On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning

This paper studies and analyzes a preconditioned Krylov solver for Helmholtz problems that are formulated with absorbing boundary layers based on complex coordinate stretching. The preconditioner problem is a Helmholtz problem where not only the coordinates in the absorbing layer have an imaginary part, but also the coordinates in the interior region. This results into a preconditioner problem that is invertible with a multigrid cycle. We give a numerical analysis based on the eigenvalues and evaluate the performance with several numerical experiments. The method is an alternative to the complex shifted Laplacian and it gives a comparable performance for the studied model problems

Mapping development and health effects of cooking with solid fuels in low-income and middle-income countries, 2000-18 : a geospatial modelling study

Background More than 3 billion people do not have access to clean energy and primarily use solid fuels to cook. Use of solid fuels generates household air pollution, which was associated with more than 2 million deaths in 2019. Although local patterns in cooking vary systematically, subnational trends in use of solid fuels have yet to be comprehensively analysed. We estimated the prevalence of solid-fuel use with high spatial resolution to explore subnational inequalities, assess local progress, and assess the effects on health in low-income and middle-income countries (LMICs) without universal access to clean fuels.Methods We did a geospatial modelling study to map the prevalence of solid-fuel use for cooking at a 5 km x 5 km resolution in 98 LMICs based on 2.1 million household observations of the primary cooking fuel used from 663 population-based household surveys over the years 2000 to 2018. We use observed temporal patterns to forecast household air pollution in 2030 and to assess the probability of attaining the Sustainable Development Goal (SDG) target indicator for clean cooking. We aligned our estimates of household air pollution to geospatial estimates of ambient air pollution to establish the risk transition occurring in LMICs. Finally, we quantified the effect of residual primary solid-fuel use for cooking on child health by doing a counterfactual risk assessment to estimate the proportion of deaths from lower respiratory tract infections in children younger than 5 years that could be associated with household air pollution.Findings Although primary reliance on solid-fuel use for cooking has declined globally, it remains widespread. 593 million people live in districts where the prevalence of solid-fuel use for cooking exceeds 95%. 66% of people in LMICs live in districts that are not on track to meet the SDG target for universal access to clean energy by 2030. Household air pollution continues to be a major contributor to particulate exposure in LMICs, and rising ambient air pollution is undermining potential gains from reductions in the prevalence of solid-fuel use for cooking in many countries. We estimated that, in 2018, 205000 (95% uncertainty interval 147000-257000) children younger than 5 years died from lower respiratory tract infections that could be attributed to household air pollution.Interpretation Efforts to accelerate the adoption of clean cooking fuels need to be substantially increased and recalibrated to account for subnational inequalities, because there are substantial opportunities to improve air quality and avert child mortality associated with household air pollution. Copyright (C) 2022 The Author(s). Published by Elsevier Ltd.Peer reviewe

Mapping geographical inequalities in access to drinking water and sanitation facilities in low-income and middle-income countries, 2000-17

Background: Universal access to safe drinking water and sanitation facilities is an essential human right, recognised in the Sustainable Development Goals as crucial for preventing disease and improving human wellbeing. Comprehensive, high-resolution estimates are important to inform progress towards achieving this goal. We aimed to produce high-resolution geospatial estimates of access to drinking water and sanitation facilities. Methods: We used a Bayesian geostatistical model and data from 600 sources across more than 88 low-income and middle-income countries (LMICs) to estimate access to drinking water and sanitation facilities on continuous continent-wide surfaces from 2000 to 2017, and aggregated results to policy-relevant administrative units. We estimated mutually exclusive and collectively exhaustive subcategories of facilities for drinking water (piped water on or off premises, other improved facilities, unimproved, and surface water) and sanitation facilities (septic or sewer sanitation, other improved, unimproved, and open defecation) with use of ordinal regression. We also estimated the number of diarrhoeal deaths in children younger than 5 years attributed to unsafe facilities and estimated deaths that were averted by increased access to safe facilities in 2017, and analysed geographical inequality in access within LMICs. Findings: Across LMICs, access to both piped water and improved water overall increased between 2000 and 2017, with progress varying spatially. For piped water, the safest water facility type, access increased from 40·0% (95% uncertainty interval [UI] 39·4–40·7) to 50·3% (50·0–50·5), but was lowest in sub-Saharan Africa, where access to piped water was mostly concentrated in urban centres. Access to both sewer or septic sanitation and improved sanitation overall also increased across all LMICs during the study period. For sewer or septic sanitation, access was 46·3% (95% UI 46·1–46·5) in 2017, compared with 28·7% (28·5–29·0) in 2000. Although some units improved access to the safest drinking water or sanitation facilities since 2000, a large absolute number of people continued to not have access in several units with high access to such facilities (>80%) in 2017. More than 253 000 people did not have access to sewer or septic sanitation facilities in the city of Harare, Zimbabwe, despite 88·6% (95% UI 87·2–89·7) access overall. Many units were able to transition from the least safe facilities in 2000 to safe facilities by 2017; for units in which populations primarily practised open defecation in 2000, 686 (95% UI 664–711) of the 1830 (1797–1863) units transitioned to the use of improved sanitation. Geographical disparities in access to improved water across units decreased in 76·1% (95% UI 71·6–80·7) of countries from 2000 to 2017, and in 53·9% (50·6–59·6) of countries for access to improved sanitation, but remained evident subnationally in most countries in 2017. Interpretation: Our estimates, combined with geospatial trends in diarrhoeal burden, identify where efforts to increase access to safe drinking water and sanitation facilities are most needed. By highlighting areas with successful approaches or in need of targeted interventions, our estimates can enable precision public health to effectively progress towards universal access to safe water and sanitation

Global, regional, and national progress towards Sustainable Development Goal 3.2 for neonatal and child health: all-cause and cause-specific mortality findings from the Global Burden of Disease Study 2019

Background Sustainable Development Goal 3.2 has targeted elimination of preventable child mortality, reduction of neonatal death to less than 12 per 1000 livebirths, and reduction of death of children younger than 5 years to less than 25 per 1000 livebirths, for each country by 2030. To understand current rates, recent trends, and potential trajectories of child mortality for the next decade, we present the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019 findings for all-cause mortality and cause-specific mortality in children younger than 5 years of age, with multiple scenarios for child mortality in 2030 that include the consideration of potential effects of COVID-19, and a novel framework for quantifying optimal child survival. Methods We completed all-cause mortality and cause-specific mortality analyses from 204 countries and territories for detailed age groups separately, with aggregated mortality probabilities per 1000 livebirths computed for neonatal mortality rate (NMR) and under-5 mortality rate (USMR). Scenarios for 2030 represent different potential trajectories, notably including potential effects of the COVID-19 pandemic and the potential impact of improvements preferentially targeting neonatal survival. Optimal child survival metrics were developed by age, sex, and cause of death across all GBD location-years. The first metric is a global optimum and is based on the lowest observed mortality, and the second is a survival potential frontier that is based on stochastic frontier analysis of observed mortality and Healthcare Access and Quality Index. Findings Global U5MR decreased from 71.2 deaths per 1000 livebirths (95% uncertainty interval WI] 68.3-74-0) in 2000 to 37.1 (33.2-41.7) in 2019 while global NMR correspondingly declined more slowly from 28.0 deaths per 1000 live births (26.8-29-5) in 2000 to 17.9 (16.3-19-8) in 2019. In 2019,136 (67%) of 204 countries had a USMR at or below the SDG 3.2 threshold and 133 (65%) had an NMR at or below the SDG 3.2 threshold, and the reference scenario suggests that by 2030,154 (75%) of all countries could meet the U5MR targets, and 139 (68%) could meet the NMR targets. Deaths of children younger than 5 years totalled 9.65 million (95% UI 9.05-10.30) in 2000 and 5.05 million (4.27-6.02) in 2019, with the neonatal fraction of these deaths increasing from 39% (3.76 million 95% UI 3.53-4.021) in 2000 to 48% (2.42 million; 2.06-2.86) in 2019. NMR and U5MR were generally higher in males than in females, although there was no statistically significant difference at the global level. Neonatal disorders remained the leading cause of death in children younger than 5 years in 2019, followed by lower respiratory infections, diarrhoeal diseases, congenital birth defects, and malaria. The global optimum analysis suggests NMR could be reduced to as low as 0.80 (95% UI 0.71-0.86) deaths per 1000 livebirths and U5MR to 1.44 (95% UI 1-27-1.58) deaths per 1000 livebirths, and in 2019, there were as many as 1.87 million (95% UI 1-35-2.58; 37% 95% UI 32-43]) of 5.05 million more deaths of children younger than 5 years than the survival potential frontier. Interpretation Global child mortality declined by almost half between 2000 and 2019, but progress remains slower in neonates and 65 (32%) of 204 countries, mostly in sub-Saharan Africa and south Asia, are not on track to meet either SDG 3.2 target by 2030. Focused improvements in perinatal and newborn care, continued and expanded delivery of essential interventions such as vaccination and infection prevention, an enhanced focus on equity, continued focus on poverty reduction and education, and investment in strengthening health systems across the development spectrum have the potential to substantially improve USMR. Given the widespread effects of COVID-19, considerable effort will be required to maintain and accelerate progress. Copyright (C) 2021 The Author(s). Published by Elsevier Ltd

Semi Analytic Solution of Hodgkin-Huxley Model by Homotopy Perturbation Method

Hodgkin-Huxley model is a system of four non-linear coupleddifferential equations which describes and explains the threshold and actionpotential by a stimulus arising in a single neuron. The solution andanalysis of Hodgkin-Huxley equations is a formidable task because of thecoupling between non-linear differential equations, lots of unknowns andtheir dependence on many physical parameters. Although this model hasbeen solved by numerical methods, finding an analytic solution is interestingdue to the challenges that the continuum model offers. In this paper,first order semi analytic solution of this model, in space-clamped situation,is derived by Homotopy Perturbation Method. We applied this techniquein piece wise manner due to the strong and complex coupling betweenthe variables in the model. Without this modification, finding an accurateanalytic solution is impossible for this neural model. Results showthat computed analytic solution has excellent agreement with higher ordernumerical solution. Robustness of the computed analytic solution in differentphysical scenarios is examined. Further, this analytic solution candescribe many key properties such as the threshold potential, the actionpotential and the refractory period. MATLAB software is used to simulatethe solution.AMS (MOS) Subject Classification Codes: 34E10 ; 34A25 ; 34A34 ; 34A45 ; 92C20KeyWords: Nonlinear coupled system of ordinary differential equations, Bio-neural model, Hodgkin-Huxley Equations, Semi analytic solution, Homotopy Perturbation Method

A preconditioned iterative solver for the scattering solutions of the Schrodinger equation

The Schr\"odinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schr\"odinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjuction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasbility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.Comment: Accepted in Communications in Computational Physic

Multigrid preconditioners based on polynomial smoothers for the Helmholtz equation with absorbing layers

Session 1International audienc

Global, regional, and national progress towards Sustainable Development Goal 3.2 for neonatal and child health : all-cause and cause-specific mortality findings from the Global Burden of Disease Study 2019

Background Sustainable Development Goal 3.2 has targeted elimination of preventable child mortality, reduction of neonatal death to less than 12 per 1000 livebirths, and reduction of death of children younger than 5 years to less than 25 per 1000 livebirths, for each country by 2030. To understand current rates, recent trends, and potential trajectories of child mortality for the next decade, we present the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019 findings for all-cause mortality and cause-specific mortality in children younger than 5 years of age, with multiple scenarios for child mortality in 2030 that include the consideration of potential effects of COVID-19, and a novel framework for quantifying optimal child survival. Methods We completed all-cause mortality and cause-specific mortality analyses from 204 countries and territories for detailed age groups separately, with aggregated mortality probabilities per 1000 livebirths computed for neonatal mortality rate (NMR) and under-5 mortality rate (USMR). Scenarios for 2030 represent different potential trajectories, notably including potential effects of the COVID-19 pandemic and the potential impact of improvements preferentially targeting neonatal survival. Optimal child survival metrics were developed by age, sex, and cause of death across all GBD location-years. The first metric is a global optimum and is based on the lowest observed mortality, and the second is a survival potential frontier that is based on stochastic frontier analysis of observed mortality and Healthcare Access and Quality Index. Findings Global U5MR decreased from 71.2 deaths per 1000 livebirths (95% uncertainty interval WI] 68.3-74-0) in 2000 to 37.1 (33.2-41.7) in 2019 while global NMR correspondingly declined more slowly from 28.0 deaths per 1000 live births (26.8-29-5) in 2000 to 17.9 (16.3-19-8) in 2019. In 2019,136 (67%) of 204 countries had a USMR at or below the SDG 3.2 threshold and 133 (65%) had an NMR at or below the SDG 3.2 threshold, and the reference scenario suggests that by 2030,154 (75%) of all countries could meet the U5MR targets, and 139 (68%) could meet the NMR targets. Deaths of children younger than 5 years totalled 9.65 million (95% UI 9.05-10.30) in 2000 and 5.05 million (4.27-6.02) in 2019, with the neonatal fraction of these deaths increasing from 39% (3.76 million [95% UI 3.53-4.021) in 2000 to 48% (2.42 million; 2.06-2.86) in 2019. NMR and U5MR were generally higher in males than in females, although there was no statistically significant difference at the global level. Neonatal disorders remained the leading cause of death in children younger than 5 years in 2019, followed by lower respiratory infections, diarrhoeal diseases, congenital birth defects, and malaria. The global optimum analysis suggests NMR could be reduced to as low as 0.80 (95% UI 0.71-0.86) deaths per 1000 livebirths and U5MR to 1.44 (95% UI 1-27-1.58) deaths per 1000 livebirths, and in 2019, there were as many as 1.87 million (95% UI 1-35-2.58; 37% [95% UI 32-43]) of 5.05 million more deaths of children younger than 5 years than the survival potential frontier. Interpretation Global child mortality declined by almost half between 2000 and 2019, but progress remains slower in neonates and 65 (32%) of 204 countries, mostly in sub-Saharan Africa and south Asia, are not on track to meet either SDG 3.2 target by 2030. Focused improvements in perinatal and newborn care, continued and expanded delivery of essential interventions such as vaccination and infection prevention, an enhanced focus on equity, continued focus on poverty reduction and education, and investment in strengthening health systems across the development spectrum have the potential to substantially improve USMR. Given the widespread effects of COVID-19, considerable effort will be required to maintain and accelerate progress. Copyright (C) 2021 The Author(s). Published by Elsevier Ltd.Peer reviewe