Nonsymmetric Operators Research Articles

The Immersed Boundary Conformal Method for Kirchhoff–Love and Reissner–Mindlin shells

This work utilizes the Immersed Boundary Conformal Method (IBCM) to analyze linear elastic Kirchhoff–Love and Reissner–Mindlin shell structures within an immersed domain framework. Immersed boundary methods involve embedding complex geometries within a background grid, which allows for great flexibility in modeling intricate shapes and features despite the simplicity of the approach. The IBCM method introduces additional layers conformal to the boundaries, allowing for the strong imposition of Dirichlet boundary conditions and facilitating local refinement. In this study, the construction of boundary layers is combined with high-degree spline-based approximation spaces to further increase efficiency. The Nitsche method, employing non-symmetric average operators, is used to couple the boundary layers with the inner patch, while stabilizing the formulation with minimal penalty parameters. High-order quadrature rules are applied for integration over cut elements and patch interfaces. Numerical experiments demonstrate the efficiency and accuracy of the proposed formulation, highlighting its potential for complex shell structures modeled through Kirchhoff–Love and Reissner–Mindlin theories. These tests include the generation of conformal interfaces, the coupling of Kirchhoff–Love and Reissner–Mindlin theories, and the simulation of a cylindrical shell with a through-the-thickness crack.

Non-symmetric over-time pooling using pseudo-grouping functions for convolutional neural networks

Convolutional Neural Networks (CNNs) are a family of networks that have become state-of-the-art in several fields of artificial intelligence due to their ability to extract spatial features. In the context of natural language processing, they can be used to build text classification models based on textual features between words. These networks fuse local features to generate global features in their over-time pooling layers. These layers have been traditionally built using the maximum function or other symmetric functions such as the arithmetic mean. It is important to note that the order of input local features is significant (i.e. the symmetry is not an inherent characteristic of the model). While this characteristic is appropriate for image-oriented CNNs, where symmetry might make the network robust to image rigid transformations, it seems counter-productive for text processing, where the order of the words is certainly important. Our proposal is, hence, to use non-symmetric pooling operators to replace the maximum or average functions. Specifically, we propose to perform over-time pooling using pseudo-grouping functions, a family of non-symmetric aggregation operators that generalize the maximum function. We present a construction method for pseudo-grouping functions and apply different examples of this family to over-time pooling layers in text-oriented CNNs. Our proposal is tested on seven different models and six different datasets in the context of engineering applications, e.g. text classification. The results show an overall improvement of the models when using non-symmetric pseudo-grouping functions over the traditional pooling function.

Inverse Problem Numerical Analysis of Forager Bee Losses in Spatial Environment without Contamination

We consider an inverse problem of recovering the mortality rate in the honey bee difference equation model, that tracks a forage honeybee leaving and entering the hive each day. We concentrate our analysis to the model without pesticide contamination in the symmetric spatial environment. Thus, the mathematical problem is formulated as a symmetric inverse problem for reaction coefficient at final time constraint. We use the overspecified information to transform the inverse coefficient problem to the forward problem with non-local terms in the differential operator and the initial condition. First, we apply semidiscretization in space to the new nonsymmetric differential operator. Then, the resulting non-local nonsymmetric system of ordinary differential equations (ODEs) is discretized by three iterative numerical schemes using different time stepping. Results of numerical experiments which compare the efficiency of the numerical schemes are discussed. Results from numerical tests with synthetic and real data are presented and discussed, as well.

Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

Let Ω⊂Rn+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R}^{n+1}$$\\end{document}, n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 2$$\\end{document}, be an open set with Ahlfors–David regular boundary satisfying the corkscrew condition. When Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is quantitatively connected (i.e., it satisfies the Harnack chain condition) it is known that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures (i.e., its membership to the class A∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\infty $$\\end{document}) is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting, it is also known that these properties are stable under Fefferman–Kenig–Pipher’s perturbations. Nonetheless, it has been an open problem to show whether, without connectivity, the previous two conditions remain equivalent or whether there is a Fefferman–Kenig–Pipher perturbation theory. In this paper we settle the question of whether, for general real elliptic operators in sets without any connectivity assumption, quantitative absolute continuity of the elliptic measure (expressed via a corona decomposition for the elliptic measure) and Carleson measure type estimates for bounded null solutions are equivalent. We show that any of these conditions is also equivalent to the fact that the Green function is comparable, in the corona sense, to the distance to the boundary. Our characterization has profound consequences. First, we extend Fefferman–Kenig–Pipher’s perturbation result to sets which are not necessarily connected, and this is the first result in this general setting. Second, in the case of the Laplacian, and more generally for L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-Kenig–Pipher non-symmetric operators with variable coefficients, our conditions characterize the uniform rectifiability of the boundary. Last, our results generalize previous work in settings where quantitative connectivity is assumed.

Spectral analysis of a family of nonsymmetric fractional elliptic operators

Spectral analysis of a family of nonsymmetric fractional elliptic operators

Refinements of the Euclidean Operator Radius and Davis–Wielandt Radius-Type Inequalities

This paper proves several new inequalities for the Euclidean operator radius, which refine some recent results. It is shown that the new results are much more accurate than the related, recently published results. Moreover, inequalities for both symmetric and non-symmetric Hilbert space operators are studied.

Heat kernels for time-dependent non-symmetric mixed Lévy-type operators

In this paper we establish the existence, uniqueness and regularity of heat kernels to a large class of time-inhomogeneous non-symmetric nonlocal operators with Dini's continuous kernels. Moreover, quantitative estimates including two-sided estimates, gradient and fractional derivative estimates of the heat kernels are obtained.

Newton–Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the bidomain model

A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton–Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations’ residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the bidomain model.

On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols

In this article we consider direct and inverse problems for α-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective “domain of dependence of the operator” and in several, adapted function spaces. This formulation allows one to avoid natural “gauges” which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only “one-sided operators” new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.

Non-reciprocal interactions enhance heterogeneity

We study a process of pattern formation for a generic model of species anchored to the nodes of a network where local reactions take place, and that experience non-reciprocal non-local long-range interactions, encoded by the network directed links. By assuming the system to exhibit a stable homogeneous equilibrium whenever only local interactions are considered, we prove that such equilibrium can turn unstable once suitable non-reciprocal non-local long-range interactions are allowed for. Stated differently, we propose sufficient conditions allowing for patterns to emerge by using a non-symmetric coupling, while initial perturbations about the homogeneous equilibrium always fade away by assuming reciprocal coupling, namely the latter is stable. The instability, precursor of the emerging spatio-temporal patterns, can be traced back, via a linear stability analysis, to the complex spectrum of an interaction non-symmetric Laplace operator. The proposed theory is then applied to several paradigmatic dynamical models largely used in the literature to study the emergence of patterns or synchronisation. Taken together, our results pave the way for the understanding of the many and heterogeneous patterns of complexity found in ecological, chemical or physical systems composed by interacting parts, once no diffusion takes place.

Conservativeness and uniqueness of invariant measures related to non-symmetric divergence type operators

ABSTRACT We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by -uniqueness and imply that a given infinitesimally invariant measure becomes an invariant measure. We explore further conditions on the coefficients of the partial differential operators that ensure the uniqueness of the invariant measure beyond the case where the corresponding semigroups are recurrent. A main observation is that for conservativeness and uniqueness of invariant measures in this article, no growth conditions are required for the partial derivatives related to the anti-symmetric matrix of functions that determine a part of the drift coefficient. As stochastic counterparts, our results can be applied to show not only the existence of a pathwise unique and strong solution up to infinity to a corresponding Itô-SDE, but also the existence and uniqueness of invariant measures for the family of strong solutions.

Harmonic extension technique for non-symmetric operators with completely monotone kernels

We identify a class of non-local integro-differential operators K in mathbb {R} with Dirichlet-to-Neumann maps in the half-plane mathbb {R}times (0, infty ) for appropriate elliptic operators L. More precisely, we prove a bijective correspondence between Lévy operators K with non-local kernels of the form nu (y - x), where nu (x) and nu (-x) are completely monotone functions on (0, infty ), and elliptic operators L= a(y) partial _{xx} + 2 b(y) partial _{x y} + partial _{yy}. This extends a number of previous results in the area, where symmetric operators have been studied: the classical identification of the Dirichlet-to-Neumann operator for the Laplace operator in mathbb {R}times (0, infty ) with -sqrt{-partial _{xx}}, the square root of one-dimensional Laplace operator; the Caffarelli–Silvestre identification of the Dirichlet-to-Neumann operator for nabla cdot (y^{1 - alpha } nabla ) with (-partial _{xx})^{alpha /2} for alpha in (0, 2); and the identification of Dirichlet-to-Neumann maps for operators a(y) partial _{xx} + partial _{yy} with complete Bernstein functions of -partial _{xx} due to Mucha and the author. Our results rely on recent extension of Krein’s spectral theory of strings by Eckhardt and Kostenko.

Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem

In this article, we discuss a solution to time‐fractional diffusion equation with the homogeneous Dirichlet boundary condition, where an elliptic operator is not necessarily symmetric. We prove that the solution is identically zero if its normal derivative with respect to the operator vanishes on an arbitrarily chosen subboundary of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition for a nonsymmetric elliptic operator. As a direct application, we prove the uniqueness result for an inverse problem on determining the spatial component in the source term by Neumann boundary data on subdoundary.

On the Dragomir Extension of Furuta’s Inequality and Numerical Radius Inequalities

In this work, some numerical radius inequalities based on the recent Dragomir extension of Furuta’s inequality are obtained. Some particular cases are also provided. Among others, the celebrated Kittaneh inequality reads: wT≤12T+T*. It is proved that wT≤12T+T*−12infx=1Tx,x12−T*x,x122, which improves on the Kittaneh inequality for symmetric and non-symmetric Hilbert space operators. Other related improvements to well-known inequalities in literature are also provided.

Non-symmetric stable operators: Regularity theory and integration by parts

We study solutions to Lu=f in Ω⊂Rn, being L the generator of any, possibly non-symmetric, stable Lévy process.On the one hand, we study the regularity of solutions to Lu=f in Ω, u=0 in Ωc, in C1,α domains Ω. We show that solutions u satisfy u/dγ∈Cε∘(Ω‾), where d is the distance to ∂Ω, and γ=γ(L,ν) is an explicit exponent that depends on the Fourier symbol of operator L and on the unit normal ν to the boundary ∂Ω.On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features.Finally, we generalize the integration by parts identities in half spaces to the case of bounded C1,α domains. We do it via a new efficient approximation argument, which exploits the Hölder regularity of u/dγ. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.

Laplacian comparison theorem on Riemannian manifolds with modified $m$-Bakry-Emery Ricci lower bounds for $m\leq1$

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Bakry-Émery Ricci curvature under $m\geq n$ ([19, 21, 27, 33]) or $m=1$ ([34, 35]) if the vector field is a gradient type. When $m<1$, our results are new in the literature.

Nonlinear acceleration of momentum and primal-dual algorithms

We describe convergence acceleration schemes for multistep optimization algorithms where the underlying fixed-point operator is not symmetric. In particular, our analysis handles algorithms with momentum terms such as Nesterov’s accelerated method or primal-dual methods. The acceleration technique combines previous iterates through a weighted sum, whose coefficients are computed via a simple linear system. We analyze performance in both online and offline modes, and we study in particular a variant of Nesterov’s method that uses nonlinear acceleration at each iteration. We use Crouzeix’s conjecture to show that acceleration performance is controlled by the solution of a Chebyshev problem on the numerical range of a non-symmetric operator modeling the behavior of iterates near the optimum. Numerical experiments are detailed on logistic regression problems.

The principal transmission condition

&lt;abstract&gt;&lt;p&gt;The paper treats pseudodifferential operators $ P = \operatorname{Op}(p(\xi)) $ with homogeneous complex symbol $ p(\xi) $ of order $ 2a &amp;gt; 0 $, generalizing the fractional Laplacian $ (-\Delta)^a $ but lacking its symmetries, and taken to act on the halfspace ${\mathbb R}^n_+$. The operators are seen to satisfy a principal $ \mu $-transmission condition relative to ${\mathbb R}^n_+$, but generally not the full $ \mu $-transmission condition satisfied by $ (-\Delta)^a $ and related operators (with $ \mu = a $). However, $ P $ acts well on the so-called $ \mu $-transmission spaces over ${\mathbb R}^n_+$ (defined in earlier works), and when $ P $ moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for $ P $, leading to regularity results with a factor $ x_n^\mu $ (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over ${\mathbb R}^n_+$ for $ P $ acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal $ \mu $-transmission condition has weaker requirements than the full $ \mu $-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.&lt;/p&gt;&lt;/abstract&gt;

An iterative residual-based procedure for solving coupled elastoplastic damage problems

In the computational damage mechanics, the coupled elastoplastic damage model involves a nonsymmetric tangent operator. This implies that a sparse matrix solver is needed, which means more computational cost to solve the problem. In the context of Lemaitre’s coupled damage model, a new iterative residual-based procedure is proposed to solve the Lemaitre’s coupled elastoplastic damage problems. The elastoplastic damaged tangent operator is decomposed into symmetric and nonsymmetric parts. The symmetric part of the tangent operator is contributed to the overall stiffness matrix, while the nonsymmetric part is considered as an additional residual force contributed to the overall force vector. The symmetry and banded format of the stiffness matrix is maintained, consequently the computational costs are largely reduced. The Lemaitre coupled damage model is implemented in ABAQUS commercial Software using a developed user material user subroutine, while the proposed procedure is implemented into a nonlinear 2 D FE model. Two different problems are analyzed to illustrate the applicability and effectiveness of the proposed iterative residual-based procedure. The analysis show that results of the Lemaitre’s coupled damage model and the proposed procedure are very close.

Distributions for Nonsymmetric Monotone and Weakly Monotone Position Operators

We study the vacuum distribution, under an appropriate scaling, of a family of partial sums of nonsymmetric position operators on weakly monotone and monotone Fock spaces, respectively. We preliminary treat the case of weakly monotone Fock space, and show that any single operator has the vacuum law belonging to the free Meixner class. After establishing some relations between the combinatorics of Motzkin and Riordan paths, we give a recursive formula for the vacuum moments of the law of any finite sum. Since the operators are monotone independent, the distribution is the monotone convolution of the free Meixner law above. We also investigate the asymptotic measure for these sums, which can be seen as “Poisson type” limit law. It turns out to belong to the free Meixner class, with an atomic and an absolutely continuous part (w.r.t. the Lebesgue measure). Finally, we briefly apply analogous considerations to the case of monotone Fock space.