We propose a reflection-free Langevin framework for sampling and optimization on compact polyhedra. The method is based on the inverse Hessian of the logarithmic barrier, which defines a Dikin-Langevin diffusion whose drift and noise adapt to the local interior-point geometry. We show that trajectories started in the interior remain feasible for all finite times almost surely, so the constrained domain is preserved without reflections or projections. For computation, we discretize the diffusion using the Euler-Maruyama scheme and apply a Metropolis-Hastings correction, yielding a sampler that targets the exact constrained distribution. We also propose an annealed interacting variant for non-convex optimization. Numerically, the Metropolis-adjusted method outperforms both the Dikin random walk and standard MALA on anisotropic box-constrained Gaussians, and the interacting optimizer escapes suboptimal basins more reliably than the non-interacting method.
We consider the problem of improving the accuracy, convergence and conditioning of univariate nonlinear function approximations using (mainly) shallow neural networks (NN) with a rectified linear unit (ReLU) activation function. The standard $L_{2}$-based approximation problem is ill conditioned and the behaviour of the optimization algorithms used in training these networks degrades rapidly as the width of the network increases. This can lead to significantly poorer approximation in practice than we would expect from the theoretical expressivity of the ReLU NN architecture. Univariate shallow ReLU NNs and traditional approximation methods, such as univariate free knot splines (FKS) span the same function space, and thus have the same theoretical expressivity. However, the FKS representation, both remains well conditioned as the number of knots increases, and can be highly accurate if the knots are correctly placed. We leverage the theory of optimal piecewise linear interpolants to improve the training procedure for both an FKS and a ReLU NN. For the FKS we propose a novel two-level training procedure. First solving the nonlinear problem of finding the optimal knot locations of the interpolating FKS using an equidistribution approach. Then solving the nearly linear, well-conditioned, problem of finding the optimal weights and knots of the FKS. The training of the FKS gives insights into how we can train a ReLU NN effectively to give an equally accurate approximation. To do this we combine the training of the ReLU NN with an equidistribution based loss to find the breakpoints of the ReLU functions, this is then combined with preconditioning the ReLU NN approximation (to take an FKS form) to find the scalings of the ReLU functions. This procedure leads to a fast, well-conditioned and reliable method of finding an accurate shallow ReLU NN approximation to a univariate target function. This method avoids spectral bias and is highly effective for a wide variety of functions. We test this method on a series of regular, singular and rapidly varying target functions and obtain good results, realizing the expressivity of the shallow ReLU network in all cases. We conclude that in the shallow case to gain full expressivity for the ReLU NN we must both find the optimal breakpoints (by equidistribution) and precondition the problem of finding the optimal coefficients. We then extend our results to more general activation functions, and to deeper networks.
This paper deals with an age-structured HIV model with antiretrowviral therapy and two infection routes (virus-to-cell and cell-to-cell). The model is first formulated as an abstract non-densely defined Cauchy problem and the existence of the equilibria is obtained under some conditions. Based on the existence of equilibria, the global asymptotical stability of the disease-free equilibrium is studied by applying theory of operator semigroups and spectral analysis. The stability and Hopf bifurcation results with two delays around the endemic equilibrium are also well described under some conditions by the geometric stability switch criteria. Finally some numerical examples are presented to illustrate the obtained results.
In this work, we develop a new biological transmission model for Chagas disease. This model, set in two juxtaposed habitats with skew Brownian motion conditions at the interface, is composed of two reaction-diffusion equations and takes into account the sylvatic transmission. We write it as an abstract perturbed Cauchy problem using operator theory. Then, we show that the main operator, which models the dispersal process, generates an analytic semigroup in an adequate Banach space.
First published in 1961, Twersky's formula provides an alternative representation for a class of Schl & ouml;milch series that is important in wave scattering theory. When Twersky's formula is used in practical applications, its rate of convergence is often accelerated using Kummer's transformation. This amounts to expanding the summand in negative powers of the index, subtracting these terms and adding back their exact sum. The first few terms can easily be obtained using a computer algebra system, but they become increasingly complicated as the expansion progresses. In this paper, we derive a general expression for the coefficients in the expansion of the summand. We also show that these coefficients can be calculated using a stable, linear recurrence relation. Using these results it is possible to create simple and very rapidly convergent representations for the Schl & ouml;milch series.
In this paper, we derive general bright-dark soliton solutions to the coupled Sasa-Satsuma (CSS) equation using the Kadomtsev-Petviashvili reduction method. Since the CSS equation is a special case of the four-component Hirota equation, our approach begins with the construction of two-bright-two-dark soliton solutions for the four-component Hirota equation. By imposing specific parameter constraints, these solutions are subsequently reduced to the bright-dark soliton solutions of the CSS equation. Finally, the dynamical behaviours of the one- and two-bright-dark soliton solutions are thoroughly analysed and illustrated.
In this paper, we propose a complex stochastic hepatitis B virus (HBV) epidemic model with vaccination strategy, where random fluctuations of the transmission dynamics of HBV is driven by Black-Karasinski process, for the first time. It is shown that Black-Karasinski process is a biologically and mathematically reasonable assumption compared with existing stochastic modelling approaches. For the deterministic model, the basic reproduction number R-0, possible equilibria and related asymptotic stability are studied. Then for the stochastic model, the existence and global positivity of the solution are proved. We further derive two stochastic critical values R-0(S) and R-0(E) related to R-0 to characterize the long-term behaviour of HBV, and it turns out that (i) the stochastic model has a stationary distribution if R-0(S) > 1; (ii) the infected individuals will go extinct exponentially fast when R-0(E) < 1; (iii) R-0(S) = R-0(E) = R-0 if there is no environmental noise. Our results reveal that random fluctuations introduced will facilitate HBV prevalence. Moreover, by analysing the stable structure of the model, we provide a complete classification and explicit approximation for the local density function of the stationary distribution. Finally, some numerical examples are performed to support our theoretical findings. The techniques and methods of analysis in this paper can be applied to many complex high-dimensional epidemic models motivated by Black-Karasinski process.
We consider the problem in synthetic aperture RADAR (SAR) of identifying and classifying objects located on the ground by means of convolutional neural networks (CNNs). Specifically, we adopt a single scattering approximation to classify the shape of the object using both simulated SAR data and reconstructed images from these data, and we compare the success of these approaches. We then identify ice types in real SAR imagery from the satellite Sentinel-1. In both experiments, we achieve a promising high classification accuracy ($\geq $85%). Our results demonstrate the effectiveness of CNNs in using SAR data for both geometric and environmental classification tasks. Our investigation also explores the effect of SAR data acquisition at different antenna heights on our ability to classify objects successfully.
We explore two approaches to proving existence and analyticity of solutions to nonlinear parabolic differential equations. One of these methods works well for more general nonlinearities, while the second method gives stronger results when the nonlinearity is simpler. The first approach uses the exponentially weighted Wiener algebra, and is related to prior work of Duchon and Robert for vortex sheets. The second approach uses two norms, one with a supremum in time and one with an integral in time, with the integral norm representing the parabolic gain of regularity. As an example of the first approach we prove analyticity of small solutions of a class of generalized one-dimensional Kuramoto-Sivashinsky equations, which model the motion of flame fronts and other phenomena. To illustrate the second approach, we prove existence and analyticity of solutions of the dissipative Constantin-Lax-Majda equation (which models vortex stretching), with and without added advection, with two classes of rough data. The classes of data treated include both data in the Wiener algebra with negative-power weights, as well as data in pseudomeasure spaces with negative-power weights.
In this paper, we consider a company can simultaneously reduce its emissions and buy carbon allowances at any time. We establish an optimal control model involving two stochastic processes with two control variables, which is a singular control problem. This model can then be converted into a Hamilton-Jacobi-Bellman equation, which is a 2D variational equality with gradient barrier, so that the free boundary is a surface. We prove the existence and uniqueness of the solution. Finally, some numerical results are shown.
We analyse periodic operators on $\mathbb{R}<^>{n}$ and small perturbations of these operators. The perturbation is periodic in $n-1$ directions and has bounded support in the remaining direction. We show that, when the perturbation has a sign, every spectral gap for the unperturbed operator is reduced by the perturbation. We develop a general theory that can be applied to elliptic operators, to systems such as that of linear elasticity and to Maxwell's equations.
This paper deals with the Keller-Segel-Stokes system with subquadratic degradation n(t) + u & centerdot; del n= Delta n-chi V & centerdot; (n del v) + rn-mu n(alpha); v(t) + u & centerdot; del v = Delta v-v + n; u(t) = Delta u + P + n del phi; del & centerdot; u = 0 in a smoothly bounded convex domain Omega subset of R3, where chi > 0, r E R,mu > 0 and alpha E (1, 2). Previous literature has asserted that for all reasonably mild initial data, an associated no-flux/no-flux/Dirichlet initial-boundary value problem possesses at least one global generalized solution whenever alpha E (1, 2), but the knowledge on the regularity properties of solutions has not yet exceeded some information on fairly basic integrability features. The present study shows that these generalized solutions become eventually smooth and bounded if alpha is an element of(5/3,2)andr <=eta & centerdot;min{mu(2/)4 alpha-alpha(2)-1, mu(2)(3-alpha)}with some eta = eta(chi, alpha, Omega) > 0. . Our result inter alia reveals that the any type of infinite-time blow-up and persisting oscillatory behaviour of solutions to 3D Keller-Segel-Stokes systems with certain subquadratic degradations will never occur for the situations in which the considered population does not spontaneously proliferate, or proliferates with an adequately small rate.
We propose an unfolded accelerated projected-gradient descent procedure to estimate model and algorithmic parameters for super-resolution and molecule localization problems in fluorescence microscopy. The variational lower-level constraint enforces sparsity of the solution and encodes different noise statistics (Gaussian, Poisson), while the upper-level cost assesses optimality w.r.t. the task considered. In more details, a standard $\ell _{2}$ cost is considered for image reconstruction (e.g. deconvolution/super-resolution, semi-blind deconvolution) problems, while a smoothed $\ell _{1}$ loss with learned binarization is employed to assess localization precision in some exemplary fluorescence microscopy problems exploiting single-molecule activation. Several numerical experiments are reported to validate the proposed approach on both synthetic and benchmark images from the ISBI datasets.
In this paper, we analyse a 3D Tropical Climate Model that potentially includes damping terms in the equations governing the barotropic and first baroclinic modes of the velocity. We establish a blow-up criterion based on the time-integrability of the spatial bounded mean oscillation-norm of the local strong solution, considering both the presence and absence of damping effects.
This paper presents an analytical model for laminar pressure-driven axial flow along a circular pipe containing an internal concentric circular grating of no-slip fins. Explicit expressions for the velocity field are derived using complex analysis techniques and conformal mappings. From these solutions, formulas for the Fanning friction factor associated with axial flow in such finned pipes are derived and, using these, the drag properties of the pipes are examined. The results offer valuable insights for optimizing finned duct designs in applications involving heat transfer, fluid flow and biofilm cultivation.
This paper is concerned with the propagation dynamics of a general non-local dispersal system with shifting habitats. By constructing appropriate upper and lower solutions, and applying the fixed-point theory, we prove the existence of a non-decreasing wave front with speed consistent with the habitat shifting speed. Moreover, using the sliding method, the uniqueness of forced waves is obtained. And, the stability of the forced waves is proved by applying the dynamical systems approach. Finally, we give an application and the corresponding numerical simulation to illustrate our analytical results.
This work investigates the steady, laminar, 2D Poiseuille flow of a Newtonian fluid through parabolic segment and lens-shaped ducts. The governing equations are solved using the finite element method. Using parabolic coordinates and the method of separation of variables, special analytical solutions are derived. For ducts with small aspect ratios, a systematic perturbation method is applied to obtain approximate solutions, while for ducts with large aspect ratios, the Maclaine-Cross formula is employed to determine the limiting value of the friction factor-Reynolds number product. The flow rate and the friction factor-Reynolds number product are computed for various aspect ratios. The numerical and analytical solutions show excellent agreement.
The dynamic behaviours of a three-degree-of-freedom collision vibration system with clearances are investigated with both theoretical and numerical methods in this paper. The periodic solution of the system is obtained through the modal superposition method. The stability of the system is studied in two scenarios: one in which the system is subjected to unilateral collisions and the other in which it is subjected to bilateral collisions. In both cases, the unilateral grazing bifurcation and symmetric periodic motion of the system are examined through the construction of a discontinuity mapping and the utilization of the composite Poincar & eacute; mapping method, respectively. Finally, numerical simulations are conducted to validate the previous theoretical analysis.
This paper studies the critical mass phenomenon in an attraction-repulsion chemotaxis system with rotation under zero-flux boundary conditions in a two-dimensional bounded domain with a smooth boundary. On one side, when attraction dominates repulsion (in a certain sense characterized by the system parameters), a threshold value of the total cellular mass is identified, which controls the finite-time blow-up or global boundedness of classical solutions. On the other side, when repulsion dominates or balances attraction, global boundedness of classical solutions is established for arbitrary initial data under radial symmetry and synchronized rotations. These are the first rigorous mathematical results concerning the critical mass phenomenon in an attraction-repulsion chemotaxis system with rotation.
We consider small-amplitude deformations of a long thin-walled elastic tube, caused by a pressure difference between the interior and exterior. The tube initially has a uniform elliptical cross-section and is subject to a large axial pre-stress. The tube length and wall thinness can be exploited to derive simplified models of the wall mechanics. Such models typically neglect effects such as axial bending, which are small over most of the tube but contain higher-order axial derivatives. The resulting models are unable to satisfy the full set of clamped boundary conditions where an elastic section of tube is joined to a rigid support. In this work, we examine the asymptotic boundary layers that arise near the clamped end of an elastic-walled tube, which allow a bulk solution to a simplified model in the interior to be matched to the boundary conditions at the tube ends. We consider the region of parameter space where the width of the thinnest bending boundary layer is small compared with the tube diameter, but still much larger than the thickness of the tube wall. Within this region, we find three distinct regimes that give rise to different sets of nested boundary layers involving different physical effects. Our matched asymptotic solutions show excellent agreement with an exact solution in a case where the full problem can be solved analytically.
