Method of characteristics and first integrals for systems of quasi-linear partial differential equations of first order

Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.     

Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial differential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of differential equations. A general existence and uniqueness theorem is established. Formal theory is then contrasted with Janet-Riquier theory in the formulation of Reid. Finally, the implications of the results for the Krichever-Novikov equation are outlined.     

Chebyshev and L1 solutions of overdetermined systems of linear equations with bounded variables

Two algorithms are here presented. The first one is for obtaining a Chebyshev solution of an overdetermined system of linear equations subject to bounds on the elements of the solution vector. The second algorithm is for obtaining an L₁ solution of an overdetermined system of linear equations subject to the same constraints. Efficient solutions are obtained using linear programming techniques. Numerical results and comments are given.     

A Boundary Value Problem for an Overdetermined Steady System in Two-Velocity Hydrodynamics

In this paper, a two-velocity steady hydrodynamic system with a single pressure and inhomogeneous divergent and boundary conditions for two velocities is investigated. This system is overdetermined. By changing the sought-for functions, the problem is reduced to a homogeneous one. The resulting system is solved by consecutively solving two boundary value problems: a Stokes problem for one velocity and the pressure and an overdetermined system for the other velocity. Generalized statements of these problems and their discrete approximations using a finite element method are presented. A new regularization method is used to solve the overdetermined problem.     

A Serrin-type symmetry result on model manifolds: An extension of the Weinberger argument

We consider the classical ‘‘Serrin's symmetry result” for the overdetermined boundary value problem related to the equation $\mathrm{\Delta }u=?1$ in a model manifold of non-negative Ricci curvature. Using an extension of the Weinberger classical argument we prove a Euclidean symmetry result under a suitable ‘‘compatibility” assumption between the solution and the geometry of the model.     

A Randomized Kaczmarz Algorithm with Exponential Convergence

The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling. T. Strohmer was supported by NSF DMS grant 0511461. R. Vershynin was supported by the Alfred P. Sloan Foundation and by NSF DMS grant 0401032.     

Infinitesimal Isometries Along Curves and Generalized Jacobi Equations

On a Riemannian manifold, a solution of the Killing equation is an infinitesimal isometry. Since the Killing equation is overdetermined, infinitesimal isometries do not exist in general. A completely determined prolongation of the Killing equation is a PDE on the bundle of 1-jets of vector fields. Restricted to a curve, this becomes an ODE that generalizes the Jacobi equation. A solution of this ODE is called an infinitesimal isometry along the curve, which we show to be an infinitesimal rigid variation of the curve. We define Killing transport to be the associated linear isometry between fibers of the bundle along the curve, and show that it is parallel translation for a connection on the bundle related to the Riemannian connection. Restricting to dimension two, we study the holonomy of this connection, prove the Gauss–Bonnet theorem by means of Killing transport, and determine the criteria for local existence of infinitesimal isometries.     

A Carleman inequality for the stationary anisotropic Maxwell system

A Carleman estimate for the stationary anisotropic Maxwell system is established. Its proof adopts a technique pioneered by Calderón to an overdetermined systems with rough coefficients. As an application, the conditional stability of the Cauchy problem is discussed.     

Variational formula and overdetermined problem for the principle eigenvalue of k-Hessian operator

We first deduce the first variational formula and some overdetermined problems for the principle eigenvalue of the k-Hessian operator, and then prove Serrin type symmetry result for our overdetermined problems.     

A nonlinear singular eigenvalue problem for a linear system of ordinary differential equations with redundant conditions

A nonlinear eigenvalue problem for a linear system of ordinary differential equations is examined on a semi-infinite interval. The problem is supplemented by nonlocal conditions specified by a Stieltjes integral. At infinity, the solution must be bounded. In addition to these basic conditions, the solution must satisfy certain redundant conditions, which are also nonlocal. A numerically stable method for solving such a singular overdetermined eigenvalue problem is proposed and analyzed. The essence of the method is that this overdetermined problem is replaced by an auxiliary problem consistent with all the above conditions.     

Newton's method for overdetermined systems of equations

Complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors. In this paper we consider overdetermined systems where there are more equations than unknowns. We study Newton's method for such a system.

     

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8], [9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy–Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.     

Arbitrariness of the general solution of the partial differential equations and its applications

Using the framework of formal theory of partial differential equations, we consider a method of computation of the bi-Hilbert polynomial (i.e. Hilbert polynomial in two variables). Furthermore, present an approach to compute the number of arbitrary functions of positive differential order in the general solution. Then, under the “AC=BD” model for mathematics mechanization developed by Hong-qing ZHANG, we present a method to reduce an overdetermined system to a well-determined one. As applications, the Maxwell equations and weakly overdetermined equations are considered.     

ON THE ACCURACY OF THE LEAST SQUARES AND THE TOTAL LEAST SQUARES METHODS

Consider solving an overdetermined system of linear algebraic equations by both the least squares method (LS) and the total least squares method (TLS). Extensive published computational evidence shows that when the original system is consistent. one often obtains more accurate solutions by using the TLS method rather than the LS method. These numerical observations contrast with existing analytic perturbation theories for the LS and TLS methods which show that the upper bounds for the LS solution are always smaller than the corresponding upper bounds for the TLS solutions. In this paper we derive a new upper bound for the TLS solution and indicate when the TLS method can be more accurate than the LS method.Many applied problems in signal processing lead to overdetermined systems of linear equations where the matrix and right hand side are determined by the experimental observations (usually in the form of a lime series). It often happens that as the number of columns of the matrix becomes larger, the ra     

广义块Broyden方法与超定方程组求解

1．引言［1］提出用块Broyden方法求解成组的线性与非线性方程组，同时证明了：若有p组n阶线性方程组则块Brorden方法具有至多2n／p步的有限终止性．这种块形式算法，对于大型成组问题的计算，在计算量和存储量方面，都会有相当的改善，并且有利于并行计算．本文推广上述结果，建立一种广义块nroxaen方法，并将它应用于成组的超定方程组的求解．我们证明了对于给出的p组。x叫x三叫的线性超定方程组其中AeRm””，x；e＊”，kEBm，广义块Broxden方法同样具有至多z。／r步的有限终止性，这表明超定方程组的纽数越多（P5…，方法所需的选代…     

Overdetermined elliptic problems in topological disks

We introduce a method, based on the Poincaré–Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.     

Radial symmetry and partially overdetermined problems in a convex cone

We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.     

Acceleration of randomized Kaczmarz method via the Johnson–Lindenstrauss Lemma

The Kaczmarz method is an algorithm for finding the solution to an overdetermined consistent system of linear equations Ax = b by iteratively projecting onto the solution spaces. The randomized version put forth by Strohmer and Vershynin yields provably exponential convergence in expectation, which for highly overdetermined systems even outperforms the conjugate gradient method. In this article we present a modified version of the randomized Kaczmarz method which at each iteration selects the optimal projection from a randomly chosen set, which in most cases significantly improves the convergence rate. We utilize a Johnson–Lindenstrauss dimension reduction technique to keep the runtime on the same order as the original randomized version, adding only extra preprocessing time. We present a series of empirical studies which demonstrate the remarkable acceleration in convergence to the solution using this modified approach.     

Two-Stage Method for Solving Systems of Nonlinear Equations and Its Applications to the Inverse Atmospheric Sounding Problem

Doklady Mathematics - For an overdetermined system of nonlinear equations, a two-stage method is suggested for constructing an error-stable approximate solution. The first stage consists in...