The Moore-Penrose metric generalized inverse T+ of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moo...The Moore-Penrose metric generalized inverse T+ of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moore-Penrose metric generalized inverse T+ is usually homogeneous and nonlinear in general. By means of the methods of geometry of Banach Space, the necessary and sufficient conditions for existence, continuitv, linearity and minimum property of the Moore-Penrose metric generalized inverse T+ will be given, and some properties of T+ will be investigated in this paper.展开更多
LetE andF be Banach spaces, andB(E,F) all of bounded linear operators onE intoF. LetT 0 ∈B( E,F) with an outer inverseT 0 # ∈B( F,E). Then a characteristic condition ofS=(I + T0 # (T-T0)-1 T0 # with T∈B(E, F) and ...LetE andF be Banach spaces, andB(E,F) all of bounded linear operators onE intoF. LetT 0 ∈B( E,F) with an outer inverseT 0 # ∈B( F,E). Then a characteristic condition ofS=(I + T0 # (T-T0)-1 T0 # with T∈B(E, F) and ∥ T0 # (T - T0 ∥ < 1, being a generalized inverse ofT, is presented, and hence, a rank theorem of operators onE intoF is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.展开更多
Let X be a topological space and A_x with closed range R(A_x) be a continuous mapping from X into B(H). It is well known that even if dim(H)<∞, the Moore-Penrose inverses A_x^+ may fail to be continuous. So it is ...Let X be a topological space and A_x with closed range R(A_x) be a continuous mapping from X into B(H). It is well known that even if dim(H)<∞, the Moore-Penrose inverses A_x^+ may fail to be continuous. So it is necessary to find sufficient and necessary conditions for A_x^+ being continuous. This will be important for the advanced analysis and its application. This paper gives these conditions in the cases that X is a topological space or X is locally compact, and A_x is a Fredholm family, respectively.展开更多
In this paper,the authors propose Neumann series neural operator(NSNO)to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions.Helmholtz equation is a crucial ...In this paper,the authors propose Neumann series neural operator(NSNO)to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions.Helmholtz equation is a crucial partial differential equation(PDE)with applications in various scientific and engineering fields.However,efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber.Recently,deep learning has shown great potential in solving PDEs especially in learning solution operators.Inspired by Neumann series in Helmholtz equation,the authors design a novel network architecture in which U-Net is embedded inside to capture the multiscale feature.Extensive experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60%lower relative L^(2)-error,especially in the large wavenumber case,and has 50%lower computational cost and less data requirement.Moreover,NSNO can be used as the surrogate model in inverse scattering problems.Numerical tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.展开更多
We consider the permeability estimation problem in two-phase porous media flow. We try to identify the permeability field by utilizing both the production data from wells as well as inverted seismic data. The permeabi...We consider the permeability estimation problem in two-phase porous media flow. We try to identify the permeability field by utilizing both the production data from wells as well as inverted seismic data. The permeability field is assumed to be piecewise constant, or can be approximated well by a piecewise constant function. A variant of the level set method, called Piecewise Constant Level Set Method is used to represent the interfaces between the regions with different permeability levels. The inverse problem is solved by minimizing a functional, and TV norm regularization is used to deal with the ill-posedness. We also use the operator-splitting technique to decompose the constraint term from the fidelity term. This gives us more flexibility to deal with the constraint and helps to stabilize the algorithm.展开更多
基金the National Natural Science Foundation of China(No.19971023)the Heilongjiang Provincial Natural Science Foundation of China.
文摘The Moore-Penrose metric generalized inverse T+ of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moore-Penrose metric generalized inverse T+ is usually homogeneous and nonlinear in general. By means of the methods of geometry of Banach Space, the necessary and sufficient conditions for existence, continuitv, linearity and minimum property of the Moore-Penrose metric generalized inverse T+ will be given, and some properties of T+ will be investigated in this paper.
文摘LetE andF be Banach spaces, andB(E,F) all of bounded linear operators onE intoF. LetT 0 ∈B( E,F) with an outer inverseT 0 # ∈B( F,E). Then a characteristic condition ofS=(I + T0 # (T-T0)-1 T0 # with T∈B(E, F) and ∥ T0 # (T - T0 ∥ < 1, being a generalized inverse ofT, is presented, and hence, a rank theorem of operators onE intoF is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.
基金Project supported by the National Natural Science Foundation of China.
文摘Let X be a topological space and A_x with closed range R(A_x) be a continuous mapping from X into B(H). It is well known that even if dim(H)<∞, the Moore-Penrose inverses A_x^+ may fail to be continuous. So it is necessary to find sufficient and necessary conditions for A_x^+ being continuous. This will be important for the advanced analysis and its application. This paper gives these conditions in the cases that X is a topological space or X is locally compact, and A_x is a Fredholm family, respectively.
基金supported by the National Science Foundation of China under Grant No.92370125the National Key R&D Program of China under Grant Nos.2019YFA0709600 and 2019YFA0709602.
文摘In this paper,the authors propose Neumann series neural operator(NSNO)to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions.Helmholtz equation is a crucial partial differential equation(PDE)with applications in various scientific and engineering fields.However,efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber.Recently,deep learning has shown great potential in solving PDEs especially in learning solution operators.Inspired by Neumann series in Helmholtz equation,the authors design a novel network architecture in which U-Net is embedded inside to capture the multiscale feature.Extensive experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60%lower relative L^(2)-error,especially in the large wavenumber case,and has 50%lower computational cost and less data requirement.Moreover,NSNO can be used as the surrogate model in inverse scattering problems.Numerical tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.
基金the Norwegian Research Council,Petromaks Programme
文摘We consider the permeability estimation problem in two-phase porous media flow. We try to identify the permeability field by utilizing both the production data from wells as well as inverted seismic data. The permeability field is assumed to be piecewise constant, or can be approximated well by a piecewise constant function. A variant of the level set method, called Piecewise Constant Level Set Method is used to represent the interfaces between the regions with different permeability levels. The inverse problem is solved by minimizing a functional, and TV norm regularization is used to deal with the ill-posedness. We also use the operator-splitting technique to decompose the constraint term from the fidelity term. This gives us more flexibility to deal with the constraint and helps to stabilize the algorithm.
