Stability Estimates Research Articles

Stability estimates for the Vlasov–Poisson system in p$p$‐kinetic Wasserstein distances

AbstractWe extend Loeper's ‐estimate (Theorem 2.9 in J. Math. Pures Appl. (9) 86 (2006), no. 1, 68–79) relating the force fields to the densities for the Vlasov–Poisson system to , with , based on the Helmholtz–Weyl decomposition. This allows us to generalize both the classical Loeper's 2‐Wasserstein stability estimate (Theorem 1.2 in J. Math. Pures Appl. (9) 86 (2006), no. 1, 68–79) and the recent stability estimate by the first author relying on the newly introduced kinetic Wasserstein distance (Theorem 3.1 in Arch Rational Mech. Anal. 244 (2022), no. 1, 27–50) to kinetic Wasserstein distances of order .

Estimation of road cut slope stability along the national and state highway of Garhwal Himalaya (India) applying slope mass rating and kinematic analysis

Slope stability assessment of different road cut slopes has been conducted along the National Highway in Garhwal Himalaya, Uttarakhand (NH-34, NH-134, NH-7, NH-107 and a state highway from Dharasu to Tilwara). Slope instability issues in the Himalayan terrain are radically increasing due to high anthropogenic activities and rapid growth of urban centers since past years. Road networks in mountainous areas are the main source of transportation and socio-economic development. In this concerned study, forty (40) road cut slopes have been selected. These sites are selected based on lithology and structure. Diverse slope stability quantification methods along with various modes of failures have been systematically analyzed. The entire region falls under the tectonically active belt or the vicinity of the active fault zones that make the slopes inherently unstable. The Geological Strength Index (GSI) and Rock Mass Rating (RMR) have been applied for determination of slope conditions. Slope mass stability has been evaluated using Slope Mass Rating (SMR) index. The RMR rating ranges from 26 to 87 and the SMR values of forty (40) roads cut slopes range from 10 to 91. The kinematic analysis shows that most of the probable slopes failures are susceptible under planar and wedge failure. This study will definitely help to reduce the loss of life and property and also helps the ongoing constructional processes along the road section.

A preserving accuracy two-grid reduced-dimensional Crank-Nicolson mixed finite element method for nonlinear wave equation

In this paper, we mainly resort to a proper orthogonal decomposition (POD) method to study the reduced-dimension of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the nonlinear wave equation and build a preserving accuracy reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method for the nonlinear wave equation. For this purpose, we first build a new TGCNMFE method for the nonlinear wave equation and demonstrate the existence, unconditional stability, and error estimates for the TGCNMFE solutions. From there, we build a preserving accuracy RDETGCNMFE method for the nonlinear wave equation by using the POD method to decrease the dimension of unknown TGCNMFE solution coefficient vectors and employ the matrix analysis to analyze the existence, unconditional stability, convergence, and errors for the RDETGCNMFE solutions. We finally make use of numerical experiments to verify our theoretical results and to show the advantages of RDETGCNMFE method.

Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator.

Nonsmooth data error estimates for fully discrete finite element approximations of semilinear parabolic equations in Banach space

Implicit–explicit (IMEX) Euler formula together with finite element methods is proposed to solve numerically rather general semilinear parabolic differential equations with initial data in Banach space Xα, 0<α≤1. The time-space regularity of solutions to this class of equations is first investigated. Stability and error estimates are provided for this fully discrete scheme by discrete semigroup method. In the special case when X is the Hilbert space L2(Ω), these error estimates bridge the gap between the existing results on this scheme for semilinear parabolic problems with initial data in L2(Ω) and in H1(Ω) under general nonlinearity. Numerical experiments for nonlinear Allen–Cahn equations verify and complement our theoretical results.

General decay of solutions for a viscoelastic porous system with Kelvin-Voigt damping

This study investigates the asymptotic stability of a viscoelastic porous system with Kelvin-Voigt damping. The system under consideration involves coupled equations describing the displacement of the solid elastic material, the volume fraction, and a viscoelastic or memory term. Previous studies have shown that introducing control to one equation of these systems can lead to uniform stability for weak solutions when the wave speeds are equal. However, when the wave propagation speeds differ, the decay rate of classical solutions becomes slower. In this work, we improve previous findings by considering the viscoelastic porous elasticity system with Kelvin-Voigt damping applied to the volume fraction equation. By employing the multiplier method in conjunction with the characteristics of convex functions, we establish a general stability estimate independent of the wave speeds. Our results provide valuable insights into the stability properties of viscoelastic porous systems. This study contributes to the existing literature on the subject and has implications for the analysis and control of such systems in various engineering applications.

Unconditional stability and error estimates of the modified characteristics FEMs for the Micropolar Navier–Stokes Equations

In this paper, the unconditional stability and optimal error estimate of the velocity, pressure and angular velocity for the modified characteristics FEMs of the unsteady Micropolar Naiver–Stokes Equations (MNSE) are presented. In this method, the nonlinear equation is linearized by the characteristic finite element method for dealing with the time derivative term and the convection term. Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the error between the numerical solution and the solution of the time-discrete system is τ-independent, where τ denotes the time stepsize. The stability results and optimal error estimates in L2 norm and H1 norm will be given. Finally, some numerical results will be provided to confirm our theoretical analysis.

Local solvability and stability for the inverse Sturm‐Liouville problem with polynomials in the boundary conditions

In this paper, we for the first time prove local solvability and stability of the inverse Sturm‐Liouville problem with complex‐valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The proof method is constructive. It is based on the reduction of the inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation of the inverse problem remains uniquely solvable. Furthermore, we derive new reconstruction formulas for obtaining the problem coefficients from the solution of the main equation and get stability estimates for the recovered coefficients.

Three‐dimensional stability analysis for a deep‐buried tunnel roof considering soil stratum strength nonlinearity

AbstractA three‐dimensional (3D) collapse mechanism was employed in this work to investigate the roof stability of deep‐buried cylindrical tunnels in soil considering strength nonlinearity. Based on the kinematic approach of limit analysis, three tunnel roof stability measures, namely, the stability number, the required support pressure, and the factor of safety solutions were derived to provide quantitative indicators for evaluating tunnel roof stability. Optimal upper bound solutions for these measures were obtained through optimization. Subsequently, the effects of soil strength nonlinearity and 3D geometry characteristics on roof stability and the profiles of collapsing blocks were investigated: The increase in the nonlinear coefficient m and the tensile strength σt, as well as the decrease in the initial cohesion c0, will lead to a deterioration in the stability of the tunnel roof, and also result in a larger range of collapse blocks of the tunnel roof. A 3D stability analysis of the tunnel roof can provide more reasonable stability estimates than a 2D stability analysis. When the length‐to‐radius ratio L/R → ∞, the 2D solution results.

Pro+: Automated protrusion and critical shear stress estimates from 3D point clouds of gravel beds

AbstractThe dimensionless critical shear stress (τ*c) needed for the onset of sediment motion is important for a range of studies from river restoration projects to landscape evolution calculations. Many studies simply assume a τ*c value within the large range of scatter observed in gravel‐bedded rivers because direct field estimates are difficult to obtain. Informed choices of reach‐scale τ*c values could instead be obtained from force balance calculations that include particle‐scale bed structure and flow conditions. Particle‐scale bed structure is also difficult to measure, precluding wide adoption of such force‐balance τ*c values. Recent studies have demonstrated that bed grain size distributions (GSD) can be determined from detailed point clouds (e.g. using G3Point open‐source software). We build on these point cloud methods to introduce Pro+, software that estimates particle‐scale protrusion distributions and τ*c for each grain size and for the entire bed using a force‐balance model. We validated G3Point and Pro+ using two laboratory flume experiments with different grain size distributions and bed topographies. Commonly used definitions of protrusion may not produce representative τ*c distributions, and Pro+ includes new protrusion definitions to better include flow and bed structure influences on particle mobility. The combined G3Point/Pro+ provided accurate grain size, protrusion and τ*c distributions with simple GSD calibration. The largest source of error in protrusion and τ*c distributions were from incorrect grain boundaries and grain locations in G3Point, and calibration of grain software beyond comparing GSD is likely needed. Pro+ can be coupled with grain identifying software and relatively easily obtainable data to provide informed estimates of τ*c. These could replace arbitrary choices of τ*c and potentially improve channel stability and sediment transport estimates.

Stability, homogeneity and measurement uncertainty estimation of PVP-I solutions for the application on oro and nasopharynx against SARS-CoV-2

Aqueous solution containing different concentration (0.5, 0.6 and 1.0%) (w/v) of Polyvinyl pyrrolodon-Iodine (PVP-I) complex, a well-known antiseptic; is prepared and the stability and homogeneity of these solution is assessed as per the ICH Guidelines and International Harmonized Protocol respectively. The solutions were found to be sufficiently homogeneous and stable for a year at 25 °C (60%RH). Measurement uncertainty of the prepared PVP-I solutions were estimated by identifying possible sources of uncertainty using Ishikawa diagram and preparing uncertainty budget based on scope of calibration laboratory. The stable and homogenized PVP-I solution is to be used in a clinical trial for the application on oro and nasopharynx against novel SARS-CoV-2 Virus.

In Silico and Chromatographic Methods for Analysis of Biotransformation of Prospective Neuroprotective Pyrrole-Based Hydrazone in Isolated Rat Hepatocytes.

In the current study, chromatographic and in silico techniques were applied to investigate the biotransformation of ethyl 5-(4-bromophenyl)-1-(2-(2-(2-hydroxybenzylidene) hydrazinyl)-2-oxoethyl)-2-methyl-1H-pyrrole-3-carboxylate (11b) in hepatocytic media. The initial chromatographic procedure was based on the employment of the conventional octadecyl stationary phase method for estimation of the chemical stability. Subsequently, a novel and rapid chromatographic approach based on a phenyl-hexyl column was developed, aiming to separate the possible metabolites. Both methods were performed on a Dionex 3000 ThermoScientific (ACM 2, Sofia, Bulgaria) device equipped with a diode array detector set up at 272 and 279 nm for analytes detection. An acetonitrile: phosphate buffer of pH 3.5: methanol (17:30:53 v/v/v) was eluted isocratically as a mobile phase with a 1 mL/min flow rate. A preliminary purification from the biological media was achieved by protein precipitation with methanol. A validation procedure was carried out, where the method was found to correspond to all ICH (Q2) and M10 set criteria. Additionally, an in silico-based approach with the online server BioTransformer 3.0 was applied in an attempt to predict the possible metabolites of the title compound 11b. It was hypothesized that four CYP450 isoforms (1A2, 2C9, 3A4, and 2C8) were involved in the phase I metabolism, resulting in the formation of 12 metabolites. Moreover, docking studies were conducted to evaluate the formation of stable complexes between 11b and the aforementioned isoforms. The obtained data indicated three metabolites as the most probable products, two of which (M9_11b and M10_11b) were synthesized by a classical approach for verification. Finally, liquid chromatography with a mass detector was implemented for comprehensive and summarized analysis, and the obtained results revealed that the metabolism of the 11b proceeds possibly with the formation of glucuronide and glycine conjugate of M11_11b.

Improved RNA stability estimation through Bayesian modeling reveals most Salmonella transcripts have subminute half-lives

RNA decay is a crucial mechanism for regulating gene expression in response to environmental stresses. In bacteria, RNA-binding proteins (RBPs) are known to be involved in posttranscriptional regulation, but their global impact on RNA half-lives has not been extensively studied. To shed light on the role of the major RBPs ProQ and CspC/E in maintaining RNA stability, we performed RNA sequencing of Salmonella enterica over a time course following treatment with the transcription initiation inhibitor rifampicin (RIF-seq) in the presence and absence of these RBPs. We developed a hierarchical Bayesian model that corrects for confounding factors in rifampicin RNA stability assays and enables us to identify differentially decaying transcripts transcriptome-wide. Our analysis revealed that the median RNA half-life in Salmonella in early stationary phase is less than 1 min, a third of previous estimates. We found that over half of the 500 most long-lived transcripts are bound by at least one major RBP, suggesting a general role for RBPs in shaping the transcriptome. Integrating differential stability estimates with cross-linking and immunoprecipitation followed by RNA sequencing (CLIP-seq) revealed that approximately 30% of transcripts with ProQ binding sites and more than 40% with CspC/E binding sites in coding or 3' untranslated regions decay differentially in the absence of the respective RBP. Analysis of differentially destabilized transcripts identified a role for ProQ in the oxidative stress response. Our findings provide insights into posttranscriptional regulation by ProQ and CspC/E, and the importance of RBPs in regulating gene expression.

Error estimates for finite element approximations of viscoelastic dynamics: The generalized Maxwell model

We prove error estimates for a finite element approximation of viscoelastic dynamics based on continuous Galerkin in space and time, both in energy norm and in L2 norm. The proof is based on an error representation formula using a discrete dual problem and a stability estimate involving the kinetic, elastic, and viscoelastic energies. To set up the dual error analysis and to prove the basic stability estimates, it is natural to formulate the problem as a first-order-in-time system involving evolution equations for the viscoelastic stress, the displacements, and the velocities. The equations for the viscoelastic stress can, however, be solved analytically in terms of the deviatoric strain velocity, and therefore, the viscoelastic stress can be eliminated from the system, resulting in a system for displacements and velocities.

Stabilization with unknown-input estimation for time-delayed systems using functional-observer-based feedback control design

This paper focuses on the stabilization and unknown-input estimation (UIE) problem of continuous-time delayed systems by virtue of functional-observer (FO)-based feedback controller. It allows us to estimate the unknown external variable that degrades the system performance with respect to the partially known input of the system. In particular, the unknown inputs are considered in system dynamics’s state as well as in output, which makes the addressed problem to be more applicable. In addition to that, the aim is then to design minimal order functional observer. Notably, the FO-based feedback control law is introduced for the first time and it is used to ensure the asymptotic stabilization with UIE of the proposed system. Whilst, the design and analysis of UIE and FO is based on Lyapunov stability theory and applying the inequality analysis technique, the sufficient stability conditions are established in the forms of matrix inequality. Eventually, the numerical examples with simulation results are provided to validate the practicability and efficacy of the developed control strategy.

Rational approximation of operator semigroups via the [formula omitted]-calculus

We improve the classical results by Brenner and Thomée on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, provide sharper stability estimates, and obtain optimal approximation rates. Moreover, we strengthen a result due to Egert-Rozendaal on subdiagonal Padé approximations of operator semigroups. Our approach is direct and based on the theory of the B- functional calculus developed recently. On the way, we elaborate a new and simple approach to construction of the B-calculus thus making the paper essentially self-contained.

Stable reconstruction of anisotropic scattering objects from electromagnetic Cauchy data

ABSTRACT This paper addresses the electromagnetic inverse scattering problem of determining the location and shape of anisotropic objects from near-field Cauchy data. We investigate both cases involving the Helmholtz equation and Maxwell's equations for this inverse problem. Our study focuses on developing efficient imaging functionals that enable a fast and stable recovery of the anisotropic object. The implementation of the imaging functionals is simple and avoids the need to solve an ill-posed problem. The resolution analysis of the imaging functionals is conducted using the Green representation formula. Furthermore, we establish stability estimates for these imaging functionals when noise is present in the data. To illustrate the effectiveness of the methods, we present numerical examples showcasing their performance.

Stable determination of an impedance obstacle by a single far-field measurement

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in R2 . The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle’s geometric configurations and the wave field’s vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition.

Role of Explicit Solvation in Computational Modeling of Chemical Reactions: Mechanism of Cu(I) Transfer Between Thiolate-Based Chelators in Water.

Solvation plays important roles in controlling the thermodynamic and kinetic aspects of chemical reactions. The conventional approaches to treat solvation via electronic structure methods are likely to become inadequate, when the reacting solutes have strong electrostatic and hydrogen bonding interactions with the solvent and undergo significant structural changes during the course of the reaction. In this article, we present evidence of such solvent and structural effects in the computational study of the Cu(I) transfer reaction between thiolate-based chelators dithiobutylamine (DTBA) and dithiotheritol (DTT) in water, inspired from biological copper trafficking phenomena. We propose a general solution to the problem by combining classical molecular dynamics (MD) simulations of the bulk system and static quantum chemistry calculations. The fluctuating solvation shell was estimated from MD, and energetics was assessed by averaging QM energies of a series of molecular clusters constructed from the MD snapshots. Applying this approach, we propose a reaction pathway with estimates of relative intermediate stabilities and barriers, which suggest the overall reaction to be reversible in nature and likely to go through both two and three coordinated intermediates, confirming previous studies of similar protein analogues. An interesting fact that emerged from our study was the strong indication that the rate-determining step is the deprotonation of initial thiol bound Cu(I) complex, without involving any Cu(I)-S bonds. The proposed method will lead to a better treatment of solvations, and these mechanistic insights will aid our understanding of biological copper(I) trafficking.

Consistency of the Bayes method for the inverse scattering problem

In this work, we consider the inverse scattering problem of determining an unknown refractive index from the far-field measurements using the nonparametric Bayesian approach. We use a collection of large ‘samples’, which are noisy discrete measurements taking from the scattering amplitude. We will study the frequentist property of the posterior distribution as the sample size tends to infinity. Our aim is to establish the consistency of the posterior distribution with an explicit contraction rate in terms of the sample size. We will consider two different priors on the space of parameters. The proof relies on the stability estimates of the forward and inverse problems. Due to the ill-posedness of the inverse scattering problem, the contraction rate is of a logarithmic type. We also show that such contraction rate is optimal in the statistical minimax sense.