Regular Theory Research Articles

Uniform estimates for conforming Galerkin method for anisotropic singularly perturbed elliptic problems

In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming Q 1 finite element method are derived, and some other results of convergence and regularity for the continuous problem are proved. The arguments are based on the tensorization trick and standard elliptic regularity theory tools.

Local composition-regular solution theory for analysis of pharmaceutical solubility in mixed-solvents

Local composition regular solution theory (LC-RST) model was developed by introducing non-equal i–j and j–i parameters into regular solution theory (RST) to account for specific molecular interactions. The LC-RST model was applied to the correlation of solubilities of active pharmaceutical ingredients (API) in hydrogen bond donor (HBD) – hydrogen bond acceptor (HBA) mixed-solvent systems. In the assessment of 32 API-mixed-solvent systems that contained 11 different APIs and 1947 data, it was determined that the LC-RST model had a logarithmic average relative deviation (ARDln) in solubility of 0.028, the Wilson model had an ARDln value of 0.24 and the purely predictive RST model had an ARDln value of 1.29. Partial molar excess enthalpies and entropies of the API in the mixed-solvents determined from the LC-RST model allowed estimation of API solubility regions where specific interactions occur between API and HBD-HBA complex molecules. In the correlation scheme, component numbers were designated as API molecule (1) – HBD molecule (2) - HBA molecule (3) and systems were grouped according to HBD molecule interactions with the API (pro-solvent, anti-solvent, synergistic with HBA molecule). Phenomenological parameter sensitivity analysis of the LC-RST model showed that dominant specific interactions tended to be 1–2 or 2–1 molecules for pro-solvent groupings, 2–3 or 2–3 molecules for anti-solvent groupings and all i–j and j-–i molecular pairs for synergistic HBD-HBA molecule groupings. Based on parameters determined from a given API-mixed-solvent system, the LC-RST model allows qualitative prediction of solubilities for other systems. The LC-RST model is simple to apply and it retains inherent predictive characteristics of regular solution theory.

A Regularity Theory of Causation

AbstractIn this paper, we propose a regularity theory of causation. The theory aims to be reductive and to align with our pre‐theoretic understanding of the causal relation. We show that our theory can account for a wide range of causal scenarios, including isomorphic scenarios, omissions, and scenarios which suggest that causation is not transitive.

Hölder estimate for a tug-of-war game with $1\lt p\lt 2$ from Krylov–Safonov regularity theory

We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the p -Laplacian with 1\lt p\lt 2 . For this version, the asymptotic Hölder continuity of solutions can be directly derived from recent Krylov–Safonov type regularity results in the singular case. Moreover, existence of a measurable solution can be obtained without using boundary corrections. We also establish a comparison principle.

Singular solutions of a fractional Dirichlet problem in a punctured domain

Let D be a bounded regular domain in Rn (n ? 3) containing 0, 0 < ? < 2, and ? < 1. We take up in this article the existence and asymptotic behavior of a positive continuous solution for the following semi-linear fractional differential equation (??|D)?2 u = a(x)u?(x) in D\{0}, with the boundary Dirichlet conditions lim |x|?0 |x|n??u(x) = 0 and lim x??D ?(x)2??u(x) = 0, where (??|D)?/2 is the fractional Laplace associated to the subordinate killed Brownian motion process in D and ?(x) = dist ( x, ?D) denotes the Euclidean distance between x and ?D. The function a is a positive continuous function in D\{0}, which may be singular at x = 0 and/or at the boundary ?D satisfying some appropriate assumption related to Karamata class. More precisely, we shall prove the existence and global asymptotic behavior of a positive continuous solution on ?D\{0}. We will use some potential theory arguments and Karamata regular variation theory tools.

Synergistic effects between a non-ionic and an anionic surfactant on the micellization process and the adsorption at liquid/air surfaces.

Predicting the behaviour of solutions with surfactants of significantly different critical micelle concentration (CMC) values remains a challenge. The study of the molecular interactions within micelles and interfaces in surfactant combinations used in everyday products is essential to understand these complex systems. In this work, the equilibrium and dynamic surface tension in the presence of mixed non-ionic (tristyrylphenol ethoxylates) and anionic (sodium benzene sulfonate with alkyl chain lengths of C10-C13) surfactants, commonly encountered as delivery systems in agrochemicals, were studied and their CMC values were determined. For the surfactant mixtures, four molar ratios were examined: nEOT/nNaDDBS = 0.01, 0.1, 1, 4 and two different cases were analysed, the premixed and the add one by one surfactant. The surface tension for single surfactants stabilised quickly, while the mixtures needed a long time to reach equilibrium; up to 15 h for the premixed mixtures and 40 min when surfactants were added one by one. The CMC values for the nEOT/nNaDDBS = 0.01, 0.1 premixed surfactant mixtures were found to be in between the CMC values of the single surfactants, but those for the nEOT/nNaDDBS = 1 and 4 mixtures were lower than the CMCs of both single surfactants. Calculations based on the regular solution theory suggested that there are attractive forces in the mixed micelles and at the interface layers, while the supramolecular assemblies in the bulk (i.e., micelles) and at interfaces (surfactant films) are preferentially enriched in EOT.

On divergence-free (form-bounded type) drifts

We develop regularity theory, including a posteriori Harnack inequality and higher integrability of the gradient of solutions, for divergence-form elliptic equations with singular divergence-free drift satisfying a broad form-boundedness-type condition (more generally, our drifts can have singular divergence). A key step in our proofs is a new iteration procedure (needed already to prove Caccioppoli's inequality), which is used in addition to the classical De Giorgi's iterations and Moser's method.

Influence of Surfactant Structure on Polydisperse Formulations of Alkyl Ether Sulfates and Alkyl Amidopropyl Betaines.

Surfactants provide detergency, foaming, and texture in personal care formulations, yet the micellization of typical industrial primary and cosurfactants is not well understood, particularly in light of the polydisperse nature of commercial surfactants. Synergistic interactions are hypothesized to drive the formation of elongated wormlike self-assemblies in these mixed surfactant systems. Small-angle neutron scattering, rheology, and pendant drop tensiometry are used to examine surface adsorption, viscoelasticity, and self-assembly structure for wormlike micellar formulations comprising cocoamidopropyl betaine, and its two major components laurylamidopropyl betaine and oleylamidopropyl betaine, with sodium alkyl ethoxy sulfates. The tail length of sodium alkyl ethoxy sulfates was related to their ability to form wormlike micelles in electrolyte solutions, indicating that a tail length greater than 10 carbons is required to form wormlike micelles in NaCl solutions, with the decyl homologue unable to form elongated micelles and maintaining a low viscosity even at 20 wt % surfactant loading with 4 wt % NaCl present. For these systems, the incorporation of a disperse ethoxylate linker does not enable shorter chain surfactants to elongate into wormlike micelles for single-component systems; however, it could increase the interactions between surfactants in mixed surfactant systems. For synergy in surfactant mixing, the nonideal regular solution theory is used to study the sulfate/betaine mixtures. Tail mismatch appears to drive lower critical micelle concentrations, although tail matching improves synergy with larger relative reductions in critical micelle concentrations and greater micelle elongation, as seen by both tensiometric and scattering measurements.

Stochastic linear regularization methods: random discrepancy principle and applications

The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.

Understanding fast adsorption in single-solute breakthrough curves

To design efficient adsorption separations and select high-performing materials, understanding the relationship between adsorption equilibria and the movement of solute waves along the fluid is critical. While approximate-analytical methods are useful to combat nonlinearities in the differential equations and efficiently investigate the myriad of materials, their utility is limited by the lack of a clear physical interpretation and mathematical relationship to well-established models. In this work, we show that the two popular methods can be understood with regular perturbation theory and boundary layer theory in small ɛ, where a dilute solute travels from fluid to solid much faster than along the column. We consider conditions associated with a popular experimental technique used to characterize solute wave movement, where the column is initially at steady state and the fluid concentrations exiting, or breaking through, are time-monitored in response to a step change in inlet concentration. Our transformations and asymptotic analyses illustrate the time scales associated with popular techniques. We generalize their applicability to equilibria possessing inflection points while considering both adsorption and desorption modes. We use numerical simulations to estimate convergence orders of asymptotic approximations of breakthrough concentrations. For the leading-order approximation of regular perturbation theory, rarefaction waves possess positive convergence orders, depending on the extent of nonlinearity, while shock waves do not. Boundary-layer theory affords a more accurate approximation of the breakthrough curve than a shock wave, without guaranteeing a positive convergence order. This lack of positive convergence order is likely due to the evolution of the boundary layer, as illustrated with numerical simulations. An improved understanding of fast adsorption sets the stage for more realistic selection of high-performing materials and design configurations from thousands of available candidates.

Partial regularity of suitable weak solution to a three-dimensional fractional parabolic-elliptic chemotaxis-Navier–Stokes system

In this paper, we investigate a fractional parabolic-elliptic chemotaxis-Navier–Stokes system in spatial dimensions three and obtain the global existence of the suitable weak solution by a contraction mapping theorem. Furthermore, we improve the regularity of the solution through a local maximal L p regularity estimate for the fractional heat equation such that the suitable weak solution is smooth away from a closed set whose one-dimensional parabolic Hausdorff measure is zero, which extends the partial regularity theory of Caffarelli, Kohn and Nirenberg [] on the Navier–Stokes equation to the fractional parabolic-elliptic chemotaxis-Navier–Stokes system.

Effect of glycerol on micellization of quasi-binary mixture of zwitterionic cocamidopropyl hydroxysultaine and cetyltrimethylammonium bromide in aqueous solution

The micellization behavior of quasi-binary mixture of zwitterionic cocamidopropyl hydroxysultaine (CAPHS) with cationic cetyltrimethylammonium bromide (CTAB) was investigated both in aqueous solution and in glycerol-water. Both the tensiometric and the conductive techniques were adopted to determine the critical micelle concentrations (cmc) of individual and mixed surfactants. In the framework of the regular solution theory and based on the pseudophase separation model, some micellization and thermodynamic parameters were estimated by the Rubingh’s and the Clint’s models. The results indicate that the nonideal mixing of CAPHS with CTAB and the synergistic interaction between them play vital roles on the reduction of mixed cmc, the amplitude of synergistic interaction depends on the composition of mixture and the presence of glycerol induces the increase in the individual and mixed cmc values and the change of the participation of CAPHS in mixed micelle with the composition in bulk solution. The electrostatic repulsion, the diversity of molecular conformation and the steric hindrance can account for the formation of mixed micelle and the difference in micellar composition. Thermodynamic parameters show that the entropy makes overwhelming contributions on the spontaneous process of micellization, the release of mixing heat makes the stable mixed micelle formed and the presence of glycerol makes for reducing the contribution of entropy and forming the more stable mixed micelle. These findings help with understanding the interaction behavior between two species and the effect of glycerol and further providing some information for the design of the related surfactant formulations.

Sharp Morrey regularity theory for a fourth order geometrical equation

Sharp Morrey regularity theory for a fourth order geometrical equation

Regularity theory for parabolic systems with Uhlenbeck structure

We establish local regularity theory for parabolic systems of Uhlenbeck type with φ-growth. In particular, we prove local boundedness of weak solutions and their gradient, and then local Hölder continuity of the gradients, providing suitable assumptions on the growth function φ. Our approach, being independent of the degeneracy of the system, allows for a unified treatment of both the degenerate and the singular case.

Regularization-Theory-Based Fast Torque Tracking Method for Interior Permanent Magnet Synchronous Machines

With the rapid growth of interior permanent magnet synchronous machines in electric vehicle applications, there is a need to generate torque tracking look-up tables, that can both track the torque command and implement maximum torque per ampere (MTPA)/maximum torque per volt (MTPV). So far, most torque tracking methods require a large amount of test points, giving rise to long test time and workloads. This paper proposes a fast torque tracking MTPA/MTPV look-up table generating method to improve the efficiency. The proposed method is based on a machine learning regularization theory, using a L1/L2 regularization to establish a data-driven torque tracking model. Then a Lagrange dual principle is introduced to solve the unknown parameters, so that the look-up tables of optimal d-q axis currents are yielded by a global optimization solver. Experimental results show that the proposed method can generate the look-up tables with the same accuracy as classical methods, but requires less test points and testing time. As a result, the testing work loads are reduced, as the time cost is only 10-15% of the classical methods.

Second-Order Estimates for Transition Layers and a Curvature Estimate for the Parabolic Allen–Cahn

Abstract The parabolic Allen–Cahn equation is a semilinear partial differential equation that is closely linked to the mean curvature flow by a singular perturbation. Motivated by the work of Wang–Wei [ 21] and Chodosh–Mantoulidis [ 3] in the elliptic setting, we initiate the corresponding regularity theory for parabolic Allen–Cahn flows. In particular, we establish an improved convergence property of parabolic Allen–Cahn flows to the mean curvature flow: if the phase-transition level sets converge in $C^{2}$, then they converge in $C^{2,\theta }$ as well. As an application, we obtain a curvature estimate for the parabolic Allen–Cahn equation, which can be seen as a diffused version of Brakke’s [ 1] and White’s [ 24] regularity theorems for mean curvature flow.

Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Impedance Boundary Conditions

AbstractThe time-harmonic Maxwell equations at high wavenumberkin domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly inkand an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of orderpon a mesh with mesh sizehis shown under thek-explicit scale resolution condition that (a)kh/pis sufficient small and (b)$$p/\ln k$$p/lnkis bounded from below.

Invertible residual networks in the context of regularization theory for linear inverse problems

Abstract Learned inverse problem solvers exhibit remarkable performance in applications like image reconstruction tasks. These data-driven reconstruction methods often follow a two-step procedure. First, one trains the often neural network-based reconstruction scheme via a dataset. Second, one applies the scheme to new measurements to obtain reconstructions. We follow these steps but parameterize the reconstruction scheme with invertible residual networks (iResNets). We demonstrate that the invertibility enables investigating the influence of the training and architecture choices on the resulting reconstruction scheme. For example, assuming local approximation properties of the network, we show that these schemes become convergent regularizations. In addition, the investigations reveal a formal link to the linear regularization theory of linear inverse problems and provide a nonlinear spectral regularization for particular architecture classes. On the numerical side, we investigate the local approximation property of selected trained architectures and present a series of experiments on the MNIST dataset that underpin and extend our theoretical findings.

Regularity theory and Green’s function for elliptic equations with lower order terms in unbounded domains

We consider elliptic operators in divergence form with lower order terms of the form Lu = -{{textrm{div}}}(A cdot nabla u + b u ) - c cdot nabla u - du, in an open set Omega subset mathbb {R}^n, n ge 3, with possibly infinite Lebesgue measure. We assume that the n times n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either b, c in L^{n,infty }_{text {loc}}({Omega }) and d in L_{text {loc}}^{frac{n}{2}, infty }(Omega ), or |b|^2, |c|^2, |d| in mathcal {K}_{text {loc}}(Omega ), where mathcal {K}_{text {loc}}(Omega ) stands for the local Stummel–Kato class. Let {mathcal {K}_{text {Dini}}}(Omega ) be a variant of mathcal {K}(Omega ) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of Lu = f - {{textrm{div}}}g, where f and |g|^2 in {mathcal {K}_{text {Dini}}}(Omega ) if, for q in [n, infty ), any of the following assumptions holds: (i) |b|^2, |d| in {mathcal {K}_{text {Dini}}}(Omega ) and either c in L^{n,q}_{text {loc}}(Omega ) or |c|^2 in mathcal {K}_{text {loc}}(Omega ); (ii) {{textrm{div}}}b +d le 0 and either b+c in L^{n,q}_{text {loc}}(Omega ) or |b+c|^2 in mathcal {K}_{text {loc}}(Omega ); (iii) -{{textrm{div}}}c + d le 0 and |b+c|^2 in {mathcal {K}_{text {Dini}}}(Omega ). We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming -{{textrm{div}}}c +d le 0, we construct the Green’s function associated with L satisfying quantitative estimates. Under the additional hypothesis |b+c|^2 in mathcal {K}'(Omega ), we show that it satisfies global pointwise bounds and also construct the Green’s function associated with the formal adjoint operator of L. An important feature of our results is that all the estimates are scale invariant and independent of Omega , while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.

Joint inversion of gravity and gravity gradient data using smoothed L0 norm regularization algorithm with sensitivity matrix compression

Improving efficiency and accuracy are critical issues in geophysical inversion. In this study, a new algorithm is proposed for the joint inversion of gravity and gravity gradient data. Based on the regularization theory, the objective function is constructed using smoothed L0 norm (SL0), then the optimal solution is obtained by the non-linear conjugate gradient method. Numerical modeling shows that our algorithm is much more efficient than the conventional SL0 based on the sparse theory, especially when inverting large-scale data, and also has better anti-noise performance while preserving its advantage of high accuracy. Compressing the sensitivity matrices has further improved efficiency, and introducing the data weighting and the self-adaptive regularization parameter has improved the convergence rate of the inversion. Moreover, the impacts of the depth weighting, model weighting, and density constraint are also analyzed. Finally, our algorithm is applied to the gravity and gravity gradient measurements at the Vinton salt dome. The inverted distribution range, thickness, and geometry of the cap rock are in good agreement with previous studies based on geological data, drilling data, seismic data, etc., validating the feasibility of this algorithm in actual geological conditions.