Local Monotonicity Research Articles

Second-order strongly nonlinear impulsive coupled systems

This paper presents sufficient conditions for the solvability of a second-order coupled system, composed of two differential equations involving different laplacians applied to fully discontinuous nonlinearities, two-point boundary conditions, and generalized impulsive effects. Applying the lower and upper solutions technique and Schauder’s fixed point theorem, it is obtained an existence and localization theorem, based on local monotone properties on the nonlinearities and the impulsive functions.

Large Deviation Principle for a Class of Stochastic Partial Differential Equations with Fully Local Monotone Coefficients Perturbed By Lévy Noise

Large Deviation Principle for a Class of Stochastic Partial Differential Equations with Fully Local Monotone Coefficients Perturbed By Lévy Noise

Wong–Zakai Approximation for a Class of SPDEs with Fully Local Monotone Coefficients and Its Application

Wong–Zakai Approximation for a Class of SPDEs with Fully Local Monotone Coefficients and Its Application

Well-posedness of stochastic partial differential equations with fully local monotone coefficients

Well-posedness of stochastic partial differential equations with fully local monotone coefficients

The critical variational setting for stochastic evolution equations

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.

Local Monotone Operator Learning Using Non-Monotone Operators: MnM-MOL

Local Monotone Operator Learning Using Non-Monotone Operators: MnM-MOL

An Incremental Gradient Method for Optimization Problems With Variational Inequality Constraints

We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinear complementarity problems arising in transportation networks. We develop an iteratively regularized incremental gradient method where at each iteration, agents communicate over a directed cycle graph to update their solution iterates using their local information about the objective and the mapping. The proposed method is single-timescale in the sense that it does not involve any excessive hard-to-project computation per iteration. We derive non-asymptotic agent-wise convergence rates for the suboptimality of the global objective function and infeasibility of the VI constraints measured by a suitably defined dual gap function. The proposed method appears to be the first fully iterative scheme equipped with iteration complexity that can address distributed optimization problems with VI constraints over cycle graphs.

Mean attractors and invariant measures of locally monotone and generally coercive SPDEs driven by superlinear noise

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α-model, convective Brinkman-Forchheimer equations, fractional (s,p)-Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others.

Analysis of the single reference coupled cluster method for electronic structure calculations: the full-coupled cluster equations

The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an L^2-type Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high-accuracy quantum chemical simulations. The existing numerical analysis of coupled cluster methods relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.

It is never too LATE: a new look at local average treatment effects with or without defiers

Summary In heterogeneous treatment effect models with endogeneity, identification of the local average treatment effect (LATE) typically relies on the availability of an exogenous instrument monotonically related to treatment participation. First, we demonstrate that a strictly weaker local monotonicity condition—invoked for specific potential outcome values rather than globally—identifies the LATEs on compliers and defiers. Second, we show that our identification results apply to subsets of compliers and defiers when imposing an even weaker local compliers-defiers assumption that allows for both types at any potential outcome value. We propose estimators that are potentially more efficient than two-stage least squares (2SLS) in finite samples, even in cases where 2SLS is consistent. Finally, we provide an empirical application to estimating returns to education using the quarter of birth instrument.

Orbital stability of the sum of N peakons for the mCH-Novikov equation

ABSTRACT This paper investigates a generalized Camassa–Holm equation with cubic nonlinearities (alias the mCH-Novikov equation), which is a generalization of some special equations. The mCH-Novikov equation possesses well-known peaked solitary waves that are called peakons. The peakons were proved orbital stable by Chen et al. in [Stability of peaked solitary waves for a class of cubic quasilinear shallow-water equations. Int Math Res Not. 2022;1–33]. We mainly prove the orbital stability of the multi-peakons in the mCH-Novikov equation. In this paper, it is proved that the sum of N fully decoupled peaks is orbitally stable in the energy space by using energy argument, combining the orbital stability of single peakons and local monotonicity of the method.

Monotonicity-Based Interval Analysis Methods of Half-Wave Wall Radomes With Thickness and Permittivity Errors

The complex relation between permittivity and transmission coefficients has been a great obstacle to the interval-based tolerance analysis of radomes. This work presents two monotonicity-based methods (global/local) to facilitate precise interval analysis (IA) of half-wave-wall radomes with thickness and permittivity errors. The global monotonicity method reveals that the key parameters of radome transmission coefficients decrease monotonously with permittivity under a loose constraint in the form of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\vert $ </tex-math></inline-formula> tan(x) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\vert \le \text {x}$ </tex-math></inline-formula> , which makes the IA of permittivity errors as simple/accurate as thickness errors. The local monotonicity method, with no premises and less accuracy, treats the two correlative parts of the key parameters independently and can be a potential supplement to the global one. Results of spherical radomes and tangent ogival radomes indicate that the monotonicity-based methods can well improve the IA precision, and the global monotonicity method applies to general cases.

Lazy Rearrangement Planning in Confined Spaces

Object rearrangement is important for many applications but remains challenging, especially in confined spaces, such as shelves, where objects cannot be accessed from above and they block reachability to each other. Such constraints require many motion planning and collision checking calls, which are computationally expensive. In addition, the arrangement space grows exponentially with the number of objects. To address these issues, this work introduces a lazy evaluation framework with a local monotone solver and a global planner. Monotone instances are those that can be solved by moving each object at most once. A key insight is that reachability constraints at the grasps for objects' starts and goals can quickly reveal dependencies between objects without having to execute expensive motion planning queries. Given that, the local solver builds lazily a search tree that respects these reachability constraints without verifying that the arm paths are collision free. It only collision checks when a promising solution is found. If a monotone solution is not found, the non-monotone planner loads the lazy search tree and explores ways to move objects to intermediate locations from where monotone solutions to the goal can be found. Results show that the proposed framework can solve difficult instances in confined spaces with up to 16 objects, which state-of-the-art methods fail to solve. It also solves problems faster than alternatives, when the alternatives find a solution. It also achieves high-quality solutions, i.e., only 1.8 additional actions on average are needed for non-monotone instances.

Higher order monotonicity and submodularity of influence in social networks: From local to global

Kempe, Kleinberg and Tardos (KKT) proposed the following conjecture about the general threshold model in social networks: local monotonicity and submodularity implies global monotonicity and submodularity. That is, if the threshold function of every node is monotone and submodular, then the spread function is monotone and submodular. The correctness of this conjecture has been proved by Mossel and Roch. In this paper, we first provide the concept AD-k (Alternating Difference-k) as a generalization of monotonicity and submodularity. Specifically, a set function f is called AD-k if all the ℓ-th order differences of f on all inputs have sign (−1)ℓ+1 for every ℓ≤k. We propose a refined version of KKT's conjecture: in the general threshold model, local AD-k implies global AD-k. We prove the correctness of our conjecture when the social graph is a DAG. Furthermore, we affirm our conjecture on general social graphs when k=∞.

Psychometric properties of the patient-reported outcomes measurement information system scale v1.2: global health (PROMIS-GH) in a Dutch general population

PurposeTo assess the psychometric properties of the Dutch-Flemish Patient-Reported Outcome Measurement Information System Scale v1.2 – Global Health (PROMIS-GH).MethodsThe PROMIS-GH (also referred to as PROMIS-10) was administered to 4370 persons from the Dutch general population. Unidimensionality (CFI ≥ 0.95; TLI ≥ 0.95; RMSEA ≤ 0.06; SRMR ≤ 0.08), local independence (residual correlations < 0.20), monotonicity (H > 0.30), model fit with the Graded Response Model (GRM, p < 0.001), internal consistency (alpha > 0.75), precision (total score information across the latent trait), measurement invariance (no Differential Item Functioning [DIF]), and cross-cultural validity (no DIF for language, Dutch vs. United States English) of its subscales, composed of four items each, Global Mental Health (GMH) and Global Physical Health (GPH), were assessed.ResultsConfirmatory factor analyses, on both subscales, revealed slight departures from unidimensionality for GMH (CFI = 0.98; TLI = 0.95, RMSEA = 0.22; SRMR = 0.04) and GPH (CFI = 0.99; TLI = 0.97; RMSEA = 0.12; SRMR = 0.03). Local independence, monotonicity, GRM model fit, internal consistency, precision and cross-cultural validity were supported. However, Global10 (emotional problems) showed misfit on the GMH subscale, while Global08 (fatigue) presented DIF for age.ConclusionThe psychometric properties of the PROMIS-GH in the Dutch population were considered acceptable. Sufficient local independence, monotonicity, GRM fit, internal consistency, measurement invariance and cross-cultural validity were found. If future studies find similar results, structural validity of the GMH could be enhanced by improving or replacing Global10 (emotional problems).

Periodic Third-Order Problems with a Parameter

This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments are based on a new type of lower and upper solutions method, not necessarily well ordered. A Nagumo growth condition and Leray–Schauder’s topological degree theory are the existence tools. Only the existence of solution is studied here and it will remain open the discussion on the non-existence and the multiplicity of solutions. Last section contains a nonlinear third-order differential model for periodic catatonic phenomena, depending on biological and/or chemical parameters.

Local Regularity for the Harmonic Map and Yang\u2013Mills Heat Flows

We establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

Preference disaggregation method for value-based multi-decision sorting problems with a real-world application in nanotechnology

We consider a problem of multi-decision sorting subject to multiple criteria. In the newly formulated decision problem, besides performances on multiple criteria, alternatives get evaluations on multiple interrelated decision attributes involving preference-ordered classes. We propose a dedicated method for dealing with such a problem, incorporating a threshold-based value-driven sorting procedure. The Decision Maker (DM) is expected to holistically evaluate a subset of reference alternatives by indicating the quality or risk level on a pre-defined scale of each decision attribute. Based on these evaluations, we construct a set of interrelated preference models, one for each decision attribute, compatible with intra- and inter-decision constraints imposed by such indirect preference information. We also formulate a new way of dealing with potentially non-monotonic criteria by discovering local monotonicity changes in different performance scale regions. The marginal value functions for criteria with unknown monotonicity are represented as a sum of two value functions assuming opposing preference directions, one non-decreasing and the other non-increasing. This permits to obtain an aggregated marginal value function with an arbitrary non-monotonic shape. The practical usefulness of the approach is demonstrated on a case study concerning risk management related to handling (i.e., production, use, manipulation, and processing) nanomaterials in different conditions. We analyze the expert judgments and discuss the inferred preference models, which can be applied to support health and safety managers in reducing the possible risk associated with the respective exposure scenario.

A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise

We show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the supremum norm. In this set-up, the usual proof using the ordinary Gronwall lemma together with the Burkholder-Davis-Gundy inequality seems impossible. In order to solve this problem, we prove a new and quite general stochastic Gronwall lemma for cadlag martingales using Lenglart's inequality.

Non-uniform Gradient Estimates for SDEs with Local Monotonicity Conditions

By using the coupling method and the localization technique, we establish non-uniform gradient estimates for Markov semigroups of diffusions or stochastic differential equations driven by pure jump Levy noises, where the coefficients only satisfy local monotonicity conditions.