To study the uncertainty quantification of resonant states in open quantum systems,we developed a Bayesian framework by integrating a reduced basis method(RBM)emulator with the Gamow coupled-channel(GCC)approach.The R...To study the uncertainty quantification of resonant states in open quantum systems,we developed a Bayesian framework by integrating a reduced basis method(RBM)emulator with the Gamow coupled-channel(GCC)approach.The RBM,constructed via eigenvector continuation and trained on both bound and resonant configurations,enables the fast and accurate emulation of resonance properties across the parameter space.To identify the physical resonant states from the emulator’s output,we introduce an overlap-based selection technique that effectively isolates true solutions from background artifacts.By applying this framework to unbound nucleus ^(6)Be,we quantified the model uncertainty in the predicted complex energies.The results demonstrate relative errors of 17.48%in the real part and 8.24%in the imaginary part,while achieving a speedup of four orders of magnitude compared with the full GCC calculations.To further investigate the asymptotic behavior of the resonant-state wavefunctions within the RBM framework,we employed a Lippmann–Schwinger(L–S)-based correction scheme.This approach not only improves the consistency between eigenvalues and wavefunctions but also enables a seamless extension from real-space training data to the complex energy plane.By bridging the gap between bound-state and continuum regimes,the L–S correction significantly enhances the emulator’s capability to accurately capture continuum structures in open quantum systems.展开更多
In this paper,we develop a new reduced basis(RB)method,named as Single Eigenvalue Acceleration Method(SEAM),for second order parabolic equations with homogeneous Dirichlet boundary conditions.The high-fidelity numeric...In this paper,we develop a new reduced basis(RB)method,named as Single Eigenvalue Acceleration Method(SEAM),for second order parabolic equations with homogeneous Dirichlet boundary conditions.The high-fidelity numerical method adopts the backward Euler scheme and conforming simplicial finite elements for the temporal and spatial discretizations,respectively.Under the assumption that the time step size is sufficiently small and time steps are not very large,we show that the singular value distribution of the high-fidelity solution matrix U is close to that of a rank one matrix.We select the eigenfunction associated to the principal eigenvalue of the matrix U>U as the basis of Proper Orthogonal Decomposition(POD)method so as to obtain SEAM and a parallel SEAM.Numerical experiments confirm the efficiency of the new method.展开更多
In the paper,a reduced basis(RB)method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed.Because of the lack of sparsity of discretized nonlocal systems c...In the paper,a reduced basis(RB)method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed.Because of the lack of sparsity of discretized nonlocal systems compared to corresponding local partial differential equation(PDE)systems,model reduction for nonlocal systems becomes more critical.The method of snapshots and greedy(MOS-greedy)algorithm of RB method is developed for nonlocal problems with random inputs,which provides an efficient and reliable approximation of the solution.A major challenge lies in the excessive influence of the time domain on the model reduction process.To address this,the Fourier transform is applied to convert the original time-dependent parabolic equation into a frequency-dependent elliptic equation,where variable frequencies are independent.This enables parallel computation for approximating the solution in the frequency domain.Finally,the proposed MOS-greedy algorithm is applied to the nonlocal diffusion problems.Numerical results demonstrate that it provides an accurate approximation of the full order problems and significantly improves computational efficiency.展开更多
In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we ...In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the ＂primal- dual＂ formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the ＂primal-dual＂ Reduced Basis approach with respect to an ＂only primal＂ one. Parametrized advection-reaction partial differential equations, Reduced Basis method, ＂primal-dual＂ reduced basis approach, Stabilized finite element method, a posteriori error estimation.展开更多
In this work,two approaches,based on the certified Reduced Basis method,have been developed for simulating the movement of nuclear reactor control rods,in time-dependent non-coercive settings featuring a 3D geometrica...In this work,two approaches,based on the certified Reduced Basis method,have been developed for simulating the movement of nuclear reactor control rods,in time-dependent non-coercive settings featuring a 3D geometrical framework.In particular,in a first approach,a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod.In the second approach,a“staircase”strategy has been adopted for simulating themovement of all the three rods featured by the nuclear reactor chosen as case study.The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion,which,in the present case,is a set of ten coupled parametrized parabolic equations(two energy groups for the neutron flux,and eight for the precursors).Both the reduced order models,developed according to the two approaches,provided a very good accuracy comparedwith high-fidelity results,assumed as“truth”solutions.At the same time,the computational speed-up in the Online phase,with respect to the fine“truth”finite element discretization,achievable by both the proposed approaches is at least of three orders of magnitude,allowing a real-time simulation of the rod movement and control.展开更多
In this article,an effective technique is developed to efficiently obtain the output responses of parameterized structural dynamic problems.This technique is based on the conception of reduced basis method and the usa...In this article,an effective technique is developed to efficiently obtain the output responses of parameterized structural dynamic problems.This technique is based on the conception of reduced basis method and the usage of linear interpolation principle.The original problem is projected onto the reduced basis space by linear interpolation projection,and subsequently an associated interpolation matrix is generated.To ensure the largest nonsingularity,the interpolation matrix needs to go through a timenode choosing process,which is developed by applying the angle of vector spaces.As a part of this technique,error estimation is recommended for achieving the computational error bound.To ensure the successful performance of this technique,the offline-online computational procedures are conducted in practical engineering.Two numerical examples demonstrate the accuracy and efficiency of the presented method.展开更多
In this paper,we propose a Static Condensation Reduced Basis Element(SCRBE)approach for the Reynolds Lubrication Equation(RLE).The SCRBEmethod is a computational tool that allows to efficiently analyze parametrized st...In this paper,we propose a Static Condensation Reduced Basis Element(SCRBE)approach for the Reynolds Lubrication Equation(RLE).The SCRBEmethod is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components.Here,we extend the methodology to allow for a more general domain decomposition,a typical example being a checkerboard-pattern assembled from similar components.To this end,we extend the formulation and associated a posteriori error bound procedure.Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE.However,the SCRBE approach presented is not limited to bearings and the RLE,but directly extends to other component-based systems.We show numerical results for plain bearings to demonstrate the validity of the proposed approach.展开更多
.A non-intrusive reduced order model(ROM)that combines a proper orthogonal decomposition(POD)and an artificial neural network(ANN)is primarily studied to investigate the applicability of the proposed ROM in recovering....A non-intrusive reduced order model(ROM)that combines a proper orthogonal decomposition(POD)and an artificial neural network(ANN)is primarily studied to investigate the applicability of the proposed ROM in recovering the solutions with shocks and strong gradients accurately and resolving fine-scale structures efficiently for hyperbolic conservation laws.Its accuracy is demonstrated by solving a high-dimensional parametrized ODE and the one-dimensional viscous Burgers’equation with a parameterized diffusion coefficient.The two-dimensional singlemode Rayleigh-Taylor instability(RTI),where the amplitude of the small perturbation and time are considered as free parameters,is also simulated.An adaptive sampling method in time during the linear regime of the RTI is designed to reduce the number of snapshots required for POD and the training of ANN.The extensive numerical results show that the ROM can achieve an acceptable accuracy with improved efficiency in comparison with the standard full order method.展开更多
基金supported by the National Key Research and Development Program(MOST 2023YFA1606404 and MOST 2022YFA1602303)the National Natural Science Foundation of China(Nos.12347106,12147101,and 12447122)the China Postdoctoral Science Foundation(No.2024M760489).
文摘To study the uncertainty quantification of resonant states in open quantum systems,we developed a Bayesian framework by integrating a reduced basis method(RBM)emulator with the Gamow coupled-channel(GCC)approach.The RBM,constructed via eigenvector continuation and trained on both bound and resonant configurations,enables the fast and accurate emulation of resonance properties across the parameter space.To identify the physical resonant states from the emulator’s output,we introduce an overlap-based selection technique that effectively isolates true solutions from background artifacts.By applying this framework to unbound nucleus ^(6)Be,we quantified the model uncertainty in the predicted complex energies.The results demonstrate relative errors of 17.48%in the real part and 8.24%in the imaginary part,while achieving a speedup of four orders of magnitude compared with the full GCC calculations.To further investigate the asymptotic behavior of the resonant-state wavefunctions within the RBM framework,we employed a Lippmann–Schwinger(L–S)-based correction scheme.This approach not only improves the consistency between eigenvalues and wavefunctions but also enables a seamless extension from real-space training data to the complex energy plane.By bridging the gap between bound-state and continuum regimes,the L–S correction significantly enhances the emulator’s capability to accurately capture continuum structures in open quantum systems.
基金supported in part by the National Natural Science Foundation of China(12171340)National Key R&D Program of China(2020YFA0714000).
文摘In this paper,we develop a new reduced basis(RB)method,named as Single Eigenvalue Acceleration Method(SEAM),for second order parabolic equations with homogeneous Dirichlet boundary conditions.The high-fidelity numerical method adopts the backward Euler scheme and conforming simplicial finite elements for the temporal and spatial discretizations,respectively.Under the assumption that the time step size is sufficiently small and time steps are not very large,we show that the singular value distribution of the high-fidelity solution matrix U is close to that of a rank one matrix.We select the eigenfunction associated to the principal eigenvalue of the matrix U>U as the basis of Proper Orthogonal Decomposition(POD)method so as to obtain SEAM and a parallel SEAM.Numerical experiments confirm the efficiency of the new method.
基金supported by the Guangdong Basic and Applied Basic Research Foundation,China(Grant 2024A1515012548)supported by the National Natural Science Foundation of China(Grant 12401567)+1 种基金by the 2023 Guangzhou Basic and Applied Basic Research Project(Grant 2023A04J0035)by the Talent Special Projects of School-level Scientific Research Programs under Guangdong Poiytechnic Normal University(Grant 2022SDKYA025).
文摘In the paper,a reduced basis(RB)method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed.Because of the lack of sparsity of discretized nonlocal systems compared to corresponding local partial differential equation(PDE)systems,model reduction for nonlocal systems becomes more critical.The method of snapshots and greedy(MOS-greedy)algorithm of RB method is developed for nonlocal problems with random inputs,which provides an efficient and reliable approximation of the solution.A major challenge lies in the excessive influence of the time domain on the model reduction process.To address this,the Fourier transform is applied to convert the original time-dependent parabolic equation into a frequency-dependent elliptic equation,where variable frequencies are independent.This enables parallel computation for approximating the solution in the frequency domain.Finally,the proposed MOS-greedy algorithm is applied to the nonlocal diffusion problems.Numerical results demonstrate that it provides an accurate approximation of the full order problems and significantly improves computational efficiency.
基金support provided thorough the "Progetto Rocca", MIT-Politecnico di Milano collaboration
文摘In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the ＂primal- dual＂ formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the ＂primal-dual＂ Reduced Basis approach with respect to an ＂only primal＂ one. Parametrized advection-reaction partial differential equations, Reduced Basis method, ＂primal-dual＂ reduced basis approach, Stabilized finite element method, a posteriori error estimation.
基金We acknowledge CINECA and Regione Lombardia LISA computational initiative,for the availability of high performance computing resources and support.G.Rozza acknowledges INDAM-GNCS national activity group and NOFYSAS program of SISSA.
文摘In this work,two approaches,based on the certified Reduced Basis method,have been developed for simulating the movement of nuclear reactor control rods,in time-dependent non-coercive settings featuring a 3D geometrical framework.In particular,in a first approach,a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod.In the second approach,a“staircase”strategy has been adopted for simulating themovement of all the three rods featured by the nuclear reactor chosen as case study.The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion,which,in the present case,is a set of ten coupled parametrized parabolic equations(two energy groups for the neutron flux,and eight for the precursors).Both the reduced order models,developed according to the two approaches,provided a very good accuracy comparedwith high-fidelity results,assumed as“truth”solutions.At the same time,the computational speed-up in the Online phase,with respect to the fine“truth”finite element discretization,achievable by both the proposed approaches is at least of three orders of magnitude,allowing a real-time simulation of the rod movement and control.
基金supported by the National Natural Science Foundation of China (10802028)the Major State Basic Research Development Program of China (2010CB832705)the National Science Fund for Distinguished Young Scholars (10725208)
文摘In this article,an effective technique is developed to efficiently obtain the output responses of parameterized structural dynamic problems.This technique is based on the conception of reduced basis method and the usage of linear interpolation principle.The original problem is projected onto the reduced basis space by linear interpolation projection,and subsequently an associated interpolation matrix is generated.To ensure the largest nonsingularity,the interpolation matrix needs to go through a timenode choosing process,which is developed by applying the angle of vector spaces.As a part of this technique,error estimation is recommended for achieving the computational error bound.To ensure the successful performance of this technique,the offline-online computational procedures are conducted in practical engineering.Two numerical examples demonstrate the accuracy and efficiency of the presented method.
基金We would like to thank Prof.A.T.Patera and Dr.J.Eftang for helpful discussions on the SCRBE method as well as Prof.G.Knoll and Dr.R.Schönen from ISTmbH for providing the specific application.This work was supported by the Excellence Initiative of the German federal and state governments and the German Research Foundation through Grant GSC 111.
文摘In this paper,we propose a Static Condensation Reduced Basis Element(SCRBE)approach for the Reynolds Lubrication Equation(RLE).The SCRBEmethod is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components.Here,we extend the methodology to allow for a more general domain decomposition,a typical example being a checkerboard-pattern assembled from similar components.To this end,we extend the formulation and associated a posteriori error bound procedure.Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE.However,the SCRBE approach presented is not limited to bearings and the RLE,but directly extends to other component-based systems.We show numerical results for plain bearings to demonstrate the validity of the proposed approach.
基金funding support of this research by the National Natural Science Foundation of China(11871443)Shandong Provincial Qingchuang Science and Technology Project(2019KJI002)the Ocean University of China for providing the startup funding(201712011)that is used in supporting this work.
文摘.A non-intrusive reduced order model(ROM)that combines a proper orthogonal decomposition(POD)and an artificial neural network(ANN)is primarily studied to investigate the applicability of the proposed ROM in recovering the solutions with shocks and strong gradients accurately and resolving fine-scale structures efficiently for hyperbolic conservation laws.Its accuracy is demonstrated by solving a high-dimensional parametrized ODE and the one-dimensional viscous Burgers’equation with a parameterized diffusion coefficient.The two-dimensional singlemode Rayleigh-Taylor instability(RTI),where the amplitude of the small perturbation and time are considered as free parameters,is also simulated.An adaptive sampling method in time during the linear regime of the RTI is designed to reduce the number of snapshots required for POD and the training of ANN.The extensive numerical results show that the ROM can achieve an acceptable accuracy with improved efficiency in comparison with the standard full order method.
