共查询到20条相似文献,搜索用时 11 毫秒
1.
This paper presents a fourth-order Cartesian grid based boundary integral
method (BIM) for heterogeneous interface problems in two and three dimensional
space, where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions. The method reformulates the
governing equation with interface conditions into boundary integral equations (BIEs)
and reinterprets the involved integrals as solutions to some simple interface problems
in an extended regular region. Solution of the simple equivalent interface problems for
integral evaluation relies on a fourth-order finite difference method with an FFT-based
fast elliptic solver. The structure of the coefficient matrix is preserved even with the
existence of the interface. In the whole calculation process, analytical expressions of
Green’s functions are never determined, formulated or computed. This is the novelty
of the proposed kernel-free boundary integral (KFBI) method. Numerical experiments
in both two and three dimensions are shown to demonstrate the algorithm efficiency
and solution accuracy even for problems with a large diffusion coefficient ratio. 相似文献
2.
An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient
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Zhiwen Zhang Xin Hu Thomas Y. Hou Guang Lin & Mike Yan 《Communications In Computational Physics》2014,16(3):571-598
In this paper, we present an adaptive, analysis of variance (ANOVA)-based
data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven
stochastic method. To handle high-dimensional stochastic problems, we investigate
the use of adaptive ANOVA decomposition in the stochastic space as an effective
dimension-reduction technique. To improve the slow convergence of the generalized
polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the
data-driven stochastic method (DSM) for speed up. An essential ingredient of the
DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary
conditions.Our ANOVA-DSM consists of offline and online stages. In the offline stage, the
original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using
the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization
approach. Multiple trial functions are used to enrich the stochastic basis and improve
the accuracy. In the online stage, we solve each stochastic subproblem for any given
forcing function by projecting the stochastic solution into the data-driven stochastic
basis constructed offline. In our ANOVA-DSM framework, solving the original high-dimensional stochastic problem is reduced to solving a series of ANOVA-decomposed
stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided
to further reduce the number of the stochastic subproblems and speed up our method.
To demonstrate the accuracy and efficiency of our method, numerical examples are
presented for one- and two-dimensional elliptic PDEs with random coefficients. 相似文献
3.
Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other.
We present a boundary integral method in two dimensions which is specially designed
to maintain second order accuracy even if boundaries are arbitrarily close. The method
uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy. For boundaries with many components we use the fast
multipole method for efficient summation. We compute electric potentials on a domain
with cells whose conductivity differs from that of the surrounding medium. We first
solve an integral equation for a source term on the cell interfaces and then find values
of the potential near the interfaces via integrals. Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region. A number of examples
are presented. We demonstrate that increased refinement is not needed to maintain
accuracy as interfaces become very close. 相似文献
4.
A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions
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In this article, we discuss a least-squares/fictitious domain method for the
solution of linear elliptic boundary value problems with Robin boundary conditions.
Let Ω and ω be two bounded domains of Rdsuch that ω⊂Ω. For a linear elliptic problem
in Ω\ω with Robin boundary condition on the boundary γ of ω, our goal here is to
develop a fictitious domain method where one solves a variant of the original problem
on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual
control type and relies on a least-squares formulation making the problem solvable by
a conjugate gradient algorithm operating in a well chosen control space. Numerical results
obtained when applying our method to the solution of two-dimensional elliptic
and parabolic problems are given; they suggest optimal order of convergence. 相似文献
5.
Mingze Qin Ruishu Wang Qilong Zhai & Ran Zhang 《Communications In Computational Physics》2023,33(2):568-595
The weak Galerkin (WG) method is a nonconforming numerical method for
solving partial differential equations. In this paper, we introduce the WG method for
elliptic equations with Newton boundary condition in bounded domains. The Newton
boundary condition is a nonlinear boundary condition arising from science and engineering applications. We prove the well-posedness of the WG scheme by the monotone operator theory and the embedding inequality of weak finite element functions.
The error estimates are derived. Numerical experiments are presented to verify the
theoretical analysis. 相似文献
6.
A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs
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Yuling Jiao Yanming Lai Dingwei Li Xiliang Lu Fengru Wang Yang Wang & Jerry Zhijian Yang 《Communications In Computational Physics》2022,31(4):1272-1295
In recent years, physical informed neural networks (PINNs) have been
shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for
the second order elliptic equations with Dirichlet boundary condition, by establishing
the upper bounds on the number of training samples, depth and width of the deep
neural networks to achieve desired accuracy. The error of PINNs is decomposed into
approximation error and statistical error, where the approximation error is given in $C^2$ norm with ReLU$^3$ networks (deep network with activation function max$\{0,x^3\}$) and
the statistical error is estimated by Rademacher complexity. We derive the bound on
the Rademacher complexity of the non-Lipschitz composition of gradient norm with
ReLU$^3$ network, which is of immense independent interest. 相似文献
7.
The Recursive Formulation of Particular Solutions for Some Elliptic PDEs with Polynomial Source Functions
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In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations. We first approximate the source function
by Chebyshev polynomials. We then focus on how to find a polynomial particular solution when the source function is a polynomial. Through the principle of the method
of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined, the coefficients of the particular solution
satisfy a triangular system of linear algebraic equations. Explicit recursive formulas
for the coefficients of the particular solutions are derived for different types of elliptic
PDEs. The method is further incorporated into the method of fundamental solutions
for solving inhomogeneous elliptic PDEs. Numerical results show that our approach
is efficient and accurate. 相似文献
8.
Dirichlet-to-Neumann Mapping for the Characteristic Elliptic Equations with Symmetric Periodic Coefficients
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Jingsu Kang Meirong Zhang & Chunxiong Zheng 《Communications In Computational Physics》2014,16(4):1102-1115
Based on the numerical evidences, an analytical expression of the Dirichlet-to-Neumann mapping in the form of infinite product was first conjectured for the one-dimensional characteristic Schrödinger equation with a sinusoidal potential in [Commun. Comput. Phys., 3(3): 641-658, 2008]. It was later extended for the general second-order characteristic elliptic equations with symmetric periodic coefficients in [J. Comp.
Phys., 227: 6877-6894, 2008]. In this paper, we present a proof for this Dirichlet-to-Neumann mapping. 相似文献
9.
A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data
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In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS)
method, which provides a numerical approximation of the Dirichlet-to-Neumann
(DtN) mapping for the Laplace equation. The BIE-WOS is a local method on the
boundary of the domain and does not require a structured mesh, and only needs a
covering of the domain boundary by patches and a local mesh for each patch for a local BIE. A new version of the BIE-WOS method with second kind integral equations is
introduced for better error controls. The effect of errors from the Feynman-Kac formula
based path integral WOS method on the overall accuracy of the BIE-WOS method is
analyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs.
For the special case of flat patches, it is shown that the second kind integral equation
of BIE-WOS method can be simplified where the local BIE solutions can be given in
closed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches of
the boundary, resulting in a complete independent computation using a large number
of cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOS
method on a large-scale cluster with 6400 CPU cores is verified for computing the DtN
mapping of exterior Laplace problems with Dirichlet data for several domains. 相似文献
10.
Wenjun Cai Huai Zhang & Yushun Wang 《Communications In Computational Physics》2016,19(5):1375-1396
This paper explores the discrete singular convolution method for Hamiltonian
PDEs. The differential matrices corresponding to two delta type kernels of the
discrete singular convolution are presented analytically, which have the properties of
high-order accuracy, band-limited structure and thus can be excellent candidates for the
spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation
and the coupled Schrödinger equations for example, we construct two symplectic
integrators combining this kind of differential matrices and appropriate symplectic
time integrations, which both have been proved to satisfy the square conservation
laws. Comprehensive numerical experiments including comparisons with the central
finite difference method, the Fourier pseudospectral method, the wavelet collocation
method are given to show the advantages of the new type of symplectic integrators. 相似文献
11.
We present a parallel Cartesian method to solve elliptic problems with complex immersed interfaces. This method is based on a finite-difference scheme and is second-order accurate in the whole domain. The originality of the method lies in the use of additional unknowns located on the interface, allowing to express straightforwardly the interface transmission conditions. We describe the method and the details of its parallelization performed with the PETSc library. Then we present numerical validations in two dimensions, assorted with comparisons to other related methods, and a numerical study of the parallelized method. 相似文献
12.
On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
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Yeonjong Shin Jé rô me Darbon & George Em Karniadakis 《Communications In Computational Physics》2020,28(5):2042-2074
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computational
science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is
obtained by minimizing a loss function in which any prior knowledge of PDEs and
data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs.As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of
PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach
and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies
the initial/boundary conditions, the convergence mode becomes $H^1$. Computational
examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs. 相似文献
13.
Rongfang Gong Xiaoliang Cheng & Weimin Han 《Communications In Computational Physics》2016,19(1):226-250
In this paper, we introduce and study a new method for solving inverse
source problems, through a working model that arises in bioluminescence tomography
(BLT). In the BLT problem, one constructs quantitatively the bioluminescence source
distribution inside a small animal from optical signals detected on the animal's body
surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization
is used to obtain stable approximate solutions. In conventional Tikhonov
regularization, it is crucial to choose a proper regularization parameter for trade off
between the accuracy and stability of approximate solutions. The new method is
based on a combination of the boundary condition and the boundary measurement
in a parameter-dependent single complex Robin boundary condition, followed by the
Tikhonov regularization. By properly adjusting the parameter in the Robin boundary
condition, we achieve two important properties for our new method: first, the regularized
solutions are uniformly stable with respect to the regularization parameter
so that the regularization parameter can be chosen based solely on the consideration
of the solution accuracy; second, the convergence order of the regularized solutions
reaches one with respect to the noise level. Then, the finite element method is used to
compute numerical solutions and a new finite element error estimate is derived for discrete
solutions. These results improve related results found in the existing literature.
Several numerical examples are provided to illustrate the theoretical results. 相似文献
14.
15.
A Fast Direct Solver for a Class of 3-D Elliptic Partial Differential Equation with Variable Coefficient
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We propose a direct solver for the three-dimensional Poisson equation with
a variable coefficient, and an algorithm to directly solve the associated sparse linear systems that exploits the sparsity pattern of the coefficient matrix. Introducing
some appropriate finite difference operators, we derive a second-order scheme for the
solver, and then two suitable high-order compact schemes are also discussed. For
a cube containing N nodes, the solver requires O(N3/2log2N) arithmetic operations
and O(NlogN) memory to store the necessary information. Its efficiency is illustrated
with examples, and the numerical results are analysed. 相似文献
16.
Hailong Guo & Xu Yang 《Communications In Computational Physics》2022,31(4):1162-1179
This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions. To capture discontinuities
of the solution caused by interfaces, we reformulate the problem as an energy minimization problem involving two weakly coupled components. This enables us to
train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo
method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method. 相似文献
17.
A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions
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S. N. Chandler-Wilde S. Langdon & M. Mokgolele 《Communications In Computational Physics》2012,11(2):573-593
We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave. 相似文献
18.
Fast Multipole Accelerated Boundary Integral Equation Method for Evaluating the Stress Field Associated with Dislocations in a Finite Medium
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Degang Zhao Jingfang Huang & Yang Xiang 《Communications In Computational Physics》2012,12(1):226-246
In this paper, we develop an efficient numerical method based on the boundary integral equation formulation and new version of fast multipole method to solve
the boundary value problem for the stress field associated with dislocations in a finite
medium. Numerical examples are presented to examine the influence from material
boundaries on dislocations. 相似文献
19.
An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs
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In this paper, we develop a novel energy-preserving wavelet collocation
method for solving general multi-symplectic formulations of Hamiltonian PDEs. Based
on the autocorrelation functions of Daubechies compactly supported scaling functions,
the wavelet collocation method is conducted for spatial discretization. The obtained
semi-discrete system is shown to be a finite-dimensional Hamiltonian system, which
has an energy conservation law. Then, the average vector field method is used for
time integration, which leads to an energy-preserving method for multi-symplectic
Hamiltonian PDEs. The proposed method is illustrated by the nonlinear Schrödinger
equation and the Camassa-Holm equation. Since differentiation matrix obtained by
the wavelet collocation method is a cyclic matrix, we can apply Fast Fourier transform
to solve equations in numerical calculation. Numerical experiments show the high
accuracy, effectiveness and conservation properties of the proposed method. 相似文献
20.
Polynomial chaos methods (and generalized polynomial chaos methods) have been extensively applied to analyze PDEs that contain uncertainties. However, this approach is rarely applied to hyperbolic systems. In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection. We consider a simple model of a scalar wave equation with random wave speed. We show that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. A consistent method of imposing the boundary conditions is proposed and its convergence is established. Numerical examples are presented to support the analysis. 相似文献