We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem.
We consider the problem of reconstruction of the basal velocity of a glacier by measurements of the velocity on glacier’s surface. The main result is a stability estimate in a near-surface region, which represents a multiplicative inequality and shows that small errors in measurements produce small errors in the velocity in this region.
In this paper we present a one-dimensional model of blood flow in a vessel segment with an elastic wall consisting of several anisotropic layers. The model involves two variables: the radial displacement of the vessels wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight segment of a blood vessel demonstrate that the model can produce realistic flow fields that may appear under normal conditions in healthy blood vessels; as well as flow that could appear during abnormal conditions. In particular we show that weakening of the elastic properties of the wall may provoke a reverse blood flow in the vessel. (C) 2015 Elsevier Inc. All rights reserved.
The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.
In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by the authors for solving this problem but the convergence has been rather slow. We demonstrate how to instead use conjugate gradient methods for accelerating the convergence. The main idea is to introduce an artificial boundary in the interior of the domain. This addition of the interior boundary allows us to derive an inner product that is natural for the application and that gives us a proper framework for implementing the steps of the conjugate gradient methods. The numerical results performed using the finite difference method show that the conjugate gradient based methods converge considerably faster than the modified alternating iterative procedure studied previously.
We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method.
The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers in the Helmholtz equation. We prove here that if we alternate Robin and Dirichlet boundary conditions instead of Neumann and Dirichlet boundary conditions, then the algorithm will converge. We present also another algorithm based on the same idea, which converges for large wavenumbers. Numerical implementations obtained using the finite difference method are presented. Numerical results illustrate that the algorithms suggested in this paper, produce a convergent iterative sequences.
Kozlov & Mazya (1989, Algebra Anal., 1, 144-170) proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems. However, in many applied problems, operators appear that do not satisfy these requirements, e.g. Helmholtz-type operators. Therefore, in this study, an alternating procedure for solving Cauchy problems for self-adjoint non-coercive elliptic operators of second order is presented. A convergence proof of this procedure is given.
This paper brings together a novel information representation model for use in signal processing and computer vision problems, with a particular algorithmic development of the Landweber iterative algorithm. The information representation model allows a representation of multiple values for a variable as well as an expression for confidence. Both properties are important for effective computation using multi-level models, where a choice between models will be implementable as part of the optimization process. It is shown that in this way the algorithm can deal with a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. While the algorithm has been applied to the solution of such problems, it has so far been used heuristically. In this paper we describe the properties and some of the peculiarities of the channel representation and optimization, and put them on firm mathematical ground. We consider the optimization a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective with currently available projected Landweber methods. An application to supervised learning is described, and the method is evaluated in an experiment involving function approximation, as well as application to transient signals. © 2006 Elsevier Ltd. All rights reserved.
This report brings together a novel approach to some computer vision problems and a particular algorithmic development of the Landweber iterative algorithm. The algorithm solves a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. The algorithm has recently been applied to these problems, but it has been used rather heuristically. In this report we describe the method and put it on firm mathematical ground. We consider a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective of currently available projected Landweber methods. The application to supervised learning is described, and the method is evaluated in a function approximation experiment.
We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.
We consider variational solutions to the Dirichlet problem for elliptic systems of arbitrary order. It is assumed that the coefficients of the principal part of the system have small, in an integral sense, local oscillations near a boundary point and other coefficients may have singularities at this point. We obtain an asymptotic representation for these solutions and derive sharp estimates for them which explicitly contain information on the coefficients. ©2006 Texas State University - San Marcos.
The dependence on the domain for the Dirichlet eigenvalues of elliptic operators considered in bounded domains is studied. The proximity of domains is measured by a norm of the difference of two orthogonal projectors corresponding to the reference domain and the perturbed one; this allows to compare eigenvalues corresponding to domains that have non-smooth boundaries and different topology. The main result is an asymptotic formula in which the remainder is evaluated in terms of this quantity. Applications of this result are given. The results are new for the Laplace operator.
We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small in L8 -norm. Copyright © 2006 Vladimir Kozlov.
Bounded solutions of the Emden-Fowler equation in a semi-cylinder are considered. For small solutions the asymptotic representations at infinity are derived. It is shown that there are large solutions whose behavior at infinity is different. These solutions are constructed when some inequalities between the dimension of the cylinder and the homogeneity of the nonlinear term are fulfilled. If these inequalities are not satisfied then it is proved, for the Dirichlet problem, that all bounded solutions tend to zero and have the same asymptotics as small solutions. © 2002 Elsevier Science (USA).
We consider the first eigenvalue of the Dirichlet-Laplacian in three cases: C1, 1-domains, Lipschitz domains, and bounded domains without any smoothness assumptions. Asymptotic formula for this eigenvalue is derived when domain subject arbitrary perturbations. For Lipschitz and arbitrary nonsmooth domains, the leading term in the asymptotic representation distinguishes from that in the Hardamard formula valid for smooth perturbations of smooth domains. For asymptotic analysis we propose and prove an abstract theorem demonstrating how eigenvalues vary under perturbations of both operator in Hilbert space and Hilbert space itself. This abstract theorem is of independent interest and has substantially broader field of applications. © 2006 Elsevier Inc. All rights reserved.
The traction boundary value problem for anisotropic elasticity is considered. For polyhedral domains in R-3, it is proved that the displacements are Holder continuous. In the n-dimensional case, n > 3, the Holder continuity is proved for domains with conic points on the boundary. The proof is based on the study of spectrum of operator pencils associated with singularities of the boundary, which is of independent interest.
Using the dimension reduction procedure, a one-dimensional model of a periodic blood flow in the artery through a small hole in a thin elastic wall to a spindle-shaped hematoma, is constructed. This model is described by a system of two parabolic and one hyperbolic equations provided with mixed boundary and periodicity conditions. The blood exchange between the artery and the hematoma is expressed by the Kirchhoff transmission conditions. Despite the simplicity, the constructed model allows us to describe the damping of a pulsating blood flow by the hematoma and to determine the condition of its growth. In medicine, the biological object considered is called a false aneurysm.
Asymptotic analysis is applied for obtaining one-dimensional models of the blood flow in narrow, thin-walled, elastic vessels. The models for arteries and veins essentially distinguish from each other, and the reason for this is the structure of their walls, as well as the operationing conditions. Although the obtained asymptotic models are simple, they explain various effects known in medical practice, in particular, describe the mechanism of vein-muscle pumping of blood.
An alternating procedure for solving a Cauchy problem for the stationary Stokes system is presented. A convergence proof of this procedure and numerical results are included. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
We investigate a mixed problem with variable lateral conditions for the heat equation that arises in modelling exocytosis, i.e. the opening of a cell boundary in specific biological species for the release of certain molecules to the exterior of the cell. The Dirichlet condition is imposed on a surface patch of the boundary and this patch is occupying a larger part of the boundary as time increases modelling where the cell is opening (the fusion pore), and on the remaining part, a zero Neumann condition is imposed (no molecules can cross this boundary). Uniform concentration is assumed at the initial time. We introduce a weak formulation of this problem and show that there is a unique weak solution. Moreover, we give an asymptotic expansion for the behaviour of the solution near the opening point and for small values in time. We also give an integral equation for the numerical construction of the leading term in this expansion.
The Poisson equation in two-dimensional case for a nonsmooth domain is considered. The geometrical domain has a cut (crack) where inequality type boundary conditions are imposed. A behavior of the solution near the crack tips is analyzed. In particular, estimates for the second derivatives in a weighted Sobolev space are obtained and asymptotics of the solution near crack tips is established. © 2007 Birkhaeuser.
The asymptotic behavior of the solution to the Poisson equations near the crack tip with nonlinear boundary conditions and second derivatives of the in weighted Sobolev spaces were determined. The internal regularity results for the equation showed that the second derivative of u belonged to L2 in the neighborhood of interior points of the crack. The results demonstrated analogy between the properties of the solution to the linear problem and the nonlinear problem. It was also found that an asymptotic representation to the problem can be constructed near the crack tip.
This Note deals with an eigenvalue problem that contains a spectral parameter in a boundary condition. The problem for the two-dimensional Laplace equation describes free, time-harmonic water waves in a canal having uniform cross-section and bounded from above by a horizontal free surface. It is shown that there exists a domain for which at least one of eigenfunctions has a nodal line with both ends on the free surface. Since Kuttler essentially used the non-existence of such nodal lines in his proof of simplicity of the fundamental sloshing eigenvalue in the two-dimensional case, we propose a new variational principle for demonstrating this latter fact. ⌐ 2003 AcadΘmie des sciences.
Steady, free-surface, vortical flows of an inviscid, incompressible, heavy fluid over a horizontal, rigid bottom are considered. All flows of constant depth are described for any Lipschitz vorticity distribution. It is shown that the values of Bernoullis constant, for which such flows exist, are greater than or equal to some critical value depending on the vorticity. For the critical value, only one flow exists and it is unidirectional. Supercritical flows exist for all values of Bernoullis constant greater than the critical one; every such flow is also unidirectional and its depth is smaller than that of the critical flow. Furthermore, at least one flow other than supercritical does exist for every value of Bernoullis constant greater than the critical one. It is found that for some vorticity distributions, the number of constant depth flows increases unrestrictedly as Bernoullis constant tends to infinity. However, all these flows, except for one or two, have counter-currents; their number depends on Bernoullis constant and increases by at least two every time when this constant becomes greater than a critical value (the above mentioned is the smallest of them), belonging to a sequence defined by the vorticity. A classification of vorticity distributions is presented; it divides all of them into three classes in accordance with the behaviour of some integral of the distribution on the interval [0, 1]. For distributions in the first class, a unidirectional subcritical flow exists for all admissible values of Bernoullis constant. For vorticity distributions belonging to the other two classes such a flow exists only when Bernoullis constant is less than a certain value. If Bernoullis constant is greater than this value, then at least one flow with counter-currents does exist along with the unidirectional supercritical flow. The second and third classes of vorticity distributions are distinguished from one another by the character of the counter-currents. If a distribution is in the second class, then a near-bottom counter-current is always present for sufficiently large values of Bernoullis constant. For distributions in the third class, a near-surface counter-current is always present for such values of the constant. Several examples illustrating the results are considered.
For the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoullis constant. (ii) If only one parallel shear flow exists for a given value of Bernoullis constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.
Two-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm-Liouville problem considered by Constantin and Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481-527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133-175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm-Liouville problem mentioned in (i) has an eigenvalue is obtained.
We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem for the two-dimensional Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a canal having uniform cross-section and bounded from above by a horizontal free surface. It is demonstrated that there exist domains such that at least one of the eigenfunctions has a nodal line or lines with both ends on the free surface (earlier, Kuttler tried to prove that there are no such nodal lines for all domains but his proof is erroneous). It is also shown that the fundamental eigenvalue is simple, and for the corresponding eigenfunction the behaviour of the nodal line is characterized. For this purpose, a new variational principle is proposed for an equivalent statement of the sloshing problem in terms of the conjugate stream function. © 2004 The Royal Society.
Necessary conditions for the existence of arbitrary bounded steady waves are proved (earlier, these conditions, that have the form of bounds on the Bernoulli constant and other wave characteristics, were established only for Stokes waves). It is also shown that there exists an exact upper bound such that if the free-surface profile is less than this bound at infinity (positive, negative, or both), then the profile asymptotes the constant level corresponding to a unform stream (supercritical or subcritical). Finally, an integral property of arbitrary steady waves is obtained. A new technique is proposed for proving these results; it is based on modified Bernoulli’s equation that along with the free surface profile involves the difference between the potential and its vertical average.
The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds on the free-surface profiles and on the total head are obtained under minimal assumptions about properties of solutions to the problem and the vorticity distribution.
The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: one for waves with fixed Bernoullis constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.
The two-dimensional free-boundary problem of steady gravity waves on water of finite depth is considered. Bounds on the free-surface profiles and on the values of Bernoullis constant are obtained under minimal assumptions about properties of solutions to the problem.
The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.
The sloshing problem is considered in a half-space covered by a rigid dock with apertures. The dependence of the fundamental sloshing frequency on the shape of the free surface region is studied. It is proved that the inequality holds between the fundamental eigenvalues corresponding to two different regions if some conditions are fulfilled. These conditions are verified for particular classes of regions of a fixed area in order to demonstrate that the disk yields the maximum of the fundamental eigenvalue for each of these classes. On the other hand, examples of regions are constructed for which the fundamental eigenfrequency is larger than that for the circular aperture of the same area and even as large as one wishes.
Spatial dynamical systems are obtained for two-dimensional steady gravity waves with vorticity on water of finite depth. These systems have Hamiltonian structure and Hamiltonian is essentially the flow-force invariant.
In 1954, Benjamin and Lighthill made a conjecture concerning the classical nonlinear problem of steady gravity waves on water of finite depth. According to this conjecture, a point of some cusped region on the (r, s)-plane (r and s are the non-dimensional Bernoullis constant and the flow force, respectively), corresponds to every steady wave motion described by the problem. Conversely, at least one steady flow corresponds to every point of the region. In the present paper, this conjecture is proved for near-critical flows (when r attains values close to one), under the assumption that the slopes of wave profiles are bounded. Another question studied here concerns the uniqueness of solutions, and it is proved that for every near-critical value of r only the following waves do exist: (i) a unique (up to translations) solitary wave; (ii) a family of Stokes waves (unique up to translations), which is parametrised by the distance from the bottom to the wave crest. The latter parameter belongs to the interval bounded below by the depth of the subcritical uniform stream and above by the distance from the bottom to the crest of solitary wave corresponding to the chosen value of r.
The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin-Lighthill conjecture is proved for these waves provided Bernoullis constant attains near-critical values. In fact this is a consequence of the following more general results. If Bernoullis constant is near-critical, then all corresponding waves have sufficiently small heights and slopes. Moreover, for every near-critical value of Bernoullis constant, there exist only the following waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave; this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch Rational Mech Anal 197, 433-488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope of this wave is also small.
We study an eignevalue problem with a spectral parameter in a boundary condition. This problem for the Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a half-space covered by a rigid dock with some apertures (an ice sheet with fishing holes). The dependence of the fundamental eigenvalue on holes' geometry is investigated. We give conditions on a plane region guaranteeing that the fundamental eigenvalue corresponding to this region is larger than the fundamental eigenvalue corresponding to a single circular hole. Examples of regions satisfying these conditions and having the same area as the unit disk are given. New results are also obtained for the problem with a single circular hole. On the other hand, we construct regions for which the fundamental eigenfrequency is larger than the similar frequency for the circular hole of the same area and even as large as one wishes. In the latter examples, the hole regions are either not connected or bounded by a rather complicated curves.
The two-dimensional problem of steady waves on water of finite depth is considered without assumptions about periodicity and symmetry of waves. A new form of Bernoulli's equation is derived, and it involves a new bifurcation parameter which is the product of the Froude number μ and the rate of flow ω. The main result obtained from this equation is the absence of waves, having sufficiently small amplitude, provided |μω| > 1. © 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.
A correct proof is given for the following assertions about the two-dimensional sloshing problem. The fundamental eigenvalue is simple and the corresponding stream function may be chosen to be non-negative in the closure of the water domain. New proof is based on stricter assumptions about the water domain; namely, it must satisfy John’s condition.
The paper deals with the classical non-linear problem of steady two-dimensional waves on water of finite depth. The problem is formulated so that it describes all waves without stagnation points on the free-surface profiles that are bounded themselves and have bounded slopes. By virtue of reducing the problem to an integro-differential equation the following three results are proved. First, there are no waves when the flow is critical. Second, there are no waves having profiles totally above the upper boundary of the uniform subcritical stream. Finally, only two types of the free-surface behaviour are possible at positive (or/and negative) infinity: the profile either oscillates infinitely many times around the upper boundary of the subcritical uniform stream or asymptotes the upper level of a uniform stream (subcritical or supercritical). The latter assertion is proved under additional assumption that the slope of the free surface is a uniformly continuous function. © 2008 Elsevier Masson SAS. All rights reserved.
We study asymptotics as t -> infinity of solutions to a linear, parabolic system of equations with time-dependent coefficients in Omega x (0, infinity), where Omega is a bounded domain. On partial derivative Omega x (0, infinity) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function kappa(t). This includes in particular situations when the coefficients may take different values on different parts of Omega and the boundaries between them can move with t but stabilize as t -> infinity. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if kappa epsilon L-1 (0, infinity), then the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.
We study asymptotics as t --> infinity of solutions to a linear, parabolic system of equations with time-dependent coefficients in Omega x (0,infinity), where Omega is a bounded domain. On partial derivative Omega x (0,infinity) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function kappa(t). This includes in particular situations when the coefficients may take different values on different parts of Omega and the boundaries between them can move with t but stabilize as t --> infinity. The main result is an asymptotic representation of solutions for large t. A consequence is that for kappa is an element of L-1(0,infinity), the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.