximization problem with a non-linear normalization restriction. We extend this existence result in Section 4 to general dimensions. In Section 5 we use the results in Sections 3 and 4 to build a deflation algorithm to approximate the solution of a parametric family of elliptic problems and we show the convergence. Section 6 explains why the method introduced is a genuine extension of both POD and PGD algorithms, and provides a theoretical analysis for the latter. Finally in Section 7 we present the main conclusions of the paper.

Statement of the problem

Let H be a separable Hilbert space endowed with the scalar product (•, •). The related norm is denoted by • . We denote by B s (H) the space of bilinear, symmetric and continuous forms in H.

Assume given a measure space (Γ, B, µ), with standard notation, so that µ is σ-finite.

Let a ∈ L ∞ (Γ, B s (H); dµ) be such that there exists α > 0 satisfying α u 2 ≤ a(u, u; γ), ∀u ∈ H, dµ-a.e. γ ∈ Γ.

(

) 1
For µ-a.e γ ∈ Γ, the bilinear form a(•, •; γ) determines a norm uniformly equivalent to the norm

• . Moreover, a ∈ B s (L 2 (Γ, H; dµ)) defined by a(v, w) = Γ a(v(γ), w(γ); γ) dµ(γ), ∀v, w ∈ L 2 (Γ, H; dµ) (2)
defines an inner product in L 2 (Γ, H; dµ) which generates a norm equivalent to the standard one in L 2 (Γ, H; dµ).

Let be given a data function f ∈ L 2 (Γ, H ; dµ). We are interested in the variational problem:

Find u(γ) ∈ H such that a(u(γ), v; γ) = f (γ), v , ∀v ∈ H, dµ-a.e. γ ∈ Γ, (3)
where •, • denotes the duality pairing between H and H. By Riesz representation theorem, problem (3) admits a unique solution for dµ-a.e. γ ∈ Γ. On the other hand, we claim that ũ solution of

ũ ∈ L 2 (Γ, H; dµ), ā(ũ, v) = Γ f (γ), v(γ) dµ(γ), ∀ v ∈ L 2 (Γ, H; dµ), (4)
also satisfies (3): Indeed taking v = vχ B , with v ∈ H fixed and B ∈ B arbitrary, implies that there exists a subset

N v ∈ B with µ(N v ) = 0 such that a(ũ(γ), v; γ) = f (γ), v , ∀ γ ∈ Γ \ N v .
The separability of H implies that N v can be chosen independent of v, which proves the claim. By the uniqueness of the solution of (3) this shows that ũ = u dµ-a.e. γ ∈ Γ.

(

) 5
This proves that u defined by (3) belongs to L 2 (Γ, H; dµ) and provides an equivalent definition of u,namely, that u is the solution of (4). Given a closed subspace Z of H, let us denote by u Z (γ) the solution of the Galerkin approximation of problem (3) on Z, which is defined as

u Z (γ) ∈ Z, a(u Z (γ), z; γ) = f (γ), z , ∀z ∈ Z, dµ-a.e. γ ∈ Γ, (6)
or equivalently as

u Z ∈ L 2 (Γ, Z; dµ), ā(u Z , z) = Γ f (γ), z(γ) dµ(γ), ∀ z ∈ L 2 (Γ, Z; dµ). (7)
For every k ∈ IN, we intend to find the best subspace W of H of dimension smaller than or equal to k that minimizes the mean error (in the norm defined by ā) between u and u W . That is, W solves min

Z∈G ≤k ā(u -u Z , u -u Z ), (8)
where G ≤k is the family of subspaces of H of dimension smaller than or equal to k. Note that G ≤k is a connected component of the Grassmaniann variety G ≤k of H, defined as

G ≤k = k≥0 G k ,
where G k is the set formed by all sub-spaces of H of dimension k. The set G k is a Hilbert manifold modeled in a particular Hilbert space (see [START_REF] Abraham | Manifolds, Tensor Analysis and Applications[END_REF][START_REF] Margalef-Roig | Differential Topology[END_REF]). Problem (8) will be proved to have a solution in Sections 3 and 4. We will then use this result in Section 5 to approximate the solution u of problem (3) by a deflation algorithm.

Note that when looking at the formulation of the minimization problem (8), it seems that solving it requires the knowledge of the solution u of (3). Such is not the case, since Proposition 2.6 below provides an equivalent formulation of (8) which does not depend on the knowledge of u but only on the data f . Let us provide some equivalent formulations of problem [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF]. First we observe that Proposition 2.1 For every closed subspace Z ⊂ H, the function u Z defined by ( 7) is also the unique solution of min

z∈L 2 (Γ,Z;dµ) ā(u -z, u -z). (9)
Moreover, for dµ-a.e. γ ∈ Γ, the vector u Z (γ) is the solution of

min z∈Z a(u(γ) -z, u(γ) -z; γ). (10)
Proof: It is a classical property of the Galerkin approximation of the variational formulation of linear elliptic problems that u Z satisfies [START_REF] Cancès | Convergence of a greedy algorithm for high-dimensional convex nonlinear problems[END_REF]. Indeed, the symmetry of ā gives

ā(u -z, u -z) = ā(u -u Z , u -u Z ) + 2ā(u -u Z , u Z -z) + ā(u Z -z, u Z -z),
for every z ∈ L 2 (Γ, H; dµ), where by ( 4), ( 5) and ( 7) the second term on the right-hand side vanishes, while the third one is nonnegative. This proves [START_REF] Cancès | Convergence of a greedy algorithm for high-dimensional convex nonlinear problems[END_REF]. The proof of ( 10) is the same by taking into account ( 3) and ( 6) instead of ( 4) and [START_REF] Azaïez | Error Bounds for POD expansions of parametrized transient temperatures[END_REF].

As a consequence of Proposition 2.1 and definition (2) of ā, we have

Corollary 2.2 A space W ∈ G ≤k is a solution of (8) if and only if it is a solution of min Z∈G ≤k min z∈L 2 (Γ,Z;dµ) ā(u -z, u -z). (11)
Moreover min

Z∈G ≤k min z∈L 2 (Γ,Z;dµ) ā(u -z, u -z) = min Z∈G ≤k Γ min z∈Z a(u(γ) -z, u(γ) -z; γ)dµ(γ). (12)
Remark 2.3 Optimization problem (11) is reminiscent of the Kolmogorov k-width related to the best approximation of the manifold (u(γ)) γ∈Γ by subspaces in H with dimension k as presented in [START_REF] Maday | Reduced basis method for the rapid and reliable solution of partial differential equations[END_REF]. In the present minimization problem, we use the norm of L 2 (Γ, H; dµ) instead of the norm of L ∞ (Γ, H; dµ) as used there. The minimization problem in [START_REF] Maday | Reduced basis method for the rapid and reliable solution of partial differential equations[END_REF] can indeed be written as

min Z∈G ≤k ess sup γ∈Γ min z∈Z a(u(γ) -z, u(γ) -z; γ), (13)
if one uses a(•, •; γ) as the inner product in H.

The analysis performed in the present paper is strongly based on the Hilbertian framework associated to the minimization in L 2 (Γ, H; dµ). To the best of our knowledge few is known about problem [START_REF] Ghanem | Stochastic finite elements: A spectral approach[END_REF], in particular there is no proof of existence of solutions. The extension to this problem of the techniques used in the present paper is far from being straightforward, and we intend to discuss this in a future paper. Indeed the L ∞ (Γ, H; dµ) framework is specially interesting whenever uniform error estimates with respect to the parameter are needed. This happens, for instance, when upper bounds for energy consumption (either mechanical, thermal, etc.) should be respected.

For a function v ∈ L 2 (Γ, H; dµ), we denote by R(v) the closure of the vectorial space spanned by v(γ) when γ belongs to Γ; more exactly, taking into account that v is only defined up to sets of zero measure, the correct definition of R(v) is given by

R(v) = µ(N )=0 Span v(γ) : γ ∈ Γ \ N . (14)
The following result proves that in ( 14) the intersection can be replaced a single closed spanned space corresponding to a single set M ∈ B. This proves in particular that it does not reduce to {0} if R(v) is not zero dµ-a.e. γ ∈ Γ:

Proposition 2.4 For every v ∈ L 2 (Γ, H; dµ) there exists M ∈ B, with µ(M ) = 0 such that R(v) = Span v(γ) : γ ∈ Γ \ M .
Proof: For every N ∈ B, we define P N as the orthogonal projection of H into

R N := Span v(γ) : γ ∈ Γ \ N .
We also define P as the orthogonal projection of H into R(v).

Let us first prove

∀ z ∈ H, ∃M z ∈ B with µ(M z ) = 0 such that P z = P Mz z. (15)
In order to prove this result, we consider N n ∈ B, with µ(N n ) = 0, such that

P Nn z → inf µ(N )=0 P N z .
Taking M z = ∪ n N n , we have that µ(M z ) = 0. Moreover, using that

N n ⊂ M z implies R Mz ⊂ R Nn , we get inf µ(N )=0 P N z ≤ P Mz z ≤ P Nn z , ∀ n ≥ 1. Therefore P Mz z = inf µ(N )=0 P N z .
Now, we use that for every N ∈ B with M z ⊂ N , µ(N ) = 0, we have

R N ⊂ R Mz , P Mz z ≤ P N z
and then

P N z = P Mz z, ∀ N ⊃ M z with µ(N ) = 0. ( 16
)
We take now an arbitrary 16), we get

N ∈ B with µ(N ) = 0. Using M z ⊂ N ∪ M z , µ(N ∪ M z ) = 0 and (
P Mz z = P N ∪Mz z ∈ R N ∪Mz ⊂ R N , ∀N ∈ B, with µ(N ) = 0,
and so, P Mz z belongs to R(v). On the other hand, observe that R(v) ⊂ R Mz and the definition of

P Mz imply z -P Mz z ≤ z -η , ∀ η ∈ R(v)
and thus, P Mz z = P z. This proves [START_REF] Hotelling | Analysis of a complex of statistical variables into principal components[END_REF].

Let us now use [START_REF] Hotelling | Analysis of a complex of statistical variables into principal components[END_REF] to prove the statement of Proposition 2.4. We consider an orthonormal basis {z k } of R(v) ⊥ . By [START_REF] Hotelling | Analysis of a complex of statistical variables into principal components[END_REF], we know that for every k ≥ 1, there exists

M z k ∈ B with µ(M z k ) = 0 such that P M k z k = 0. Then, we define M = k≥1 M z k ,
let us prove that M satisfies the thesis of the Proposition. Clearly µ(M ) = 0, moreover, [START_REF] Holmes | Coherent structures, dynamical systems and symmetry[END_REF] and M z k ⊂ M for every k ≥ 1, imply

P M e k = 0, ∀ k ≥ 1.

This shows P

M z = 0 for every z ∈ R(v) ⊥ and then R(v) ⊥ ⊂ R ⊥ M or equivalently R M ⊂ R(v).
Since the other contention is immediate, we have then proved R M = R(v), which finishes the proof.

Taking into account [START_REF] Falcó | A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach[END_REF], a new formulation of ( 8) is given by

Proposition 2.5 If W is a solution of (8), then u W is a solution of min v∈L 2 (Γ,H;dµ) dim R(v)≤k ā(u -v, u -v). (17)
Reciprocally, if û is a solution of (17), then R(û) is a solution of ( 8) and û = u R(û) .

As announced above, the next Proposition provides an equivalent formulation for (8) which does not depend on the knowledge of the solution u of (3), but only on the data f . Proposition 2. [START_REF] Brézis | Functional analysis, Sobolev spaces, and partial differential equations[END_REF] The subspace W ∈ G ≤k solves problem [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF] if and only if it is a solution of the problem max

Z∈G ≤k Γ f (γ), u Z (γ) dµ(γ), (18)
where u Z is defined by [START_REF] Azaïez | Error Bounds for POD expansions of parametrized transient temperatures[END_REF].

Proof: As in the proof of the first part of Proposition 2.1, one deduces from (4), ( 5) and ( 7) that ā(u -u Z , z) = 0, ∀z ∈ L 2 (Γ, Z; dµ).

Using the symmetry of ā, we then have

ā(u -u Z , u -u Z ) = ā(u, u) -a(u Z , u) = ā(u, u) -ā(u Z , u Z ) = ā(u, u) - Γ f (γ), u Z (γ) dµ(γ).
Thus W solves [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF] if and only if it solves (18).

One-dimensional approximations

In Section 4 we shall show the existence of the solution of problem [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF] for any arbitrary k. However a particularly interesting case from the point of view of the applications is k = 1. We dedicate this section to this special case. Observe that for Z ∈ G 1 , there exists z ∈ H \ {0} such that Z = Span{z}. The problem to solve can be reformulated as follows.

Lemma 3.1 Assume f ≡ 0. Then, the subspace W ∈ G 1 solves problem [START_REF] Kahlbacher | Galerkin proper orthogonal decomposition methods for parameter depending elliptic systems[END_REF] if and only if W = Span{w}, where w is a solution of

max z∈H z =0 Γ f (γ), z 2 a(z, z; γ) dµ(γ). (19)
Proof: Let Z ∈ G 1 . Then Z = Span{z}, for some z ∈ H \ {0}, and there exists a function ϕ : Γ → R such that u Z (γ) = ϕ(γ) z, dµ-a.e. γ ∈ Γ.

As z = 0, then, as u Z (γ) is the solution to the variational equation ( 6), we derive that

ϕ(γ) = f (γ), z a(z, z; γ)
, dµ-a.e. γ ∈ Γ.

Using this formula we obtain that

Γ f, u Z (γ) dγ = Γ f (γ), z 2 a(z, z; γ) dµ(γ). (20)
If the maximum in ( 18) is obtained by a space of dimension one, then formula (20) proves the desired result.

In contrast, if the maximum in ( 18) is obtained by the null space, then the maximum in G 1 is equal to zero. Therefore the right-hand side of ( 20) is zero for every z ∈ H, which implies that f = 0 dµ-a.e. in Γ, in contradiction with the assumption f ≡ 0. Remark 3.2 Since the integrand which appears in [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF] is homogenous of degree zero in z, problem [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF] is equivalent to

max z∈H z =1 Γ f (γ), z 2 a(z, z; γ) dµ(γ).
We now prove the existence of a solution to problem [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF]. [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF] admits at least a solution.

Theorem 3.3 Assume f ≡ 0. Problem (
Note that if f ≡ 0, then, every vector w ∈ H \ {0} is a solution of [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF].

Proof: Define

M * := sup z∈H z =1 Γ f (γ), z 2 a(z, z; γ) dµ(γ), (21)
and consider a sequence w n ⊂ H, with w n = 1 such that

lim n→∞ Γ f (γ), w n 2 a(w n , w n ; γ) dµ(γ) = M * . ( 22
)
Up to a subsequence, we can assume the existence of w ∈ H, such that w n converges weakly in H to w. Taking into account that f (γ) ∈ H , a(•, •, γ) ∈ B s (H) dµ-a.e. γ ∈ Γ and ( 1) is satisfied, we get lim

n→∞ f (γ), w n = f (γ), w , dµ-a.e. γ ∈ Γ, (23)
lim inf n→∞ a(w n , w n ; γ) ≥ a(w, w; γ), dµ-a.e. γ ∈ Γ.

On the other hand, we observe that (1) and w n = 1 imply

| f (γ), w n | ≤ f (γ) H , 1 a(w n , w n ; γ) ≤ 1 α dµ-a.e. γ ∈ Γ. ( 25
)
If w = 0, then ( 23), ( 25) and Lebesgue's dominated convergence theorem imply

lim n→∞ Γ f (γ), w n 2 a(w n , w n ; γ) dµ(γ) = 0,
which by ( 22) is equivalent to M * = 0. Taking into account (1) and the definition (21) of M * , this is only possible if f ≡ 0 is the null function. As we are assuming f ≡ 0, we conclude that w is different of zero. Then, [START_REF] Margalef-Roig | Differential Topology[END_REF] proves

0 ≤ f (γ) 2 H α - f (γ), w n 2 a(w n , w n ; γ)
, dµ-a.e. γ ∈ Γ, while ( 23) and ( 24) prove

lim inf n→∞ f (γ) 2 H α - f (γ), w n 2 a(w n , w n ; γ) ≥ f (γ) 2 H α - f (γ), w 2 a(w, w; γ) , dµ-a.e. γ ∈ Γ. ( 26
)
Using ( 22), Fatou's lemma implies

Γ f (γ) 2 H α - f (γ), w 2 a(w, w; γ) dµ(γ) ≤ lim inf n→∞ Γ f (γ) 2 H α - f (γ), w n 2 a(w n , w n ; γ) dµ(γ) = Γ f (γ) 2 H α dµ(γ) -M * ,
or equivalently

M * ≤ Γ f (γ), w 2 a(w, w; γ) dµ(γ). (27)
By definition [START_REF] Lass | Adaptive POD basis computation for parametrized nonlinear systems using optimal snapshot location[END_REF] of M * , this proves that the above inequality is an equality and that w is a solution of [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF].

Remark 3.4 Actually, in place of [START_REF] Muller | On the POD method. An abstract investigation with applications to reduced-order modeling and suboptimal control[END_REF], one has the stronger result

lim inf n→∞ f (γ) 2 H α - f (γ), w n 2 a(w n , w n ; γ) = f (γ) 2 H α - f (γ), w 2 lim inf n→∞ a(w n , w n ; γ))
, dµ-a.e. γ ∈ Γ, which by the proof used to prove [START_REF] Nouy | A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations[END_REF] shows

M * ≤ Γ f (γ), w 2 lim inf n→∞ a(w n , w n ; γ) dµ(γ).
Combined with

M * = Γ f (γ), w 2 a(w, w; γ) dµ(γ)
and [START_REF] Maday | Reduced basis method for the rapid and reliable solution of partial differential equations[END_REF], this implies a(w, w; γ) = lim inf n→∞ a(w n , w n ; γ) dµ-a.e. γ ∈ Γ such that f (γ), w = 0.

By (1) and f ≡ 0, this proves the existence of a subsequence of w n which converges strongly to w a. e. γ.

Since this proof can be carried out by replacing w n by any subsequence of w n , we conclude that the whole sequence w n (which we extracted just after [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF] assuming that it converges weakly to some w) actually converges strongly to w.

The above result may be used to build a computable approximation of a solution of (19). Indeed, for f ≡ 0, let {H n } n≥1 be an internal approximation of H, that is a sequence of subspaces of finite dimension of H such that

lim n→∞ inf ψ∈Hn z -ψ = 0, ∀z ∈ H.
and consider a solution w n of

max z∈Hn z =1 Γ f (γ), z 2 a(z, z; γ) dµ(γ).
The existence of such a w n can be obtained by reasoning as in the proof of Theorem

lim n→∞ Γ f (γ), w n 2 a(w n , w n ; γ) dµ(γ) = Γ f (γ), w 2 a( w, w; γ) dµ(γ) = M * .
This proves that the sequence w n satisfies [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF]. Therefore any subsequence of w n which converges weakly to some w converges strongly to w which is a solution of (19).

Higher-dimensional approximations

This section is devoted to the proof of the existence of an optimal subspace which is solution of (8) when k ≥ 1 is any given number.

Theorem 4.1 For any given k ≥ 1, problem (8) admits at least one solution.

Proof: As in the proof of Theorem 3.3, we use the direct method of the Calculus of Variations.

Denoting by

m k m k = inf Z∈G ≤k ā(u -u Z , u -u Z ), (28)
we consider a sequence of spaces

W n ∈ G ≤k such that w n := u Wn satisfies lim n→∞ ā(u -w n , u -w n ) = m k . ( 29
)
Taking into account that by Proposition 2.1

Z ⊂ Z =⇒ ā(u -u Z , u -u Z ) ≤ ā(u -u Z , u -u Z ), (30)
we can assume that the dimension of W n is equal to k. Moreover, we observe that (29) implies that w n is bounded in L 2 (Γ, H; dµ).

Let (z 1 n , • • • , z k n ) be an orthonormal basis of W n . It holds w n (γ) = k j=1 (w n (γ), z j n ) z j n , dµ-a.e. γ ∈ Γ. ( 31
)
Since the norm of the vectors z j n is one, there exists a subsequence of n and k vectors z j ∈ H such that

z j n z j in H, ∀ j ∈ {1, • • • , k}. (32)
Using also

|(w n (γ), z j n )| ≤ w n (γ) , dµ-a.e γ ∈ Γ,
we get that (w n , z j n ) is bounded in L 2 (Γ, H; dµ) for every j and thus, there exists a subsequence of n and k functions p j ∈ L 2 (Γ; dµ) such that

(w n , z j n ) p j in L 2 (Γ, H; dµ), ∀ j ∈ {1, • • • , k}. (33)
We claim that

w n w := n j=1 p j z j in L 2 (Γ; dµ). (34)
Indeed, taking into account that w n is bounded in L 2 (Γ, H; dµ) and [START_REF] Quarteroni | Reduced order methods for modeling and computational reduction[END_REF], it is enough to show

lim n→∞ Γ (w n , z j n )z j n , ϕ v dµ(γ) = Γ (p j z j , ϕ v) dµ(γ), ∀ ϕ ∈ L 2 (Γ; dµ), ∀ v ∈ H. (35)
This is a simple consequence of

Γ (w n , z j n )z j n , ϕ v dµ(γ) = (z j n , v) Γ (w n , z j n ) ϕ dµ(γ),
combined with ( 32) and [START_REF] Torquato | Optimal Design of Heterogeneous Materials[END_REF].

From the continuity and convexity of the quadratic form associated to ā, as well as from ( 34) and ( 29), we have

ā(u -w, u -w) ≤ lim n→∞ ā(u -w n , u -w n ) = m k . ( 36
)
Using that W = Span{z 1 , • • • , z k } ∈ G ≤k , and that (see Proposition 2.1)

ā(u -u W , u -u W ) ≤ ā(u -w, u -w), (37)
we conclude that W is a solution of ( 8).

ā(u -w, u -w) = m k = lim n→∞ ā(u -w n , u -w n ),
which combined with [START_REF] Volkwein | Proper orthogonal decomposition: Theory and reduced-order Modelling[END_REF] proves that w n converges strongly to w in L 2 (Γ, H; dµ). As in Remark 3.4, this can be used to build a strong approximation of a solution of (8) by using an internal approximation of H.

An iterative algorithm by deflation

In the previous section, for any given k ≥ 1, we have proved the existence of an optimal subspace for problem [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF]. We use here this fact to build an iterative approximation of the solution of (3) by a deflation approach. We build recursive approximations on finite-dimensional optimal subspaces by minimizing the mean parametric error of the current residual, similar to the one introduced in [START_REF] Falcó | A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach[END_REF]. Let us denote

Π k (v) = v W | W solves min Z∈G ≤k ā(v -v Z , v -v Z ), , ∀ v ∈ L 2 (Γ, H; dµ). ( 38
)
where v Z is defined by [START_REF] Azaïez | Error Bounds for POD expansions of parametrized transient temperatures[END_REF] The deflation algorithm is as follows

• Initialization: u 0 = 0 (39) • Iteration: Assuming u i-1 ∈ H known for i = 1, 2, • • • , set e i-1 = u -u i-1 , choose s i ∈ Π k (e i-1
), and define

u i = u i-1 + s i . ( 40
)
Remark 5.1 Note that s i (and therefore u i ) in general is not defined in a unique way. Note also that the algorithm (40) does not need the knowledge of the solution u of ( 4), since e i-1 = u -u i-1 is directly defined from f and u i-1 by

     e i-1 ∈ L 2 (Γ, H; dµ), ā(e i-1 , v) = Γ f (γ), v(γ) dµ(γ) -ā(u i-1 , v), ∀ v ∈ L 2 (Γ, H; dµ). (41)
Then Proposition 2.6 applied to the case where f is replaced by the function f i defined by

Γ f i (γ), v(γ) dµ(γ) = Γ f (γ), v(γ) dµ(γ) -ā(u i-1 , v), ∀ v ∈ L 2 (Γ, H; dµ), (42)
proves that

s i ∈ Π k (e i-1 ) ⇐⇒ s i = (e i-1 ) W i ,
where W i is a solution of

max Z∈G ≤k Γ f (γ), (e i-1 ) Z (γ) dµ(γ) -ā(u i-1 , (e i-1 ) Z ) , (43)
where, in accordance to [START_REF] Azaïez | Error Bounds for POD expansions of parametrized transient temperatures[END_REF], (e i-1 ) Z denotes the solution of

     (e i-1 ) Z ∈ L 2 (Γ, Z; dµ), ā (e i-1 ) Z , z = Γ f (γ), z(γ) dµ(γ) -ā(u i-1 , z), ∀ z ∈ L 2 (Γ, Z; dµ). ( 44
)
This observation allows one to carry out the iterative process without knowing the function u. Note also that

u i = i j=1 s j ,
namely that u i is the partial sum of the series j≥1 s j .

Remark 5.2 In this remark we take k = 1. Then every space of G ≤1 is spanned by an element of H, and in particular W i = Span{w i } for some w i ∈ H, then u i (γ) = i j=1 Φ j (γ) w j , where

Φ j (γ) ∈ L 2 (Γ, dµ
) is defined by s j = Φ j (γ) w j . Note also that if w i = 0 for some i ≥ 0, then

f i = 0 and thus u ≡ u i-1 ,
The convergence of the algorithm is given by the following theorem. Its proof follows the ideas of [START_REF] Falcó | A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach[END_REF]. When k = 1, Remark 5.2 implies that the series j≥1 Φ j (γ) w j converges in L 2 (Γ, H; dµ) to this parametrized solution.

Proof: By (40) and Proposition 2.5 applied to the case where u is replaced by e i-1 , we have that s i is a solution of min

v∈L 2 (Γ,H;dµ) dim R(v)≤k ā(e i-1 -v, e i-1 -v). (45)
This proves in particular that s i is a solution of min

v∈L 2 (Γ,H;dµ) R(v)⊂R(s i ) ā(e i-1 -v, e i-1 -v),
and therefore

ā(e i-1 -s i , v) = 0, ∀ v ∈ L 2 (Γ, H; dµ) with R(v) ⊂ R(s i ). But (40) implies that e i-1 -s i = e i , (46)
which gives ā(e i , v) = 0, ∀ v ∈ L 2 (Γ, H; dµ) with R(v) ⊂ R(s i ). (47)
Taking v = s i and using again (46) we get ā(e i-1 , e i-1 ) = ā(s i , s i ) + ā(e i , e i ),

∀ i ≥ 1, (48) and therefore
ā(e i , e i ) + i j=1 ā(s j , s j ) = ā(e 0 , e 0 ), ∀ i ≥ 1.

(49) Thus, we have

e i is bounded in L 2 (Γ, H; dµ), (50)
∞ j=1 ā(s j , s j ) ≤ ā(e 0 , e 0 ). (51)
By (50), there exists a subsequence e in of e i and e ∈ L 2 (Γ, H; dµ), such that e in e in L 2 (Γ, H; dµ).

On the other hand, since s in+1 is a solution of (45) with i -1 replaced by i n , we get

ā(e in -s in+1 , e in -s in+1 ) ≤ ā(e in -v, e in -v) = ā(e in , e in ) -2ā(e in , v) + ā(v, v), ∀ v ∈ L 2 (Γ, H; dµ), dim R(v) ≤ k, (53)
and then

ā(e in -s in+1 , e in -s in+1 )-ā(e in , e in ) ≤ -2ā(e in , v)+ ā(v, v), ∀ v ∈ L 2 (Γ, H; dµ), dim R(v) ≤ k,
or in other terms

-2ā(e in , s in+1 ) + ā(s in+1 , s in+1 ) ≤ -2ā(e in , v) + ā(v, v), ∀ v ∈ L 2 (Γ, H; dµ), dim R(v) ≤ k.
Thanks to (50) and ( 51), the left-hand side tends to zero when n tends to infinity, while in the right-hand side we can pass to the limit by (52). Thus, we have

2ā(e, v) ≤ ā(v, v), ∀ v ∈ L 2 (Γ, H; dµ), dim R(v) ≤ k.
Replacing in this equality v by tv with t > 0, dividing by t, letting t tend to zero and writing the resulting inequality for v and -v, we get

ā(e, v) = 0, ∀ v ∈ L 2 (Γ, H; dµ), dim R(v) ≤ k.
Taking v = wϕ, with w ∈ H, ϕ ∈ L 2 (Γ; dµ), and recalling definition (2) of ā we deduce

Γ a(e(γ), w; γ) ϕ(γ) dµ(γ) = 0, ∀ z ∈ H, ∀ ϕ ∈ L 2 (Γ; dµ),
and then for any w ∈ H, there exists a subset N w ∈ B with µ(N w ) = 0 such that a(e(γ), w; γ) = 0, ∀ γ ∈ Γ \ N w .

The separability of H implies that N w can be chosen independent of w, and then we have

a(e(γ), w; γ) = 0, ∀ w ∈ H, dµ-a.e. γ ∈ Γ,
and therefore e(γ) = 0 dµ-a.e. γ ∈ Γ.

This proves that e does not depend on the subsequence in (52) and that

e i 0 in L 2 (Γ, H; dµ). (55)
Let us now prove that in (55) the convergence is strong in L 2 (Γ, H; dµ). We use that thanks to (46), we have

e i = - i j=1 s j + e 0 , ∀ i ≥ 1,
and so, ā(e i , e i ) = -i j=1 ā(e i , s j ) + ā(e i , e 0 ), ∀ i ≥ 1.

(56)

In order to estimate the right-hand side of the latest equality, we introduce, for i, j ≥ 1, the function z i,j as the solution of

z i,j ∈ L 2 (Γ, R(s j ); dµ), ā(z i,j , v) = ā(e i-1 , v), ∀ v ∈ L 2 (Γ, R(s j ); dµ). (57)
We have ā(e i-1 , s j ) = ā(z i,j , s j ) ≤ ā(z i,j , z i,j )

1 2 ā(s j , s j ) 1 2 . ( 58
)
Using ( 48), ( 46), the fact that s i is a solution of (45) and dim R(s j ) ≤ k ā(e i-1 , e i-1 ) -ā(s i , s i ) = ā(e i-1 -s i , e i-1 -s i ) ≤ ā(e i-1 -z i,j , e i-1 -z i,j ).

Expending the right-hand side and using v = z i,j in (57) this gives ā(z i,j , z i,j ) ≤ ā(s i , s i ), which combined with (58) provides the estimate ā(e i-1 , s j ) ≤ ā(s i , s i )

1 2 ā(s j , s j ) 1 2 , ∀ i, j ≥ 1.
Using the latest estimate in (56) and then Cauchy-Schwarz's inequality, we get

                
ā(e i , e i ) ≤ ā(s i+1 , s i+1 )

1 2 i j=1
ā(s j , s j )

1 2 + ā(e i , e 0 ) ≤ ā(s i+1 , s i+1 ) 1 2 i 1 2   ∞ j=1 ā(s j , s j )   1 2
+ ā(e i , e 0 ), ∀ i ≥ 1.

(59)

But the criterion of comparison of two series with nonnegative terms and the facts that (see (51))

∞ i=1 1 i = ∞, ∞ i=1 ā(s i , s i ) < ∞, prove that lim inf i→∞ ā(s i+1 , s i+1 ) i = lim inf i→∞ ā(s i+1 , s i+1 ) 1 i = 0.
Since ā(e i , e i ) is a decreasing sequence by (48) and since (55) asserts that e i converges weakly to zero, we can pass to the limit in (59), to deduce lim i→∞ ā(e i , e i ) = lim inf i→∞ ā(e i , e i )

≤ lim inf i→∞    ā(s i+1 , s i+1 ) 1 2 i 1 2   ∞ j=1 ā(s j , s j )   1 2 + ā(e i , e 0 )    = 0.
This proves that e i converges strongly to zero in L 2 (Γ, H; dµ). Since e i = u -u i this finishes the proof of Theorem 5.3.

Remark 5.5 In many cases the corrections s i decrease exponentially in the sense that:

|||s i ||| = O(ρ -i ) as i → +∞, for some ρ > 1,
where |||•||| denotes the norm in L 2 (Γ, H; dµ). This occurs in particular for the standard POD expansion when Γ is an open set of R N , µ is the Lebesgue measure and the function f = f (γ) is analytic with respect to γ (see [START_REF] Azaïez | Error Bounds for POD expansions of parametrized transient temperatures[END_REF]). Then |||s i ||| is a good estimator for the error |||u -u i |||.

Relationship with POD and PGD methods

The "intrinsic" PGD method developed in the previous sections is a genuine extension of both POD and PGD method. Indeed, to analyze the connexions with the POD method, let us consider the problem studied in [START_REF] Falcó | A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach[END_REF], namely (P k ) min

Z∈G ≤k Γ (u(γ) -u Z (γ), u(γ) -u Z (γ)) H dµ(γ), (60)
where (•, •) H is an inner product on H. In this case a solution of (P k ) is the space generated by the first k eigenfunctions of the POD operator P : H → H, which is given by (see below)

P(v) = Γ (u(γ), v) H u(γ) dµ(γ), ∀v ∈ H.
In the present case, due the dependence of a with respect to γ, it does not seem that the problem can be reduced to a spectral problem.

As an example, from now on we fix in this Section

k = 1.
Then problem [START_REF] Jollife | Principal component analysis[END_REF] can be written as

min v∈H, ϕ∈L 2 (Γ;dµ) Γ a(u(γ) -ϕ(γ)v, u(γ) -ϕ(γ)v; γ)dµ(γ). (61)
Note that problem (61) has at least a solution (see Section 3 above). So, taking the derivative of the functional

(v, ϕ) ∈ H × L 2 (Γ; dµ) → Γ a(u(γ) -ϕ(γ)v, u(γ) -ϕ(γ)v; γ) dµ(γ),
we deduce that if (w, ψ) ∈ H × L 2 (Γ; dµ) is a solution of (61), with w = 0, then ψ(γ) = a(u(γ), w; γ) a(w, w; γ) , dµ-a.e. γ ∈ Γ,

and w is a solution of the non-linear variational problem

Γ a(u(γ), w; γ) a(w, w; γ) a(u(γ), v; γ)dµ(γ) = Γ a(u(γ), w; γ) 2 a(w, w; γ) 2 a(w, v; γ)dµ(γ), ∀ v ∈ H. ( 63
) Note that if w = 0, Γ a(u(γ)-ϕ(γ)v, u(γ)-ϕ(γ)v; γ) dµ(γ) ≥ Γ a(u(γ), u(γ); γ) dµ(γ), ∀v ∈ H, ∀ϕ ∈ L 2 (Γ, dµ).
This implies that u = 0 and therefore f = 0. If a does not depend on γ, statement (63) can be written as

a Γ a(u(γ), w) u(γ)dµ(γ), v = a     Γ a(u(γ), w) 2 dµ(γ) a(w, w) w, v     , ∀ v ∈ H, f i (γ), v(γ) dµ(γ) = Γ f (γ), v(γ) dµ(γ) -ā( u i-1 , v), ∀ v ∈ L 2 (Γ, H; dµ), (65)
with

u i-1 (γ) = i-1 j=1 Φ j (γ) w j , for i ≥ 2. ( 66
)
If problem (64) admits a solution such that w i = 0, then the second equation in (64) is equivalent to, a( Φ i (γ) w i , w i ; γ) = f i (γ), w i dµ -a. e. γ ∈ Γ, which in turn is equivalent to

Φ i (γ) = f i (γ), w i a( w i , w i ; γ) dµ -a. e. γ ∈ Γ. ( 67
)
Then the first equation in (64), is equivalent to the non-linear variational problem

w i ∈ H, Γ f i (γ), w i a( w i , w i ; γ) f i (γ), v dµ(γ) = Γ f i (γ), w i 2 a( w i , w i ; γ) 2 a( w i , v; γ) dµ(γ), ∀ v ∈ H. (68)
Note that this problem is just problem (63) with w replaced by w i and f replaced by f i . Conversely, if problem (68) admits a solution, then the pair ( w i , Φ i ), with Φ defined by (67), is a solution on the PGD problem (64).

Consequently the sequence (Φ i , w i ) provided by the deflation algorithm (39)-( 40) is also a solution of the PGD algorithm (64)-( 65)-(66), with fi = f i for all i ≥ 1. Thus, if the PGD algorithm is computed in such a way that at each step Φ i w i = Φ i w i (= s i ), the analysis developed in Section 5 proves that the sequence u i converges in L 2 (Γ, H; dµ) to the parametric solution u(γ) of problem [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF].

However there is the possibility that problem (64) admits several solutions and that some of these do not provide a solution of the optimization problem (45). Then the convergence properties studied in Section 5 may be lost. It is then convenient to solve the PGD problem (64) ensuring that the solution does provide an optimal sub-space.

The previous analysis presents some differences with preceding works on the analysis of convergence of PGD methods applied to the solution of PDEs and optimization problems. Let us describe some of them. In [START_REF] Ammar | On the convergence of a Greedy Rank-One Update Algorithm for a class of Linear Systems[END_REF] the authors prove the convergence of the PGD for finite dimensional linear systems Ax = b where A ∈ R N ×N is an invertible high dimensional matrix, i.e. N = N 1 N 2 • • • N n . The solution is searched as a series of rank-one summands, belonging to

R N = R N 1 ⊗ a R N 2 ⊗ a • • • ⊗ a R Nn (
where ⊗ a denotes the algebraic tensor product). Also, in [START_REF] Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF] the authors prove the convergence of the PGD algorithm applied to the Laplace problem in a tensor product domain,

-∆u = f in Ω x × Ω y , u| ∂Ωx×Ωy = 0,
where Ω x ⊂ R and Ω y ⊂ R are two bounded domains. They solve the problem on the tensor space H 1 0 (Ω x ) ⊗ a H 1 0 (Ω y ) which is dense in H 1 0 (Ω x × Ω y ) for the norm of H 1 0 (Ω x × Ω y ). The work [START_REF] Cancès | Convergence of a greedy algorithm for high-dimensional convex nonlinear problems[END_REF] proves the convergence of the PGD for the optimization problem: Find u ∈ L 2 (Ω, H 1 (I)) such that u ∈ arg min v∈L 2 (Ω,H 1 (I))

E(v)

, where E is a strongly convex functional, with Lipschitz gradient on bounded sets. This method can be used for high-dimensional nonlinear convex problems. Further, in [START_REF] Falcó | Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces[END_REF], the authors prove the convergence of a PGD-like algorithm, where the set of rank-one tensors in a tensor space is substituted by a closed cone Σ, to solve the variational problem: Find u ∈ X such that u ∈ arg min v∈X E(v), where E : X -→ R is a convex functional defined over a reflexive Banach space X. Moreover, in [START_REF] Falcó | A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach[END_REF] the authors prove the convergence of the PGD for elliptic PDEs in the form Au = f where u and f belong to a Hilbert tensor space

H = H 1 ⊗ a H 2 ⊗ a • • • ⊗ a H n |||•||| .
Here the norm ||| • ||| 2 = •, • is given by

•, • = •, • 1 •, • 2 • • • •,
• n . This is a generalization of Eckart and Young theorem.

The results reported in the present paper are a generalization of this last work [START_REF] Falcó | A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart Young approach[END_REF] when the operator A depends on a parameter.

Conclusion

In this paper we have introduced an iterative deflation algorithm to solve parametric symmetric elliptic equations. It is a Proper Generalized Decomposition algorithm as it builds a tensorized representation of the parameterized solutions, by means of optimal subspaces that minimize the residual in mean quadratic norm. It is intrinsic in the sense that in each deflation step the residual is minimized in the "natural" parametric norm generated by the parametric elliptic operator. It is conceptually close to the Proper Orthogonal Decomposition with the difference that in the POD the residual is minimized with respect to a fixed mean quadratic norm. Due to this difference, spectral theory cannot be applied.

We have proved the existence of the optimal subspaces of dimension less than or equal to a fixed number, as required in each iteration of the deflation algorithm, with a specific analysis for the one-dimensional case. Also, we have proved the strong convergence in the parametric elliptic norm of the deflation algorithm for quite general parametric elliptic operators.

We have further proved that the method introduced is a genuine extension of both POD and PGD methods, and that in particular it provides a theoretical analysis of the PGD method, when this method is applied in such a way that it provides the optimal sub-spaces : The PGD expansion is strongly convergent to the targeted solution in parametric elliptic norm.

We will next focus our research on the analysis of convergence rates of the intrinsic PGD expansion that we introduced. We will analyze wether the standard PGD provides the optimal sub-spaces, and compare the convergence rates with those of the POD expansion, to determine whether the use of optimal modes provides improved convergence rates. We will also work on the use of optimization techniques as an alternative way to compute the optimal modes, rather than the Power Iteration method that is common in PGD computations.

All the results obtained in the present paper refer to a symmetric. In a future work we will consider the non-symmetric case.

Remark 4 . 2

 42 From (36), (37), definition[START_REF] Nouy | Generalized spectral decomposition method for stochastic non linear problems[END_REF] of m k and Proposition 2.1, we have that w = u W in the proof of Theorem 4.1. Moreover,

Theorem 5 . 3 Remark 5 . 4

 5354 The sequence u i provided by the least-squares PGD algorithm (39)-(40) strongly converges in L 2 (Γ, H; dµ) to the parameterized solution γ ∈ Γ → u(γ) ∈ H of problem[START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF]. In view of the last assertion of Remark 5.1, Theorem 5.3 proves that the series j≥1 s j converges in L 2 (Γ, H; dµ) to the parametrized solution γ ∈ Γ → u(γ) ∈ H of problem (3).

     Γ a( Φ i (γ) w i , Φ i (γ) v; γ) dµ(γ) = Γ f i (γ), Φ i (γ) v dµ(γ), ∀v ∈ H, Γ a( Φ i (γ) w i , Φ * (γ) w i ; γ) dµ(γ) = Γ f i (γ), Φ * (γ) w i dµ(γ), ∀ Φ * ∈ L 2 (Γ, dµ),(64)where f 1 = f , and f i is defined by Γ

Acknowledgements

The work of Mejdi Azaiez and Tomás Chacón has been partially supported by the Spanish Government -Feder EU grant MTM2015-64577-C2-1-R.

The work of Juan Casado-Díaz and François Murat has been partially supported by the Spanish Government -Feder EU grant MTM2014-53309-P.

which implies that Γ a(u(γ), w) u(γ)dµ(γ) = Γ a(u(γ), w) 2 dµ(γ) a(w, w) w, which proves that w is an eigenvector of the operator

for the eigenvalue

a(w, w) .

In contrast, when a depends on γ it does not seem that problem (63) corresponds to an eigenvalue problem.

To analyze the relationship with the PGD method, let us remember that this method approximates the solution u of problem ( 8) by a series similar to that provided by the deflation algorithm introduced in Section 5, namely

where the pair ( Φ i , w i ) ∈ L 2 (Γ, dµ) × H is recursively obtained as a solution of the non-linear coupled problems