A convergent stochastic scalar auxiliary variable method

We discuss an extension of the scalar auxiliary variable approach which was originally introduced by Shen et al.~([Shen, Xu, Yang, J.~Comput.~Phys., 2018]) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable, this approach allows to derive a linear scheme, while still maintaining unconditional stability. Our extension augments the approximation of the evolution of this scalar auxiliary variable with higher order terms, which enables its application to stochastic partial differential equations. Using the stochastic Allen--Cahn equation as a prototype for nonlinear stochastic partial differential equations with multiplicative noise, we propose an unconditionally energy stable, linear, fully discrete finite element scheme based on our stochastic scalar auxiliary variable method. Recovering a discrete version of the energy estimate and establishing Nikolskii estimates with respect to time, we are able to prove convergence of appropriate subsequences of discrete solutions towards pathwise unique martingale solutions by applying Jakubowski's generalization of Skorokhod's theorem. A generalization of the Gy\"ongy--Krylov characterization of convergence in probability to quasi-Polish spaces finally provides convergence of fully discrete solutions towards strong solutions of the stochastic Allen--Cahn equation

A convergent SAV scheme for Cahn--Hilliard equations with dynamic boundary conditions

The Cahn-Hilliard equation is one of the most common models to describe phase separation processes in mixtures of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for this equation have been proposed. Recently, a family of models using Cahn-Hilliard-type equations on the boundary of the domain to describe adsorption processes was analysed (cf. Knopf, Lam, Liu, Metzger, ESAIM: Math. Model. Numer. Anal., 2021). This family of models includes the case of instantaneous adsorption processes studied by Goldstein, Miranville, and Schimperna (Physica D, 2011) as well as the case of vanishing adsorption rates which was investigated by Liu and Wu (Arch. Ration. Mech. Anal., 2019). In this paper, we are interested in the numerical treatment of these models and propose an unconditionally stable, linear, fully discrete finite element scheme based on the scalar auxiliary variable approach. Furthermore, we establish the convergence of discrete solutions towards suitable weak solutions of the original model. Thereby, when passing to the limit, we are able to remove the auxiliary variables introduced in the discrete setting completely. Finally, we present simulations based on the proposed linear scheme and compare them to results obtained using a stable, non-linear scheme to underline the practicality of our scheme

Hyperstaticity and loops in frictional granular packings

The hyperstatic nature of granular packings of perfectly rigid disks is analyzed algebraically and through numerical simulation. The elementary loops of grains emerge as a fundamental element in addressing hyperstaticity. Loops consisting of an odd number of grains behave differently than those with an even number. For odd loops, the latent stresses are exterior and are characterized by the sum of frictional forces around each loop. For even loops, the latent stresses are interior and are characterized by the alternating sum of frictional forces around each loop. The statistics of these two types of loop sums are found to be Gibbsian with a "temperature" that is linear with the friction coefficient mu when mu<1.Comment: 4 pages; Powders and Grains 2009, Golden, Colorado, US

Changes of smooth muscle contractile filaments in small bowel atresia

AIM: To investigate morphological changes of intestinal smooth muscle contractile fibres in small bowel atresia patients. METHODS: Resected small bowel specimens from small bowel atresia patients (n = 12) were divided into three sections (proximal, atretic and distal). Standard histology hematoxylin-eosin staining and enzyme immunohistochemistry was performed to visualize smooth muscle contractile markers α-smooth muscle actin (SMA) and desmin using conventional paraffin sections of the proximal and distal bowel. Small bowel from age-matched patients (n = 2) undergoing Meckel's diverticulum resection served as controls. RESULTS: The smooth muscle coat in the proximal bowel of small bowel atresia patients was thickened compared with control tissue, but the distal bowel was unchanged. Expression of smooth muscle contractile fibres SMA and desmin within the proximal bowel was slightly reduced compared with the distal bowel and control tissue. There were no major differences in the architecture of the smooth muscle within the proximal bowel and the distal bowel. The proximal and distal bowel in small bowel atresia patients revealed only minimal differences regarding smooth muscle morphology and the presence of smooth muscle contractile filament markers. CONCLUSION: Changes in smooth muscle contractile filaments do not appear to play a major role in postoperative motility disorders in small bowel atresia

Acute kidney injury prediction in cardiac surgery patients by a urinary peptide pattern: a case-control validation study

Background Acute kidney injury (AKI) is a prominent problem in hospitalized patients and associated with increased morbidity and mortality. Clinical medicine is currently hampered by the lack of accurate and early biomarkers for diagnosis of AKI and the evaluation of the severity of the disease. In 2010, we established a multivariate peptide marker pattern consisting of 20 naturally occurring urinary peptides to screen patients for early signs of renal failure. The current study now aims to evaluate if, in a different study population and potentially various AKI causes, AKI can be detected early and accurately by proteome analysis. Methods Urine samples from 60 patients who developed AKI after cardiac surgery were analyzed by capillary electrophoresis-mass spectrometry (CE-MS). The obtained peptide profiles were screened by the AKI peptide marker panel for early signs of AKI. Accuracy of the proteomic model in this patient collective was compared to that based on urinary neutrophil gelatinase-associated lipocalin (NGAL) and kidney injury molecule-1 (KIM-1) ELISA levels. Sixty patients who did not develop AKI served as negative controls. Results From the 120 patients, 110 were successfully analyzed by CE-MS (59 with AKI, 51 controls). Application of the AKI panel demonstrated an AUC in receiver operating characteristics (ROC) analysis of 0.81 (95 % confidence interval: 0.72–0.88). Compared to the proteomic model, ROC analysis revealed poorer classification accuracy of NGAL and KIM-1 with the respective AUC values being outside the statistical significant range (0.63 for NGAL and 0.57 for KIM-1)

Phase-field dynamics with transfer of materials: The Cahn--Hillard equation with reaction rate dependent dynamic boundary conditions

The Cahn--Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn--Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein, Miranville and Schimperna (Physica D, 2011) and the model by Liu and Wu (Arch.~Ration.~Mech.~Anal., 2019). Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model