Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute th...Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute the elements of the matrix is derived for a given list ∧ and dependency parameters. Explicit computations are performed for n≤5 and r≤4 to illustrate the result.展开更多
In this article,we discuss singular Hermitian matrices of rank greater or equal to four for an inverse eigenvalue problem.Specifically,we look into how to generate n by n singular Hermitian matrices of ranks four and ...In this article,we discuss singular Hermitian matrices of rank greater or equal to four for an inverse eigenvalue problem.Specifically,we look into how to generate n by n singular Hermitian matrices of ranks four and five from a prescribed spectrum.Numerical examples are presented in each case to illustrate these scenarios.It was established that given a prescribed spectral datum and it multiplies,then the solubility of the inverse eigenvalue problem for n by n singular Hermitian matrices of rank r exists.展开更多
This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs...This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided.展开更多
In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the ...In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.展开更多
In cooperative game theory, a central problem is to allocate fairly the win of the grand coalition to the players who agreed to cooperate and form the grand coalition. Such allocations are obtained by means of values,...In cooperative game theory, a central problem is to allocate fairly the win of the grand coalition to the players who agreed to cooperate and form the grand coalition. Such allocations are obtained by means of values, having some fairness properties, expressed in most cases by groups of axioms. In an earlier work, we solved what we called the Inverse Problem for Semivalues, in which the main result was offering an explicit formula providing the set of all games with an a priori given Semivalue, associated with a given weight vector. However, in this set there is an infinite set of games for which the Semivalues are not coalitional rational, perhaps not efficient, so that these are not fair practical solutions of the above fundamental problem. Among the Semivalues, coalitional rational solutions for the Shapley Value and the Banzhaf Value have been given in two more recent works. In the present paper, based upon a general potential basis, relative to Semivalues, for a given game and a given Semivalue, we solve the connected problem: in the Inverse Set, find out a game with the same Semivalue, which is also coalitional rational. Several examples will illustrate the corresponding numerical technique.展开更多
In this paper,a class of inverse boundary value problems for(λ,1)bi-analytic functions is given.Using the method of Riemann boundary value problem for analytic functions,the conditions of solvability and the expressi...In this paper,a class of inverse boundary value problems for(λ,1)bi-analytic functions is given.Using the method of Riemann boundary value problem for analytic functions,the conditions of solvability and the expression of the solutions for the inverse problems are obtained.展开更多
In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players. The Inverse Set relative to the S...In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players. The Inverse Set relative to the Shapley Value of a game is a set of games in which the Shapley Value is the same as the initial one. In the Inverse Set, we determined a family of games for which the Shapley Value is also a coalitional rational value. The Egalitarian Allocation of the game is efficient, so that in the set called the Inverse Set relative to the Shapley Value, the allocation is the same as the initial one, but may not be coalitional rational. In this paper, we shall find out in the same family of the Inverse Set, a subfamily of games with the Egalitarian Allocation is also a coalitional rational value. We show some relationship between the two sets of games, where our values are coalitional rational. Finally, we shall discuss the possibility that our procedure may be used for solving a very similar problem for other efficient values. Numerical examples show the procedure to get solutions for the efficient values.展开更多
In the paper an inverse boundary value problem for a fourth order elliptic equation with an integral condition of the first kind is investigated. First, the given problem is reduced to an equivalent problem in a certa...In the paper an inverse boundary value problem for a fourth order elliptic equation with an integral condition of the first kind is investigated. First, the given problem is reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the system of integral equations. The existence and uniqueness of a solution to the system of integral equation is proved by the contraction mapping principle. This solution is also the unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence and uniqueness of a classical solution to the given problem is proved.展开更多
For the 2-D wave inverse problems introduced from geophysical exploration, in this paper, the author presents integration-characteristic method to solve the velocity parameter, and then applies it to common shotpoint ...For the 2-D wave inverse problems introduced from geophysical exploration, in this paper, the author presents integration-characteristic method to solve the velocity parameter, and then applies it to common shotpoint model data, in noise-free case. The accuracy is quite good.展开更多
The following inverse problem is solved—given the eigenvalues and the potential b(n) for a difference boundary value problem with quadratic dependence on the eigenparameter, λ, the weights c(n) can be uniquely ...The following inverse problem is solved—given the eigenvalues and the potential b(n) for a difference boundary value problem with quadratic dependence on the eigenparameter, λ, the weights c(n) can be uniquely reconstructed. The investi-gation is inductive on m where represents the number of unit intervals and the results obtained depend on the specific form of the given boundary conditions. This paper is a sequel to [1] which provided an algorithm for the solution of an analogous inverse problem, where the eigenvalues and weights were given and the potential was uniquely reconstructed. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in [1], an additional spectrum is required more often than was the case in [1].展开更多
Accurate estimation of fracture density and orientation is of great significance for seismic characterization of fractured reservoirs.Here,we propose a novel methodology to estimate fracture density and orientation fr...Accurate estimation of fracture density and orientation is of great significance for seismic characterization of fractured reservoirs.Here,we propose a novel methodology to estimate fracture density and orientation from azimuthal elastic impedance(AEI)difference using singular value decomposition(SVD).Based on Hudson's model,we first derive the AEI equation containing fracture density in HTI media,and then obtain basis functions and singular values from the normalized AEI difference utilizing SVD.Analysis shows that the basis function changing with azimuth is related to fracture orientation,fracture density is the linearly weighted sum of singular values,and the first singular value contributes the most to fracture density.Thus,we develop an SVD-based fracture density and orientation inversion approach constrained by smooth prior elastic parameters.Synthetic example shows that fracture density and orientation can be stably estimated,and the correlation coefficient between the true value and the estimated fracture density is above 0.85 even when an S/N ratio of 2.Field data example shows that the estimated fracture orientation is consistent with the interpretation of image log data,and the estimated fracture density reliably indicates fractured gas-bearing reservoir,which could help to guide the exploration and development of fractured reservoirs.展开更多
A necessary and sufficient condition for the existence of simultaneous (M,N)singular value decomposition of matrices is given.Some properties about the weighted partial ordering are discussed with the help of the deco...A necessary and sufficient condition for the existence of simultaneous (M,N)singular value decomposition of matrices is given.Some properties about the weighted partial ordering are discussed with the help of the decomposition.展开更多
Utilizing the properties of the smallest singular value of a matrix, we propose a new, efficient and reliable algorithm for solving nonsymmetric matrix inverse eigenvalue problems, and compare it with a known method. ...Utilizing the properties of the smallest singular value of a matrix, we propose a new, efficient and reliable algorithm for solving nonsymmetric matrix inverse eigenvalue problems, and compare it with a known method. We also present numerical experiments which illustrate our results.展开更多
We first define the pseudo almost periodic functions in a more general setting.Then we show the existence,uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of bounda...We first define the pseudo almost periodic functions in a more general setting.Then we show the existence,uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of boundary value problems.展开更多
The application of the singular boundary method(SBM),a relatively new meshless boundary collocation method,to the inverse Cauchy problem in threedimensional(3D)linear elasticity is investigated.The SBM involves a coup...The application of the singular boundary method(SBM),a relatively new meshless boundary collocation method,to the inverse Cauchy problem in threedimensional(3D)linear elasticity is investigated.The SBM involves a coupling between the non-singular boundary element method(BEM)and the method of fundamental solutions(MFS).The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS.Due to the boundary-only discretizations and its semi-analytical nature,the method can be viewed as an ideal candidate for the solution of inverse problems.The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique,while the optimal regularization parameter is determined by the L-curve criterion.Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.展开更多
The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2...The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.展开更多
A one-phase Stefan problem for the nonhomogeneous heat equation with the source term depending on an unknown parameter p(t) is considered. The existence and uniqueness of the solution (p, s, u) are also demonstrated.
文摘Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute the elements of the matrix is derived for a given list ∧ and dependency parameters. Explicit computations are performed for n≤5 and r≤4 to illustrate the result.
文摘In this article,we discuss singular Hermitian matrices of rank greater or equal to four for an inverse eigenvalue problem.Specifically,we look into how to generate n by n singular Hermitian matrices of ranks four and five from a prescribed spectrum.Numerical examples are presented in each case to illustrate these scenarios.It was established that given a prescribed spectral datum and it multiplies,then the solubility of the inverse eigenvalue problem for n by n singular Hermitian matrices of rank r exists.
文摘This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided.
基金Supported by the Key Discipline Construction Project of Tianshui Normal University
文摘In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.
文摘In cooperative game theory, a central problem is to allocate fairly the win of the grand coalition to the players who agreed to cooperate and form the grand coalition. Such allocations are obtained by means of values, having some fairness properties, expressed in most cases by groups of axioms. In an earlier work, we solved what we called the Inverse Problem for Semivalues, in which the main result was offering an explicit formula providing the set of all games with an a priori given Semivalue, associated with a given weight vector. However, in this set there is an infinite set of games for which the Semivalues are not coalitional rational, perhaps not efficient, so that these are not fair practical solutions of the above fundamental problem. Among the Semivalues, coalitional rational solutions for the Shapley Value and the Banzhaf Value have been given in two more recent works. In the present paper, based upon a general potential basis, relative to Semivalues, for a given game and a given Semivalue, we solve the connected problem: in the Inverse Set, find out a game with the same Semivalue, which is also coalitional rational. Several examples will illustrate the corresponding numerical technique.
基金Supported by the Natural Science Foundation of Fujian Province(2020J01322)
文摘In this paper,a class of inverse boundary value problems for(λ,1)bi-analytic functions is given.Using the method of Riemann boundary value problem for analytic functions,the conditions of solvability and the expression of the solutions for the inverse problems are obtained.
文摘In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players. The Inverse Set relative to the Shapley Value of a game is a set of games in which the Shapley Value is the same as the initial one. In the Inverse Set, we determined a family of games for which the Shapley Value is also a coalitional rational value. The Egalitarian Allocation of the game is efficient, so that in the set called the Inverse Set relative to the Shapley Value, the allocation is the same as the initial one, but may not be coalitional rational. In this paper, we shall find out in the same family of the Inverse Set, a subfamily of games with the Egalitarian Allocation is also a coalitional rational value. We show some relationship between the two sets of games, where our values are coalitional rational. Finally, we shall discuss the possibility that our procedure may be used for solving a very similar problem for other efficient values. Numerical examples show the procedure to get solutions for the efficient values.
文摘In the paper an inverse boundary value problem for a fourth order elliptic equation with an integral condition of the first kind is investigated. First, the given problem is reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the system of integral equations. The existence and uniqueness of a solution to the system of integral equation is proved by the contraction mapping principle. This solution is also the unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence and uniqueness of a classical solution to the given problem is proved.
文摘For the 2-D wave inverse problems introduced from geophysical exploration, in this paper, the author presents integration-characteristic method to solve the velocity parameter, and then applies it to common shotpoint model data, in noise-free case. The accuracy is quite good.
文摘The following inverse problem is solved—given the eigenvalues and the potential b(n) for a difference boundary value problem with quadratic dependence on the eigenparameter, λ, the weights c(n) can be uniquely reconstructed. The investi-gation is inductive on m where represents the number of unit intervals and the results obtained depend on the specific form of the given boundary conditions. This paper is a sequel to [1] which provided an algorithm for the solution of an analogous inverse problem, where the eigenvalues and weights were given and the potential was uniquely reconstructed. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in [1], an additional spectrum is required more often than was the case in [1].
基金sponsorship of the National Natural Science Foundation of China(41674130,U19B2008)the Postgraduate Innovation Project in China University of Petroleum(East China)(YCX2021016)for their funding this research。
文摘Accurate estimation of fracture density and orientation is of great significance for seismic characterization of fractured reservoirs.Here,we propose a novel methodology to estimate fracture density and orientation from azimuthal elastic impedance(AEI)difference using singular value decomposition(SVD).Based on Hudson's model,we first derive the AEI equation containing fracture density in HTI media,and then obtain basis functions and singular values from the normalized AEI difference utilizing SVD.Analysis shows that the basis function changing with azimuth is related to fracture orientation,fracture density is the linearly weighted sum of singular values,and the first singular value contributes the most to fracture density.Thus,we develop an SVD-based fracture density and orientation inversion approach constrained by smooth prior elastic parameters.Synthetic example shows that fracture density and orientation can be stably estimated,and the correlation coefficient between the true value and the estimated fracture density is above 0.85 even when an S/N ratio of 2.Field data example shows that the estimated fracture orientation is consistent with the interpretation of image log data,and the estimated fracture density reliably indicates fractured gas-bearing reservoir,which could help to guide the exploration and development of fractured reservoirs.
基金The Guangxi Science Foundation(0575032,06400161)the support program for 100 Young and Middle-aged Disciplinary Leaders in Guangxi Higher Education Institutions
文摘A necessary and sufficient condition for the existence of simultaneous (M,N)singular value decomposition of matrices is given.Some properties about the weighted partial ordering are discussed with the help of the decomposition.
文摘Utilizing the properties of the smallest singular value of a matrix, we propose a new, efficient and reliable algorithm for solving nonsymmetric matrix inverse eigenvalue problems, and compare it with a known method. We also present numerical experiments which illustrate our results.
基金supported by the National Natural Science Foundation of China (Grant No. 10671046)
文摘We first define the pseudo almost periodic functions in a more general setting.Then we show the existence,uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of boundary value problems.
基金The work described in this paper was supported by the National Natural Science Foundation of China(Nos.11402075,11401332,71571108)Projects of International(Regional)Cooperation and Exchanges of NSFC(No.71611530712)+2 种基金the Natural Science Foundation of Shandong Province of China(Nos.ZR2017BA003,ZR2015GZ007,ZR2017JL004)the Research Grants Council of the Hong Kong Special Administrative Region(No.CityU 11204414)the Science and Technology Innovation Commission of Shenzhen Municipality(No.JCYJ20160229165310679).
文摘The application of the singular boundary method(SBM),a relatively new meshless boundary collocation method,to the inverse Cauchy problem in threedimensional(3D)linear elasticity is investigated.The SBM involves a coupling between the non-singular boundary element method(BEM)and the method of fundamental solutions(MFS).The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS.Due to the boundary-only discretizations and its semi-analytical nature,the method can be viewed as an ideal candidate for the solution of inverse problems.The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique,while the optimal regularization parameter is determined by the L-curve criterion.Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.
基金supported in part by the Social Science Foundation of Ministry of Education(07JJD790154)the National Science Foundation for Young Scholars (60803076)+2 种基金the Natural Science Foundation of Zhejiang Province (Y6090211)Foundation of Education Department of Zhejiang Province (20070590)the Young Talent Foundation of Zhejiang Gongshang University
文摘The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.
文摘A one-phase Stefan problem for the nonhomogeneous heat equation with the source term depending on an unknown parameter p(t) is considered. The existence and uniqueness of the solution (p, s, u) are also demonstrated.
