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.     

New methods for localizing discontinuities of a noisy function

For a problem of localizing singularities (discontinuities of the first kind) of a noisy function in L _p (1 ≤ p < ∞), new classes of regularizing methods are constructed. The methods determine the number of discontinuities and approximate their positions. The upper and lower bounds of the localizing singularities and the separability threshold are also obtained. It is proved that the methods are order-optimal by accuracy and separability on some classes of functions with discontinuities.     

Approximation of discontinuity lines of a noisy function of two variables

In the paper, the methods are constructed and investigated of localization (determination of the location) of the line in a neighborhood of which a measured function of two variables is smooth, whereas it has a discontinuity of the first kind on this line. Instead of the exact function, there is available some of its approximations in L ₂ and the perturbation level. The problem under consideration belongs to the class of nonlinear ill-posed problems, and to solve it some regularization algorithms are needed. A simplified theoretical approach is suggested to the problem of localization of the discontinuity line of a noisy function, when the conditions on the exact function are imposed on an arbitrary small strip intersecting the discontinuity line. Some averaging methods are constructed, and estimates for the precision of the line localization are obtained.     

Asymptotic stability of viscous contact discontinuity to an inflow problem for compressible Navier-Stokes equations

This paper is concerned with an initial-boundary value problem for one-dimensional full compressible Navier-Stokes equations with inflow boundary conditions in the half space R₊=(0,+∞). The asymptotic stability of viscous contact discontinuity is established under the conditions that the initial perturbations and the strength of contact discontinuity are suitably small. Compared with the free-boundary and the initial value problems, the inflow problem is more complicated due to the additional boundary effects and the different structure of viscous contact discontinuity. The proofs are given by the elementary energy method.     

Singular points of a self-similar function of spectral order zero: Self-similar stieltjes string

In the present paper, we introduce the notion of self-similar function of spectral order zero and study its properties. Such functions have at most countably many discontinuity points, and these points are discontinuity points of the first kind except possibly for a single point, which is a singular point. We derive a formula for calculating the coordinates of this point from the parameters of the self-similar function. We also study the behavior of the self-similar function near the singular point. A nondecreasing function f of spectral order zero belonging to the space L ₂[0, 1] generates a self-similar Stieltjes string, namely, a spectral problem of the form $$ - y'' - \lambda \rho y = 0,y(0) = y(1) = 0 $$ where ρ is a function from the space $ \mathop W\limits^ \circ _{_2^{ - 1} } \left[ {0,1} \right] $ and f′ = ρ. Such a function f that is not of a fixed sign leads to the notion of self-similar indefinite Stieltjes string.     

On the convergence of the Dirichlet grid problem with a singularity for a singularly perturbed convection–diffusion equation

The Dirichlet problem for a singulary perturbed convection–diffusion equation in a rectangle when a discontinuity at the flow exit the first derivative of the boundary condition gives rise to an inner layer for the solution. On piecewise-uniform Shishkin grids that condense near regular and characteristic layers, the solution obtained using the classical five-point difference scheme with a directed difference is shown to converge with respect to the small parameter to solve the original problem in the grid norm L _∞ ^h almost with the first order. This theoretical result is confirmed via numerical analysis.     

Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows

In our previous work, we have established the existence of transonic characteristic discontinuities separating supersonic flows from a static gas in two-dimensional steady compressible Euler flows under a perturbation with small total variation of the incoming supersonic flow over a solid right wedge. It is a free boundary problem in Eulerian coordinates and, across the free boundary (characteristic discontinuity), the Euler equations are of elliptic–hyperbolic composite-mixed type. In this paper, we further prove that such a transonic characteristic discontinuity solution is unique and L ¹–stable with respect to the small perturbation of the incoming supersonic flow in Lagrangian coordinates.     

On the basis property of root elements of ordinary differential operators in the space L 2,n (a, b)

We study a spectral problem for a system of linear ordinary differential operators in the vector function space L _2,n (a, b) with parameter-dependent boundary conditions. We prove a theorem stating that the system of root functions of the problem is a basis with parentheses in L _2,n (a, b). Corollaries of the theorem are considered.     

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.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> spline fits via sliding window process: continuous and discrete cases

The best L ₁ approximation of the Heaviside function and the best ? ₁ approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using L ₁ spline fits which are the best L ₁ approximations in an appropriate spline space obtained by the union of L ₁ interpolation splines. We prove here the existence of L ₁ spline fits for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude, but within a linear complexity on the number of spline nodes.     

A simple and space-efficient fragment-chaining algorithm for alignment of DNA and protein sequences

In the segment-based approach to sequence alignment, nucleic acid, and protein sequence alignments are constructed from fragments, i.e., from pairs of ungapped segments of the input sequences. Given a set F of candidate fragments and a weighting function w : F → R⁺₀, the score of an alignment is defined as the sum of weights of the fragments it consists of, and the optimization problem is to find a consistent collection of pairwise disjoint fragments with maximum sum of weights. Herein, a sparse dynamic programming algorithm is described that solves the pairwise segment-alignment problem in O(L + N_max) space where L is the maximum length of the input sequences while N_max ≤ #F holds. With a recently introduced weighting function w, small sets F of candidate fragments are sufficient to obtain alignments of high quality. As a result, the proposed algorithm runs in essentially linear space.     

Nonlinear stability of viscous contact wave for the one‐dimensional compressible fluid models of Korteweg type

This paper is concerned with the large time behavior of solutions of the Cauchy problem to the one‐dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave corresponding to the contact discontinuity is asymptotically stable provided that the strength of contact discontinuity and the initial perturbation are suitably small. The analysis is based on the elementary L²‐energy method together with continuation argument. Copyright © 2013 John Wiley & Sons, Ltd.     

Square function and maximal function estimates for operators beyond divergence form equations

We prove square function estimates in L ₂ for general operators of the form B ₁ D ₁ + D ₂ B ₂, where D _i are partially elliptic constant coefficient homogeneous first-order self-adjoint differential operators with orthogonal ranges, and B _i are bounded accretive multiplication operators, extending earlier estimates from the Kato square root problem to a wider class of operators. The main novelty is that B ₁ and B ₂ are not assumed to be related in any way. We show how these operators appear naturally from exterior differential systems with boundary data in L ₂. We also prove non-tangential maximal function estimates, where our proof needs only off-diagonal decay of resolvents in L ₂, unlike earlier proofs which relied on interpolation and L _p estimates.     

A dual control problem and application to marketing

The general approach to adaptive and dual control is to formulate an optimal stochastic control problem. However, for such an approach only mathematical representations of the solution are available which allow little insight into the structure of the optimal controller. Here, an alternative deterministic approach is presented based upon determining a control in which a disturbance attenuation function remains bounded for all allowable (L₂ functions) disturbances. The disturbance attenuation function is composed of the ratio of an L₂ function of the desired outputs over an L₂ function of the disturbance inputs. This disturbance attenuation problem is converted to a differential game. For this game, the optimal control law, in a closed-form, is obtained by performing a minmax operation with respect to a quadratic cost function subjected to a bilinear system. The resulting controller is time-varying and depends nonlinearly on the state and the parameter estimates vector, and on an associated Riccati-type matrix. We provide insights into the structure of the resulting dual controller and illustrate the method by two examples. One of the examples is an application to marketing, to set promotional spending of a company, considering that the effect of promotional effort on sales is unknown.     

Control problems governed by Sobolev partial differential equations

Let G be a bounded subset of Rⁿ with a smooth boundary and Q = G × (0, T]. We consider a control problem governed by the Sobolev initial-value problem My_t(u) + Ly(u) = u in L²(Q), y(·, 0; u) = 0 in L₂(G), where M = M(x) and L = L(x) are symmetric uniformly strongly elliptic operators of orders 2m and 2l, respectively. The problem is to find the control u₀ of L²(Q)-norm at most b that steers to within a prescribed tolerance ? of a given function Z in L²(G) and that minimizes a certain energy functional. Our main results establish regularity properties of u₀. We also give results concerning the existence and uniqueness of the optimal control, the controllability of Sobolev initial-value problems, and properties of the Lagrange multipliers associated with the problem constraints.     

Optimal control of a system governed by hyperbolic operator with an infinite number of variables

A distributed control problem for the hyperbolic operator with an infinite number of variables is considered. The controls are allowed to be in the space L₂(0, T; L₂(R^∞)) = L₂(Q). The necessary and sufficient condition for the control to be optimal is obtained and the set of inequalities that characterize this condition is also obtained.     

Convex approximations in stochastic programming by semidefinite programming

The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in L ₁, L _?? and L ₂ norm. Extensive numerical experiments to investigate the behavior of the proposed methods are also performed.     

On the Navier problem for the stationary Navier-Stokes equations

The Navier problem is to find a solution of the steady-state Navier-Stokes equations such that the normal component of the velocity and a linear combination of the tangential components of the velocity and the traction assume prescribed value a and s at the boundary. If Ω is exterior it is required that the velocity converges to an assigned constant vector u₀ at infinity. We prove that a solution exists in a bounded domain provided ‖a_‖L²(∂Ω) is less than a computable positive constant and is unique if ‖a_‖W1/2,2(∂Ω)+‖s_‖L²(∂Ω) is suitably small. As far as exterior domains are concerned, we show that a solution exists if ‖a_‖L²(∂Ω)+‖a−u₀⋅n_‖L²(∂Ω) is small.     

Narrow and ℓ2-strictly singular operators from L p

In the first part of the paper we prove that for 2 < p, r < ∞ every operator T: L _p → ? _r is narrow. This completes the list of sequence and function Lebesgue spaces X with the property that every operator T : L _p → X is narrow. Next, using similar methods we prove that every ?₂-strictly singular operator from L _p , 1 < p < ∞, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H. P. Rosenthal asserts that if an operator T from L ₁[0, 1] to itself satisfies the assumption that for each measurable set A ? [0, 1] the restriction \(T{|_{{L_1}(A)}}\) is not an isomorphic embedding, then T is narrow. (Here L ₁(A) = {x ∈ L ₁ : supp x ? A}.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being ?₂-strictly singular, for operators from L _p [0, 1] to itself, 1 < p < 2, to be narrow. We define a notion of a “gentle” growth of a function and we prove that for 1 < p < 2 every operator T from L _p to itself which, for every A ? [0, 1], sends a function of “gentle” growth supported on A to a function of arbitrarily small norm is narrow.