Camilla Cadorin
email: camilla.cadorin@univr.it

Introduction

The COVID-19 pandemic has posed a serious health risk, especially in vulnerable populations. Even before the pandemic, people with mental disorders had worse physical health outcomes compared to the general population. This umbrella review investigated whether having a pre-pandemic mental disorder was associated with worse physical health outcomes due to the COVID-19 pandemic.

Methods Following a pre-registered protocol available on the Open Science Framework platform, we searched Ovid MEDLINE All, Embase (Ovid), PsycINFO (Ovid), CINAHL, and Web of Science up to the 6th of October 2021 for systematic reviews on the impact of COVID-19 on people with pre-existing mental disorders. The following outcomes were considered: risk of contracting the SARS-CoV-2 infection, risk of severe illness, COVID-19 related mortality risk, risk of long-term physical symptoms after COVID-19. For meta-analyses, we considered adjusted odds ratio (OR) as effect size measure. Screening, data extraction and quality assessment with the AMSTAR 2 tool have been done in parallel and duplicate.

Results

We included five meta-analyses and four narrative reviews. The meta-analyses reported that people with any mental disorder had an increased risk of SARS-CoV-2 infection (OR: 1.71, 95% CI 1.09-2.69), severe illness course (OR from 1.32 to 1.77, 95%CI between 1.19-1.46 and 1.29-2.42, respectively) and COVID-19 related mortality (OR from 1.38 to 1.52, 95%CI between 1.15-1.65 and 1.20-1.93, respectively) as compared to the general population. People with anxiety disorders had an increased risk of SAR-CoV-2 infection, but not increased mortality. People with mood and schizophrenia spectrum disorders had an increased COVID-19 related mortality but without evidence of increased risk of severe COVID-19 illness. Narrative reviews were consistent with findings from the meta-analyses.

Risk of SARS-CoV-2 infection, severe COVID-19 illness and COVID-19 mortality in people with pre-existing mental disorders: an umbrella review Federico Bertolini 1 , Anke B Witteveen 2 , Susanne Young 2 , Pim Cuijpers 2,3 , Jose Luis Ayuso-Mateos 4,5,6 , Corrado Barbui 1 , María Cabello 5,6 , Camilla Cadorin 1* , Naomi Downes 7 , Daniele Franzoi 2 , Michael Elizabeth Gasior 2 , Brandon Gray 9 , Ann John 8 , Maria Melchior 7 , Mark van Ommeren 9 , Christina Palantza 2 , Marianna Purgato 1 , Judith Van der Waerden 7 , Siyuan Wang 2 and Marit Sijbrandij 2

Background

The COVID-19 pandemic has affected most of the population worldwide. Disadvantaged population groups are thought to have suffered the most, with possibly among those, people affected by mental disorders. The physical health repercussions have been a particular point of concern, as many known risk factors for severe COVID-19 illness course and COVID-19 related mortality are common in people with mental disorders [START_REF] Booth | Population risk factors for severe disease and mortality in COVID-19: a global systematic review and meta-analysis[END_REF][START_REF] Singh | Prevalence of co-morbidities and their association with mortality in patients with COVID-19: a systematic review and meta-analysis[END_REF][START_REF] Ssentongo | Association of cardiovascular disease and 10 other pre-existing comorbidities with COVID-19 mortality: a systematic review and meta-analysis[END_REF]. Firstly, it is unclear whether this population, or specific subgroups, had an increased risk of contracting the SARS-CoV-2 virus than the general population [START_REF] Lee | Association between mental illness and COVID-19 susceptibility and clinical outcomes in South Korea: a nationwide cohort study[END_REF][START_REF] Taquet | Bidirectional associations between COVID-19 and psychiatric disorder: retrospective cohort studies of 62 354 COVID-19 cases in the USA[END_REF]. Shared living accommodations represented contexts in which adhering to social distancing might have been not possible or difficult. Secondly, there has been a focus on the disease course in people with mental disorders. Obesity, diabetes mellitus and cardiovascular disease are common in this population [START_REF] Dickerson | Obesity among individuals with serious mental illness[END_REF][START_REF] Suvisaari | Metabolic syndrome among persons with schizophrenia and other psychotic disorders in a general population survey[END_REF]. In addition to that, many psychotropic medications have the potential of a detrimental effect on the respiratory function [START_REF] Wang | Association between Antipsychotic Agents and Risk of Acute Respiratory failure in patients with Chronic Obstructive Pulmonary Disease[END_REF][START_REF] Brandt | Benzodiazepines and Z-Drugs: an updated review of major adverse outcomes reported on in epidemiologic research[END_REF]. Given these premises, it seems highly relevant to assess if and how the COVID-19 pandemic has affected the physical health of people with pre-existing mental disorders. Against this background, an umbrella review was performed aiming to summarize evidence from systematic reviews (SR) and meta-analyses (MAs). This work builds on a wider project of evidence synthesis on the impact of the COVID-19 pandemic on mental health, performed to inform a WHO Scientific Brief [START_REF]COVID-19: Early evidence of the pandemic's impact: Scientific Brief[END_REF].

Methods

Umbrella review design

We followed published guidelines on performing umbrella reviews [START_REF] Ioannidis | Integration of evidence from multiple meta-analyses: a primer on umbrella reviews, treatment networks and multiple treatments metaanalyses[END_REF][START_REF] Solmi | The role of meta-analyses and umbrella reviews in assessing the harms of psychotropic medications: beyond qualitative synthesis[END_REF][START_REF] Fusar-Poli | Ten simple rules for conducting umbrella reviews[END_REF]. An umbrella review consists of a systematic search and assessment of systematic reviews on a specific research question. This allows for the comparison of results of clinical outcomes and provides a clear picture on broad healthcare areas, possibly revealing whether the evidence base is consistent or contradictory. We registered a protocol to define the research methodology to inform the WHO Scientific brief that is publicly available on the Open Science Framework platform [START_REF] Covid-19 | An umbrella review of systematic reviews with or without meta-analyses[END_REF]. The present work is based on the 'Question 3 -Are people living with pre COVID-19 existing mental health disorders at (increased) risk of severe illness and mortality and/or of contracting SARS-CoV-2 compared to any other population?' of the protocol, with the addition of focusing on estimates that have adjusted for confounding.

Literature search and study selection

An information specialist performed a systematic search on Ovid MEDLINE All, Embase (Ovid), PsycINFO (Ovid), CINAHL, and Web of Science, from December 31, 2019 until October 6, 2021, using a search strategy based on a combination of keywords and text words for reviews, COVID-19, and mental disorders or problems. The search strategy can be found in the supplementary material. This search was designed to inform a wide panel of different research questions on COVID-19 and mental health. We deduplicated search results using Endnote software [15]. Entries were divided into groups, each screened in parallel and independently by two different authors using Rayyan [START_REF] Ouzzani | Rayyan-a web and mobile app for systematic reviews[END_REF]. In case of disagreement, records were conserved. The full text of the remaining articles was assessed against the inclusion and exclusion criteria, again in duplicate and independently. Disagreement was resolved by discussion or with a senior review author.

Eligibility criteria and data extraction

We considered for inclusion reviews that: (1) were systematic, as defined by having a systematic search on at least one bibliographic database, explicitly reporting primary studies selection criteria and providing a list and synthesis of included studies; (2) considered as primary studies either cohort or cross-sectional or case-control studies; (3) have been published in a peer-reviewed journal; (4) compared people with pre-existing mental disorders (i.e., with a diagnosis of mental disorder established before the pandemic onset) with those without a preexisting mental disorder; and (5) assessed at least one of the following outcomes: (a) risk of contracting SARS-Cov-2 infection, (b) risk of severe COVID-19 illness, (c) risk of long-term physical symptoms after COVID-19 illness, (d) mortality from SARS-CoV-2 infection. For the outcome 'risk of severe illness' , we considered definitions as provided by original systematic reviewers, or proxy outcomes such as hospitalization. We did not exclude reviews on the basis of language of publication. In the case of reviews considering several pandemics, we planned to include only those that reported separated Discussion and conclusions As compared to the general population, there is strong evidence showing that people with pre-existing mental disorders suffered from worse physical health outcomes due to the COVID-19 pandemic and may therefore be considered a risk group similar to people with underlying physical conditions. Factors likely involved include living accommodations with barriers to social distancing, cardiovascular comorbidities, psychotropic medications and difficulties in accessing high-intensity medical care.

Keywords Umbrella review, Covid-19, Sars-CoV-2, Mental health, Mortality, Pre-existing mental health disorders data for the COVID-19 pandemic. We considered systematic reviews independently of whether a meta-analysis was performed. Two authors independently and in parallel extracted the following data: author, date of publication, timeframe covered by the search, population and comparator, design of included studies, inclusion and exclusion criteria, outcomes, risk of bias assessment strategy, how exposure to SARS-CoV-2 was established, number of included studies and participants, and outcomes of interest with I 2 statistic as a measure of heterogeneity. In terms of meta-analysis, we aimed to summarize effect size in terms of adjusted Odds Ratio (aOR), if that was not available, we considered the most adjusted model available.

Assessment of quality

Two independent authors assessed the quality of the included systematic reviews using the AMSTAR 2 tool [START_REF] Shea | AMSTAR 2: a critical appraisal tool for systematic reviews that include randomised or non-randomised studies of healthcare interventions, or both[END_REF]. For systematic reviews that did not perform a metaanalysis, items 11, 12 and 15 were not considered. Also, small adaptations to some items were performed to make them more suitable to score (see the supplementary material). For an overall rating, we used the proposed scheme by Shea and colleagues that provides judgments of high, moderate, low or critically low confidence [START_REF] Shea | AMSTAR 2: a critical appraisal tool for systematic reviews that include randomised or non-randomised studies of healthcare interventions, or both[END_REF].

Results

Selection and inclusion of systematic reviews

The systematic search provided 46,284 record entries, reduced to 31,559 after deduplication. 939 records were selected for full text retrieval by screening titles and abstracts. Of these, nine were finally included after comparing full texts against inclusion and exclusion criteria. Figure 1 reports the details of the study selection process.

Characteristics of included systematic reviews

Of the nine included reviews, five performed a metaanalysis on at least one of the outcomes of interest [START_REF] Ceban | Association between Mood Disorders and Risk of COVID-19 infection, hospitalization, and death: a systematic review and Meta-analysis[END_REF][START_REF] Liu | Mental and neurological disorders and risk of COVID-19 susceptibility, illness severity and mortality: a systematic review, meta-analysis and call for action[END_REF][START_REF] Vai | Mental disorders and risk of COVID-19-related mortality, hospitalisation, and intensive care unit admission: a systematic review and meta-analysis[END_REF][START_REF] Toubasi | A metaanalysis: the mortality and severity of COVID-19 among patients with mental disorders[END_REF][START_REF] Fond | Association between Mental Health Disorders and Mortality among patients with COVID-19 in 7 countries: a systematic review and Meta-analysis[END_REF], while the remaining four were of a narrative nature [START_REF] Lemieux | Management of COVID-19 for persons with Mental Illness in Secure units: a Rapid International Review to inform practice in Québec[END_REF][START_REF] Karaoulanis | Do patients with schizophrenia have higher infection and mortality rates due to COVID-19? A systematic review[END_REF][START_REF] Fornaro | Implications of the COVID-19 pandemic for people with bipolar disorders: a scoping review[END_REF][START_REF] Murphy | The impact of the COVID-19 pandemic and its related restrictions on people with pre-existent mental health conditions: a scoping review[END_REF]. Table 1 summarises the characteristics of the included reviews.

Systematic reviews with meta-analyses (MAs) generally had a search covering up to the first quarter of 2021. They included primary studies with mainly a cohort design, with the notable exceptions of the review by Liu and colleagues which considered case-series too and the review by Fond and colleagues which considered populationbased cohort studies. MAs included a mean of 45 studies and a median of 21, the review by Liu and colleagues being an outlier with 149 included studies. Most of the MAs considered people with any mental disorder, possibly providing subgroups for specific disorders groups, while the review by Ceban and colleagues only focused on people with a mood disorder. Included studies covered a wide range of countries but with a general lack of representation from low-and middle-income countries. Infection by SARS-CoV-2 was defined by laboratory testing, ICD-10, electronic health records, or clinical judgment in the review by Ceban and colleagues and by laboratory results and diagnosis in conjunction with clinical presentation in the review by Liu and colleagues. Severe illness was defined as ICU admission, mechanical ventilatory support, oxygen therapy, extracorporeal membrane oxygenation, acute respiratory distress syndrome, and/or cardiopulmonary resuscitation by Ceban and colleagues and as hospitalization, ICU admission, or requirement for other special treatment (including oxygen therapy) by Liu and colleagues. For the review by Vai and colleagues, which does not have a "severe illness" outcome, we considered the outcome 'hospitalisation' . For the review by Toubasi and colleagues, we considered the pooled mortality and severe illness outcome, as a separate estimate for mortality only was not available. Notably, no review was available to inform on the long-term physical symptoms after SARS-CoV2 infection. Four reviews reported estimates based on pooling adjusted ORs only [START_REF] Ceban | Association between Mood Disorders and Risk of COVID-19 infection, hospitalization, and death: a systematic review and Meta-analysis[END_REF][START_REF] Liu | Mental and neurological disorders and risk of COVID-19 susceptibility, illness severity and mortality: a systematic review, meta-analysis and call for action[END_REF][START_REF] Toubasi | A metaanalysis: the mortality and severity of COVID-19 among patients with mental disorders[END_REF][START_REF] Fond | Association between Mental Health Disorders and Mortality among patients with COVID-19 in 7 countries: a systematic review and Meta-analysis[END_REF]. The review by Vai and colleagues reported a fully adjusted model only when considering people with any mental disorder, for the diagnostic groups we have then considered the 'partially adjusted model' , where review authors considered aORs pooled together with crude ORs when an adjusted figure was not available from primary studies.

The four narrative reviews varied in their specific design, with two scoping reviews [START_REF] Fornaro | Implications of the COVID-19 pandemic for people with bipolar disorders: a scoping review[END_REF][START_REF] Murphy | The impact of the COVID-19 pandemic and its related restrictions on people with pre-existent mental health conditions: a scoping review[END_REF], one rapid review [START_REF] Lemieux | Management of COVID-19 for persons with Mental Illness in Secure units: a Rapid International Review to inform practice in Québec[END_REF], and one systematic review without metaanalysis [START_REF] Karaoulanis | Do patients with schizophrenia have higher infection and mortality rates due to COVID-19? A systematic review[END_REF]. They considered a wide range of study designs with the review by Lemieux and colleagues considering opinion pieces and other reviews; as for population of interest, they considered people with bipolar disorder [START_REF] Fornaro | Implications of the COVID-19 pandemic for people with bipolar disorders: a scoping review[END_REF], people with schizophrenia [START_REF] Karaoulanis | Do patients with schizophrenia have higher infection and mortality rates due to COVID-19? A systematic review[END_REF], people with mental illness in secure settings [START_REF] Lemieux | Management of COVID-19 for persons with Mental Illness in Secure units: a Rapid International Review to inform practice in Québec[END_REF], and generally people with mental disorders [START_REF] Murphy | The impact of the COVID-19 pandemic and its related restrictions on people with pre-existent mental health conditions: a scoping review[END_REF].

Quality of included reviews

The AMSTAR 2 rated level of quality for all the MAs was "low", with the exception of the review by Liu and colleagues with "high" and the review by Fond and colleagues with "critically low". The review by Liu and colleagues did not have any weaknesses in critical items, while all other MAs did not report the list of excluded studies with reasons for exclusion; the review by Fond and colleagues also did not account for the impact of risk of bias in primary studies on the results. The level of quality for the narrative reviews was "critically low" with the exception of the review by Fornaro and colleagues ("low"), mainly because of a lack of risk of bias assessment and protocol registration. See the supplementary material for detailed AMSTAR 2 evaluation of the included reviews.

Risk of SARS-CoV-2 infection

Two MAs informed on the association between Sars-CoV-2 infection and having a pre-existing mental disorder compared to not having a pre-existing mental disorder (Fig. 2) [START_REF] Ceban | Association between Mood Disorders and Risk of COVID-19 infection, hospitalization, and death: a systematic review and Meta-analysis[END_REF][START_REF] Liu | Mental and neurological disorders and risk of COVID-19 susceptibility, illness severity and mortality: a systematic review, meta-analysis and call for action[END_REF].

For people with any mental disorder, Liu and colleagues found a statistically significant positive association (OR: 1.71, 95% CI 1.09-2.69). For people with anxiety disorders, Liu 2021 and colleagues in a partially adjusted model found a statistically significant positive association although this effect size relies on two studies only (OR 1.63, 95%CI 1.44-1.85). For people with mood disorders, Liu 2021 and colleagues in a partially adjusted model found a statistically significant positive association (OR: 2.02, 95%CI 1.08-3.76), while Ceban and colleagues did not find a statistically significant association (OR: 1.50, 95%CI 0.75-2.99). For people with schizophrenia spectrum disorders, Liu and colleagues, in a partially adjusted model, did not find a statistically significant association (OR: 1.72, 95%CI 0.62-4.77). The level of statistical heterogeneity was been generally very high, with most I 2 statistics over 95%, with the exception of the estimate for people with mood disorders by Liu and colleagues (0%). I 2 was not reported in Ceban et al., 2021.

Risk of severe illness

Three MAs informed on the association between a severe course of COVID-19 and having a pre-existing mental disorder compared to not having a pre-existing mental disorder (Fig. 3) [START_REF] Ceban | Association between Mood Disorders and Risk of COVID-19 infection, hospitalization, and death: a systematic review and Meta-analysis[END_REF][START_REF] Liu | Mental and neurological disorders and risk of COVID-19 susceptibility, illness severity and mortality: a systematic review, meta-analysis and call for action[END_REF][START_REF] Vai | Mental disorders and risk of COVID-19-related mortality, hospitalisation, and intensive care unit admission: a systematic review and meta-analysis[END_REF].

For people with any mental disorder, both the review by Liu and colleagues and Vai and colleagues found a statistically significant positive association (OR: 1.32, 95%CI 1.19-1.46 and OR: 1.77, 95%CI 1.29-2.42, respectively).

No study informed on people with anxiety disorders. For people with a mood disorder, Ceban and colleagues found no association (OR: 0.99, 95%CI: 0.80-1.24), Liu and colleagues in a partially adjusted model found a statistically significant positive association (OR: 1.34, 95%CI 1.08-1.67) while Vai and colleagues did not find a statistically significant association (OR 1.27, 95%CI 0.64-2.50). For people with a schizophrenia spectrum disorder, both the reviews by Liu and colleagues and Vai and colleagues did not find a statistically significant association (OR: 1.22, 95%CI 0.70-2.13 and OR: 1.38, 95%CI 0.61-2.94, respectively). The level of statistical heterogeneity was moderate to very high, with an I 2 statistic between 65 and 100%, but low for the estimate for people with mood disorders by Vai and colleagues (23%). I 2 was not reported in Ceban et al., 2021.

COVID-19 related mortality

Four MAs informed on the association between mortality and having a pre-existing mental disorder compared to not having a pre-existing mental disorder (Fig. 4) [START_REF] Liu | Mental and neurological disorders and risk of COVID-19 susceptibility, illness severity and mortality: a systematic review, meta-analysis and call for action[END_REF][START_REF] Vai | Mental disorders and risk of COVID-19-related mortality, hospitalisation, and intensive care unit admission: a systematic review and meta-analysis[END_REF][START_REF] Toubasi | A metaanalysis: the mortality and severity of COVID-19 among patients with mental disorders[END_REF][START_REF] Fond | Association between Mental Health Disorders and Mortality among patients with COVID-19 in 7 countries: a systematic review and Meta-analysis[END_REF].

For people with any mental disorder all four MAs found a statistically significant positive association, ranging from an OR of 1.38 (95%CI: 1.15-1.65) for the review by Fond and colleagues to 1.52 (95%CI: 1.20-1.93) for the review by Toubasi and colleagues (which however considered in this outcome severe illness cases as well). For people with anxiety disorders both the reviews by Liu and colleagues, in a partially adjusted model, and by Vai and colleagues did not find a statistically significant association (OR: 1.16, 95%CI 0.75-1.79 and OR: 1.01, 95%CI 0.77-1.32, respectively). For people with mood disorders, all three informing MAs found a statistically significant positive association with ORs ranging from 1.36 (95%CI: 1.15-1.79, Liu and colleagues, partially adjusted model) to 1.57 (95%CI: 1.26-1.95 Vai and colleagues). For people with schizophrenia spectrum disorders both the reviews by Liu and colleagues, in a partially adjusted model, and by Vai and colleagues found a positive association (OR 2.28, 95%CI 1.40-3.73, and OR 1.68, 95%CI 1.29-2.18, respectively). The level of statistical heterogeneity has been generally moderate to considerable, with an I 2 statistic between 60% and 81.4%, but for the review by Vai and colleagues for the estimate for people with any mental disorder (39%), and low for the estimate for people with anxiety disorders (0%) and mood disorders (22%). The I 2 has not been reported in [START_REF] Ceban | Association between Mood Disorders and Risk of COVID-19 infection, hospitalization, and death: a systematic review and Meta-analysis[END_REF].

Narrative reviews

The narrative reviews corroborated meta-analytic findings by indicating that patients with serious pre-COVID-19 mental disorders show adverse health Fig. 2 Risk of contracting SARS-CoV-2 infection. CI: confidence interval; K: number of included studies; OR: Odds ratio; n: total number of included participants; NR: not reported; *: partially adjusted model outcomes related to COVID-19 infection in terms of higher severity and mortality. Fornaro and colleagues [START_REF] Fornaro | Implications of the COVID-19 pandemic for people with bipolar disorders: a scoping review[END_REF] performed a scoping review on clinical and public health themes for people with bipolar disorder. They identified four major themes from the 14 included papers, among which was the impact of having bipolar disorder on the risk of contracting the SARS-CoV-2 infection. For this theme, one study reported an increased risk of infection contraction for people with bipolar disorder [START_REF] Wang | Increased risk of COVID-19 infection and mortality in people with mental disorders: analysis from electronic health records in the United States[END_REF]. This study was considered by the MAs previously reported. Karaoulanis and Christodoulou performed a systematic review without meta-analysis on infection rates and mortality in people with schizophrenia spectrum disorders. The included studies suggest an increased infection and mortality risk; these studies have been included in the MAs previously reported. Lemieux and colleagues performed a rapid review on the management of COVID-19 for people with mental illness in secure settings. They considered a wide range of publications including opinion pieces and other reviews and report greater morbidity and mortality. Murphy and colleagues conducted a scoping review on the impact of COVID-19 and related restrictions on people with pre-existing mental disorders. They reported an increased infection risk in this population.

Discussion

To our knowledge this is the first umbrella review aimed at summarizing evidence on the impact of the COVID-19 pandemic on health outcomes in people with pre-existing mental disorders. Compared to the individual systematic reviews previously published, this umbrella review contextualizes the single piece of evidence, and provides an overview for different mental disorders, while systematic reviews have so far focused on specific diagnostic groups or considered only some of the outcomes of interest on the physical health repercussions of people with mental disorders. Two reviews considered several diagnostic groups and outcomes, however one employed only partially adjusted models within diagnostic groups [START_REF] Liu | Mental and neurological disorders and risk of COVID-19 susceptibility, illness severity and mortality: a systematic review, meta-analysis and call for action[END_REF], and the other could include a considerably smaller number of studies [START_REF] Vai | Mental disorders and risk of COVID-19-related mortality, hospitalisation, and intensive care unit admission: a systematic review and meta-analysis[END_REF]. Overall, we found consistent results across the various reviews. Of interest, we found no previous reviews exploring the long-term effects of SARS-CoV-2 infection. Having any mental disorder was found to be associated with a higher likelihood of contracting the SARS-CoV-2 infection, a more severe COVID-19 illness, and higher mortality. We found different risks for different disorder groups. People with pre-existing anxiety disorders had an increased risk of contracting the SARS-CoV-2 infection; for people with mood disorders there was conflicting evidence of increased risk for severe COVID-19 course, and evidence of increased mortality; people with schizophrenia spectrum disorders had an increased risk of mortality, but there was no clear evidence of increased severe illness course for people with schizophrenia spectrum disorders. Notably, we found no review assessing the association between long-term physical symptoms after COVID-19 and having a pre-existing mental disorder.

Having any mental disorder has been associated with an increased risk of contracting the SARS-CoV-2 infection. Looking at the data for the specific diagnostic groups, however, we observe that the association is confirmed for people with an anxiety disorder (and in an estimate based on only two studies) only, while for people with mood disorders the evidence is not consistent across reviews, and for people with a schizophrenia spectrum disorder the association is not statistically significant with a very wide confidence interval, making it hard to draw clear conclusions. Many people with a mental disorder live in shared households, nursing homes, therapeutic communities, or are inpatients. It has been noted that such settings pose challenges in putting into practice infection control measures [START_REF] De Girolamo | Mental Health in the Coronavirus Disease 2019 emergency-the italian response[END_REF]. In addition to that, people in an acute phase of a mental disorder might find it difficult to understand the need for and adhere to behavioural means of social distancing [START_REF] Muruganandam | COVID-19 and severe Mental illness: impact on patients and its relation with their awareness about COVID-19[END_REF]. Still, these considerations regard mostly people with serious mental illness such as bipolar disorder or schizophrenia, adding difficulty to the interpretation of these results.

Having a mental disorder was associated with having a more severe illness course and increased mortality. This could be partially explained by the increased risk of contracting the SARS-CoV-2 infection, but as the three disorder groups showed differential patterns, we argue that additional factors may have influenced this outcome. In particular, anxiety disorders were not associated with increased mortality despite an increased infection risk, and indeed the results are compatible with a random distribution in terms of effect sizes and confidence intervals. Mood disorders and schizophrenia spectrum disorders have been associated with increased mortality from COVID-19 without conclusive evidence of increased infection risk. Several factors might come into play in determining such a negative outcome. People with severe mental illness, such as bipolar disorder and schizophrenia, more frequently present with high BMI, diabetes mellitus, generally limited exercise tolerance, and are more likely to also smoke and have substance abuse disorders [START_REF] Dickerson | Obesity among individuals with serious mental illness[END_REF][START_REF] Suvisaari | Metabolic syndrome among persons with schizophrenia and other psychotic disorders in a general population survey[END_REF][START_REF] Hunt | Prevalence of comorbid substance use in schizophrenia spectrum disorders in community and clinical settings, 1990-2017: systematic review and meta-analysis[END_REF]. All of these are known risk factors for severe COVID-19 and for COVID-19 related mortality [START_REF] Singh | Prevalence of co-morbidities and their association with mortality in patients with COVID-19: a systematic review and meta-analysis[END_REF][START_REF] Ssentongo | Association of cardiovascular disease and 10 other pre-existing comorbidities with COVID-19 mortality: a systematic review and meta-analysis[END_REF][START_REF] Wang | COVID-19 risk and outcomes in patients with substance use disorders: analyses from electronic health records in the United States[END_REF]. Although the use of adjusted odds ratios should have mitigated the impact of these risk factors in the estimate of effect sizes there was high heterogeneity in terms of adjusted factors used by primary studies. Therefore, it is possible that some factors, such as BMI, might not have been appropriately accounted for [START_REF] Luykx | Are psychiatric disorders risk factors for COVID-19 susceptibility and severity? A two-sample, bidirectional, univariable, and multivariable mendelian randomization study[END_REF]. Many people with these disorders would have received an antipsychotic (and potentially benzodiazepines), with a possible negative impact on respiratory function [START_REF] Wang | Association between Antipsychotic Agents and Risk of Acute Respiratory failure in patients with Chronic Obstructive Pulmonary Disease[END_REF]. However, there is limited evidence to support these hypotheses and future research is needed to fill this gap in knowledge. Moreover, we should take into consideration that socioeconomic factors and stigma might have influenced the access to medical care of these persons [START_REF] Fond | Disparities in Intensive Care Unit Admission and Mortality among patients with Schizophrenia and COVID-19: a National Cohort Study[END_REF]. Many primary studies were conducted during the initial phases of the pandemic when medical resources were scarce compared to needs, access to intensive care units was limited and subject to stringent triage, and no vaccine was available yet. The finding that for people with schizophrenia spectrum disorders there was no increased risk for a more severe disease course, but higher risk for COVID-19 related mortality, is somewhat puzzling. The reviews have used slightly different definitions of "severe illness", but all considered it as an operational composite outcome where many different events qualified, including oxygen therapy. Oxygen therapy has been a widespread need in COVID-19 patients, as arterial hypoxemia is a major feature of the disease [START_REF] Tobin | Basing Respiratory Management of COVID-19 on physiological principles[END_REF]. It is possible that the definition of severe illness was therefore excessively sensitive and did not allow for the identification of differences between people with and without a pre-existing mental disorder. Another possible explanation is that people with schizophrenia might have been disproportionately affected by sudden death events. We know that cardiac sudden death events have been shown to be associated with COVID-19 [START_REF] Tan | The prevalence and Associated Death of Ventricular Arrhythmia and Sudden Cardiac death in hospitalized patients with COVID-19: a systematic review and Meta-analysis[END_REF] and that people with schizophrenia have a higher frequency of cardiovascular disease which may predispose to such events [START_REF] Leucht | Physical illness and schizophrenia: a review of the literature[END_REF]. However, there is no direct evidence, and the use of adjusted estimates should have compensated for the increased cardiovascular burden. Overall, the mismatch between the risk of severe disease and mortality reinforces the need to better investigate factors associated with the increased mortality risk.

The findings of this umbrella review should be put into the context of some limitations. For the various diagnostic groups, the number of included primary studies has been limited, especially for anxiety disorders. Mood disorders include both depression and bipolar disorder, two disorders with different pharmacological approaches and neuro-inflammatory profiles. Additionally, information on the disease status of participants (remission or relapse) has generally not been considered. Low-and middle-income countries have been scarcely represented in the primary studies included by the reviews. The publication timeframe covered by the meta-analyses span to the first quarter of 2021; as such, these findings depict the first year of the pandemic. The general landscape has since changed, thanks to health care systems adaptations, improved COVID-19 treatments and importantly, the introduction of vaccines. There is, however, conflicting evidence regarding vaccine uptake rates in people with mental disorders, possibly due to regional differences [START_REF] Mazereel | COVID-19 vaccination rates in a Cohort study of patients with Mental Illness in Residential and Community Care[END_REF][START_REF] Nilsson | Vaccination against SARS-CoV-2 infection among vulnerable and marginalised population groups in Denmark: a nationwide populationbased study[END_REF]. The reviews considered different study designs for inclusion; notably, Liu and colleagues considered case series. The review by Vai and colleagues informed on many diagnostic groups, but in only partially adjusted models. There has been considerable heterogeneity across all outcomes, which the meta-analyses struggled to explain. Heterogeneity likely reflects methodological and qualitative differences among the primary studies. Moreover, this high heterogeneity and the generally wide confidence intervals limit the accuracy of the estimates.

In light of these findings, there are relevant questions for future research. The new Omicron variant of the SARS-CoV-2 virus spreads more easily but usually causes less severe illness [START_REF] Madhi | Population Immunity and Covid-19 severity with Omicron variant in South Africa[END_REF]. Assessing if this holds true for people with a pre-exiting mental disorder as a risk factor would be valuable. In parallel, there is still limited evidence on vaccine hesitancy and uptake rate in this population. Moreover, the topic of long-term physical symptoms after COVID-19 in people with mental disorders remains scarcely investigated. Addressing these three points would allow for more effective health care planning and possibly targeted intervention to address vaccine hesitancy.

Conclusion

The COVID-19 pandemic has affected people with preexisting mental disorders more severely than people without in terms of physical health. People with preexisting mental disorders, and especially those with mood or schizophrenia spectrum disorders, should have been considered at risk of severe course and increased mortality from COVID-19, similar to other identified risk groups such as patients with somatic health conditions.

Fig. 1

 1 Fig. 1 PRISMA flowchart of studies selection

Fig. 3

 3 Fig. 3 Risk of severe COVID-19 illness. CI: confidence interval; K: number of included studies; OR: Odds ratio; n: total number of included participants; NR: not reported; *: partially adjusted model

Fig. 4

 4 Fig. 4 COVID-19 related mortality risk. CI: confidence interval; K: number of included studies; OR: Odds ratio; n: total number of included participants; NR: not reported; *: partially adjusted model; **: considers mortality and severe illness together

Table 1

 1 Characteristics of included systematic reviews

	Author,	Design	Timeframe	Design of included	Population /	Countries	Effect size	Num-	Outcomes
	year		covered by	studies	comparator		estimation	ber of
			bibliograph-				methods	included
			ic search					Studies
	Ceban	Systematic	From incep-	Case-control and	People with a	USA, South	Random ef-	21	1-Susceptibility
	2021	review with	tion to Febru-	cohort studies	mood disorder /	Korea, Spain,	fect model	(15 aOR,	(laboratory testing,
		meta-analysis	ary 1, 2021		people without	Italy, Turkey, UK,	pooling	6 crude	ICD-10, EHR, and/or
					mood disorders	Israel	adjusted	ORs)	clinical judgment)
							and crude		2-Hospitalization
							ORs. Prefer-		3-Occurrence
							ence given		of severe events
							to adjusted		(includes oxygen
							ORs when		therapy)
							available.		4-Death
							Subgroups
							analyses ac-
							cording to
							adjustment
							status
	Fond	Systematic	From incep-	Population-based	People with any	Denmark,	Random	21	1-Mortality
	2021	review with	tion to Febru-	cohort studies	mental disorder	France, Israel,	effects		2-ICU admission
		meta-analysis	ary 12, 2021	based on medico-	(includes sub-	South Korea,	models,
				administrative	stance abuse	Spain, UK, USA	separate
				health databases or	disorders and		pooling of
				a health care data	ADHD) / people		aORs and
				warehouse	without a men-		crude ORs
					tal disorder
	Fornaro	Scoping	From incep-	Cross-sectional,	People with	Italy, UK, Aus-		14	Identified themes:
	2021	review (no	tion up to	prospective cohort,	Bipolar disorder	tralia, USA, the			1-Impact of COVID-
		meta-analysis)	April 24, 2021	retrospective	/ NA	Netherlands,			19-related stressors
				cohort studies, case		India, China			on BD
				reports/series					2-Impact of COVID-
									19 on mental health
									service utilization
									among people with
									BD
									3-Impact of BD on
									the risk of acquiring
									SARS-cov-2 infection
									4-Engagement in
									preventative behav-
									iors among people
									with BD
	Karaou-	Systematic	From incep-	Unreported	People with	USA, Korea,		7	1-Susceptibility
	lanis	review without	tion up to 21		Schizophrenia /	France, Israel			(being infected)
	2021	meta-analysis	March 2021		people without				2-Mortality
					Schizophrenia
	Lemieux	Rapid review	From Decem-	Opinion pieces by	People with	USA, China,		69	Narrative review
	2020		ber 2019 up	experts (editori-	mental illness in	Italy, UK, France,			on strategies,
			18 August	als, letters to the	secure settings	Germany,			challenges and
			2020	editor, position	/ NA	Ireland, Sin-			recommendations
				papers and other		gapore, Brazil,			for dealing with the
				correspondences),		Canada, Spain,			COVID-19 outbreak
				rapid literature re-		Australia, India,			in secure settings for
				views, narrative (i.e.,		Israel, Poland,			persons with mental
				Non-systematic)		New Zealand,			illness
				literature reviews,		South Korea,
				empirical studies		Scandinavia,
				(surveys and		Switzerland
				descriptive/case
				studies)

aORs: adjusted Odds Ratios, ADHD: Attention-deficit hyperactivity disorder, BP: Bipolar disorder, ICU: intensive care unit, NA: Not applicable, ORs: Odds Ratios, UK: United Kingdom, USA: United States of America,

Table 1 (continued)

 1

Acknowledgements

Not applicable.

Funding

Funding for this work was provided by the World Health Organization. The funding body The founding institution provided a list of research questions and methodological requirements, had no further role in protocol development and carrying on the review, including data collection, analysis, and interpretation and in writing the manuscript.

Data availability

all data generated or analysed during this study are included in this published article and its supplementary information file and the original studies' publications.

Supplementary Information

The online version contains supplementary material available at https://doi. org/10.1186/s12888-023-04641-y.

Supplementary Material 1

Authors' contributions ABW, PC, CB, MS, conceptualized the umbrella review design; all authors commented on the protocol; FB, SY, JLAM, MC, CC, ND, DF, MEG, CP, JvdW, SW performed the screening, data extraction and AMSTAR-2 assessment; all authors contributed to the interpretation of data; FB drafted the manuscript, CC submitted the manuscript, all authors revised the draft and approved the final manuscript.

Declarations Disclaimer

The authors alone are responsible for the views expressed in this article and they do not necessarily represent the views, decisions, policy, or views of WHO or the institutions with which the authors are affiliated.

Ethics approval and consent to participate not applicable.

Consent for publication not applicable.

Competing interests the authors declare that they have no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


R Abdelkhalek
email: rached.abdelkhalek@total.com

H Calandra

O Coulaud

G Latu

J Roman

Fast Seismic Modeling and Reverse Time Migration on a GPU Cluster

Keywords: Seismic modeling, Finite Difference, Reverse Time Migration, GPGPU, CUDA

We have designed a fast parallel simulator that solves the acoustic wave equation on a GPU cluster. Solving the acoustic wave equation in an oil exploration industrial context aims at speeding up seismic modeling and Reverse Time Migration. We consider a finite difference approach on a regular mesh, in both 2D and 3D cases. The acoustic wave equation is solved in either a constant density or a variable density domain. All the computations are done in single precision, since double precision is not required in our context. We use CUDA to take advantage of the GPUs computational power. We study different implementations and their impact on the application performance. We obtain a speed up of 10 for Reverse Time Migration and up to 30 for the modeling application over a sequential code running on general purpose CPU.

INTRODUCTION

The extraordinary challenge that the oil and gas industry must face for hydrocarbon exploration requires the development of leading edge technologies to recover an accurate representation of the subsurface. Seismic modeling and Reverse Time Migration (RTM) based on the full wave equation discretization, are tools of major importance since they give an accurate representation of complex wave propagation areas.

Unfortunately, they are highly compute intensive. Advances in High Performance Computing technologies resulted in renewed attention from the seismic community to these techniques.

The recent development in GPU technologies with unified architecture and general-purpose languages coupled with the high and rapidly increasing performance throughput of these components made General Purpose Processing on Graphics Processing Units (GPGPU) an attractive solution to speed up diverse applications [START_REF] Owens | GPU Computing[END_REF].

In this work, we will discuss how seismic modeling and RTM can take advantage of GPUs to achieve massive computation capacity. The remainder of this paper is organized as follows: in Section 2 we begin by describing the context, the sequential algorithm and the CPU implementation of the modeling and RTM applications. We also introduce the GPU architecture and programming model. The GPU implementations and optimizations are described in Section 3. In Section 4 we provide and analyse performance results. We conclude in Section 5 by validating the numerical results.

BACKGROUND 2.1 Context 2.1.1 Seismic Surveys

Seismic surveys are the first step in oil and gas exploration and production projects. A typical seismic survey consists in generating a seismic wave by sources placed at the surface such as air guns in marine surveys or dynamite in land surveys. The seismic wave which propagates downward in the subsurface is refracted and/or reflected at the interfaces of the geological layers. The refracted and/or reflected wave propagates upward and is recorded by sensors called geophones. Each geophone records time series of events which are called seismic traces or seismograms.

The recorded wave field is used as an initial condition or boundary condition to determine the earth's subsurface structure. The refraction and the reflection of seismic energy at geological interfaces exactly follow the same laws that govern the refraction and reflection of light through prisms.

Seismic Modeling

Numerical seismic modeling aims at simulating seismic wave propagation in a geological midum in order to generate synthetic seismograms that are the seismograms that a set of sensors would record, given an assumed structure of the subsurface. Among the numerous approaches to seismic modeling, direct methods based on approximating the geological model by a numerical mesh are of particular interest. Infact, this approch can give very accurate results. However, a disadvantage of this approach is its high computational demand [START_REF] Carcione | Seismic modeling[END_REF].

Reverse Time Migration

Reverse Time Migration (RTM) is a technique for creating seismic images in areas of complex wave propagation, providing imaging of so called turning and prismatic waves. RTM was introduced in the 1980's [START_REF] Baysal | Reverse Time Migration[END_REF] but despite showing promising imaging capabilities, its high computational cost prevented it from being used in practice, until recent advances in HPC technologies. RTM is based on the simulation of waves propagation: both source and receivers wave fields are propagated respectively forward and back-ward in time. These wave fields are then compared using an imaging condition for corresponding time steps in order to form the subsoil image. Figure 1 (extract from [START_REF] Biondi | 3D SEISMIC IMAGING, ser. Investigations in Geophysics[END_REF]) depicts the RTM backward sweep, for times t = 1.20s (a), 0.75s (b) and 0.30s (c): source wave field (left) and receiver wave field (center) are propagated backward in time. the imaging condition is then applied to form the subsurface image (right).

Figure 1: Reverse Time Migration Backward Sweep.

Governing Equations

The three dimensional acoustic wave equation ( 1) links the pressure field u(x, y, z, t) to the density ρ(x, y, z) and the velocity c(x, y, z) [START_REF] Claerbout | IMAGING THE EARTH'S INTE-RIOR[END_REF].

1 c 2 ρ ∂ 2 u ∂t 2 = ∇.( 1 ρ ∇u) (1)
Using finite difference (FD) methods to solve the wave equation is one way among others to tackle direct methods. The way this equation is derived, among a regularly meshed domain, is described in [START_REF] Etgen | Computational methods for large-scale 3d acoustic finite-difference modeling: A tutorial[END_REF]: we write a cascaded first order spatial difference expression to compute the second time difference of the wave field:

∂ 2 u ∂t 2 = K[ ∂- ∂x ( 1 ρ ∂+u ∂x ) + ∂- ∂y ( 1 ρ ∂+u ∂y ) + ∂- ∂z ( 1 ρ ∂+u ∂z )] (2)
where K = c 2 ρ is the bulk modulus. The ∂ -and ∂ + symbols denote the spatial difference operators that are centered halfway between grid points either forward or backward in the direction of the spatial difference. We use operators with a 3D stencil width of 8. So for example, the first derivative of u with respect to x evaluated at (i + 1 2 )∆x is written

∂+ ∂x u((i + 1 2 )∆x) = 3 X n=0 an[u((i + 1 + n)∆x) -u((i -n)∆x)] (3)
The a n coefficients are the 8th order finite difference operator optimized coefficients [START_REF] Holberg | Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena[END_REF].

When the density is considered to be constant in all the domain, equation ( 1) is simplified to: This approximation is especially done during migration process. The discretization of this equation is done with a second order in time leap-frog scheme and an 8th-order centred difference scheme in space with either Taylor or optimized coefficients [START_REF] Ye | Dispersion-relation-preserving finite difference operators: derivation and application[END_REF], which leads in 2D to:

1 c 2 ∂ 2 u ∂t 2 = ∆u (4)
U n+1 i,j = 2U n i,j -U n-1 i,j + c 2 ∆t 2 » 1 ∆x 2 4 X m=-4 bmU n i+m,j + 1 ∆y 2 4 X m=-4 bmU n i,j+m - (5)
with U n i,j = u(i∆x, j∆y, n∆t).

Modeling and RTM on Homogeneous Processors

We describe in this section our CPU code translation of discretizations of (1) and ( 4). The latter is easily inferred from [START_REF] Claerbout | IMAGING THE EARTH'S INTE-RIOR[END_REF]. At each grid point the stencil is applied to compute the Laplacian value. Thus, to update one grid point, 25 data reads are performed from the wave field's last update. Data are contiguous along X axis. When a domain is too large to fit into the CPU cache size, which is the case almost all the time, data accesses along Y and Z axes become expensive. Cache blocking [START_REF] Kamil | Implicit and explicit optimizations for stencil computations[END_REF], along Y axis is used to increase data reuse and to exploit the memory hierarchy.

No cache strategy is implemented for variable density domains. In this case, we need to loop over all the grid points twice as depicted in the algorithm below. During first sweep, three 3D arrays are filled with the forward first derivatives along each axis. Then, backward first derivatives are computed using these arrays and the density array. The stencil is then twice as large as in the density case. Figure 2 depicts the data dependency along X axis. save seismogram planes in the variable density case.

In practice, since the simulated domain cannot extend infinitely, damping zones are added at the borders of the domain to avoid reflections. In this zones we use Perfectly Matched Layers [START_REF] Cohen | HIGHER-ORDER NUMERICAL METH-ODS FOR TRANSIENT WAVE EQUATIONS, ser. Scientific Computation[END_REF] to simulate non reflecting boundaries. for all domain grid points do 14:

apply imaging condition

The Reverse Time Migration algorithm is listed in Algorithm 2. The imaging condition applied during backward recursion consists in the accumulation of the crosscorrelation between source and receiver wave fields over the time iterations.

GPGPU: Architecture and Programming Model

In this section, we briefly describe the S1070 GPUs architecture and programming model. Even though other models exist, we focus on the Compute Unified Device Architecture (CUDA) that we use for our implementations. We also focus on architectural features that are most relevant to our use. We invite the reader to refer to [START_REF]NVIDIA CUDA PROGRAMMING GUIDE[END_REF] for a more detailed description.

The S1070 blades are composed of 4 T10 graphics processing units (GPU) embedded with 4GB of memory per GPU. Each pair is connected to a host node via a PCIe 2.0 bus, acting as coprocessors. PCIe gen 2.0 connection delivers a theoretical peak throughput of 8 GB/s one way. In practice, we have been able to reach a sustained 5.6 GB/s bandwidth when using DMA transfers. Since 2 GPUs share the same PCIe bus, the data transfer rate is reduced to 3,3 GB/s for each GPU, when performing intensive memcopies between host and GPUs in the same direction. Writing to one GPU while reading from the other seems to give higher throughput (measured average of 4.3 GB/s).The limited PCIe Bandwidth -compared to the 102 GB/s memory bandwidth-and its latency (16 µs), makes it the main bottleneck when tackling small problems in terms of data sizes or problems of low computing intensity.

The T10 GPU can be seen as a set of 30 multiprocessors. Each multiprocessor is composed of 8 streaming processors running in a Single Instruction Multiple Data (SIMD) like way. These processors can execute 3 single precision floating point operations per clock cycle. Our T10 GPUs have a clock rate of 1.44 GHz, providing thus a theoretical peak performance of 30 × 8 × 3 × 1.44 = 1 TFlops per GPU and 4 TFlops per S1070 blade.

The processors inside a multiprocessor have access to 16,384 32-bit registers (divided among processors), and to 16 kB shared memory space that can be seen as a cache memory with very low latency (4 clock cycles per read/write if no conflict). All the GPU's processors have read/write access to the off-ship global memory (nearly 4GB) but with a higher latency varying from 400 to 600 clock cycles.

To harness the GPU computing power and make it available for non graphics programmers, NVIDIA introduced CUDA. CUDA is based on a Single Program Multiple Data (SPMD)1 paradigm: the programmer writes a portion of code (the kernel ) to be executed by several threads in parallel on different data. CUDA defines a thread hierarchy depicted in Figure 4 in order to organize threads in a geometric topology. Thus threads are grouped into 1D, 2D or 3D thread blocks. These blocks are arranged into 1D or 2D grids. This topology matches the thread organization to the GPU physical structure. In fact, threads of the same block are run on the same multiprocessor, making them able to have access to the multiprocessor's shared resources (same shared memory space, texture and constant memory cache) and to coordinate their activities using a barrier synchronization function. Each thread is then identified according to its thread ID and the block ID of the block it belongs to. Threads of consecutive thread IDs are grouped into warps of 32 threads.

GPU IMPLEMENTATIONS

To measure and to understand GPU code performance, CUDA introduces new terms and metrics. The concept of occupancy is of major importance when designing CUDA kernels. Occupancy is defined as the ratio of active warps per multiprocessor to the maximum number of active warps (32 for the T10 GPU). The number of active warps is defined by the availability of shared resources inside the multiprocessor.

We start by considering modeling the wave propagation in a constant density domain. This case has been the first example we tackled due to its simplicity and its high regularity and parallelism. We expose the learning curve we followed in order to improve the application performance. Then we describe the variable density modeling and the RTM implementation and performance results.

Constant Density Modeling

The equations and computations are described in 2.3. We have studied different steps to optimize this application on both the CUDA kernel and the host-GPU communication sides.

Host-GPU Communication

As described in 2.3, to update pressure field at a given iteration t n+1 , pressure field updates at iterations t n-1 and t n are required. Data in the GPU global memory are persistent across different kernel launches. Time evolution of the wave field is then performed in the GPU memory by swapping t n and t n+1 arrays. Only the ghost nodes need then to be exchanged between host and GPU at each time step. Exchanging only ghost nodes instead of the whole domain, although requiring more programming effort, largely reduces the amount of data exchanged and thus computing time. For example, transferring a whole 3D 528 × 254 × 1067 domain to/from GPU takes approximately 0.1s, that is equivalent to the time needed for a single time iteration.

CUDA Kernel Implementation

Our first approach has been to use global memory and to dedicate a CUDA thread for each grid point in the domain. Although it reached a 66% occupancy on N80 GPUs, this direct implementation was of poor performance essentially because it was memory bounded: each thread makes 25 global memory read accesses and 1 global memory write access to the wave field arrays.

To limit memory accesses, we used shared memory: data are first copied to shared memory (texture fetches in 2D, global memory in 3D), then read by threads of the same block. This classical technique reduces global memory accesses for the same data, but increases shared memory use. Since shared memory is limited to 16KB for a multiprocessor, using 16 × 4 × 4 thread blocks requires at least 9kB of shared memory for each thread block. That means that only one thread block of 256 threads can run at a given time on one multiprocessor instead of running 768 possible threads on devices of compute capabilities2 under 1.2 (33% occupancy) and 1024 threads for other devices (25% occupancy).

To limit shared memory use, and thus increase occupancy, we adopted a novel strategy: instead of dedicating a thread to each grid point, we used a sliding window algorithm, sliding over planes in the z direction. We used shared memory to store the (x, y, z = z current ) plane to be updated, so that all the threads can compute Laplacian value in the x and y directions. For the z direction, each thread loads into float4 registers the 4 pressure values under and over the current plane. At each loop iteration, corresponding to a shift along z direction, wave field values are shifted as shown in Figure 5. Thus, only the (x, y, z = z current+4 ) plane is read from global memory. This kernel (Basic) showed to be the most efficient. Adding padding to ensure global memory coalescing added 15% speed up on GPUs with compute capability under 1.2.

We used the same approach to implement the wave propagation in the damping zones (Damping). This 1 we sum up the kernels resources use and occupancy.

To solve the problem over a whole subdomain, we can use two different strategies. Corresponding times are reported in Figure 6 for one time iteration on a domain of size 528 × 254 × 1067 including 5 damping zones. We can either apply the damping kernel to all the domain (2) or apply it on the borders (or not, depending on the domain position) and apply next the basic kernel inside the domain (1,3 and 4). In (1) we limited the registers use to 21 using a compiler option. This increases occupancy but induces use of local memory which reduces the kernel's performance. In (3) Damping kernel was used in the damping zones while in (4) we used WF kernel and PML kernel. The latter strategy is the most efficient one.

Variable Density Modeling

The communication scheme remains the same, except that ghost node width is doubled. The kernel radically changes. In fact, as described in 2.3, the stencil is twice as large as in

Reverse Time Migration

Our current implementation of the RTM propagates the source wave field on the GPU during the forward sweep. Then, during the backward sweep, the receiver wave field propagation is accelerated using GPU while the host CPU takes in charge the source wave field retro-propagation and the imaging condition. Thus, Reverse Time Migration induces more CPU-GPU communications especially during the backward sweep where the whole receiver wave field has to be sent back to CPU to apply the imaging condition. With CUDA, we can overlap communications with computations using CUDA asynchronous communications and kernel launches. The domain to be updated on the GPU is decomposed into 4 streams. Each stream consists in a data transfer from host to GPU, a kernel launch, and a data transfer from GPU to CPU. Thus while executing a stream's kernel, the data corresponding to the next stream are transferred.

PERFORMANCE ANALYSIS

In this section we report and analyse some performance results obtained using GPUs and compare them with the CPU implementation. We focus on the scalability of each solution. The reference time is the time to process the entire domain without using any subdomain decomposition. Thus, the CPU reference time is given by running a sequential implementation on a single CPU core; and the GPU reference time is given by using one GPU. This implies that a GPU is considered as a processing unit, even though it comprises several multiprocessors. We also focus on the ratio of the execution time of the CPU-based implementation to that of the GPU-based one depending on the dataset size.

We consider a 3D test case (hereafter referred to as

GPU Cluster Testbed

Our GPU cluster testbed is composed of 10 Xeon bi-socket quad-core nodes coupled with 5 NVIDIA TESLA S1070 servers. Each node is connected via one PCIe gen2 bus to the TESLA server. The TESLA server is composed of 4 T10 GPUs. Each pair is connected via a switch to a PCIe 2.0 connection. Thus each node has access to 2 GPUs via the same bus. Table 2 reports the detailed technical specifications of the system used in our experiments.

CPU Scalability

We compare the scalability of the CPU application, using cores of the same processor socket, and using different processors, for the 3DModel test case. Results are summarized in Table 3. They show that the memory access is the bottleneck when using Harpertown processors.

Constant Density Modeling

We study the speed up obtained with the implementation described below for the 3DModel. All the GPU computing times indicated hereafter, unless explicitly stated, take into account the kernel execution time as well as the CPU-GPU communication needed to initiate and finish the kernel.

Results reported in Table 4 show that the CPU/GPU time ratio decreases with the number of subdomains increasing because the computing time decreases and the communication time becomes more predominant In Figure 7 we report the averaged computing times in seconds for one time iteration on domains of increasing sizes and the corresponding CPU/GPU time ratio for a single domain. This figure shows that the ratio increases with the size of the domain increasing. This is due to the fact that for a domain of size n 3 , CPU-GPU communication cost grows as Θ(n 2 ) while computing time grows as Θ(n 3 ). Yet, the CPU/GPU time ratio can be written as

R = t CP U [θ(n 3 )] tcomm[θ(n 2 )]+t GP U [θ(n 3 )] .

Variable Density Modeling

The same statements can be made about variable density modeling as shown in Table 5 and Figure 8. In Table 6 reported data transfer times reinforce that the overhead due to MPI communications increases with the number of subdomains. When using 8 GPUs, 40% of one iteration time is spent in communications which significantly reduces the speed-up over the CPU implementation.

We evaluate the difference between a configuration with 2 GPUs accessible on the same node via the same PCI-X 2.0 connection and 2 GPUs hosted by two distinct nodes. As stated in 2.4 the data transfer rate is reduced when accessing 2 GPUs through the same PCI-X connection. Communication between the two MPI processes is impacted as well: in one case shared memory is used, in the other case high performance infiniband network. Thus communication times are expected to be quite different in these two configurations. Results are reported in Table 7 and show that even though communication times are significantly different, they remain quite low compared to the computation time. Then, the overall simulation time is slightly impacted. The host-GPU, the host memory and the network bandwidths are not a bottleneck in this particular case.

Reverse Time Migration

We report in Figure 9 times and speedup for the Reverse Time Migration. The RTM implementation involves more host-GPU communications than the modeling. This reduces the obtained CPU/GPU time ratio very significantly.

CONCLUSION

We ported a seismic modeling application and Reverse Time Migration on a GPU cluster. The difference between synthetic seismograms produced using the CPU and the GPU implementations for a realistic data set used in production process is proportional to the wave field amplitude but remains very low (0.1% in terms of percentage error). This validates our implementation. We compared our GPU implementation to the original CPU version on several test cases and obtained a performance increase of 10x for RTM and up to 30x for modeling. Results may still be improved by recovering MPI communications by GPU computations. This shows that GPGPU solutions are worth being used in a large-scale industrial context, especially when double precision is not required.

Figure 2 :

 2 Figure 2: Data Dependency along X Axis.

Figure 3 :Algorithm 2 6 :

 326 Figure 3: 2D Domain Decomposition and Ghost Node Exchanges.

Figure 4 :

 4 Figure 4: CUDA Programming Model: Grids of Thread Blocks.

Figure 5 :

 5 Figure 5: 3D Modeling Kernel

Figure 6 :

 6 Figure 6: 3D Constant Density Modeling Strategies

  ρ = ct) 3.90 2.60 1.45 N Sockets (ρ = ct) 3.90 1.59 0.94 1 Socket (ρ(x, y, z)) 9.23 7.55 4.96 N Sockets (ρ(x, y, z)) 9.23 4.54 1.59 3DModel) of grid dimensions 521 × 254 × 1067 with ∆x = 12.5, ∆y = 12.5 and ∆z = 10 in meters. Thus this model covers a surface of 20.7km 2 and goes 10.7km in deep.

Figure 7 :

 7 Figure 7: 3D Constant Density Modeling Times.

Figure 8 :

 8 Figure 8: 3D Variable Density Modeling Times.

Figure 9 :

 9 Figure 9: Reverse Time Migration Times in Seconds for one Iteration on a 288 × 118 × 338 Test Case.

Table 1 :

 1 Modeling CUDA Kernels, Resources and Occupancy

	Kernel	Shared Memory Registers Occupancy
	Basic	2400	20	75%
	Damping	2432	28	50%
	WF	2432	20	75%
	PML	2720	17	75%
	induces more complex computations, leading to more
	memory accesses and registers needs. This makes the
	occupancy limited by registers used by each thread.
	We use thread blocks of size 256(16 × 16). To achieve
	a 100% occupancy, each thread must use less than 16
	(16, 384/256 × 4) registers. Using up to 21 registers
	achieves a 75% occupancy. The full implementation of
	the damping kernel uses 32 registers leading to a 50%
	occupancy. To limit register use, we split the kernel
	into 2 separate ones. The first kernel (WF), updates
	the wave field array. The second one (PML) updates
	the damping function array. In Table

Table 2 :

 2 Technical Specifications of the System Used in our Experiments.

	Blade	CPU Node NVIDIA S1070
	Number	10	5
	Processor	Xeon 5405	T10
	Sockets × cores	2 × 4	4 × 240
	Clock Frequency (GHz)	2.00	1.44
	Memory per Blade (Gbytes)	16	16 (4 × 4)
	Cache/Shared Memory	4 × 6M B	30 × 16KB
	Peak Performance (GFlops)	64	1036

Table 3 :

 3 Constant and Variable Density Modeling Averaged Times in Seconds on 3DModel.

Table 4 :

 4 Constant Density Modeling Averaged Time in Seconds for one Iteration on 3DModel

	Number of subdomains GPU time CPU time Ratio
	1	0.146	3.9	26.71
	2	0.083	1.59	19.15
	4	0.046	0.94	20.43
	8	0.034	0.47	15.16
	(both CPU-GPU and MPI communications). Yet, only
	computing time is reduced when using GPUs.

Table 5 :

 5 Variable Density Modeling Averaged Time in Seconds for one Iteration on 3DModel

	Number of subdomains GPU time CPU time Ratio
	1	0.342	9.90	28.90
	2	0.183	4.75	25.96
	4	0.11	2.45	22.34
	8	0.073	1.22	16.66

Table 6 :

 6 Averaged Time in Seconds and Speedup (in Parenthesis) for one Iteration on 3DModel

	Nb GPU.	2	4	8
	Data transfer	0.0045	0.0133	0.029
	GPU kernel

0.1702 (2.00) 0.0847(4.02) 0.0441 (7.72) Total time 0.1831 (1.87) 0.1097(3.12) 0.0735 (4.66)

Table 7 :

 7 Averaged Time in Seconds for one Iteration on 3DModel

	Configuration Shared Memory InfiniBand
	GPU kernel	0.1708	0.1702
	Other comp.	0.0163	0.0129

NVIDIA introduced the SIMT acronym standing for Single Instruction Multiple Thread.

The compute capability of a device indicates its core architecture and the features it supports.

ACKNOWLEDGEMENTS

The authors would like to thank Bertrand Denel from Total and Alexandre Louvet from SGI for their extensive contribution in the implementation and optimization of the considered CPU version of the RTM application. Also, the authors acknowledge Georges-Emmanuel Moulard and Romain Dolbeau from CAPS for their help on the GPU kernel optimizations.


Olivier Lézoray

Christophe Charrier

Abderrahim Elmoataz

Learning complete lattices for manifold mathematical morphology

Keywords:

The extension of lattice based operators to manifolds is still a challenging theme in mathematical morphology. In this paper, we propose to explicitly construct complete lattices and replace each element of a manifold by its rank suitable for classical morphological processing. Manifold learning is considered as the basis for the construction of a complete lattice.

Introduction

Mathematical Morphology (MM) is a nonlinear approach to image processing that relies on a fundamental structure, the complete lattice L [START_REF] Ronse | Why mathematical morphology needs complete lattices[END_REF] (a nonempty set equipped with an ordering relation). With the complete lattice theory, it is possible to define morphological operators for any type of data once a proper ordering is established [START_REF] Aptoula | A comparative study on multivariate mathematical morphology[END_REF]. If Mathematical Morphology is well defined for binary and gray scale images, there exist no general extension that permits to perform basic operations on manifolds (multivariate data) since there is no natural ordering on vectors. In this paper, we propose to use a rank transformation with Manifold Learning for complete lattice creation.

Rank transform

A manifold is considered as a mapping f : Ω → R p where p is the dimensionality of the vectors living on the manifold. One way to define an ordering relation between vectors is to use a transform [START_REF] Goutsias | Morphological operators for image sequences[END_REF] h : R p → R q , with q ≪ p followed by a conditional ordering on each dimension of R q . Then, ∀(

x i , x j ) ∈ R p × R p , x i ≤ x j ⇔ h(x i ) ≤ h(x j ).
From this, it is easy to show the following equivalence (complete lattice on R p )⇔(bijective application h : R p → R q )⇔(rank transform on R p ) [START_REF] Garcia | Multivariate mathematical morphology and bayesian classifier application to colour and medical images[END_REF][START_REF] Lezoray | Mathematical morphology in any color space[END_REF]. This implies that, to induce a complete lattice, the vectors' values are not important but only their position in the lattice: this corresponds to a rank transform defined by the mapping h : R p → N. Ordering comparisons involved in morphological operations are performed directly on ranks and one obtains a common framework valid for data of arbitrary dimensions.

Mathematical Morphology on Graphs

A graph is a couple G = (V, E) where V is a finite set of vertices and E is a set of edges included in a subset of

V × V . Two vertices are adjacent if the edge (u, v) ∈ E. u ∼ v denotes the set of vertices u connected to the vertex v via the edges (u, v) ∈ E. A graph is weighted if it is associated with a weight function k : E → R + satisfying k(u, v) > 0 if (u, v) ∈ E, and k(u, v) = 0 if (u, v) /
∈ E. We now introduce several definitions. The neighborhood set of vertices N (G, v) of a vertex v is defined as:

N (G, v) = {u ∈ V : (u, v) ∈ E} ∪ {v}.
The set of edges A(G, v) connecting any vertices in N (G, v) is defined as:

A(G, v) = {(u, w) ∈ E : u ∈ N (G, v), w ∈ N (G, v)}. A structuring element S(G, v) at a given vertex v is a sub- graph of G defined as: S(G, v) = (N (G, v), A(G, v)).
With these definitions, the erosion ǫ of a function f on a graph G at a vertex v is defined by:

ǫ(G, f, v) = {f (u) : h(f (u)) = ∧h(f (w)), w ∈ N (G, v)}.
If we compare this definition with the usual definition of an erosion, the structuring element is directly expressed by the graph topology and the lattice is defined by the use of the rank transform h. Similar definitions can be found in [START_REF] Heijmans | Graph morphology[END_REF]. With this definition, the graph topology never changes, but only vectors associated to vertices. We can reformulate the erosion as a contracting erosion that modifies the graph topology. To that aim, we define the erosion at a vertex v in terms of vertex preservation:

ǫ V (G, f, v) = {u : h(f (u)) = ∧h(f (w)), w ∈ N (G, v)}.
Then, one can define the vertex erosion ǫ V (G, f ) and the edge erosion ǫ E (G, f ) of a graph as:

ǫ V (G, f ) = V ∩ {ǫ V (G, f, v), ∀v ∈ V } and ǫ E (G, f ) = {(u, v) ∈ E, u ∈ ǫ V (G, f ), v ∈ ǫ V (G, f )}.

Finally a contracting erosion ǫ

C (G, f ) is an operation that produces a new graph (ǫ V (G, f ), ǫ E (G, f )) that is a sub- graph of G.
Similar definitions apply for dilation.

In the previous definitions of Mathematical Morphology on Graphs, the complete lattice is assumed to be known and expressed by the rank transform h. However, the construction of such a rank transform is a difficult problem. To perform this, we consider manifold learning methods that enable to perform dimensionality reduction. This is equivalent to a rank transform when the dimension of the projected is space is equal to one. Graph-based methods have recently emerged as powerful tools for nonlinear dimensionality reduction. Among the existing methods, we consider Laplacian Eigenmaps [START_REF] Belkin | Laplacian eigenmaps for dimensionality reduction and data representation[END_REF]. Let . To have a parameterless Gaussian kernel, σ is estimated by σ = max v∈V,u∼v f (v)f (u) . Then, one seeks to minimize

{x 1 , x 2 , • • • , x n } ∈ R p
1 2 ij W ij y i -y j 2 = T r(Y T ∆Y) with ∆ = D-W
that represents the un-normalized Laplacian (D is the degree matrix). The solution of the previous minimization problem can be found by solving ∆y = λDy. The eigenvectors of this equation corresponding to the smallest non zero eigenvalues form the manifold representation. To perform a complete lattice learning with manifold learning, a vertex is associated to each input vector data and a neighborhood graph is constructed. Then, we consider only the first non-zero eigenvector of the obtained Manifold representation and re-arrange the initial vectors increasingly according to their value in the first non-zero eigenvector: this defines the rank transform. Manifold learning, although being attractive, is a time consuming step for the complete lattice construction when the amount of data is large: complexity is O(n 3 ). To overcome this, several strategies can be considered that rely on the same idea: to reduce the size of the data on which the complete lattice construction is performed. We propose two strategies in the sequel.

Data Quantization

Local lattice learning

A second strategy can consist in performing locally the complete lattice creation. The rank transform h is defined on sub-graphs of the initial graph: the structuring elements S(G, v). This comes to define the rank transform only on a reduced set of vertices: N (G, v).The manifold learning is therefore applied on the data set {f (u), u ∈ N (G, v)}.

With this strategy, the complete lattice is not available for the whole manifold but only one sub-manifold defined, for a local processing, by S(G, v).

Results

In this Section, we illustrate the two above-mentioned strategies for complete lattice learning. First, we consider Vector Quantization with Manifold Learning. Figure 1 illustrates this principle on a color image (f : Z 2 → R 3 ) represented by a (k 2 -1)-adjacency graph that means using a k × k square structuring element. The image is quantized into 512 colors and the obtained codebook is re-ordered by Manifold Learning to construct the complete lattice of the 512 colors. A rank image is created by assigning to each pixel its rank on the complete lattice of the codebook. Then, morphological operations are performed on the rank image and the final color images are obtained by reconstruction with the codebook defining the lattice. As it can be seen in Figure 1, the induced morphological operations enable an accurate processing of the image. To show that our formalism is easily applicable to any multivariate image, Figure 2 presents a morphological processing of a multispectral image (f : Z 2 → R 20 ) with a 1024 codebook. The segmentation of the image is performed with a watershed on the morphological gradient of an Alternate Sequential Filter of the rank image. Second, we illustrate the use of local manifold learning for morphological processing. Figure 3 presents such a processing on an image manifold of 99 images (f : Ω → R 16×16 ). In this case, structuring elements are defined by adjacency rela-tionships in the graph associated to the data: a 5 nearest neighbor graph (5-nn) is considered with an Euclidean distance. Such a processing is interesting for manifold simplification (flat zones are created). Moreover, contracting morphological operations results are also provided to illustrate the abilities of the method to reduce a dataset to its main elements. Finally, Figure 4 presents the results of two successive contracting openings on the iris data set. Applying a k-means (with k = 3) on both these data sets (original and after morphological processing) gives 88.7% and 100% of classification rate. This illustrates the interest of contracting morphological operations for data mining purposes.

Conclusion

In this paper, we have considered the general case of morphological processing of manifolds

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  be a set of n initial vectors. Manifold learning aims at searching for a new representation {y 1 , y 2 , • • • , y n } with y i ∈ R n . From a neighborhood graph G built from the initial data set, an adjacency matrix W is considered and weighted by a Gaussian kernel W ij = e -||xi-xj || 2 σ 2

A

  first strategy can consist in reducing the input data size by Vector Quantization (VQ). Given an initial data set of size n, VQ: R p → R p is applied to construct a codebook C : N → R p and an encoder I : R p → N. An index h : Ω → N can be deduced from D and I by applying h(x) = I(f (x)) to each vector f (x) = x of the original data set. The initial data set can be reconstructed with loss from the index and the codebook by C(h(x)): the obtained data set is an approximation of the initial data set with only 2 k elements. The codebook being of reduced size, one can apply manifold learning on the complete graph associated to the codebook. This enables to construct the complete lattice (the ordering of the codebook) and to define the rank transform (obtained with the function h).

  f : Z 2 → R 3 C • h : Z 2 → R 3Rank ImageComplete Lattice of the codebookh = I • f : Z 2 → N from topleft to right down with I : R 3 → N C : N → R 3 Dilation (11 × 11) Erosion (11 × 11) Closing (11 × 11) Opening (11 × 11) Opening by reconstruction Alternate Sequential Filter (11 × 11) (4 iterations with a square) Gradient (3 × 3) Watershed of the gradient with gradient minima as markers.

Figure 1 :Figure 2 :

 12 Figure 1: Processing examples with a rank image obtained from Manifold Learning with Vector Quantization.

Figure 3 :

 3 Figure 3: Processing examples with a local complete lattice learning. A 5-nn graph is considered.