Multigrid and saddle-point preconditioners for unfitted finite element modelling of inclusions

In this work, we consider the modeling of inclusions in the material using an unfitted finite element method. In the unfitted methods, structured background meshes are used and only the underlying finite element space is modified to incorporate the discontinuities, such as inclusions. Hence, the unfitted methods provide a more flexible framework for modeling the materials with multiple inclusions. We employ the method of Lagrange multipliers for enforcing the interface conditions between the inclusions and matrix, this gives rise to the linear system of equations of saddle point type. We utilize the Uzawa method for solving the saddle point system and propose preconditioning strategies for primal and dual systems. For the dual systems, we review and compare the preconditioning strategies that are developed for FETI and SIMPLE methods. While for the primal system, we employ a tailored multigrid method specifically developed for the unfitted meshes. Lastly, the comparison between the proposed preconditioners is made through several numerical experiments

A Multigrid Preconditioner for Jacobian-free Newton-Krylov Methods

In this work, we propose a multigrid preconditioner for Jacobian-free Newton-Krylov (JFNK) methods. Our multigrid method does not require knowledge of the Jacobian at any level of the multigrid hierarchy. As it is common in standard multigrid methods, the proposed method also relies on three building blocks: transfer operators, smoothers, and a coarse level solver. In addition to the restriction and prolongation operator, we also use a projection operator to transfer the current Newton iterate to a coarser level. The three-level Chebyshev semi-iterative method is employed as a smoother, as it has good smoothing properties and does not require the representation of the Jacobian matrix. We replace the direct solver on the coarsest level with a matrix-free Krylov subspace method, thus giving rise to a truly Jacobian-free multigrid preconditioner. We will discuss all building blocks of our multigrid preconditioner in detail and demonstrate the robustness and the efficiency of the proposed method using several numerical examples

Nonlinear Schwarz preconditioning for nonlinear optimization problems with bound constraints

We propose a nonlinear additive Schwarz method for solving nonlinear optimization problems with bound constraints. Our method is used as a "right-preconditioner" for solving the first-order optimality system arising within the sequential quadratic programming (SQP) framework using Newton's method. The algorithmic scalability of this preconditioner is enhanced by incorporating a solution-dependent coarse space, which takes into account the restricted constraints from the fine level. By means of numerical examples, we demonstrate that the proposed preconditioned Newton methods outperform standard active-set methods considered in the literature

Comparative BER Analysis of Free Space Optical System using Wavelength Diversity over Exponentiated Weibull channel

Atmospheric turbulence is considered as major threat to Free Space Optical (FSO) communication as it causes irradiance and phase fluctuations of the transmitted signal which degrade the performance of FSO system. Wavelength diversity is one of the techniques to mitigate these effects. In this paper,the wavelength diversity technique is applied to FSO system to improve the performance under different turbulence conditions which are modeled using Exponentiated Weibull (EW) channel. In this technique, the data was communicated through 1.55 µm, 1.31 µm, and 0.85 µm carrier wavelengths. Optimal Combining (OC) scheme has been considered to receive the signals atreceiver. Mathematical equation for average BER is derived for wavelength diversity based FSO system. Results are obtained for the different link length under different turbulence conditions. The obtained average BER results for different turbulence conditions characterized by EW channel is compared with the published result of average BER for different turbulence which is presented by classical channel model. A comparative BER analysis shows that maximum advantage of wavelength diversity technique is obtained when different turbulence conditions are modeled by EW channel

Multilevel solution strategies for unfitted finite element methods

In the unfitted finite element methods, traditionally we can use Nitsche's method or the method of Lagrange multipliers to enforce the boundary/interface conditions. In this work, we present tailored multilevel methods for solving the problems stemming from either of these discretizations. Generally, multigrid methods require a hierarchy of finite element (FE) spaces which can be created geometrically using a hierarchy of nested meshes. However, in the unfitted FE framework, standard multigrid methods might demonstrate poor convergence properties if the hierarchy of FE spaces employed is not nested. We design a prolongation operator for the multigrid methods in such a way that it can accommodate the arbitrary shape of the boundaries/interfaces and recursively induces a nested FE space hierarchy. The prolongation operator is constructed using so-called pseudo-$L^2$-projections; as common, the adjoint of the prolongation operator is employed as the restriction operator. We employ this transfer operator in our multigrid method and solve the linear system of equations that arise from using Nitsche's method. In the numerical experiments, we show that our multigrid method is robust with respect to highly varying coefficients and the number of interfaces in a domain. It shows level independent convergence rates when applied to different variants of Nitsche's method. Additionally, we present a generalized multigrid method for solving the problems stemming from the discretization of the interface conditions using Lagrange multipliers. This method can be used to solve the quadratic minimization problems with linear equality/inequality constraints, efficiently. The essential component of this multigrid method is the technique that decouples the linear constraints by projecting them into a new basis. The decoupled constraints are then handled by a modified version of the projected Gauss-Seidel method. By means of several numerical experiments, we exhibit the robustness of our multigrid method for the boundary or interface conditions with respect to varying coefficients. In addition, we demonstrate that this multigrid method can also handle the inequality constraints arising from the contact problems in the unfitted framework

Global, regional, and national burden of diabetes from 1990 to 2021, with projections of prevalence to 2050: a systematic analysis for the Global Burden of Disease Study 2021

Background: Diabetes is one of the leading causes of death and disability worldwide, and affects people regardless of country, age group, or sex. Using the most recent evidentiary and analytical framework from the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD), we produced location-specific, age-specific, and sex-specific estimates of diabetes prevalence and burden from 1990 to 2021, the proportion of type 1 and type 2 diabetes in 2021, the proportion of the type 2 diabetes burden attributable to selected risk factors, and projections of diabetes prevalence through 2050. Methods: Estimates of diabetes prevalence and burden were computed in 204 countries and territories, across 25 age groups, for males and females separately and combined; these estimates comprised lost years of healthy life, measured in disability-adjusted life-years (DALYs; defined as the sum of years of life lost [YLLs] and years lived with disability [YLDs]). We used the Cause of Death Ensemble model (CODEm) approach to estimate deaths due to diabetes, incorporating 25 666 location-years of data from vital registration and verbal autopsy reports in separate total (including both type 1 and type 2 diabetes) and type-specific models. Other forms of diabetes, including gestational and monogenic diabetes, were not explicitly modelled. Total and type 1 diabetes prevalence was estimated by use of a Bayesian meta-regression modelling tool, DisMod-MR 2.1, to analyse 1527 location-years of data from the scientific literature, survey microdata, and insurance claims; type 2 diabetes estimates were computed by subtracting type 1 diabetes from total estimates. Mortality and prevalence estimates, along with standard life expectancy and disability weights, were used to calculate YLLs, YLDs, and DALYs. When appropriate, we extrapolated estimates to a hypothetical population with a standardised age structure to allow comparison in populations with different age structures. We used the comparative risk assessment framework to estimate the risk-attributable type 2 diabetes burden for 16 risk factors falling under risk categories including environmental and occupational factors, tobacco use, high alcohol use, high body-mass index (BMI), dietary factors, and low physical activity. Using a regression framework, we forecast type 1 and type 2 diabetes prevalence through 2050 with Socio-demographic Index (SDI) and high BMI as predictors, respectively. Findings: In 2021, there were 529 million (95% uncertainty interval [UI] 500-564) people living with diabetes worldwide, and the global age-standardised total diabetes prevalence was 6·1% (5·8-6·5). At the super-region level, the highest age-standardised rates were observed in north Africa and the Middle East (9·3% [8·7-9·9]) and, at the regional level, in Oceania (12·3% [11·5-13·0]). Nationally, Qatar had the world's highest age-specific prevalence of diabetes, at 76·1% (73·1-79·5) in individuals aged 75-79 years. Total diabetes prevalence-especially among older adults-primarily reflects type 2 diabetes, which in 2021 accounted for 96·0% (95·1-96·8) of diabetes cases and 95·4% (94·9-95·9) of diabetes DALYs worldwide. In 2021, 52·2% (25·5-71·8) of global type 2 diabetes DALYs were attributable to high BMI. The contribution of high BMI to type 2 diabetes DALYs rose by 24·3% (18·5-30·4) worldwide between 1990 and 2021. By 2050, more than 1·31 billion (1·22-1·39) people are projected to have diabetes, with expected age-standardised total diabetes prevalence rates greater than 10% in two super-regions: 16·8% (16·1-17·6) in north Africa and the Middle East and 11·3% (10·8-11·9) in Latin America and Caribbean. By 2050, 89 (43·6%) of 204 countries and territories will have an age-standardised rate greater than 10%. Interpretation: Diabetes remains a substantial public health issue. Type 2 diabetes, which makes up the bulk of diabetes cases, is largely preventable and, in some cases, potentially reversible if identified and managed early in the disease course. However, all evidence indicates that diabetes prevalence is increasing worldwide, primarily due to a rise in obesity caused by multiple factors. Preventing and controlling type 2 diabetes remains an ongoing challenge. It is essential to better understand disparities in risk factor profiles and diabetes burden across populations, to inform strategies to successfully control diabetes risk factors within the context of multiple and complex drivers. Funding: Bill & Melinda Gates Foundation