Solution Map Research Articles

Global dynamics of a diffusive viral infection model with spatial heterogeneity

To explore the joint impact of cell-free infection and cytokine-enhanced viral infection, we propose a PDE model with spatial heterogeneity by taking into account a general cell reproduction function, free-virus infection function and cytokine-enhanced viral infection function. Mathematical challenges lie in the facts that (i) the solution map of the model system loses its compactness; and (ii) the definition of the basic reproduction number and the global asymptotic stability of the infection-free steady state become challenging since the linear system at the infection-free steady state is constituted by three equations. We define the basic reproduction number R0 as the spectral radius of the sum of two linear operators corresponding to cell-free infection and cytokine-enhanced viral infection, and prove its threshold role: if R0<1, the infection-free steady state is globally asymptotically stable; if R0=1, the infection-free steady state is locally asymptotically stable; and if R0>1, the model system is uniformly persistent. A special case is given to show the global attractiveness of the infection steady state.

Sparse Polynomial Approximations for Affine Parametric Saddle Point Problems

In this work we study convergence properties of sparse polynomial approximations for a class of affine parametric saddle point problems, including the Stokes equations for viscous incompressible flow, mixed formulation of diffusion equations for groundwater flow, time-harmonic Maxwell equations for electromagnetics, etc. Due to the lack of knowledge or intrinsic randomness, the (viscosity, diffusivity, permeability, permittivity, etc.) coefficients of such problems are uncertain and can often be represented or approximated by high- or countably infinite-dimensional random parameters equipped with suitable probability distributions, and the coefficients affinely depend on a series of either globally or locally supported basis functions, e.g., Karhunen–Loève expansion, piecewise polynomials, or adaptive wavelet approximations. We consider sparse polynomial approximations of the parametric solutions, in particular sparse Taylor approximations, and study their convergence properties for these parametric problems. Under suitable sparsity assumptions on the parametrization of the random coefficients, we show the algebraic convergence rates O(N−r) for the sparse polynomial approximations of the parametric solutions based on the results for affine parametric elliptic PDEs (Cohen, A. et al.: Anal. Appl. 9, 11–47, 2011), (Bachmayr, M., et al.: ESAIM Math. Model. Numer. Anal. 51, 321–339, 2017), (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015), (Chkifa, A., et al.: J. Math. Pures Appl. 103, 400–428, 2015), (Chkifa, A., et al.: ESAIM Math. Model. Numer. Anal. 47, 253–280, 2013), (Cohen, A., Migliorati, G.: Contemp. Comput. Math., 233–282, 2018), with the rate r depending only on a sparsity parameter in the parametrization, not on the number of active parameter dimensions or the number of polynomial terms N. We note that parametric saddle point problems were considered in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2) with the anticipation that the same results on the approximation of the solution map obtained for elliptic PDEs can be extended to more general saddle point problems. In this paper, we consider a general formulation of saddle point problems, different from that presented in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2), and obtain convergence rates for the two variables, e.g., velocity and pressure in Stokes equations, which are different for the case of locally supported basis functions.

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result, specific for ideal efficient solutions to unperturbed problem and enhanced by related error bounds, is discussed. Some connections with the concept of robustness in multi-objective optimization are also sketched.

Positive solutions for a class of nonlocal problems with possibly singular nonlinearity

We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term f which is nonlocal depending on the \(L^{p}\)-norm of the unknown function. The nonlinearity f can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map, to produce multiplicity of classical positive solutions with ordered norms.

Several Dynamic Properties for the gkCH Equation

In this paper, we focus on a generalized Camassa–Holm equation (also known as a gkCH equation), which includes both the Camassa–Holm equation and Novikov equation as two special cases. Because of the potential applications in physics, we will further investigate the properties of the equation from a mathematical point of view. More precisely, firstly, we give a new wave-breaking phenomenon. Then, we present the theorem of existence and uniqueness of global weak solutions for the equation, provided that the initial data satisfy certain sign conditions. Finally, we prove the Hölder continuity of a solution map for the equation.

Stable circular and double-cell lean hydrogen-air premixed flames in quasi two-dimensional channels

In this work, a computational study on nearly two-dimensional reacting fronts under canonical configurations is proposed. Specifically, a quasi-2D description of the conservation equations with a one-step irreversible Arrhenius chemical model is used to simulate ultra-lean hydrogen-air premixed flames that slowly propagate over streams of reactants that are confined between plates and are subject to conductive heat losses. Under these conditions, recent previous experiments identified two unprecedented stable flame configurations: two-headed and circular isolated flames. This study identified buoyancy and the relative importance of conductive heat losses as the two parameters controlling the formation of these isolated flames. The numerical evaluation of the different terms ascertains the physical mechanisms that drive the formation of two-headed or circular flames. Additionally, the systematic variation of the controlling parameters depicts the maps of stable solutions that determine the capability of the two flame configurations to emerge. In particular, the existence of a range of parametric values in which both flame configurations are stable proves, also, that initial conditions are crucial to determine which of the two configurations prevails.

Nonlocal kernel network (NKN): A stable and resolution-independent deep neural network

Neural operators [1–5] have recently become popular tools for designing solution maps between function spaces in the form of neural networks. Differently from classical scientific machine learning approaches that learn parameters of a known partial differential equation (PDE) for a single instance of the input parameters at a fixed resolution, neural operators approximate the solution map of a family of PDEs [6,7]. Despite their success, the uses of neural operators are so far restricted to relatively shallow neural networks and confined to learning hidden governing laws. In this work, we propose a novel nonlocal neural operator, which we refer to as nonlocal kernel network (NKN), that is resolution independent, characterized by deep neural networks, and capable of handling a variety of tasks such as learning governing equations and classifying images. Our NKN stems from the interpretation of the neural network as a discrete nonlocal diffusion reaction equation that, in the limit of infinite layers, is equivalent to a parabolic nonlocal equation, whose stability is analyzed via nonlocal vector calculus. The resemblance with integral forms of neural operators allows NKNs to capture long-range dependencies in the feature space, while the continuous treatment of node-to-node interactions makes NKNs resolution independent. The resemblance with neural ODEs, reinterpreted in a nonlocal sense, and the stable network dynamics between layers allow for generalization of NKN's optimal parameters from shallow to deep networks. This fact enables the use of shallow-to-deep initialization techniques [8]. Our tests show that NKNs outperform baseline methods in both learning governing equations and image classification tasks and generalize well to different resolutions and depths.

Asymptotic profiles of a nonlocal dispersal SIR epidemic model with treat-age in a heterogeneous environment

Infectious diseases have a significant impact on human life, and additional efforts are required to contain them. Treatment measure is very helpful to contain the epidemic and protect infected persons from disease-related mortality. Therefore, we consider a SIR epidemic model with nonlocal diffusion modeled by a convolution operator and treat-age effect. We show the well-posedness of the solution to the problem that is existence, uniqueness and positivity, and boundedness. Next, we determine the corresponding basic reproduction number R0 that depends on the structure of studied bounded domain Ω∈RN, and we show its threshold role in determining the asymptotic profiles of the solution. Moreover, it is proved that the solution map has a global compact attractor. Indeed, for R0<1, there exist a Lipschitz functions Sp such that (Sp,0,0) is globally stable, which is related to the extinction scenario of the epidemic. However, for R0>1, we show the solution map is uniformly persistent and there exists a unique endemic steady state denoted (S∗,I∗,T∗) that is globally stable. The influence of the treat-age on the spatiotemporal and threshold profiles is discussed through the research.

On the initial value problem for the hyperbolic Keller-Segel equations in Besov spaces

In this paper, we first show by constructing a special initial data that the solution map for the one dimensional hyperbolic Keller-Segel equations (HKSE) starting from u0 is discontinuous at t=0 in the metric of B2,∞s(R), s>32. Then, we establish the Hadamard local well-posedness result for the high dimensional HKSE in the larger Besov spaces Bp,11+dp(Rd), 1≤p<∞, which improves the local theory proved by [Zhou, Zhang & Mu, J. Differ. Equ., 302(2021), pp.662-679]. Moreover, we investigate the inviscid limit of the Keller-Segel equations with small diffusivity ϵΔu as ϵ→0 in the same topology of Besov spaces as the initial data. Finally, we establish two kinds of blow-up criteria for strong solutions in Besov spaces by means of the Littlewood-Paley theory.

Diffusive host-pathogen model revisited: Nonlocal infections, incubation period and spatial heterogeneity

We formulate and analyze a general diffusive host-pathogen model with an incubation period and nonlocal infections in a spatially heterogeneous environment. The model system is partially degenerate and the solution map is not compact. We first prove that the solutions of the model exist globally and are ultimately bounded. Next, we define the basic reproduction number R0 as the spectral radius of the sum of two next generation operators corresponding to direct and indirect infection modes, and prove that R0 is decreasing with respect to the incubation period and the diffusion coefficient of infectious hosts under some conditions. Finally, we demonstrate that the model system possesses a global attractor, and explore the global dynamics of the system. Especially, we show that the infection-free steady state is globally asymptotically stable if R0≤1. On the other hand, if R0>1, then the infection will persist and the model system admits at least one positive steady state. For spatial homogeneous system, global asymptotic stability of the positive steady state is proved via Lyapunov functional technique. Numerical simulation is conducted to explore singular perturbation phenomenon when the diffusion coefficients approach zero. We observe that the diffusion will not only spread the infection to the low-risk region, but it may also increase the infection level in the high-risk region.

Detection of sub-surface fractures based on filtering, modeling, and interpreting aeromagnetic data in the Deng Deng – Garga Sarali area, Eastern Cameroon

Abstract The aeromagnetic anomalies existing in the Deng Deng – Garga Sarali region in Eastern Cameroon were filtered for a structural study of the region’s subsoil. This study presents the results of lineaments and fractures extracted by aeromagnetic image processing methods and compared with scientific data to obtain potential terrain models in the study area. The methodological approach used is based on the filtering of aeromagnetic image by using the analytical signal, the Euler deconvolution, and 2D3/4 modeling, to establish the maps of the lineaments and faults of the study area and their characteristics, and also to propose three models from three previously chosen profiles. Analytical signal and Euler deconvolution techniques have been applied to aeromagnetic anomalies to highlight the relationship between the depth and the source of magnetic anomalies, two parameters whose importance in geoexploration and modeling of the body is essential. We identified the potential contacts by interpretation of the deep Euler anomalies, these are highlighted on the basis of a certain similarity between the maps of the total magnetic field, the map of the analytical signal, the map of the maxima of the gradient horizontal, and geological map. Euler’s map of solutions correlates well with the edges of certain superficial and deep causative bodies.

Determination of Average Coulombic Efficiency for Rechargeable Magnesium Metal Anodes in Prospective Electrolyte Solutions.

The design of electrolyte solutions that permit reversible and efficient Mg metal electrodeposition is one of the most important tasks in the development of rechargeable Mg batteries. Several types of electrolyte solutions for Mg metal anodes have been developed and explored over the last two decades. These investigations have contributed to a better understanding of the Mg deposition and stripping processes. However, the Coulombic efficiency (CE) for reversible electrodeposition reported for these various systems and their performance in comparison to one another remained unclear. We used rigorous electrochemical methods to accurately quantify the average CE of the major electrolyte solutions considered for secondary Mg metal batteries. We demonstrated how changes in the experiential protocols influence CE measurements, resulting in inconsistent reports. Even though exceptional efficiency has been reported for a variety of systems, we discovered that the only candidate that currently meets the 99% CE benchmark during a prolonged cycling procedure is the dichloro-complex, which is a first-generation Grignard-based electrolyte solution. Second- and third-generation Grignard-free and chloride-free solutions showed reasonable CE only when the deposition currents densities were lowered. This comprehensive and systematic investigation will help to create a more accurate treasure map for potential electrolyte solutions for rechargeable Mg metal anodes.

The augmented Lagrangian method can approximately solve convex optimization with least constraint violation

There are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the nonlinear optimization problem as the one optimizing the objective function over the set of points with the least constraint violation. This leads to the study of the shifted problem. This paper focuses on the constrained convex optimization problem. The sufficient condition for the closedness of the set of feasible shifts is presented and the continuity properties of the optimal value function and the solution mapping for the shifted problem are studied. Properties of the conjugate dual of the shifted problem are discussed through the relations between the dual function and the optimal value function. The solvability of the dual of the optimization problem with the least constraint violation is investigated. It is shown that, if the least violated shift is in the domain of the subdifferential of the optimal value function, then this dual problem has an unbounded solution set. Under this condition, the optimality conditions for the problem with the least constraint violation are established in term of the augmented Lagrangian. It is shown that the augmented Lagrangian method has the properties that the sequence of shifts converges to the least violated shift and the sequence of multipliers is unbounded. Moreover, it is proved that the augmented Lagrangian method is able to find an approximate solution to the problem with the least constraint violation and it has linear rate of convergence under an error bound condition. The augmented Lagrangian method is applied to an illustrative convex second-order cone constrained optimization problem with least constraint violation and numerical results verify our theoretical results.

Quantitative Sensitivity Bounds for Nonlinear Programming and Time-Varying Optimization

Inspired by classical sensitivity results for nonlinear optimization, we derive and discuss new quantitative bounds to characterize the solution map and dual variables of a parametrized nonlinear program. In particular, we derive explicit expressions for the local and global Lipschitz constants of the solution map of nonconvex or convex optimization problems, respectively. Our results are geared towards the study of time-varying optimization problems, which are commonplace in various applications of online optimization, including power systems, robotics, signal processing, and more. In this context, our results can be used to bound the rate of change of the optimizer. To illustrate the use of our sensitivity bounds we generalize existing arguments to quantify the tracking performance of continuous-time, monotone running algorithms. Furthermore, we introduce a new continuous-time running algorithm for time-varying constrained optimization, which we model as a so-called perturbed sweeping process. For this discontinuous scheme we establish an explicit bound on the asymptotic solution tracking for a class of convex problems.

A Staggered Discontinuous Galerkin Method for Quasi-Linear Second Order Elliptic Problems of Nonmonotone Type

Abstract In this paper, we present a priori and a posteriori analysis of a staggered discontinuous Galerkin (DG) method for quasi-linear second order elliptic problems of nonmonotone type. First, existence is proved by using Brouwer’s fixed point argument and uniqueness is verified utilizing Lipschitz continuity of the discrete solution map. Next, optimal a priori error estimates for both potential and flux variables are derived. Then the residual based a posteriori error estimates on the potential energy error and the flux L 2 {L^{2}} error, respectively, are proposed. The flux error estimator makes use of a Helmholtz-type decomposition for the nonlinear system, which relies on appropriate choice of an auxiliary problem. While a priori error analysis is based on the observation that the staggered DG method can be viewed as a nonconforming approximation of the primal mixed formulation of the problem, a posteriori error estimation takes advantage of the primal formulation which can be obtained from the primal mixed formulation by eliminating the flux variable in the continuous setting. Finally, the theoretical findings are illustrated by numerical experiments.

Stability Analysis for Semi-Infinite Vector Optimization Problems under Functional Perturbations

This paper aims to study the stability of a class of semi-infinite vector optimization problems (SVOP) under functional perturbations. By using an important hypothesis a necessary and sufficient condition of Hausdorff continuity for weak efficient solution mappings and certain sufficient conditions for Painlevé-Kuratowski convergence of weak efficient solution sets for SVOP are established under the perturbations of both constraint sets and objective functions.

Rigorous numerics for nonlinear heat equations in the complex plane of time

In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

On second-order radial-asymptotic proto-differentiability of the borwein perturbation maps

This paper deals with second-order sensitivity analysis of parameterized vector optimization problems. We prove that the Borwein efficient solution map and the Borwein efficient perturbation map of a parametric vector optimization problem are second-order radial-asymptotic proto-differentiable under some suitable qualification conditions. Some examples are also given for illustrating the obtained results.

On the Cauchy problem for the two-component Novikov system with peakons

ABSTRACT Considered herein is the Cauchy problem of the two-component Novikov system with peakons. Based on the local well-posedness results for this problem, it is shown that the solution map of this problem on the line is not uniformly continuous in Besov spaces with through the method of approximate solutions. Next, in the periodic case, the non-uniform continuity of the solution map in Besov spaces and Hölder spaces with is discussed. Finally, the Hölder continuity of this solution map in Sobolev spaces is investigated.

Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces

We consider the compressible Navier–Stokes system in the critical Besov spaces. It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces of the homogeneous Besov spaces $\dot{B}^{n/p}_{p,1} \times \dot{B}^{n/p-1}_{p,1}$ for all $1 \leq p < 2n$. However, if the data is in a larger scaling invariant class such as $p > 2n$, then the system is not well-posed. In this paper, we demonstrate that for the critical case $p = 2n$ the system is ill-posed by showing that a sequence of initial data is constructed to show discontinuity of the solution map in the critical space. Our result indicates that the well-posedness results due to Danchin and Haspot are indeed sharp in the framework of the homogeneous Besov spaces.