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

A bioassay for Heterodera spp. without counting cysts or larvae

RESP-469

The good character at work: an initial study on the contribution of character strengths in identifying healthy and unhealthy work-related behavior and experience patterns

Purpose: Positive psychological functioning has been related to various positive work-related outcome variables, such as job satisfaction or work engagement. The aim of the present study was to examine the relations between morally positively valued traits (i.e., strengths of character) and work-related behaviors. Method: A sample of 887 adult women completed the Values in Action Inventory of Strengths (VIA-IS) and the Work-related Behavior and Experience Patterns Questionnaire (AVEM) in an online survey. Results: Those assigned to healthy work-related behavior and experience patterns differed in their strengths profiles from those that demonstrated unhealthy patterns (i.e., burnout type) in a predictable way. Especially the strengths of zest, persistence, hope, and curiosity seemed to play a key role in healthy and ambitious work behavior. Conclusions: The study underlines the relevance of character strengths in work settings and suggests that interventions based on character strengths could substantiate interventions already existing at the workplace in order to enhance positive work outcomes further (e.g., work satisfaction, engagement

A Time-Dependent Dirichlet-Neumann Method for the Heat Equation

We present a waveform relaxation version of the Dirichlet-Neumann method for parabolic problem. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, each subdomain problem is now in space and time, and the interface conditions are also time-dependent. Using a Laplace transform argument, we show for the heat equation that when we consider finite time intervals, the Dirichlet-Neumann method converges, similar to the case of Schwarz waveform relaxation algorithms. The convergence rate depends on the length of the subdomains as well as the size of the time window. In this discussion, we only stick to the linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and Engineering, Vol. 98, Springer-Verlag 201

Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning

Iterative substructuring Domain Decomposition (DD) methods have been extensively studied, and they are usually associated with nonoverlapping decompositions. It is less known that classical overlapping DD methods can also be formulated in substructured form, i.e., as iterative methods acting on variables defined exclusively on the interfaces of the overlapping domain decomposition. We call such formulations substructured domain decomposition methods. We introduce here a substructured version of Restricted Additive Schwarz (RAS) which we call SRAS. We show that RAS and SRAS are equivalent when used as iterative solvers, as they produce the same iterates, while they are substantially different when used as preconditioners for GMRES. We link the volume and substructured Krylov spaces and show that the iterates are different by deriving the least squares problems solved at each GMRES iteration. When used as iterative solvers, SRAS presents computational advantages over RAS, as it avoids computations with matrices and vectors at the volume level. When used as preconditioners, SRAS has the further advantage of allowing GMRES to store smaller vectors and perform orthogonalization in a lower dimensional space. We then consider nonlinear problems, and we introduce SRASPEN (Substructured Restricted Additive Schwarz Preconditioned Exact Newton), where SRAS is used as a preconditioner for Newton’s method. In contrast to the linear case, we prove that Newton’s method applied to the preconditioned volume and substructured formulation produces the same iterates in the nonlinear case. Next, we introduce two-level versions of nonlinear SRAS and SRASPEN. Finally, we validate our theoretical results with numerical experiments

Convergence and Psychometric Properties of Character Strengths Measures: The VIA-IS and the VIA-IS-R.

This study compares the German versions of the original measure of character strengths (VIA-IS) with its latest revision (VIA-IS-R) regarding reliability and convergent, discriminant, and criterion-related validity. A sample of 499 German-speaking adults (79% women, mean age: 33.3 years) provided self-reports of character strengths (VIA-IS, VIA-IS-R) and several criteria: Core virtues, thriving, and moral behaviors. Results suggested that both measures showed satisfactory internal consistency and converged well in a multitrait-multimethod analysis. Further, both measures were comparable regarding their relationships with the criteria. Overall, the results of the current study suggest that both questionnaires are reliable and valid, and that findings based on these instruments can be considered highly comparable

Fourier Method for Approximating Eigenvalues of Indefinite Stekloff Operator

We introduce an efficient method for computing the Stekloff eigenvalues associated with the Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition repeatedly. We propose solving the related constant coefficient Helmholtz equation with Fast Fourier Transform (FFT) based on carefully designed extensions and restrictions of the equation. The proposed Fourier method, combined with proper eigensolver, results in an efficient and clear approach for computing the Stekloff eigenvalues.Comment: 12 pages, 4 figure

Importance of single or blended polymer types for controlled in vitro release and plasma levels of a somatostatin analogue entrapped in PLA/PLGA microspheres.

The aim of the work was to develop biodegradable microspheres for controlled delivery of the somatostatin analogue vapreotide and maintenance of sustained plasma levels over 2–4 weeks after a single injection in rats. Vapreotide was microencapsulated into end-group capped and uncapped low molecular weight poly(lactide) (PLA) and poly(lactide-co-glycolide) (PLGA) by spray-drying and coacervation. Microspheres were prepared from single and blended (1:1) polymer types. The microparticles were characterized for peptide loading, in vitro release and pharmocokinetics in rats. Spray-drying and coacervation produced microspheres in the size range of 1–15 and 10–70 μm, respectively, and with encapsulation efficiencies varying between 46% and 87%. In vitro release of vapreotide followed a regular pattern and lasted more than 4 weeks, time at which 40–80% of the total dose were released. Microspheres made of 14-kDa end-group uncapped PLGA50:50 or 1:1 blends of this polymer with 35 kDa end-group uncapped PLGA50:50 gave the best release profiles and yielded the most sustained plasma levels above a pre-defined 1 ng/ml over approximately 14 days. In vitro/in vivo correlation analyses showed for several microsphere formulations a linear correlation between the mean residence time in vivo and the mean dissolution time (r=0.958) and also between the amount released between 6 h and 14 days and the AUC6h–14d (r=0.932). For several other parameters or time periods, no in vitro/in vivo correlation was found. This study demonstrates that controlled release of the vapreotide is possible in vivo for a duration of a least 2 weeks when administered i.m. to rats. These results constitute a step forward towards a twice-a-month or once-a-month microsphere-formulation for the treatment of acromegaly and neuroendocrine tumors

Sequential administration of temozolomide and fotemustine: Depletion of O6-alkyl guanine-DNA transferase in blood lymphocytes and in tumours

Background: The DNA repair protein O6-alkylguanine-DNA alkyl transferase (AT) mediates resistance to chloroethylnitro-soureas. Agents depleting AT such as DTIC and its new analogue temozolomide (TMZ) can reverse resistance to chloro-ethylnitrosoureas. We report the results of a dose finding study of TMZ in association with fotemustine. Patients and methods: Twenty-four patients with metastatic melanoma or recurrent glioma were treated with escalating dose of oral or intravenous TMZ ranging from 300 to 700 mg/m2, divided over two days. Fotemustine 100 mg/m2 was given intravenously on day 2, 4 hours after TMZ. AT depletion was measured in peripheral blood mononuclear cells (PBMCs) and in selected cases in melanoma metastases and was compared to TMZ pharmacokinetics. Results: The maximum tolerated dose (MTD) of TMZ was 400 mg/m2 (200 mg/m2/d) when associated with fotemustine the 2nd day with myelosuppression as dose limiting toxicity. The decrease of AT level in PBMCs was progressive and reached 34% of pretreatment values on day 2. There was however wide interindividual variability. AT reduction was neither dose nor route dependent and did not appear to be related to TMZ systemic exposure (AUC). In the same patients, AT depletion in tumour did not correlate with the decrease of AT observed in PBMCs. Conclusions: PBMCs may not be used as a surrogate of tumour for AT depletion. Further study should concentrate on the pharmacokinetic pharmacodynamic relationship in tumour to provide the basis for individually tailored therap