A TRUST REGION METHOD FOR SOLVING DISTRIBUTED PARAMETER IDENTIFICATION PROBLEMS

This paper is concerned with the ill-posed problems of identifying a parameter in an elliptic equation which appears in many applications in science and industry. Its solution is obtained by applying trust region method to a nonlinear least squares error problem.Trust region method has long been a popular method for well-posed problems. This paper indicates that it is also suitable for ill-posed problems, Numerical experiment is given to compare the trust region method with the Tikhonov regularization method. It seems that the trust region method is more promising.     

THE L~2-NORM ERROR ESTIMATE OF NONCONFORMING FINITE ELEMENT METHOD FOR THE 2ND ORDER ELLIPTIC PROBLEM WITH THE LOWEST REGULARITY

1. IntroductionThis paper is concerned with the uniformly LZ-- norm error estimate of the nonconforming finite method for the second order elliptic problem with the lowest regularity3i.e., in the case that the solution u E H'(fl) only, but not in H'(fl).For the conforming finite element method of the second order elliptic problem, it iswell known that using the Aubin-Nitsche lemma obtained the LZ-- norm error bound,which is one order of h, the parameter of triangulation, higher than the HI…     

AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.     

A MINIMIZATION PROBLEM ON EXTERIOR DOMAIN

In this paper, using the decay estimates for the solution of semilinear elliptic equation, we obtain the existence of a minimun for a minimization problem with nonzero data in the exterior domain.     

An extended Galerkin analysis for elliptic problems

A general analysis framework is presented in this paper for many different types of finite element methods(including various discontinuous Galerkin methods). For the second-order elliptic equation-div(α▽u) = f, this framework employs four different discretization variables, uh, p_h, uhand p_h, where uh and phare for approximation of u and p =-α▽u inside each element, and uhand phare for approximation of the residual of u and p·n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.     

The Nehari Manifold and Application to a Quasilinear Elliptic Equation with Multiple Hardy-Type Terms

In this paper, by using the Nehari manifold and variational methods, we study the existence and multiplicity of positive solutions for a multi-singular quasilinear elliptic problem with critical growth terms in bounded domains. We prove that the equation has at least two positive solutions when the parameters A belongs to a certain subset of JR.     

Additive hazards regression under generalized case-cohort sampling

Case-cohort design usually requires the disease rate to be low in large cohort study,although it has been extensively used in practice.However,the disease with high rate is frequently observed in many clinical studies.Under such circumstances,it is desirable to consider a generalized case-cohort design,where only a fraction of cases are sampled.In this article,we propose the inference procedure for the additive hazards regression under the generalized case-cohort sampling.Asymptotic properties of the proposed estimators for the regression coefcients are established.To demonstrate the efectiveness of the generalized case-cohort sampling,we compare it with simple random sampling in terms of asymptotic relative efciency.Furthermore,we derive the optimal allocation of the subsamples for the proposed design.The fnite sample performance of the proposed method is evaluated through simulation studies.     

Efficient implementation of the Barnes-Hut octree algorithm for Monte Carlo simulations of charged systems

Computer simulation with Monte Carlo is an important tool to investigate the function and equilibrium properties of many biological and soft matter materials solvable in solvents.The appropriate treatment of long-range electrostatic interaction is essential for these charged systems,but remains a challenging problem for large-scale simulations.We develop an efficient Barnes-Hut treecode algorithm for electrostatic evaluation in Monte Carlo simulations of Coulomb many-body systems.The algorithm is based on a divide-and-conquer strategy and fast update of the octree data structure in each trial move through a local adjustment procedure.We test the accuracy of the tree algorithm,and use it to perform computer simulations of electric double layer near a spherical interface.It is shown that the computational cost of the Monte Carlo method with treecode acceleration scales as log N in each move.For a typical system with ten thousand particles,by using the new algorithm,the speed has been improved by two orders of magnitude from the direct summation.     

Homogenization of Elliptic Problems with Quadratic Growth and Nonhomogenous Robin Conditions in Perforated Domains

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L~2(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator.Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.     

Speeding up elliptic curve discrete logarithm computations with point halving

Pollard rho method and its parallelized variants are at present known as the best generic algorithms for computing elliptic curve discrete logarithms. We propose new iteration function for the rho method by exploiting the fact that point halving is more efficient than point addition for elliptic curves over binary fields. We present a careful analysis of the alternative rho method with new iteration function. Compared to the previous r-adding walk, generally the new method can achieve a significant speedup for computing elliptic curve discrete logarithms over binary fields. For instance, for certain NIST-recommended curves over binary fields, the new method is about 12–17% faster than the previous best methods.     

Discrete Logarithms: The Past and the Future

The first practical public key cryptosystem to be published, the Diffie–Hellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the presumed security of a variety of other public key schemes. While there have been substantial advances in discrete log algorithms in the last two decades, in general the discrete log still appears to be hard, especially for some groups, such as those from elliptic curves. Unfortunately no proofs of hardness are available in this area, so it is necessary to rely on experience and intuition in judging what parameters to use for cryptosystems. This paper presents a brief survey of the current state of the art in discrete logs.     

The State of Elliptic Curve Cryptography

Since the introduction of public-key cryptography by Diffie and Hellman in 1976, the potential for the use of the discrete logarithm problem in public-key cryptosystems has been recognized. Although the discrete logarithm problem as first employed by Diffie and Hellman was defined explicitly as the problem of finding logarithms with respect to a generator in the multiplicative group of the integers modulo a prime, this idea can be extended to arbitrary groups and, in particular, to elliptic curve groups. The resulting public-key systems provide relatively small block size, high speed, and high security. This paper surveys the development of elliptic curve cryptosystems from their inception in 1985 by Koblitz and Miller to present day implementations.     

Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the gaussian integer method

In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundred digit range. We also illustrate our results by successfully computing discrete logarithms with GNFS in a large prime field.

     

Certain properties of root vectors of operators with discrete spectrum

We establish a connection between the root vectors of operators with discrete spectrum over a Banach space and the vectors of exponential type for these operators. The result is illustrated using the example of operators with discrete spectrum generated by boundary-value problems for equations of elliptic type. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 36–38.     

Computation of discrete logarithms in prime fields

The presumed difficulty of computing discrete logarithms in finite fields is the basis of several popular public key cryptosystems. The secure identification option of the Sun Network File System, for example, uses discrete logarithms in a field GF(p) with p a prime of 192 bits. This paper describes an implementation of a discrete logarithm algorithm which shows that primes of under 200 bits, such as that in the Sun system, are very insecure. Some enhancements to this system are suggested.     

On the asymptotic error estimate for a discrete problem of fourth-order accuracy

The leading term of the error of eigenvalues of a discrete analog of the eigenvalue problem for an elliptic operator with variable coefficients is obtained. A method for refining eigenvalues by evaluating a correction with the help of a discrete problem of second-order accuracy is proposed. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 153–160.     

Interpolation Formulas and Auxiliary Functions

We prove an interpolation formula for “semi-cartesian products” and use it to study several constructions of auxiliary functions. We get in this way a criterion for the values of the exponential map of an elliptic curve E defined over Q. It reduces the analogue of Schanuel's conjecture for the elliptic logarithms of E to a statement of the form of a criterion of algebraic independence. We also consider a construction of auxiliary function related to the four exponentials conjecture and show that it is essentially optimal. For analytic functions vanishing on a semi-cartesian product, we get a version of the Schwarz lemma in which the exponent involves a condition of distribution reminiscent of the so-called technical hypotheses in algebraic independence. We show by two examples that such a condition is unavoidable.     

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.     

Computing a logarithm of a unitary matrix with general spectrum

We analyze an algorithm for computing a skew‐Hermitian logarithm of a unitary matrix and also skew‐Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software, and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near ? 1, lead to very non‐Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force skew‐Hermitian output creates accuracy issues, which are avoided by the considered algorithm. A modification is introduced to deal properly with the J‐skew‐symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed. Copyright © 2014 John Wiley & Sons, Ltd.