Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method

For linear problems, domain decomposition methods can be used directly as iterative solvers, but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration, and thus converges much faster. We show in this paper that also for non-linear problems, domain decomposition methods can either be used directly as iterative solvers, or one can use them as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two level non-linear iterative domain decomposition method and a two level RASPEN non-linear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a non-linear diffusion problem

Discretization of the viscous dissipation term with the MAC scheme

International audienceWe propose a discretization for the MAC scheme of the viscous dissipation term τ (u) : ∇u (where τ (u) stands for the shear stress tensor associated to the velocity field u), which is suitable for the approximation of this term in a conservation equation for a scalar variable. This discretization enjoys the property that the integral over the computational domain Ω of the (discrete) dissipation term is equal to what is obtained when taking the inner product of the (discrete) momentum balance equation by u and integrating over Ω. As a consequence, it may be used as an ingredient to obtain an unconditionally stable scheme for the compressible Navier-Stokes equations. It is also shown, in some model cases, to ensure the strong convergence in L1 of the dissipation term

CONNECTION FORMULAS AND REPRESENTATIONS OF LAGUERRE POLYNOMIALS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS

In this paper, we introduce the notion of Oε-classical orthogonal polynomials, where Oε := I + εD (ε 6= 0). It is shown that the scaled Laguerre polynomial sequence {a −nL (α) n (ax)}n>0, where a = −ε −1 , is actually the only Oε-classical sequence. As an illustration, we deal with some representations of Laguerre polynomials L (0) n (x) in terms of the action of linear differential operators on the Laguerre polynomials L (m) n (x). The inverse connection problem of expanding Laguerre polynomials L (m) n (x) in terms of L (0) n (x) is also considered. Furthermore, some connection formulas between the monomial basis {x n}n>0 and the shifted Laguerre basis {L (m) n (x + 1)}n>0 are deduced

A Microcontroller-Based Adaptive Model Predictive Control Platform for Process Control Applications

Model predictive control (MPC) schemes employ dynamic models of a process within a receding horizon framework to optimize the behavior of a process. Although MPC has many benefits, a significant drawback is the large computational burden, especially in adaptive and constrained situations. In this paper, a computationally efficient self-tuning/adaptive MPC scheme for a simple industrial process plant with rate and amplitude constraints on the plant input is developed. The scheme has been optimized for real-time implementation on small, low-cost embedded processors. It employs a short (2-step) control horizon with an adjustable prediction horizon, automatically tunes the move suppression (regularization) parameter to achieve well-conditioned control, and presents a new technique for generating the reference trajectory that is robust to changes in the process time delay and in the presence of any inverse response. In addition, the need for a full quadratic programming procedure to handle input constraints is avoided by employing a quasi-analytical solution that optimally fathoms the constraints. Preliminary hardware-in-the-loop (HIL) test results indicate that the resulting scheme performs well and has low implementation overhead

Staggered discretizations, pressure correction schemes and all speed barotropic flows

International audienceWe present in this paper a class of schemes for the solution of the barotropic Navier- Stokes equations. These schemes work on general meshes, preserve the stability properties of the continuous problem, irrespectively of the space and time steps, and boil down, when the Mach number vanishes, to discretizations which are standard (and stable) in the incompressible framework. Finally, we show that they are able to capture solutions with shocks to the Euler equation

The semiclassical--Sobolev orthogonal polynomials: a general approach

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$_S= +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr D}r}>,$$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time.Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theor

Composición en ácidos grasos de fosfolípidos y triacilgliceroles de la carne del salmonete gris de labios gruesos (Chelon labrosus) que vive en agua geotérmica y agua de mar tunecina: un estudio comparativo

This study was conducted to elucidate the effects of rearing conditions on the composition of different phospholipid (PLs) classes and triacylglycerols (TAG) of the thick-lipped grey mullet (Chelon labrosus), a muscle originating from seawater and geothermal water. The major fatty acids in the examined lipid classes of the two fish groups were palmitic acid (C16:0), stearic acid (C18:0), oleic acid (C18:1n-9), linoleic acid (C18:2n-6),&nbsp;arachidonic acid (C20:4n-6), eicosapentaenoic acid (C20:5n-3), and docosahexaenoic acid (C22:6n-3). The analyses demonstrated that the fatty acid profiles of the PL classes in the seawater fish group were characterized by the predominance of n-3 polyunsaturated fatty acids (PUFA). By contrast, in geothermal fish, the distribution of PUFA series proportions differed between the phospholipid fractions. It was found PUFA n-3 was particularly abundant in PS and PI, while the n-6 series dominated the PC and PE PUFA group. Nonetheless, it was found that neutral lipid fatty acids were characterized by saturated fatty acids (SFA) followed by monounsaturated fatty acids (MUFA) in the seawater fish and by PUFA in the geothermal fish. The results presented here give useful information on the role of lipid classes in the physiological adaptation of&nbsp;C. labrosus&nbsp;which can serve for the optiminzation of these aquaculture systems.Este estudio se llevó a cabo para dilucidar los efectos de las condiciones de cría sobre la composición de diferentes clases de fosfolípidos (PL) y triacilgliceroles (TAG) del músculo de salmonetes de labios gruesos (Chelon labrosus) procedentes de agua de mar y de agua geotérmica. Los principales ácidos grasos en las clases de lípidos examinados de los dos grupos de peces fueron, palmítico (C16:0), esteárico (C18:0), oleico (C18:1n-9), linoleico (C18:2n-6), araquidónico (C20:4n-6), eicosapentaenoico (C20:5n-3) y ácido docosahexaenoico (C22:6n-3). Las determinaciones mostraron que los perfiles de ácidos grasos de los PL, en el grupo de peces de agua de mar, se caracterizaron por el predominio de ácidos grasos poliinsaturados n-3 (PUFA). Por el contrario, en los peces geotérmicos, la distribución de las proporciones de las series de PUFA difirió entre las fracciones de fosfolípidos. Se encontró que los PUFA n-3 eran particularmente abundantes en PS y PI, mientras que la serie n-6 dominaba el grupo de PUFA PC y PE. No obstante, se encontró que en lipidos neutros, los mayoritarios son los ácidos grasos saturados (SFA) seguidos de los ácidos grasos monoinsaturados (MUFA) en el pescado de agua de mar y los PUFA en el pescado geotérmico. Los resultados actuales brindan información útil sobre el papel de las clases de lípidos en la adaptación fisiológica de&nbsp;C. labrosus&nbsp;que puede servir para la optimización de estos sistemas de acuicultura

Decoding the genetic relationship between Alzheimer’s disease and type 2 diabetes: potential risk variants and future direction for North Africa

IntroductionAlzheimer’s disease (AD) and Type 2 diabetes (T2D) are both age-associated diseases. Identification of shared genes could help develop early diagnosis and preventive strategies. Although genetic background plays a crucial role in these diseases, we noticed an underrepresentation tendency of North African populations in omics studies.Materials and methodsFirst, we conducted a comprehensive review of genes and pathways shared between T2D and AD through PubMed. Then, the function of the identified genes and variants was investigated using annotation tools including PolyPhen2, RegulomeDB, and miRdSNP. Pathways enrichment analyses were performed with g:Profiler and EnrichmentMap. Next, we analyzed variant distributions in 16 worldwide populations using PLINK2, R, and STRUCTURE software. Finally, we performed an inter-ethnic comparison based on the minor allele frequency of T2D-AD common variants.ResultsA total of 59 eligible papers were included in our study. We found 231 variants and 363 genes shared between T2D and AD. Variant annotation revealed six single nucleotide polymorphisms (SNP) with a high pathogenic score, three SNPs with regulatory effects on the brain, and six SNPs with potential effects on miRNA-binding sites. The miRNAs affected were implicated in T2D, insulin signaling pathways, and AD. Moreover, replicated genes were significantly enriched in pathways related to plasma protein binding, positive regulation of amyloid fibril deposition, microglia activation, and cholesterol metabolism. Multidimensional screening performed based on the 363 shared genes showed that main North African populations are clustered together and are divergent from other worldwide populations. Interestingly, our results showed that 49 SNP associated with T2D and AD were present in North African populations. Among them, 11 variants located in DNM3, CFH, PPARG, ROHA, AGER, CLU, BDNF1, CST9, and PLCG1 genes display significant differences in risk allele frequencies between North African and other populations.ConclusionOur study highlighted the complexity and the unique molecular architecture of North African populations regarding T2D-AD shared genes. In conclusion, we emphasize the importance of T2D-AD shared genes and ethnicity-specific investigation studies for a better understanding of the link behind these diseases and to develop accurate diagnoses using personalized genetic biomarkers

Effect of salinity on the biosynthesis of n-3 long-chain polyunsaturated fatty acids in silverside Chirostoma estor

The genus Chirostoma (silversides) belongs to the family Atherinopsidae, which contains around 150 species, most of which are marine. However, Mexican silverside (Chirostoma estor) is one of the few representatives of freshwater atherinopsids and is only found in some lakes of the Mexican Central Plateau. However, studies have shown that C. estor has improved survival, growth and development when cultured in water conditions with increased salinity. In addition, C. estor displays an unusual fatty acid composition for a freshwater fish with high docosahexaenoic acid (DHA) : eicosapentaenoic acid (EPA) ratios. Freshwater and marine fish species display very different essential fatty acid metabolism and requirements and so the present study investigated long-chain polyunsaturated fatty acid (LC-PUFA) biosynthesis to determine the capacity of C. estor for endogenous production of EPA and DHA, and the effect that salinity has on these pathways. Briefly, C. estor were maintained at three salinities (0, 5 and 15 ppt) and the metabolism of 14C-labelled 18:3n-3 determined in isolated hepatocyte and enterocyte cells. The results showed that C. estor has the capacity for endogenous biosynthesis of LC-PUFA from 18-carbon fatty acid precursors, but that the pathway was essentially only active in saline conditions with virtually no activity in cells isolated from fish grown in freshwater. The activity of the LCPUFA biosynthesis pathway was also higher in cells isolated from fish at 15 ppt compared to fish at 5 ppt, The pathway was around 5-fold higher in hepatocytes compared to enterocytes, although the majority of 18:3n-3 was converted to 18:4n-3 and 20:4n-3 in hepatocytes whereas the proportions of 18:3n-3 converted to EPA and DHA were higher in enterocytes. The data were consistent with the hypothesis that conversion of EPA to DHA could contribute, at least in part, to the generally high DHA:EPA ratios observed in the tissue lipids of C. estor

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

International audienceWe assess in this paper the capability of a pressure correction scheme to compute shock solutions of the homogeneous model for barotropic two-phase flows. This scheme is designed to inherit the stability properties of the continuous problem: the unknowns (in particular the density and the dispersed phase mass fraction y) are kept within their physical bounds, and the entropy of the system is conserved, thus providing an unconditional stability property. In addition, the scheme keeps the velocity and pressure constant through contact discontinuities. These properties are obtained by coupling the mass balance and the transport equation for y in an original pressure correction step. The space discretization is staggered; the numer- ical schemes which are considered are the Marker-And Cell (MAC) finite volume scheme and the nonconforming low-order Rannacher-Turek and Crouzeix-Raviart finite element approximation. In either case, a finite volume technique is used for all convection terms. Numerical experiments performed here show that, provided that a sufficient dissipation is introduced in the scheme, it converges to the (weak) solution of the continuous hyperbolic system. Observed orders of convergence for 1D Riemann problems as a function of the mesh and time step at constant CFL number vary with the studied case, and the CFL number and on the regularity of the solution. They range from 0.5 to greater than 1 for the velocity and the pressure; in most cases, the density and mass fraction converge with a 0.5 order. Finally, the scheme shows a satisfactory behaviour up to large CFL numbers