Minimal Smoothness Research Articles

A second-order structure-preserving discretization for the Cahn-Hilliard/Allen-Cahn system with cross-kinetic coupling

We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.

Propagation of chaos for topological interactions by a coupling technique

We consider a system of particles that interact through a jump process. The jump intensities are functions of the proximity rank of the particles, a type of interaction referred to as topological in the literature. Such interactions have been shown relevant for the modelling of bird flocks. We show that, in the large number of particles limit and under minimal smoothness assumptions on the data, the model converges to a kinetic equation which was derived in earlier works both formally and rigorously under more stringent regularity assumptions. The proof relies on the coupling method which assigns to the particle and limiting processes a joint process posed on the cartesian product of the two configuration spaces of the former processes. By appropriate estimates in a suitable Wasserstein metric, we show that the distance between the two processes tends to zero as the number of particles tends to infinity, with an error typical of the law of large numbers.

The commutator of the Bergman projection on strongly pseudoconvex domains with minimal smoothness

Consider a bounded, strongly pseudoconvex domain D⊂Cn with minimal smoothness (namely, the class C2) and let b be a locally integrable function on D. We characterize boundedness (resp., compactness) in Lp(D),p>1, of the commutator [b,P] of the Bergman projection P in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on b. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

On the Effect of Irregularity of the Domain Boundary on the Solution of a Boundary Value Problem for the Laplace Equation

We consider an inhomogeneous boundary value problem with mixed boundary conditions for the Laplace equation in a domain representing a perturbationof a rectangle where one of its sides is replaced by some curve of minimal smoothness. An estimate is obtained for the difference between the solutions of the perturbed and unperturbed problems in the norm of the Sobolev space on their common domain.

Analysis of a mixed discontinuous Galerkin method for the time-harmonic Maxwell equations with minimal smoothness requirements

AbstractAn error analysis of a mixed discontinuous Galerkin (DG) method with lifting operators as numerical fluxes for the time-harmonic Maxwell equations with minimal smoothness requirements is presented. The key difficulty in the error analysis for the DG method is that due to the low regularity the tangential trace of the exact solution is not well defined on the faces of the computational mesh. This difficulty is addressed by adopting the face-to-cell lifting introduced by Ern & Guermond (2021, Quasi-optimal nonconforming approximation of elliptic PDEs with contrasted coefficients and $H^{1+r}$, $r>0$, regularity. Found. Comput. Math., 1–36). To obtain optimal local interpolation estimates, we introduce Scott–Zhang-type interpolations that are well defined for $H(\textrm {curl})$ and $H(\textrm {div})$ functions with minimal regularity requirements. As a by-product of penalizing the lifting of the tangential jumps, an explicit and easily computable stabilization parameter is given.

Semicontinuity of automorphism groups in one complex variable

We prove a version of the Greene-Krantz theorem on semicontinuity of automorphism groups in the context of one complex variable with minimal boundary smoothness.

Weighted endpoint bounds for the Bergman and Cauchy-Szego projections on domains with near minimal smoothness

Weighted endpoint bounds for the Bergman and Cauchy-Szego projections on domains with near minimal smoothness

Small defects reconstruction in waveguides from multifrequency one-side scattering data

<p style='text-indent:20px;'>Localization and reconstruction of small defects in acoustic or electromagnetic waveguides is of crucial interest in nondestructive evaluation of structures. The aim of this work is to present a new multi-frequency inversion method to reconstruct small defects in a 2D waveguide. Given one-side multi-frequency wave field measurements of propagating modes, we use a Born approximation to provide a <inline-formula><tex-math id="M1">\begin{document}$ \text{L}^2 $\end{document}</tex-math></inline-formula>-stable reconstruction of three types of defects: a local perturbation inside the waveguide, a bending of the waveguide, and a localized defect in the geometry of the waveguide. This method is based on a mode-by-mode spacial Fourier inversion from the available partial data in the Fourier domain. Indeed, in the available data, some high and low spatial frequency information on the defect are missing. We overcome this issue using both a compact support hypothesis and a minimal smoothness hypothesis on the defects. We also provide a suitable numerical method for efficient reconstruction of such defects and we discuss its applications and limits.</p>

AVERAGE DENSITY ESTIMATORS: EFFICIENCY AND BOOTSTRAP CONSISTENCY

This paper highlights a tension between semiparametric efficiency and bootstrap consistency in the context of a canonical semiparametric estimation problem, namely the problem of estimating the average density. It is shown that although simple plug-in estimators suffer from bias problems preventing them from achieving semiparametric efficiency under minimal smoothness conditions, the nonparametric bootstrap automatically corrects for this bias and that, as a result, these seemingly inferior estimators achieve bootstrap consistency under minimal smoothness conditions. In contrast, several “debiased” estimators that achieve semiparametric efficiency under minimal smoothness conditions do not achieve bootstrap consistency under those same conditions.

Local finite element approximation of Sobolev differential forms

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.

H$$^2$$-Korn’s Inequality and the Nonconforming Elements for The Strain Gradient Elastic Model

We establish a new H\(^2\)-Korn’s inequality and its discrete analog, which greatly simplify the construction of nonconforming elements for a linear strain gradient elastic model. The Specht triangle (Specht in Int J Numer Methods Eng 28:705–715, 1988) and the NZT tetrahedron (Wang et al. in Numer Math 106:335–347, 2007) are analyzed as two typical representatives for robust nonconforming elements in the sense that the rate of convergence is independent of the small material parameter. We construct the regularized interpolation operators and the enriching operators for both elements, and prove the error estimates under minimal smoothness assumption on the solution. Numerical results for the smooth solution, and the solution with boundary layer are consistent with the corresponding theoretical prediction.

Weighted Lp estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness

We prove the weighted Lp regularity of the ordinary Bergman and Cauchy-Szegő projections on strongly pseudoconvex domains D in Cn with near minimal smoothness for appropriate generalizations of the Bp/Ap classes. In particular, the Bp/Ap Muckenhoupt type condition is expressed relative to balls in a quasi-metric that arises as a space of homogeneous type on either the interior or the boundary of the domain D.

A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only

The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without boundary information. We provide a general approach based on the weak definition of the stiffness-toforce operator which conduces to see the problem as a linear system. We prove that in the case of shear modulus reconstruction, we have an L 2-stability with only one measurement under minimal smoothness assumptions. This stability result is obtained though the proof that the linear operator to invert has closed range. We then describe a direct discretization which provides stable reconstructions of both isotropic and anisotropic stiffness tensors.

Total variation multiscale estimators for linear inverse problems

AbstractEven though the statistical theory of linear inverse problems is a well-studied topic, certain relevant cases remain open. Among these is the estimation of functions of bounded variation ($BV$), meaning $L^1$ functions on a $d$-dimensional domain whose weak first derivatives are finite Radon measures. The estimation of $BV$ functions is relevant in many applications, since it involves minimal smoothness assumptions and gives simplified, interpretable cartoonized reconstructions. In this paper, we propose a novel technique for estimating $BV$ functions in an inverse problem setting and provide theoretical guaranties by showing that the proposed estimator is minimax optimal up to logarithms with respect to the $L^q$-risk, for any $q\in [1,\infty )$. This is to the best of our knowledge the first convergence result for $BV$ functions in inverse problems in dimension $d\geq 2$, and it extends the results of Donoho (1995, Appl. Comput. Harmon. Anal., 2, 101–126) in $d=1$. Furthermore, our analysis unravels a novel regime for large $q$ in which the minimax rate is slower than $n^{-1/(d+2\beta +2)}$, where $\beta$ is the degree of ill-posedness: our analysis shows that this slower rate arises from the low smoothness of $BV$ functions. The proposed estimator combines variational regularization techniques with the wavelet-vaguelette decomposition of operators.

Spectral Characteristics of the Sturm-Liouville Operator under Minimal Restrictions on Smoothness of Coefficients

In this paper we consider the Sturm-Liouville problem in general form with Dirichlet boundary conditions under the minimal smoothness assumptions for the coefficients. We obtain the asymptotics formulas for eigenvalues and eigenfunctions of this problem. Under the assumption that the Lp-norm of eigenfunctions is equal to 1, we get uniform estimates of the Chebyshev norm.

ADMM-Based Distributed Optimization of Hybrid MTDC-AC Grid for Determining Smooth Operation Point

Accompanied by the rising fashion of distributed energy resources requiring distributed optimization is becoming more prevalent among power system. However, research for distributed optimization to determine smooth operation point (SOP) has attracted less attention in hybrid multi-terminal high voltage direct current (MTDC) and thus, it is still a core issue. The proposed method mainly designed to deal with SOP, and this goal can be mainly achieved by taking minimal line losses and smoothness objectives (MLLSO), which is formulated as an MLLSO optimization model. Then, the MLLSO optimization is transformed to single-objective distributed optimization through weight normalization of both objective functions. A minimal line loss objective and smoothness objective-based optimal power flow (MLLOSOBOPF) model is built at first to effectively handle prevailing constraints for the power systems parameters. Then, the alternating direction method of multipliers (ADMM) is utilized to solve this model by breaking it into smaller pieces based on OPF, each of which is easier to handle. In addition, different sized power systems are integrated with the 14-bus and 30-bus hybrid MTDC-AC systems to increase the complexity of test systems. Finally, based on 14-bus and 30-bus hybrid MTDC-AC systems, the computational output and comparison of the proposed method with centralized optimization centralized MTDC show that the proposed method is feasible and effective for determining hybrid MTDC grid SOP.

Выявление сингулярности у обобщенного решения задачи Дирихле для уравнения типа эйконала в условиях минимальной гладкости границы краевого множества

Выявление сингулярности у обобщенного решения задачи Дирихле для уравнения типа эйконала в условиях минимальной гладкости границы краевого множества

Rigorous cubical approximation and persistent homology of continuous functions

The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function.

The Hörmander multiplier theorem, II: The bilinear local $$L^2$$ L 2 case

We use wavelets of tensor product type to obtain the boundedness of bilinear multiplier operators on $$\mathbb R^n\times \mathbb R^n$$ associated with Hormander multipliers on $$\mathbb R^{2n}$$ with minimal smoothness. We focus on the local $$L^2$$ case and we obtain boundedness under the minimal smoothness assumption of n / 2 derivatives. We also provide counterexamples to obtain necessary conditions for all sets of indices.

Fundamental Solution of the Cauchy Problem for the Shilov-Type Parabolic Systems with Coefficients of Bounded Smoothness

Under the condition of minimal smoothness of the coefficients, we construct the fundamental solution of the Cauchy problem and study the principal properties of this solution for a special class of linear parabolic systems with bounded variable coefficients covering the class of Shilov-type parabolic systems of nonnegative kind.