We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic–elliptic chemotaxis-growth systems of the form ...ut=Δu−∇⋅(u∇v)+κ(x)u−μ(x)u2,0=Δv−v+u in smoothly bounded domains Ω⊂R2. Assuming that the coefficient functions satisfy κ,μ∈C0(Ω¯) with μ≥0 we prove that finite-time blow-up of the classical solution can only occur in points where μ is zero, i.e. that the blow-up set B is contained in {x∈Ω¯∣μ(x)=0}.Moreover, we show that whenever μ(x0)>0 for some x0∈Ω¯, then one can find an open neighbourhood U of x0 in Ω¯ such that u remains bounded in U throughout evolution.
We present a new optimization-based property-preserving algorithm for passive tracer transport. The algorithm utilizes a semi-Lagrangian approach based on incremental remapping of the mass and the ...total tracer. However, unlike traditional semi-Lagrangian schemes, which remap the density and the tracer mixing ratio through monotone reconstruction or flux correction, we utilize an optimization-based remapping that enforces conservation and local bounds as optimization constraints. In so doing we separate accuracy considerations from preservation of physical properties to obtain a conservative, second-order accurate transport scheme that also has a notion of optimality. Moreover, we prove that the optimization-based algorithm preserves linear relationships between tracer mixing ratios. We illustrate the properties of the new algorithm using a series of standard tracer transport test problems in a plane and on a sphere.
Copulas are becoming an essential tool in analyzing data thus encouraging interest in related questions. In the early stage of exploratory data analysis, say, it is helpful to know local copula ...bounds with a fixed value of a given measure of association. These bounds have been computed for Spearman’s rho, Kendall’s tau, and Blomqvist’s beta. The importance of another two measures of association, Spearman’s footrule and Gini’s gamma, has been reconfirmed recently. It is the main purpose of this paper to fill in the gap and present the mentioned local bounds for these two measures as well. It turns out that this is a quite non-trivial endeavor as the bounds are quasi-copulas that are not copulas for certain values of the two measures. We also give relations between these two measures of association and Blomqvist’s beta.
We prove that microstructures in shape-memory alloys have a self-similar refinement pattern close to austenite-martensite interfaces, working within the scalar Kohn-Müller model. The latter is based ...on nonlinear elasticity and includes a singular perturbation representing the energy of the interfaces between martensitic variants. Our results include the case of low-hysteresis materials in which one variant has a small volume fraction. Precisely, we prove asymptotic self-similarity in the sense of strong convergence of blow-ups around points at the austenite-martensite interface. Key ingredients in the proof are pointwise estimates and local energy bounds. This generalizes previous results by one of us to various boundary conditions, arbitrary rectangular domains, and arbitrary volume fractions of the martensitic variants, including the regime in which the energy scales as
ε
2
/
3
as well as the one where the energy scales as
ε
1
/
2
.
The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in ...space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized.
•Continuous finite element schemes for hyperbolic systems.•Nodal variational limiting scheme for nonlinear scalars and hyperbolic system.•High order background dissipation for phase errors.•Implicit and explicit time integrators.
This paper examines the application of optimization and control ideas to the formulation of feature-preserving numerical methods, with particular emphasis on the conservative and bound-preserving ...remap (constrained interpolation) and transport (advection) of a single scalar quantity. We present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization-based remap (OBR) of mass density. The latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity, subject to a system of linear inequality constraints. The approximation of an exact mass update operator defines the target quantity, which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds. The latter are enforced by the system of linear inequalities. In so doing, the generic OBR formulation separates accuracy considerations from the enforcement of physical properties.
We proceed to show how the generic OBR formulation yields the recently introduced flux-variable flux-target (FVFT) 1 and mass-variable mass-target (MVMT) 2 formulations of remap and then follow with a formal examination of their relationship. Using an intermediate flux-variable mass-target (FVMT) formulation we show the equivalence of FVFT and MVMT optimal solutions.
To underscore the scope and the versatility of the generic OBR formulation we introduce the notion of adaptable targets, i.e., target quantities that reflect local solution properties, extend FVFT and MVMT to remap on the sphere, and use OBR to formulate adaptable, conservative and bound-preserving optimization-based transport algorithms for Cartesian and latitude/longitude coordinate systems. A selection of representative numerical examples on two-dimensional grids demonstrates the computational properties of our approach.
A clear understanding of the behavior of error probability (EP) as a function of signal-to-noise ratio (SNR) and other system parameters is fundamental for assessing the design of digital wireless ...communication systems. We propose an analytical framework based on the log-concavity property of the EP which we prove for a wide family of multidimensional modulation formats in the presence of Gaussian disturbances and fading. Based on this property, we construct a class of local bounds for the EP that improve known generic bounds in a given region of the SNR and are invertible, as well as easily tractable for further analysis. This concept is motivated by the fact that communication systems often operate with performance in a certain region of interest (ROI) and, thus, it may be advantageous to have tighter bounds within this region instead of generic bounds valid for all SNRs. We present a possible application of these local bounds, but their relevance is beyond the example made in this paper.
The purpose of this paper is to prove local upper and lower bounds for weak solutions of semilinear elliptic equations of the form −Δ
u
=
cu
p
, with 0 <
p
<
p
s
= (
d
+ 2)/(
d
- 2), defined on ...bounded domains of
, without reference to the boundary behaviour. We give an explicit expression for all the involved constants. As a consequence, we obtain local Harnack inequalities with explicit constants, as well as gradient bounds.
We present best bounds on the deviation between univariate polynomials, tensor product polynomials, Bézier triangles, univariate splines, and tensor product splines and the corresponding control ...polygons and nets. Both pointwise estimates and bounds on the
L
p
-norm are given in terms of the maximum of second differences of the control points. The given estimates are sharp for control points corresponding to arbitrary quadratic polynomials in the univariate case, and to special quadratic polynomials in the bivariate case.
In this study, the distributed optimisation for solving resource allocation problem with both local bound and equality constraints is studied. A continuous-time multi-agent system with communication ...time-delay is proposed for the resource allocation. First, to reduce communication cost, a new modified multi-agent system is proposed inspired by previous work. Next, based on the system, the communication time-delay is considered to reduce the communication cost and match real situation. Then, by choosing proper parameters, sufficient conditions are derived for convergence to the optimal solution of the distributed optimisation. Moreover, delay-free case is also considered, which proves that the system is convergent based on directed communication graph by choosing proper parameters. Finally, simulation results demonstrate the characteristics of the system.
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