Identification of a spacewise dependent heat source

The inverse problem of determining the temperature of a heat conductor together with an unknown spacewise dependent heat source from measured final data or time-average temperature observation is studied. The weak solution theory is applied for calculating the gradient of the least-squares functional that is minimized. For the general case when the heat source is the product between a known function h(x,t)$h(x\text{,}t)$ and the unknown source function f(x)$f(x)$ new explicit formulae, derived via the solution of the corresponding adjoint problem, are obtained. Numerical results obtained using the conjugate gradient method are presented and discussed.     

Inverse indefinite Sturm-Liouville problems with three spectra

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a,b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a,b] and two right-definite problems defined on [a,0] and [0,b], where point 0 lies in (a,b) and is the turning point of the weight function w(x).     

On the General Algebraic Inverse Eigenvalue Problems

A number of new results on sufficient conditions for the solvability and numerical algorithms of the following general algebraic inverse eigenvalue problem are obtained: Given $n+1$ real $n\times n$ matrices $A=(a_{ij}),A_k=(a_{ij}^{(k)})(k=1,2,\cdots,n)$ and $n$ distinct real numbers $\lambda_1,\lambda_2,\cdots,\lambda_n,$ find $n$ real number $c_1,c_2,\cdots,c_n$ such that the matrix $A(c)=A+\sum\limits_{k=1}^{n}c_k A_k$ has eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n.$     

Increasing efficiency of inverse iteration

Inverse iteration is simple but not very efficient method for computing few eigenvalues with minimal absolute values and corresponding eigenvectors of a symmetric matrix. The idea is to increase its efficiency by technique similar to multigrid methods used for solving linear systems. This approach is not new, but until now multigrid was mostly used for solving linear system which appear in Rayleigh quotient iteration, inverse iteration and related iterative methods. Instead of choosing appropriate coordinates (grids), our algorithm performs inverse iteration on a sequence of subspaces with decreasing dimensions (multispace). Block Lanczos method is used for the selection of a smaller subspace. This will produce a banded matrix, which makes inverse iteration even faster in the smaller dimensions.      

线性流形上的两类矩阵最佳逼近问题

1．引言设R”””表示所有实mxn阶矩阵的全体,OR”””表示所有n阶正交阵的全体,对于A＝（ail）eR”””,B＝（b;j）eRP”’,用A＠BeR’”“”’表示矩阵A与B的Kronecker积,用A二（all,ala,…,al。,aal,…aa。,…;a＿l,…,a＿＊“表m矩阵A拉直算子,l］·IF表示矩阵的Frobenius范数,11·11。表示向量的2一范数．设S＝｛X,X6R”””f（X）二llAIXBI—Dlll》＋IIAZXBZ—D。股一Zill｝其中AlER”X”,BIERPXq,DIER”XqAZERtX”BZERPXIDZERtXI考虑下列两类问题：问题I．给定CIERll“”．FIE…     

Inverse mode problems for the finite element model of a vibrating rod

The inverse mode problems for the finite element model of an axially vibrating rod are formulated and solved. It is known that for the finite element model, based on linear shape functions, of the rod, the mass and stiffness matrices are both tridiagonal. It is shown that the finite element model of the rod can be constructed from two eigenvalues, their corresponding eigenvectors and the total mass of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive mass and stiffness elements from two eigenpairs and the total mass of the rod are established. If these conditions are satisfied, then the construction of the model is unique.     

Global existence for some transport equations with nonlocal velocity

In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of a one dimensional model of the surface quasi-geostrophic equation and the incompressible porous media equation, and one dimensional and n dimensional models of the dissipative quasi-geostrophic equations and the dissipative incompressible porous media equation.     

Determination of a differential pencil from interior spectral data

In this paper, inverse spectra problems for a differential pencil are studied. By using the approach similar to those in Hochstadt and Lieberman (1978) [14] and Ramm (2000) [26], we prove that (1) if p(x) (or q(x)) is full given on the interval [0,π], then a set of values of eigenfunctions at the mid-point of the interval [0,π] and one spectrum suffice to determine q(x) (or p(x)) on the interval [0,π] and all parameters in the boundary conditions; (2) if p(x) (or q(x)) is full given on the interval [0,π], then some information on eigenfunctions at some internal point and parts of two spectra suffice to determine q(x) (or p(x)) on the interval [0,π] and all parameters in the boundary conditions.     

Note on a nonlocal Sturm–Liouville problem with both right and left fractional derivatives

In this paper, we research the geometric multiplicity of eigenvalues for a nonlocal Sturm–Liouville eigenvalue problem. To this end, we study the uniqueness of solutions for a nonlocal Sturm–Liouville problem under some initial value conditions.     

一类奇型Sturm-Liouville算子的逆问题

研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x).     

THE SOLVABILITY CONDITIONS FOR INVERSE EIGENVALUE PROBLEM OF ANTI-BISYMMETRIC MATRICES

AbstractThis paper is mainly concerned with solving the following two problems: Problem I. Given X Cnxm, A = diag( 1, 2, ..... , m) Cmxm . Find A ABSRnxn such thatAX = XAwhere ABSRnxn is the set of all real n x n anti-bisymmetric matrices. Problem II. Given A RnXn. Find A SE such thatwhere || || is Frobenius norm, and SE denotes the solution set of Problem I.The necessary and sufficient conditions for the solvability of Problem I have been studied. The general form of SB has been given. For Problem II the expression of the solution has been provided.     

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.     

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.     

一类非局部反应扩散问题

本文研究了一类非局部反应扩散问题．利用上、下解，讨论了相应问题解的存在唯一性及其渐近性态．     

Spectral estimates of least energy solutions to the Brezis-Nirenberg problem with a variable coefficient

Let uε be a least energy solution to the Brezis-Nirenberg problem:

     

On the complexity of the multivariate Sturm–Liouville eigenvalue problem

We study the complexity of approximating the smallest eigenvalue of -Δ+q with Dirichlet boundary conditions on the d-dimensional unit cube. Here Δ is the Laplacian, and the function q is non-negative and has continuous first order partial derivatives. We consider deterministic and randomized classical algorithms, as well as quantum algorithms using quantum queries of two types: bit queries and power queries. We seek algorithms that solve the problem with accuracy . We exhibit lower and upper bounds for the problem complexity. The upper bounds follow from the cost of particular algorithms. The classical deterministic algorithm is optimal. Optimality is understood modulo constant factors that depend on d. The randomized algorithm uses an optimal number of function evaluations of q when d≤2. The classical algorithms have cost exponential in d since they need to solve an eigenvalue problem involving a matrix with size exponential in d. We show that the cost of quantum algorithms is not exponential in d, regardless of the type of queries they use. Power queries enjoy a clear advantage over bit queries and lead to an optimal complexity algorithm.     

The inverse solution of the atomic mixing equations by an operator-splitting method

The quantification problem of recovering the original material distribution from secondary ion mass spectrometry (SIMS) data is considered in this paper. It is an inverse problem, is ill-posed and hence it requires a special technique for its solution. The quantification problem is essentially an inverse diffusion or (classically) a backward heat conduction problem. In this paper an operator-splitting method (that is proposed in a previous paper by the first author for the solution of inverse diffusion problems) is developed for the solution of the problem of recovering the original structure from the SIMS data. A detailed development of the quantification method is given and it is applied to typical data to demonstrate its effectiveness.     

退化椭圆型障碍问题弱解的边界正则性

本文研究包含于R~N的有Lipschitz边界的有界区域Ω上涉及到 p-Laplacian算子的退化椭圆障碍问题弱解的边界正则性,得到了C_(loc)~(1,α)边界正则性。