Degenerate Problems Research Articles

On a degenerate boundary value problem to relativistic magnetohydrodynamics with a general pressure law

On a degenerate boundary value problem to relativistic magnetohydrodynamics with a general pressure law

Regularity of the stress field for degenerate and/or singular elliptic problems

Regularity of the stress field for degenerate and/or singular elliptic problems

A Combined Quasi-Newton-Type Method Using 4th-Order Directional Derivatives for Solving Degenerate Problems of Unconstrained Optimization

Introduction. Methods of unconstrained optimization play a significant role in machine learning [1–6]. When solving practical problems in machine learning, such as tuning nonlinear regression models, the extremum point of the chosen optimality criterion is often degenerate, which significantly complicates its search. Therefore, degenerate problems are the most challenging in optimization. Known numerical methods for solving the general unconstrained optimization problem, up to the second order, have very low convergence rates when solving degenerate problems [7, 8]. This is explained by the fact that for a significant improvement in convergence rate in this case, it is necessary to use higher-order derivatives in the method than the second order [10]. The purpose of the paper is to develop an efficient quasi-Newton method for solving degenerate unconstrained optimization problems, the idea of which (unlike regularization) involves dividing the entire space into the sum of two orthogonal subspaces. This idea was introduced in [23]. The space division at each iteration of the method is based on the spectral decomposition of the matrix approximating the Hessian of the objective function using the BFGS formula [3]. Each subspace exhibits its own behavior of the objective function, and therefore, an appropriate minimization method is applied on it. Results. A combined quasi-Newton method is presented for solving degenerate unconstrained optimization problems, based on orthogonal decomposition of the Hessian approximation matrix and division of the entire space into the sum of two orthogonal subspaces. On one subspace (the kernel of the Hessian approximation matrix), a method is applied where derivatives in the direction of the 4th order are computed, while on the orthogonal complement to it, a quasi-Newtonian method is applied. A separate one-dimensional search is applied on each of these subspaces to determine the step multiplier in the respective direction. The effectiveness of the presented combined method is confirmed by numerical experiments conducted on widely accepted test functions for unconstrained optimization problems. The proposed method allows obtaining fairly accurate solutions to test tasks in case of degeneracy of the minimum point with significantly lower computational costs for gradient calculations compared to optimization procedures of well-known mathematical packages. Conclusions. The idea of dividing the entire space into the sum of two (or possibly more) orthogonal subspaces when solving complex optimization problems is quite promising in terms of applying a combination of different numerical methods on separate subspaces. In the future, it is planned to conduct theoretical research on the convergence rate of the presented combined method in solving degenerate unconstrained optimization problems. Keywords: unconstrained optimization, quasi-Newton methods, degenerate minimum point, spectral matrix decomposition, Machine Learning.

Bounded solutions in anisotropic degenerate parabolic problems with a singular term

Bounded solutions in anisotropic degenerate parabolic problems with a singular term

Inverse source problem for a one-dimensional degenerate hyperbolic problem

Inverse source problem for a one-dimensional degenerate hyperbolic problem

The Cauchy problem for a class of linear degenerate evolution equations on the torus

We study, in the periodic setting, the well-posedness of the Cauchy problem associated to the operator P(t,Dx,Dt)=Dt−a2(t)Δx+∑j=1Na1,j(t)Dxj+a0(t), with T>0, t∈[0,T] and a2,a1,1,…,a1,N,a0∈C([0,T];C). Using Fourier analysis techniques, we obtain a complete characterization for the well-posedness of a class of degenerate initial-value problems in the Sobolev, Smooth, Gevrey and Real-Analytic frameworks.

Isogeometric method for a nonlocal degenerate parabolic problem

In this article, we apply NURBS-based isogeometric method to solve a nonlocal degenerate parabolic problem. The isogeometric method has advantages over classical finite element method in terms of exact geometry representation and higher order smooth basis functions. We consider the NURBS-based semi-discretization of the problem. The second-order Crank–Nicolson method is applied for the complete discretization of the problem. Error estimates of the semi-discrete and fully-discrete problems are derived. Few numerical examples are provided to verify the theoretical findings.

Existence results for nonlinear degenerate elliptic problems involving singular and supercritical nonlinearities

Existence results for nonlinear degenerate elliptic problems involving singular and supercritical nonlinearities

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Physicist's view on the unbalanced k-cardinality assignment problem.

The k-cardinality unbalanced assignment problem asks for assigning k "agents" to k "tasks" on a one-to-one basis, while minimizing the total cost associated with the assignment, with the total number of agents N and the total number of tasks M possibly different and larger than k. While many exact algorithms have been proposed to find such an optimal assignment, these methods are computationally prohibitive when the problem is large. We propose an approach to solving the k-cardinality assignment problem using techniques adapted from statistical physics. This paper provides a full description of this formalism, including all the proofs of its main claims. We derive a strongly concave free-energy function that captures the constraints of the k-assignment problem at a finite temperature. We prove that this free energy decreases monotonically as a function of β, the inverse of temperature, to the optimal assignment cost, providing a robust framework for temperature annealing. We also prove that for large enough β values the exact solution to the k-assignment problem can be derived using simple round-off to the nearest integer of the elements of the computed assignment matrix. We show that this framework can be adapted to handle degenerate k-assignment problems. We describe a computer implementation of our framework that is optimized for the GPU parallel architecture, using the library CUDA. This implementation is found to be as efficient as state-of-the-art implementations of parallel Hungarian algorithms on generic assignment problems, and orders of magnitude faster than those algorithms for pathological assignment cases.

Inviscid Burgers as a degenerate elliptic problem

We demonstrate the feasibility of a scheme to obtain approximate weak solutions to the (inviscid) Burgers equation in conservation and Hamilton-Jacobi form, treated as degenerate elliptic problems. We show different variants recover non-unique weak solutions as appropriate, and also specific constructive approaches to recover the corresponding entropy solutions.

Precise error bounds for numerical approximations of fractional HJB equations

Abstract We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.

Decoupling numerical method based on deep neural network for nonlinear degenerate interface problems

Many practical problems, including modeling composite materials, nuclear waste disposal, oil reservoir simulations, and flows in porous medium, commonly involve interface problems. However, the solution to interface problems with discontinuous coefficients of PDEs using fully decoupled numerical methods is challenging. The main objective is to solve the interface problems with fully decoupled numerical methods. This paper proposes an efficient decoupled numerical method for solving degenerate interface problems with double singularities. First, we divide the whole domain into singular and regular subdomains. Then, we use the Deep Neural Network (DNN) to find the solution on the singular subdomain and approximate the solution on the regular subdomain using the finite difference method. The scheme combines the solutions of singular and regular subdomains, which is an exciting idea. The key to the new approach is to split nonlinear degenerate partial differential equations with an interface into two independent boundary value problems based on deep learning. In this way, the expansion of the solution on the singular domain does not contain undetermined parameters, and two independent boundary value problems can be solved with any well-known traditional numerical methods. The main advantage of the proposed scheme is that we not only get the order of convergence of the degenerate interface problems on the whole domain, but we also can calculate VERY BIG jump ratio (such as 1012:1 or 1:1012) for the interface problems including degenerate and non-degenerate cases. Finally, with examples, we demonstrate the efficiency and accuracy of methods for 1 and 2D problems. It is also interesting that the proposed method is valid for the interface problems with degenerate and non-degenerate cases, we show it with some examples.

Combined methods for solving degenerate unconstrained optimization problems

UDC 519.853.6 : 519.613.2 We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems. The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order derivatives. Our approach is based on the decomposition of ℝ n into the direct sum of the kernel of a Hessian matrix and its orthogonal complement. The fourth-order method is applied to the kernel of the Hessian matrix, whereas the Newton method is applied to its orthogonal complement. This method proves to be efficient in the case of a one-dimensional kernel of the Hessian matrix. In order to get the second-order method, Newton's method is combined with the steepest-descent method. We study the efficiency of these methods and analyze their convergence rates. We also propose a new adaptive combined quasi-Newton-type method (ACQNM) based on the use of the second- and fourth-order methods in the degenerate case. The efficiency of ACQNM is demonstrated by analyzing an example of some most common test functions.

Discovery of seed germinating fungi (Mycetinis scorodonius) from Gastrodia elata Bl. f. glauca S. chow in Changbai Mountain and examination of their germination ability

Multi-generational asexual reproduction of Gastrodia elata Bl. will cause seedling species degeneration. Sexual reproduction of Gastrodia elata Bl. seed is an effective method to solve the problem of degeneration. The development of Gastrodia elata Bl. seeds cannot be separated from the germination fungus. However, there are few strains of germination fungus in production, and there is also the problem of species degradation in application for many years. It is very important for the sexual reproduction of Gastrodia elata Bl. to isolate more new strains of excellent germination fungus from the origin. This study used the Gastrodia elata Bl. f. glauca S. chow seeds germination vegetative propagation corms capture method to isolate its symbiotic germination fungus, and comprehensively identified the species of germination fungus by colony morphology, ITS, sporocarps regeneration and germination function, and compared the growth characteristics and germination ability with other germination fungus (Mycena purpureofusca, Mycena dendrobii and Mycena osmundicola). The germination fungus was isolated from the vegetative propagation corms of Gastrodia elata Bl. f. glauca S. chow seeds and named GYGL-1. After comprehensive identification, GYGL-1 was Mycetinis scorodonius. Compared with other germination fungus, GYGL-1 has fast germination speed, vigorous growth, and high germination ability for Gastrodia elata Bl. f. glauca S. chow seeds. Innovated the isolation method of Gastrodia elata Bl. seeds germination fungus, obtained the regenerated sporocarps of the germination fungus, and discovered that Mycetinis scorodonius has a new function of germinating Gastrodia elata Bl. f. glauca S. chow seeds, enriching the resource library of Gastrodia elata Bl. germination fungus.

Monotonicity of solutions to degenerate p-Laplace problems with a gradient term in half-spaces

Monotonicity of solutions to degenerate p-Laplace problems with a gradient term in half-spaces

Multi-bump Solutions for a Strongly Degenerate Problem with Exponential Growth in $$\mathbb {R}^N$$

Multi-bump Solutions for a Strongly Degenerate Problem with Exponential Growth in $$\mathbb {R}^N$$

Radial solutions of a pseudo-differential equations associated with a p-adic non-local ultradiffusion operator

This paper is dedicated to study some pseudo-differential equations related to the operator Dd1,d2α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{\\alpha }_{d_1,d_2}$$\\end{document} with symbol Ad1,d2α(x)=max|x|pd1,|x|pd2α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}^{\\alpha }_{d_1,d_2}(x)=\\max \\left\\{ |x|_p^{d_1},\\ |x|_p^{d_2}\\right\\} ^\\alpha $$\\end{document}, acting on a radial function φ(|x|p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi (|x|_p)$$\\end{document}. The importance of this symbol is that it is a generalization of Vladimirov and Bessel operators. Our research provides conditions for the existence of its right inverse and the explicit formulae for both operators, as well as a practical application. The inverse allows solving a Cauchy problem with the operator, which can be reduced to an integral equation whose properties are similar to those of the classical Volterra equation. We also find conditions for the existence and uniqueness of a mild solution for a degenerate nonlinear Cauchy problem.

Metastructure based broadband structural stealth with material-structure-function integration

Broadband microwave absorption is difficult to be realized in traditional coating form as multiple electromagnetic resonances are difficult to be generated even with dielectric-magnetic loss composites. Narrow absorption band confines the practical usage of the novel electromagnetic nano composites. Herein, the structural stealth concept is proposed to overcome the narrow-band absorption problem with the effect carrier of metastructure. The gradient honeycomb metastructure (GHM) was designed and optimized with module stack large mutation genetic algorithms. The structural design, fabrication, experimental verification and parameter adjustment were included in the closed loop with material-structure-function integration. Multiple resonance effects were introduced in the gradient designs. GHM achieved −10 dB absorption bandwidth in 1.92–17.6 GHz and three deep absorption peaks were introduced by three layers of electromagnetic resonant honeycomb. Broadband absorption in oblique incidence from 30° to grazing angle 85° was achieved to overcome the oblique absorption degeneration problems of traditional nano lossy composites. The structural mechanical performance was high with the maximum equivalent tensile strength of 108.6 MPa and the maximum flexural load of 0.873 kN. The results showed the importance of structural stealth with material-structure-function integration to design metastructure for broadband microwave absorption, which provided a promising approach to achieve broadband microwave absorption.

Regularizing effect of absorbtion term in singular and degenerate elliptic problems

Dans cet article, nous étudions l’existence et la régularité des solutions au problème singulier suivant\\\begin{equation}\left\{\begin{array}{lll}-\displaystyle\mbox{div} \big(a(x,u)\vert\nabla u\vert^{p-2}\nabla u\big) + \vert u\vert^{s-1}u =h(u)f &\mbox{ in } \Omega \\&u\geq 0 &\mbox{ in }\Omega \\&u=0 &\mbox{ on } \delta\Omega\\\end{array}\right.\end{equation} prouvant que le terme d’ordre inférieur $u\vert u\vert^{s-1}$ a des effets régularisants sur les solutions dans le cas d’un opérateur elliptique à coercivité dégénérée.