3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic m...3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic monitoring and tomographic imaging.In recent years,many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media.However,there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media.In this paper,we propose an adaptive finite-difference(AFD)method to numerically solve the 3D eikonal equation.The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes,and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle.Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media,our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts.Several numerical examples show the good performance of the AFD method,which will be beneficial to many scientific applications.展开更多
Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of ...Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.展开更多
In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic ...In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.展开更多
The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute s...The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators.Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.展开更多
The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to...The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to practically determine earthquake locations.However,first-arrival traveltimes are not sensitive to focal depths.Moreover,they cannot accurately constrain focal depths.To improve the accuracy,researchers have analyzed the depth phases of earthquake locations.The traveltimes of depth phases are sensitive to focal depths,and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations.Generally,researchers can determine earthquake locations in layered models.Because layered models can only represent the first-order feature of subsurface structures,the advantages of joint inversion are not fully explored if layered models are used.To resolve the issue of current joint inversions,we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models.The three seismic phases used in this study are the first P-,sPg-and PmP-waves.We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations.To verify the accuracy of the earthquake location method by the inversion of three seismic phases,we take the 2021 M_(S)6.4 Yangbi,Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data.Numerical tests demonstrate that the eikonal equation-based earthquake location method,which involves the inversion of multiple phase arrivals,can effectively improve earthquake location accuracy.展开更多
We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind sche...We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.展开更多
We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.Mor...We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.More specifically,the homotopy method is utilized to decrease the viscosity coefficient gradually,while Newton’s method is applied to compute the solution for each viscosity coefficient.Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples,but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids.Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.展开更多
We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inho...We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.展开更多
Two mathematical models for combined refraction-diffraction of regular and irregular waves on non-uniform current in water of slowly varying topography are presented in this paper. Model I is derived by wave theory an...Two mathematical models for combined refraction-diffraction of regular and irregular waves on non-uniform current in water of slowly varying topography are presented in this paper. Model I is derived by wave theory and variational principle separately. It has two kinds of expressions including the dissipation term. Model n is based on the energy conservation equation with energy flux through the wave crest lines in orthogonal curvilinear coordinates and the wave kinematic conservation equation. The analysis and comparison and special cases of these two models are also given.展开更多
A new numerical finite difference iteration method for refraction-diffraction of waves ia water of slowly varying current and topography is developed in this paper. And corresponding theoretical model including the di...A new numerical finite difference iteration method for refraction-diffraction of waves ia water of slowly varying current and topography is developed in this paper. And corresponding theoretical model including the dissipation term is briefly described, together with some analysis and comparison of computational results of the model with measurements in a hydraulic scale model (Berkhoff et al., 1982). An example of practical use of the method is given, showing that the present model is useful to engineering practice.展开更多
To the most of velocity fields, the traveltimes of the first break that seismic waves propagate along rays can be computed on a 2-D or 3-D numerical grid by finite-difference extrapolation. Under ensuring accuracy, t...To the most of velocity fields, the traveltimes of the first break that seismic waves propagate along rays can be computed on a 2-D or 3-D numerical grid by finite-difference extrapolation. Under ensuring accuracy, to improve calculating efficiency and adaptability, the calculation method of first-arrival traveltime of finite-difference is de- rived based on any rectangular grid and a local plane wavefront approximation. In addition, head waves and scat- tering waves are properly treated and shadow and caustic zones cannot be encountered, which appear in traditional ray-tracing. The testes of two simple models and the complex Marmousi model show that the method has higher accuracy and adaptability to complex structure with strong vertical and lateral velocity variation, and Kirchhoff prestack depth migration based on this method can basically achieve the position imaging effects of wave equation prestack depth migration in major structures and targets. Because of not taking account of the later arrivals energy, the effect of its amplitude preservation is worse than that by wave equation method, but its computing efficiency is higher than that by total Green′s function method and wave equation method.展开更多
This paper deals with a coupled system consisting of a scalar conservation law and an eikonal equation, called the Hughes model. Introduced in [24], this model attempts to describe the motion of pedestrians in a dense...This paper deals with a coupled system consisting of a scalar conservation law and an eikonal equation, called the Hughes model. Introduced in [24], this model attempts to describe the motion of pedestrians in a densely crowded region, in which they are seen as a ＇thinking＇ （continuum） fluid. The main mathematical difficulty is the discontinuous gradient of the solution to the eikonal equation appearing in the flux of the conservation law. On a one dimensional interval with zero Dirichlet conditions （the two edges of the interval are interpreted as ＇targets＇）, the model can be decoupled in a way to consider two classical conservation laws on two sub-domains separated by a turning point at which the pedestrians change their direction. We shall consider solutions with a possible jump discontinuity around the turning point. For simplicity, we shall assume they are locally constant on both sides of the discontinuity. We provide a detailed description of the local- in-time behavior of the solution in terms of a ＇global＇ qualitative property of the pedestrian density （that we call ＇relative evacuation rate＇）, which can be interpreted as the attitude of the pedestrians to direct towards the left or the right target. We complement our result with explicitly computable examples.展开更多
In this paper, we establish a mathematical model of the forest fire spread process based on a partial differential equation. We describe the distribution of time field and velocity field in the whole two-dimensional s...In this paper, we establish a mathematical model of the forest fire spread process based on a partial differential equation. We describe the distribution of time field and velocity field in the whole two-dimensional space by vector field theory. And we obtain a continuous algorithm to predict the dynamic behavior of forest fire spread in a short time. We use the algorithm to interpolate the fire boundary by cubic non-uniform rational B-spline closed curve. The fire boundary curve at any time can be simulated by solving the Eikonal equation. The model is tested in theory and in practice. The results show that the model has good accuracy and stability, and it’s compatible with most of the existing models, such as the elliptic model and the cellular automata model.展开更多
The velocity of the electromagnetic radiation in a perfect dielectric, containing no charges and no conduction currents, is explored and determined on making use of the Lorentz transformations. Besides the idealised b...The velocity of the electromagnetic radiation in a perfect dielectric, containing no charges and no conduction currents, is explored and determined on making use of the Lorentz transformations. Besides the idealised blackbody radiation, whose vacuum propagation velocity is the universal constant c, being this value independent of the observer, there is another behaviour of electromagnetic radiation, we call inertial radiation, which is characterized by an electromagnetic inertial density , and therefore, it happens to be described by a time-like Poynting four-vector field which propagates with velocity . is found to be a relativistic invariant expressible in terms of the relativistic invariants of the electromagnetic field. It is shown that there is a rest frame, where the Poynting vector is equal to zero. Both phase and group velocities of the electromagnetic radiation are evaluated. The wave and eikonal equations for the dynamics of the radiation field are formulated.展开更多
The surface-flattening scheme is a mathematical method that flattens irregular surfaces by transforming Cartesian coordinates into curvilinear ones.However,its application is limited to first arrivals in the context o...The surface-flattening scheme is a mathematical method that flattens irregular surfaces by transforming Cartesian coordinates into curvilinear ones.However,its application is limited to first arrivals in the context of the topography-dependent eikonal equation(TDEE).Here,we introduce a multi-block surface-flattening scheme that simultaneously transforms Earth's surface and subsurface interfaces in Cartesian coordinates into horizontal interfaces in curvilinear coordinates,while adaptively adjusting the grid according to the arbitrary geometry of each layer.This scheme allows the recovery of complex seismic velocity structures by joint tomographic inversion using multi-type phase arrivals,including converted and reflected waves.Forward modeling is performed using a multi-stage locking sweeping method with high-order finite-difference stencils,in which first arrivals are computed with a factored TDEE solver,and reflected waves are tracked by restarting the TDEE solver from reflective points on an irregular interface.An adjoint-state method formulated in curvilinear coordinates is used to estimate the preconditioned gradient,avoiding both ray tracing and explicit computation of the derivative matrix.Synthetic tests confirm the operability and effectiveness of the proposed approach for imaging complex layered velocity models.Furthermore,we apply the proposed method to wide-angle seismic data acquired in northeastern(NE)Tibetan Plateau,using refracted and reflected arrivals,to image the upper crust.The agreement between the regional tectonic division and the discernible velocity characteristics of the upper crustal structure demonstrates the good resolution achieved with the implemented algorithm.展开更多
The semiclassical approximation is an efficient approach for studying the standard Schrödinger equation(SE)both analytically and numerically,where the wavefunction is approximated by an ansatz such that its phase...The semiclassical approximation is an efficient approach for studying the standard Schrödinger equation(SE)both analytically and numerically,where the wavefunction is approximated by an ansatz such that its phase and amplitude are determined through Hamilton-Jacobi type partial differential equations(PDEs)that can be derived using the standard rules of standard derivatives.However,for the space fractional Schrödinger equation(FSE),the introduction of the fractional differential operators makes it challenging to derive relevant semiclassical approximations,because not only the problem becomes non-local,but also the rules for the standard derivatives generally do not hold for the fractional derivatives.In this work,we first attempt to derive the semiclassical approximation in the Wentzel-Kramers-Brillouin-Jeffreys(WKBJ)form for the space FSE based on the quantum Riesz fractional operators.We find that the phase and amplitude can also be determined by local Hamilton-Jacobi type PDEs even though the space FSE is non-local,the Hamiltonian for the phase is consistent with that in the classical Hamilton-Jacobi approach for the space FSE,and the semiclassical approximation reduces to that for the standard SE when the fractional order becomes integer order.We then compute the numerical solutions for the space FSE through the semiclassical approximation by solving the local Hamilton-Jacobi type PDEs with well-established numerical schemes.Numerical experiments are presented to verify the accuracy and efficiency of the derived semiclassical formulations.展开更多
In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed...In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.展开更多
We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze t...We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the first-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose a fifth-order Weighted PowerENO sweeping scheme for static Hamilton-Jacobi equations with convex Hamiltonians and present numerical examples that validate the effectiveness of the new stopping criterion.展开更多
The fast sweeping method is an efficient iterative method for hyperbolic problems. It combines Gauss-Seidel iterations with alternating sweeping orderings. In this paper several parallel implementations of the fast sw...The fast sweeping method is an efficient iterative method for hyperbolic problems. It combines Gauss-Seidel iterations with alternating sweeping orderings. In this paper several parallel implementations of the fast sweeping method are presented. These parallel algorithms are simple and efficient due to the causality of the underlying partial different equations. Numerical examples are used to verify our algorithms.展开更多
This paper introduces a neural network approach for solving two-dimensional traveltime tomography(TT)problems based on the eikonal equation.The mathematical problem of TT is to recover the slowness field of a medium b...This paper introduces a neural network approach for solving two-dimensional traveltime tomography(TT)problems based on the eikonal equation.The mathematical problem of TT is to recover the slowness field of a medium based on the boundary measurement of the traveltimes of waves going through the medium.This inverse map is high-dimensional and nonlinear.For the circular tomography geometry,a perturbative analysis shows that the forward map can be approximated by a vectorized convolution operator in the angular direction.Motivated by this and filtered backprojection,we propose an effective neural network architecture for the inverse map using the recently proposed BCR-Net,with weights learned from training datasets.Numerical results demonstrate the efficiency of the proposed neural networks.展开更多
基金The authors thank the funds supported by the China National Nuclear Corporation under Grants Nos.WUQNYC2101 and WUHTLM2101-04National Natural Science Foundation of China(42074132,42274154).
文摘3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic monitoring and tomographic imaging.In recent years,many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media.However,there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media.In this paper,we propose an adaptive finite-difference(AFD)method to numerically solve the 3D eikonal equation.The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes,and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle.Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media,our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts.Several numerical examples show the good performance of the AFD method,which will be beneficial to many scientific applications.
文摘Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.
基金supported by the National Science Foundation(No.DMS-0913982)
文摘In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.
基金supported by the National Science Foundation(No.DMS-0913982)
文摘The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators.Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.
基金supported by the National Natural Science Foundation of China(Grant Nos.42174111 and42064004)the Beijing Natural Science Foundation(Grant No.8222033)the Ningxia Science and Technology Leading Talent Training Program(Grant No.2022GKLRLX04)。
文摘The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to practically determine earthquake locations.However,first-arrival traveltimes are not sensitive to focal depths.Moreover,they cannot accurately constrain focal depths.To improve the accuracy,researchers have analyzed the depth phases of earthquake locations.The traveltimes of depth phases are sensitive to focal depths,and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations.Generally,researchers can determine earthquake locations in layered models.Because layered models can only represent the first-order feature of subsurface structures,the advantages of joint inversion are not fully explored if layered models are used.To resolve the issue of current joint inversions,we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models.The three seismic phases used in this study are the first P-,sPg-and PmP-waves.We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations.To verify the accuracy of the earthquake location method by the inversion of three seismic phases,we take the 2021 M_(S)6.4 Yangbi,Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data.Numerical tests demonstrate that the eikonal equation-based earthquake location method,which involves the inversion of multiple phase arrivals,can effectively improve earthquake location accuracy.
基金partially supported by ONR Grant N00014-02-1-0090ARO MURI Grant W911NF-07-1-0185NSF Grant DMS0811254
文摘We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.
基金supported by Natural Science Foundation of Jiangsu Province(Nos.KFR21026,PAF20042)National Natural Science Foundation of China(Nos.GBA20029,GCA20004)+2 种基金Science Challenge Project(No.TZ2018002)National Science and Technology Major Project(No.J2019-II-0007-0027)WH is supported by NSF DMS-1818769.
文摘We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.More specifically,the homotopy method is utilized to decrease the viscosity coefficient gradually,while Newton’s method is applied to compute the solution for each viscosity coefficient.Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples,but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids.Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.
文摘We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.
基金This work was financially supported by the Science Foundation of National Education Committee of China
文摘Two mathematical models for combined refraction-diffraction of regular and irregular waves on non-uniform current in water of slowly varying topography are presented in this paper. Model I is derived by wave theory and variational principle separately. It has two kinds of expressions including the dissipation term. Model n is based on the energy conservation equation with energy flux through the wave crest lines in orthogonal curvilinear coordinates and the wave kinematic conservation equation. The analysis and comparison and special cases of these two models are also given.
基金Science Foundation of National Education Committee of China
文摘A new numerical finite difference iteration method for refraction-diffraction of waves ia water of slowly varying current and topography is developed in this paper. And corresponding theoretical model including the dissipation term is briefly described, together with some analysis and comparison of computational results of the model with measurements in a hydraulic scale model (Berkhoff et al., 1982). An example of practical use of the method is given, showing that the present model is useful to engineering practice.
基金National Natural Science Foundation of China (49894190-024) and Geophysical Prospecting Key Laboratory Foundation of China National Petroleum Corporation.
文摘To the most of velocity fields, the traveltimes of the first break that seismic waves propagate along rays can be computed on a 2-D or 3-D numerical grid by finite-difference extrapolation. Under ensuring accuracy, to improve calculating efficiency and adaptability, the calculation method of first-arrival traveltime of finite-difference is de- rived based on any rectangular grid and a local plane wavefront approximation. In addition, head waves and scat- tering waves are properly treated and shadow and caustic zones cannot be encountered, which appear in traditional ray-tracing. The testes of two simple models and the complex Marmousi model show that the method has higher accuracy and adaptability to complex structure with strong vertical and lateral velocity variation, and Kirchhoff prestack depth migration based on this method can basically achieve the position imaging effects of wave equation prestack depth migration in major structures and targets. Because of not taking account of the later arrivals energy, the effect of its amplitude preservation is worse than that by wave equation method, but its computing efficiency is higher than that by total Green′s function method and wave equation method.
基金supported by the grant MTM 2011-27739-C04-02 of the Spanish Ministry of Science and Innovation,and supported by the‘Ramon y Cajal’Sub-programme (MICINN-RYC) of the Spanish Ministry of Science and Innovation,Ref. RYC-2010-06412
文摘This paper deals with a coupled system consisting of a scalar conservation law and an eikonal equation, called the Hughes model. Introduced in [24], this model attempts to describe the motion of pedestrians in a densely crowded region, in which they are seen as a ＇thinking＇ （continuum） fluid. The main mathematical difficulty is the discontinuous gradient of the solution to the eikonal equation appearing in the flux of the conservation law. On a one dimensional interval with zero Dirichlet conditions （the two edges of the interval are interpreted as ＇targets＇）, the model can be decoupled in a way to consider two classical conservation laws on two sub-domains separated by a turning point at which the pedestrians change their direction. We shall consider solutions with a possible jump discontinuity around the turning point. For simplicity, we shall assume they are locally constant on both sides of the discontinuity. We provide a detailed description of the local- in-time behavior of the solution in terms of a ＇global＇ qualitative property of the pedestrian density （that we call ＇relative evacuation rate＇）, which can be interpreted as the attitude of the pedestrians to direct towards the left or the right target. We complement our result with explicitly computable examples.
文摘In this paper, we establish a mathematical model of the forest fire spread process based on a partial differential equation. We describe the distribution of time field and velocity field in the whole two-dimensional space by vector field theory. And we obtain a continuous algorithm to predict the dynamic behavior of forest fire spread in a short time. We use the algorithm to interpolate the fire boundary by cubic non-uniform rational B-spline closed curve. The fire boundary curve at any time can be simulated by solving the Eikonal equation. The model is tested in theory and in practice. The results show that the model has good accuracy and stability, and it’s compatible with most of the existing models, such as the elliptic model and the cellular automata model.
文摘The velocity of the electromagnetic radiation in a perfect dielectric, containing no charges and no conduction currents, is explored and determined on making use of the Lorentz transformations. Besides the idealised blackbody radiation, whose vacuum propagation velocity is the universal constant c, being this value independent of the observer, there is another behaviour of electromagnetic radiation, we call inertial radiation, which is characterized by an electromagnetic inertial density , and therefore, it happens to be described by a time-like Poynting four-vector field which propagates with velocity . is found to be a relativistic invariant expressible in terms of the relativistic invariants of the electromagnetic field. It is shown that there is a rest frame, where the Poynting vector is equal to zero. Both phase and group velocities of the electromagnetic radiation are evaluated. The wave and eikonal equations for the dynamics of the radiation field are formulated.
基金supported by the National Key R&D Program of China(Grant No.2023YFF0803500)the Strategic Priority Program(B)of the Chinese Academy of Sciences(Grant No.XDB24000000)。
文摘The surface-flattening scheme is a mathematical method that flattens irregular surfaces by transforming Cartesian coordinates into curvilinear ones.However,its application is limited to first arrivals in the context of the topography-dependent eikonal equation(TDEE).Here,we introduce a multi-block surface-flattening scheme that simultaneously transforms Earth's surface and subsurface interfaces in Cartesian coordinates into horizontal interfaces in curvilinear coordinates,while adaptively adjusting the grid according to the arbitrary geometry of each layer.This scheme allows the recovery of complex seismic velocity structures by joint tomographic inversion using multi-type phase arrivals,including converted and reflected waves.Forward modeling is performed using a multi-stage locking sweeping method with high-order finite-difference stencils,in which first arrivals are computed with a factored TDEE solver,and reflected waves are tracked by restarting the TDEE solver from reflective points on an irregular interface.An adjoint-state method formulated in curvilinear coordinates is used to estimate the preconditioned gradient,avoiding both ray tracing and explicit computation of the derivative matrix.Synthetic tests confirm the operability and effectiveness of the proposed approach for imaging complex layered velocity models.Furthermore,we apply the proposed method to wide-angle seismic data acquired in northeastern(NE)Tibetan Plateau,using refracted and reflected arrivals,to image the upper crust.The agreement between the regional tectonic division and the discernible velocity characteristics of the upper crustal structure demonstrates the good resolution achieved with the implemented algorithm.
基金support was received from the Simons Foundation 714376.
文摘The semiclassical approximation is an efficient approach for studying the standard Schrödinger equation(SE)both analytically and numerically,where the wavefunction is approximated by an ansatz such that its phase and amplitude are determined through Hamilton-Jacobi type partial differential equations(PDEs)that can be derived using the standard rules of standard derivatives.However,for the space fractional Schrödinger equation(FSE),the introduction of the fractional differential operators makes it challenging to derive relevant semiclassical approximations,because not only the problem becomes non-local,but also the rules for the standard derivatives generally do not hold for the fractional derivatives.In this work,we first attempt to derive the semiclassical approximation in the Wentzel-Kramers-Brillouin-Jeffreys(WKBJ)form for the space FSE based on the quantum Riesz fractional operators.We find that the phase and amplitude can also be determined by local Hamilton-Jacobi type PDEs even though the space FSE is non-local,the Hamiltonian for the phase is consistent with that in the classical Hamilton-Jacobi approach for the space FSE,and the semiclassical approximation reduces to that for the standard SE when the fractional order becomes integer order.We then compute the numerical solutions for the space FSE through the semiclassical approximation by solving the local Hamilton-Jacobi type PDEs with well-established numerical schemes.Numerical experiments are presented to verify the accuracy and efficiency of the derived semiclassical formulations.
基金supported by the NSF (Grant No.DMS-1753581)supported by NSFC (Grant No.12071392).
文摘In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.
基金supported by DGICYT MTM2008-03597Ramon y Cajal Programsupported by NSF DMS # 0810104
文摘We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the first-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose a fifth-order Weighted PowerENO sweeping scheme for static Hamilton-Jacobi equations with convex Hamiltonians and present numerical examples that validate the effectiveness of the new stopping criterion.
基金This work is partially supported by Sloan FoundationNSF DMS0513073+1 种基金ONR grant N00014-02-1-0090DARPA grant N00014-02-1-0603
文摘The fast sweeping method is an efficient iterative method for hyperbolic problems. It combines Gauss-Seidel iterations with alternating sweeping orderings. In this paper several parallel implementations of the fast sweeping method are presented. These parallel algorithms are simple and efficient due to the causality of the underlying partial different equations. Numerical examples are used to verify our algorithms.
基金partially supported by the U.S.Department of Energy,Office of Science,Office of Advanced Scientific Computing Research,Scientific Discovery through Advanced Computing(SciDAC)programpartially supported by the National Science Foundation under award DMS-1818449.
文摘This paper introduces a neural network approach for solving two-dimensional traveltime tomography(TT)problems based on the eikonal equation.The mathematical problem of TT is to recover the slowness field of a medium based on the boundary measurement of the traveltimes of waves going through the medium.This inverse map is high-dimensional and nonlinear.For the circular tomography geometry,a perturbative analysis shows that the forward map can be approximated by a vectorized convolution operator in the angular direction.Motivated by this and filtered backprojection,we propose an effective neural network architecture for the inverse map using the recently proposed BCR-Net,with weights learned from training datasets.Numerical results demonstrate the efficiency of the proposed neural networks.
