Stability Estimates Research Articles

Multiscale mortar mixed finite element methods for the Biot system of poroelasticity

We develop a mixed finite element domain decomposition method on non-matching grids for the Biot system of poroelasticity. A displacement–pressure vector mortar function is introduced on the interfaces and utilized as a Lagrange multiplier to impose weakly continuity of normal stress and normal velocity. The mortar space can be on a coarse scale, resulting in a multiscale approximation. We establish existence, uniqueness, stability, and error estimates for the semi-discrete continuous-in-time formulation under a suitable condition on the richness of the mortar space. We further consider a fully-discrete method based on the backward Euler time discretization and show that the solution of the algebraic system at each time step can be reduced to solving a positive definite interface problem for the composite mortar variable. A multiscale stress–flux basis is constructed, which makes the number of subdomain solves independent of the number of iterations required for the interface problem, and weakly dependent on the number of time steps. We present numerical experiments verifying the theoretical results and illustrating the multiscale capabilities of the method for a heterogeneous benchmark problem.

On Stability and State-Norm Estimation of Switched Systems Under Restricted Switching

On Stability and State-Norm Estimation of Switched Systems Under Restricted Switching

Coercivity-based analysis and its application to an inverse source problem for a subdiffusion equation with time-dependent principal parts

Abstract In this article, we investigate an inverse source problem for subdiffusion equations with time-dependent principal parts using distributed observations. We analyze coercivity properties of fractional derivatives. As applications, we use a coercivity property to establish a Lipschitz stability estimate for an inverse problem of determining the temporal component in the source term, which is a new result to the best knowledge of the authors, Numerically, we transform the inverse source problem into an optimization framework, incorporating Tikhonov regularization to mitigate the ill-posedness of the inverse problem. We then propose an iterative thresholding algorithm to solve this optimization problem efficiently. Finally, we present several numerical experiments to validate the accuracy and the efficiency of our proposed algorithm.

Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov–Poisson

In this paper we establish almost-optimal stability estimates in quantum optimal transport pseudometrics for the semiclassical limit of the Hartree dynamics to the Vlasov–Poisson equation, in the regime where the solutions have bounded densities. We combine Golse and Paul’s method from [Arch Ration Mech Anal 223:57–94, 2017], which uses a semiclassical version of the optimal transport distance and which was adapted to the case of the Coulomb and gravitational interactions by the second author in [J Stat Phys 177:20–60, 2019], with a new approach developed by the first author in [Arch Ration Mech Anal 244:27–50, 2022] to quantitatively improve stability estimates in kinetic theory.

Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces

In this work, we present and study Continuous Generative Neural Networks (CGNNs), namely, generative models in the continuous setting: the output of a CGNN belongs to an infinite-dimensional function space. The architecture is inspired by DCGAN, with one fully connected layer, several convolutional layers and nonlinear activation functions. In the continuous L 2 setting, the dimensions of the spaces of each layer are replaced by the scales of a multiresolution analysis of a compactly supported wavelet. We present conditions on the convolutional filters and on the nonlinearity that guarantee that a CGNN is injective. This theory finds applications to inverse problems, and allows for deriving Lipschitz stability estimates for (possibly nonlinear) infinite-dimensional inverse problems with unknowns belonging to the manifold generated by a CGNN. Several numerical simulations, including signal deblurring, illustrate and validate this approach.

Estimation of long-term stability of precision inductance measures using different methods

Abstract Inductance is a physical quantity that characterizes the ability of a conductor to accumulate magnetic field energy when electric current flows. Considering the small number of standards of a similar level of accuracy in the world, it is interesting to study the behavior of individual characteristics of such standards using known research methods. The long-term stability of measures is an important metrological characteristic that characterizes the change in the nominal value of a standard or measure over time. Regular metrological research of the long-term stability of precision inductance measures with nominal values of 10 mH and 100 mH have been carried out since 2009. Analysis of the long-term behavior of precision inductance measures included checking the primary measurement data for randomness and outliers using Abbe’s and Grubb’s tests. The detection of anomalous results significantly affects the estimation of the random error component and the standard uncertainty of the reproduction of the inductance measure. There is a significant change in the inductance of both measures over the entire observation period, which is confirmed by Abbe’s test. At the same time, this change is most significant in the first years, and then the change in inductance occurs slowly. The analysis of the functional change in inductance using the Grabb’s test showed that, depending on the research object itself, different analytical dependencies can be used for describing the functional trend. The criterion for choosing the best approximation can be both the “criterion of the reliability of the approximation” and the standard deviation of the “cleaned” measured data. The practical value of this work is the possibility of accumulating and integrating the obtained experimental data for standards of a similar accuracy class in different countries, which makes it possible to further improve the methods of their construction, stabilization of reproducible characteristics and operational features.

Direct and inverse scattering in an optical waveguide*

Abstract In this paper, we study the direct and inverse scattering of the Schrödinger equation in a three-dimensional optical planar waveguide. For the direct problem, we derive a resonance-free region and resolvent estimates for the resolvent of the Schrödinger operator in such a geometry. Based on the analysis of the resolvent, several inverse problems are investigated. First, given the potential function, we prove the uniqueness of the inverse source problem with multi-frequency data. We also develop a Fourier-based method to reconstruct the source function. The capability of this method is numerically illustrated by examples. Second, the uniqueness and increased stability of an inverse potential problem from data generated by incident waves are achieved in the absence of the source function. To derive the stability estimate, we use an argument of quantitative analytic continuation in complex theory. Third, we prove the uniqueness of simultaneously determining the source and potential by active boundary data generated by incident waves. In these inverse problems, we only use the limited lateral Dirichlet boundary data at multiple wavenumbers within a finite interval.

Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh–Stokes equations

In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh–Stokes equations. The proposed two-grid FEM uses the L2-1σ scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The L2-norm and H1-norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy H=h12 and H=hr2r+2 respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2-1σ scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.

Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions

Abstract Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.

Improved Estimates of Folding Stabilities and Kinetics with Multiensemble Markov Models.

Markov State Models (MSMs) have been widely applied to understand protein folding mechanisms by predicting long time scale dynamics from ensembles of short molecular simulations. Most MSM estimators enforce detailed balance, assuming that trajectory data are sampled at an equilibrium. This is rarely the case for ab initio folding studies, however, and as a result, MSMs can severely underestimate protein folding stabilities from such data. To remedy this problem, we have developed an enhanced-sampling protocol in which (1) unbiased folding simulations are performed and sparse tICA is used to obtain features that best capture the slowest events in folding, (2) umbrella sampling along this reaction coordinate is performed to observe folding and unfolding transitions, and (3) the thermodynamics and kinetics of folding are estimated using multiensemble Markov models (MEMMs). Using this protocol, folding pathways, rates, and stabilities of a designed α-helical hairpin, Z34C, can be predicted in good agreement with experimental measurements. These results indicate that accurate simulation-based estimates of absolute folding stabilities are within reach, with implications for the computational design of folded miniproteins and peptidomimetics.

Stability of Schur’s iterates and fast solution of the discrete integrable NLS

We prove a sharp stability estimate for Schur iterates of contractive analytic functions in the open unit disk. We then apply this result in the setting of the inverse scattering approach and obtain a fast algorithm for solving the discrete integrable nonlinear Schrödinger equation (Ablowitz–Ladik equation) on the integer lattice, \mathbb{Z} . We also give a self-contained introduction to the theory of the nonlinear Fourier transform from the perspective of Schur functions and orthogonal polynomials on the unit circle.

Forced stability studies and estimation of encapsulated ellagic acid in nano-formulations using UV-spectroscopy

Abstract There is a growing interest in dietary materials to explore their therapeutic activities. Ellagic acid is considered as a dietary supplement and is naturally present in fruits and other foods. It has anticancer, antimalarial, antiviral, antioxidant, and anti-inflammatory activities. These activities can be enhanced with nanoformulations, which can increase its oral bioavailability. However, there is a need for an economical, simple, sensitive, and robust analytical method for the estimation of entrapped ellagic acid in the nanoformulations. Therefore, the present study presents the development and validation of a UV–visible spectroscopy method for the estimation of EA in nanoformulations. The phosphate buffer (pH 7.2) and the detection wavelength of 253.5 nm were used for the method development, and its validation was performed according to the ICH Q2A (R1) guidelines. The coefficient of determination value of the developed method was found to be 0.9988 in the concentration range of 1 μg mL−1 to 6 μg mL−1. The method was found to be linear, precise, sensitive, and robust. This method can be used for the estimation of EA in nanoformulations, bulk dosage forms, and other pharmaceutical formulations.

A novel dimension reduction model based on POD and two-grid Crank–Nicolson mixed finite element methods for 3D nonlinear elastodynamic sine–Gordon problem

Earthquake, petroleum and gas extraction, and underground nuclear test all could cause nonlinear tectonic deformation. It is of great significance to effectively predict and evaluate the disasters and impacts of these geological structure deformation. In this end, a three-dimension (3D) nonlinear elastodynamic sine–Gordon model that can be used to describe nonlinear tectonic deformation including 3D displacement vector, 3 × 3 symmetric stress tensor matrix, nonlinear sin term, and singular initial value functions is first proposed. Then, a new time semi-discrete mixed Crank–Nicolson (TSDMCN) scheme for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TSDMCN solutions are proved. Next, a new two-grid mixed finite element Crank–Nicolson (TGMFECN) method with unconditional stability for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TGMFECN solutions are proved. Thenceforth, it is the most important thing is that a novel reduced-dimensional iterative TGMFECN (RDITGMFECN) method in matrix form is established by resorting to proper orthogonal decomposition only to lower the unknown TGMFECN solution coefficient vectors and keep TGMFECN basis functions unchanged, which can ensure that the RDITGMFECN method has the same accuracy as the usual TGMFECN method, but can greatly lower the dimension of the unknown TGMFECN solution coefficient vectors so as to mitigate calculated workload, save CPU operating-time, improve computing efficiency, and improve real-time calculating accuracy. In theory, the existence, stability, and error estimates of RDITGMFECN solutions are demonstrated by matrix analysis such that the theoretical analysis becomes very intuitive and easy to be understood by the public, which is a new attempt of theoretical analysis. In application, two numerical examples are used to simulate the 3D nonlinear tectonic deformation caused by earthquake and to verify the correctness of our theoretical results and the effectiveness of the RDITGMFECN method.

Discontinuous Galerkin methods for magnetic advection-diffusion problems

We devise and analyze a class of the primal discontinuous Galerkin methods for magnetic advection-diffusion problems based on the weighted-residual approach. In addition to the upwind stabilization, we explore some terms related to convection under the vector case that provides more flexibility in constructing the schemes. Under a degenerate Friedrichs system, we show the stability and optimal error estimate, which boil down to two ingredients – the weight function and the special projection – that contain information of advection. Numerical experiments are provided to verify the theoretical results.

Numerical analysis of a 1/2-equation model of turbulence

The recent 1/2-equation model of turbulence is a simplification of the standard Kolmogorov–Prandtl 1-equation URANS model. In tests, the 1/2-equation model produced comparable velocity statistics to a full 1-equation model with lower computational complexity. There is little progress in the numerical analysis of URANS models due to the difficulties in treating the coupling between equations and the nonlinearities in highest-order terms. The numerical analysis herein on the 1/2-equation model has independent interest and is also a first numerical analysis step to address the couplings and nonlinearities in a full 1-equation model. This report develops a complete numerical analysis of the 1/2-equation model. Stability, convergence, and error estimates are proven for a semi-discrete and fully discrete approximation. Finally, numerical tests are conducted to validate the predictions of the convergence theory.

Stability estimate for a time-dependent coefficient identification problem in parabolic equations

Stability estimate for a time-dependent coefficient identification problem in parabolic equations

Assessing Drug Product Shelf Life Using the Accelerated Stability Assessment Program: A Case Study of a GLPG4399 Capsule Formulation.

Objective: To evaluate and project the shelf life of GLPG4399, an early-phase clinical drug formulation by applying the Accelerated Stability Assessment Program (ASAP) approach. Methods: Forced degradation conditions were implemented to identify the stability-limiting degradation product. The drug and its degradation products were separated using a validated liquid chromatography method. Then, the selected clinical capsule formulation was placed in a glass vial and exposed to accelerated short-term conditions of combinations of high- and low-level heat and humidity in an open state for 5 weeks. The liquid chromatography results were evaluated using the ASAP, which is based on the moisture-modified Arrhenius principle. The resulting data were fitted using a suitable diffusion kinetics method. Results: The developed model was applied to predict the shelf life of the drug product when using clinically appropriate primary packaging (high-density polyethylene container). The derived stability parameters of the moisture-modified Arrhenius equation were the Arrhenius collision frequency, activation energy, and humidity sensitivity constant. The goodness of fit parameters R2 (>0.95) and goodness of prediction Q2 (>0.80) parameters for the selected model were acceptable. The results of the accelerated, short-term stability study were verified against real-time, long-term 12-month data. Conclusions: We demonstrated the application of the ASAP approach to evaluate the shelf life of a GLPG4399 solid capsule formulation. The studied ASAP approach can be extended to evaluate the stability and shelf-life estimations of other early-phase clinical formulations.

A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller–Segel model

Abstract We study a finite volume scheme approximating a parabolic-elliptic Keller–Segel system with power-law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^{\infty }(0,T;H^{1}(\varOmega ))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behavior of the error estimator under changes of $\gamma $.

Stability estimates for the sharp spectral gap bound under a curvature-dimension condition

Stability estimates for the sharp spectral gap bound under a curvature-dimension condition

Non-Fickian diffusion enhanced by temperature

In this paper we present a novel mathematical model to describe the permeation of a fluid through a polymeric matrix, loaded with drug molecules, followed by its subsequent desorption. Both phenomena are enhanced by temperature. We deduce energy estimates and stability estimates for the weak solution of the model, showing that this solution of the problem is stable in bounded time intervals. Numerical simulations illustrate how the coupling effects, of viscoelastic properties and thermal external assistance, can have a central role in the design of drug delivery devices.