Localization method for lines of discontinuity of an approximately defined function of two variables

A function of two variables with lines of discontinuity of the first kind is considered. It is assumed that outside the discontinuity lines the function is smooth and has a bounded partial derivative. An approximation to the function in L ₂ and a perturbation level are known. The problem in question belongs to a class of nonlinear ill-posed problems, which are solved by constructing some regularizing algorithms. We propose a simple theoretical approach to solving the problem of localizing the discontinuity lines of a function that is noisy in the space L ₂. Some conditions on the exact function are imposed ??in the small.?? Methods of averaging are constructed, and error estimates of localizing the lines (in the small) are obtained.     

On the localization of discontinuities of the first kind for a function of bounded variation

Methods of the localization (detection of positions) of discontinuities of the first kind for a univariate function of bounded variation are constructed and investigated. Instead of an exact function, its approximation in L ₂(?∞,+∞) and the error level are known. We divide the discontinuities into two sets, one of which contains discontinuities with the absolute value of the jump greater than some positive Δ^min; the other set contains discontinuities satisfying a smallness condition for the value of the jump. It is required to find the number of discontinuities in the former set and localize them using the approximately given function and the error level. Since the problem is ill-posed, regularizing algorithms should be used for its solution. Under additional conditions on the exact function, we construct regular methods for the localization of discontinuities and obtain estimates for the accuracy of localization and for the separability threshold, which is another important characteristic of the method. The (order) optimality of the constructed methods on the classes of functions with singularities is established.     

Towards a Mathematical Theory of Super‐resolution

This paper develops a mathematical theory of super‐resolution. Broadly speaking, super‐resolution is the problem of recovering the fine details of an object—the high end of its spectrum—from coarse scale information only—from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex‐valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff f_c. We show that one can super‐resolve these point sources with infinite precision—i.e., recover the exact locations and amplitudes—by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/f_c. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super‐resolved signal is expected to degrade when both the noise level and the super‐resolution factor vary. © 2014 Wiley Periodicals, Inc.     

Investigation of a nonlinear heat equation with a quadratic source

We consider a particular case of the nonlinear heat equation on a straight line. A family of exact solutions of the form p(t) + q(t) cos (x/ ) is constructed, where p(t) and q(t) satisfy some dynamical system. A detailed analysis of the system is given. The existence of blowup solutions as well as solutions that decay to a nonzero background is proved for the Cauchy problem for the given equation. Part of the solutions from this family are close in a certain sense to the analytical solution of the nonlinear equation with power nonlinearities evolving in the S-regime. Profiles of various solutions are constructed and localization is investigated numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 5–23, 2006.     

One Approach to synthesizing predicate circuits on the basis of generalized variables

The asymptotic behavior of the Shannon’s function L _B(n) is studied for complexity of n-variable predicate implementation with the use of predicate circuits over arbitrary complete basis B. A new definition of the reduced weight of the predicate is introduced, regarding it as a solution of a specific linear programming problem based on a system of predicate’s generalized variables. New, more exact upper elements for L _B(n) in a number of bases are acquired by the means of special decompositions of initial predicates using universal sets of predicates constructed for circuits consisting of bases elements with minimal reduced weight.     

Discretization of a New Method for Localizing Discontinuity Lines of a Noisy Two-Variable Function

We consider the ill-posed problem of localizing (finding the position of) discontinuity lines of a noisy function of two variables. New regularizing methods of localization are constructed in a discrete form. In these methods, the smoothing kernel is varying, which simplifies the implementation of the algorithms. We obtain bounds for the localization error of the methods and for their separability threshold, which is another important characteristic.     

On a Dulac function for the Kukles system

To estimate the number and location of limit cycles of Kukles systems in a strip of the phase plane xOy, we develop a method for constructing the Dulac function in the form of a polynomial in the phase variable y with coefficients depending on the second phase variable x. The suggested method is regular for h ₂(x) = 0. The proof of the fact that the constructed function is a Dulac function is reduced to finding a specific number of positive functions that are linear combinations of known functions of the single phase variable x with arbitrary constants. To construct the Dulac function, we use a solution of the corresponding linear programming problem. In addition, we show that the presented approach is efficient from the practical viewpoint and permits one to obtain a global exact estimate for the number of limit cycles of above-mentioned systems in some cases.     

A discrete algorithm for localizing the discontinuity lines of a function of two variables

We consider an ill-posed problem of localizing the discontinuity lines of a function of two variables. It is assumed that, instead of a precisely given function f, the values are available of the averages on the square of the perturbed function f^δ at the points of a uniform grid as well as the error level δ so that \({\left\| {f - {f^\delta }} \right\|_{{L_2}}}{(_\mathbb{R}}^2)\) ≤ δ. An algorithm is constructed for localizing the discontinuity lines, its convergence is proved with the estimates of the approximation accuracy, which coincide in the order of magnitude with the estimates obtained earlier by the authors for the case when, instead of the average values of the function f^δ, the function itself is given. Also, we substantiate the estimates for an important characteristic of localization methods, i.e. separability threshold.     

The Exactness Property of the Vector Exact l1 Penalty Function Method in Nondifferentiable Invex Multiobjective Programming

In this article, the vector exact l₁ penalty function method used for solving nonconvex nondifferentiable multiobjective programming problems is analyzed. In this method, the vector penalized optimization problem with the vector exact l₁ penalty function is defined. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto optimal solutions of the considered nondifferentiable multiobjective programming problem and of the associated vector penalized optimization problem with the vector exact l₁ penalty function. This equivalence is established for nondifferentiable invex vector optimization problems. Some examples of vector optimization problems are presented to illustrate the results established in the article.     

Theory of the hypersonic viscous shock layer at high Reynolds numbers and intensive injection of foreign gases : PMM vol. 38, n 6, 1974, pp. 1015–1024

The hypersonic flow around smooth blunted bodies in the presence of intensive injection from the surface of these is considered. Using the method of external and internal expansions the asymptotics of the Navier-Stokes equations is constructed for high Reynolds numbers determined by parameters of the oncoming stream and of the injected gas. The flow in the shock layer falls into three characteristic regions. In regions adjacent to the body surface and the shock wave the effects associated with molecular transport are insignificant, while in the intermediate region they predominate. In the derivation of solution in the first two regions the surface of contact discontinuity is substituted for the region of molecular transport (external problem). An analytic solution of the external problem is obtained for small values of parameters ₁ = ρ_∞/ρs^* and δ = ρ_ω^*1/2ν_ω^*/ρ_∞1/2ν_∞, in the form of corresponding series expansions in these parameters. Asymptotic formulas are presented for velocity profiles, temperatures, and constituent concentration across the shock layer and, also, the shape of the contact discontinuity and of shock wave separation. The derived solution is compared with numerical solutions obtained by other authors. The flow in the region of molecular transport is defined by equations of the boundary layer with asymptotic conditions at plus and minus infinity, determined by the external solution (internal problem). A numerical solution of the internal problem is obtained taking into consideration multicomponent diffusion and heat exchange. The problem of multicomponent gas flow in the shock layer close to the stagnation line was previously considered in [1] with the use of simplified Navier-6tokes equations.The supersonic flow of a homogeneous inviscid and non-heat-conducting gas around blunted bodies in the presence of subsonic injection was considered in [2–7] using Euler's equations. An analytic solution, based on the classic solution obtained by Hill for a spherical vortex, was derived in [2] for a sphere on the assumption of constant but different densities in the layers between the shock wave and the contact discontinuity and between the latter and the body. Certain results of a numerical solution of the problem of intensive injection at the surface of axisymmetric bodies of various forms, obtained by Godunov's method [3], are presented. Telenin's method was used in [4] for numerical investigation of flow around a sphere; the problem was solved in two formulations: in the first, flow parameters were determined for the whole of the shock layer, while in the second this was done for the sutface of contact discontinuity, which was not known prior to the solution of the problem, with the pressure specified by Newton's formula and flow parameters determined only in the layer of injected gases. The flow with injection over blunted cones was numerically investigated in [5] by the approximate method proposed by Maslen. The flow in the shock layer in the neighborhood of the stagnation line was considered in [6, 8], and intensive injection was investigated by methods of the boundary layer theory in [8–12].     

The nonlinear programming method of Wilson,Han, and Powell with an augmented Lagrangian type line search function

Summary The paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms. This study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present. The purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail. In Part 1, the exactL ₁-penalty function of Han and Powell is replaced by a differentiable augmented Lagrange function for the line search computation to be able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution. In Part 2, the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency, and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblem.     

Regularizing algorithms for localization of breakpoints of noisy functions

The problem of localizing the singularities (breakpoints) of functions that are noisy in the spaces L _p , 1 < p < ∞, or C is considered. A wide class of smoothing algorithms that determine the number and location of breakpoints is constructed. In addition, for the case when a function is noisy in C, a finite-difference method is constructed. For the proposed methods, convergence theorems are proved and approximation accuracy estimates for the location of breakpoints are obtained. The lower estimates obtained in this paper show the order-optimality of the methods. For all the methods constructed, their capacity of separating close breakpoints is investigated.     

Gibbs' phenomenon and arclength

The arclengths of the graphs Γ(s_N(f)) of the partial sums s_N(f) of the Fourier series of a piecewise smooth function f with a jump discontinuity grow at the rate O(logN). This problem does not arise if f is continuous, and can be removed by using the standard summability methods.     

On Smoothing l1 Exact Penalty Function for Constrained Optimization Problems

This article introduces a smoothing technique to the l₁ exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm.     

Engouffrement symplectique et intersections lagrangiennes

Let }L _t{,t ∈ [0, 1], be a path of exact Lagrangian submanifolds in an exact symplectic manifold that is convex at infinity and of dimension ≥6. Under some homotopy conditions, an engulfing problem is solved: the given path }L _t{ is conjugate to a path of exact submanifolds inT ^*L_o. This impliesL _t must intersectL _o at as many points as known by the generating function theory. Our Engulfing theorem depends deeply on a new flexibility property of symplectic structures which is stated in the first part of this work.

     

On the propagation of perturbations for essentially nonautonomous quasilinear first-order equations

This paper deals with generalized solutions of the Cauchy problem for the equationu _t + [A(t, x, u)] _x +B(t, x, u) = 0 (t, x) ∈ ℝ₊ × ℝ. Here A, B may depend essentially on t, x; for example, they may tend to zero or to infinity as t becomes infinite. Sufficient conditions are obtained for the presence and the absence of finite time extinction and space localization. These phenomena have been studied earlier mainly for degenerate parabolic equations. In the case of first-order equations the situation is more complicated due to the discontinuity of solutions. The essential dependence of the coefficients on t, x gives rise to a threshold phenomenon: the presence of the finite time extinction depends on the maximum of the modulus of the initial function. Bibliography: 29 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 89–117, 1994.     

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

We study the convergence properties of an algorithm for the inverse problem of electrical impedance tomography, which can be reduced to a partial differential equation (PDE) constrained optimization problem. The direct problem consists of the potential equation div(??u) = 0 in a circle, with Neumann condition describing the behavior of the electrostatic potential in a medium with conductivity given by the function ?(x, y). We suppose that at each time a current ψ _i is applied to the boundary of the circle (Neumann's data), and that it is possible to measure the corresponding potential ? _i (Dirichlet data). The inverse problem is to find ?(x, y) given a finite number of Cauchy pairs measurements (? _i , ψ _i ), i = 1,…, N. The problem is formulated as a least squares problem, and the developed algorithm solves the continuous problem using descent iterations in its corresponding finite element approximations. Wolfe's conditions are used to ensure the global convergence of the optimization algorithm for the continuous problem. Although exact data are assumed, measurement errors in data and regularization methods shall be considered in a future work.     

Absence of localization of the Laplace series on the sphere for functions of the Nikol'skii class H 1 1 (S2)

In this article a function is constructed belonging to the class H ₁ ¹ (S²) and having a singularity at a definite point on the sphere, as a consequence of which localization fails for the Laplace series of this function at the diametrically opposite point. The constructed example shows that the sufficient condition of localization in H _a ^p of the spectral expansions in the class of all elliptic differential operators on an n-dimensional paracompact manifold cannot be improved (see [1]).Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 517–523, October, 1977.     

On the behavior of solutions of equations for double waves in the neighborhood of the quiescent region : PMM vol. 39, n 6, 1975, pp. 1043–1050

The structure of solutions of gasdynamic equations is investigated in the case of unsteady double waves in the neighborhood of the quiescent region. A general concept of double waves is presented in the form of special series with logarithmic terms. Results of numerical computations are given.The problem of determining the flow of plane and three-dimensional waves separated from the quiescent region by a weak discontinuity was considered in [1–3], where approximate solutions were derived for that neighborhood, and the formulation of boundary value problems required for solving the equation for the analog of the velocity potential in the hodograph plane was investigated.The more general problem (without the assumption of the degeneration of motion) of arbitrary potential flows of polytropic gas adjacent to the quiescent region and separated by a weak discontinuity was considerd in [4–8]. Solution of that problem was obtained in the form of special series in powers of the mo dulus of the velocity vector r in the space of the time hodograph. The value r = 0 corresponds to the surface of weak discontinuity that separates the perturbed motion region from that at rest. Some applications of derived solutions to problems such as the motion of a convex piston and the propagation of weak shock waves were also investigated in those papers. Convergence in the small of obtained series was proved in [9]. However the attempts of constructing series in powers of r, which were used in [4–8] for the presentation of equations of double waves in the neighborhood of the quiescent region, proved to be unsuccessful.Although parts of expansions in series in powers of r (accurate to within 0 (r²)), were constructed in [1–3], it was found that the coefficient at r⁸ in equations for double waves cannot be determined owing to the insolvability of its equation. This is related to the fact that the surface r = 0in the case of equations for double waves is simultaneously a line of parabolic degeneration and a characteristic.The object of the present note is the formulation of solutions of equations for plane unsteady double waves in the neighborhood of the quiescent region. Parts of the derived series, which generally are nonanalytic functions of r, can be used for defining flows at small r in particular those downstream of two-dimensional normal detonation waves [10] or in problems of angular pistons [11]. The method used for the derivation of series can be also applied in investigations of threedimensional self-similar flows with variables x₁/x₃ and x₂/x₃ (steady flows) or x₁/t, x₂/t and x₃/t (unsteady flows). However it was not possible to obtain in such cases regular series in powers of r.     

All meromorphic solutions of an ordinary differential equation and its applications

Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions w_r,2(z) and simply periodic solutions w_s,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd.