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1.
In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in , . Both the two-dimensional (2D) and three-dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large , or lattice, respectively. The finite element method discretization procedure on a 3D uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo-inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third-order tensor of size . The latter is approximated by a low-rank canonical tensor by using the multigrid Tucker-to-canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right-hand side. The computational characteristics of the presented solver in terms of a lattice parameter and the grid-size, , in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many-particle dynamics, protein docking problems or stochastic modeling. 相似文献
2.
Li-Ping Zhang Zi-Cai Li Ming-Gong Lee Hung-Tsai Huang 《Numerical Linear Algebra with Applications》2023,30(2):e2466
Consider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain . In the standard MFS, the source nodes are located on a closed contour outside the domain boundary , which is called pseudo-boundary. For circular, elliptic, and general closed pseudo-boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial-lines outside . Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde-wise matrices on nodes with . The Vandermonde-wise matrix is called in this article if it can be decomposed into the standard Vandermonde matrix. New lower and upper bounds of Cond are first derived for the standard Vandermonde matrix, and then for new algorithms of the MFS using two pseudo radial-lines. Both lower and upper bounds of Cond are intriguing in the stability study for the MFS. Numerical experiments are carried out to verify the stability analysis made. Since the fundamental solutions (as ) are the basis functions of the MFS, new Vandermonde-wise matrices are found. Since the nodes with may come from approximations and interpolations by the Laurent polynomials with singular part, the conclusions in this article are important not only to the MFS but also to matrix analysis. 相似文献
3.
The present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix as goes to infinity, with a continuous and real-valued symbol having a power singularity of degree with , at one point. The resulting matrix is dense and its entries decrease slowly to zero when moving away from the main diagonal, we apply the so called simple-loop (SL) method for constructing and justifying a uniform asymptotic expansion for all the eigenvalues. Note however, that the considered symbol does not fully satisfy the conditions imposed in previous works, but only in a small neighborhood of the singularity point. In the present work: (i) We construct and justify the asymptotic formulas of the SL method for the eigenvalues with , where the eigenvalues are arranged in nondecreasing order and is a sufficiently small fixed number. (ii) We show, with the help of numerical calculations, that the obtained formulas give good approximations in the case . (iii) We numerically show that the main term of the asymptotics for eigenvalues with , formally obtained from the formulas of the SL method, coincides with the main term of the asymptotics constructed and justified in the classical works of Widom and Parter. 相似文献
4.
Frank Uhlig 《Numerical Linear Algebra with Applications》2023,30(6):e2513
This paper describes and develops a fast and accurate path following algorithm that computes the field of values boundary curve for every conceivable complex or real square matrix . It relies on the matrix flow decomposition algorithm that finds a proper block-diagonal flow representation for the associated hermitean matrix flow under unitary similarity if that is possible. Here is the 1-parameter-varying linear combination of the real and skew part matrices and of . For indecomposable matrix flows, has just one block and the ZNN based field of values algorithm works with directly. For decomposing flows , the algorithm decomposes the given matrix unitarily into block-diagonal form with diagonal blocks whose individual sizes add up to the size of . It then computes the field of values boundaries separately for each diagonal block using the path following ZNN eigenvalue method. The convex hull of all sub-fields of values boundary points finally determines the field of values boundary curve correctly for decomposing matrices . The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices due to possible eigencurve crossings of . Tests and numerical comparisons are included. Our ZNN based method is coded for sequential and parallel computations and both versions run very accurately and fast when compared with Johnson's Francis QR eigenvalue and Bendixson rectangle based method and compute global eigenanalyses of for large discrete sets of angles more slowly. 相似文献
5.
A general, rectangular kernel matrix may be defined as where is a kernel function and where and are two sets of points. In this paper, we seek a low-rank approximation to a kernel matrix where the sets of points and are large and are arbitrarily distributed, such as away from each other, “intermingled”, identical, and so forth. Such rectangular kernel matrices may arise, for example, in Gaussian process regression where corresponds to the training data and corresponds to the test data. In this case, the points are often high-dimensional. Since the point sets are large, we must exploit the fact that the matrix arises from a kernel function, and avoid forming the matrix, and thus ruling out most algebraic techniques. In particular, we seek methods that can scale linearly or nearly linearly with respect to the size of data for a fixed approximation rank. The main idea in this paper is to geometrically select appropriate subsets of points to construct a low rank approximation. An analysis in this paper guides how this selection should be performed. 相似文献
6.
In this paper, we study the following coupled Choquard system in : where and , in which denotes if and if . The function is a Riesz potential. By using Nehari manifold method, we obtain the existence of a positive ground state solution in the case of bounded potential and periodic potential, respectively. In particular, the nonlinear term includes the well-studied case and , and the less-studied case and . Moreover, it seems to be the first existence result for the case . 相似文献
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8.
This paper deals with the following slightly subcritical Schrödinger equation: where is a nonnegative smooth function, , , , . Most of the previous works for the Schrödinger equations were mainly investigated for power-type nonlinearity. In this paper, we will study the case when the nonlinearity is a non-power nonlinearity. We show that, for ε small enough, there exists a family of single-peak solutions concentrating at the positive stable critical point of the potential . 相似文献
9.
In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor-train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix , generated by the first column of the form admits a QTT representation with the QTT ranks bounded by . Under certain assumptions on the entries of , we also derive an explicit QTT representation of . The latter can be used, for instance, to overcome stability issues arising when numerically solving differential equations with periodic boundary conditions in the QTT format. 相似文献
10.
The following kind of Klein–Gordon–Maxwell system is investigated where is a parameter, and V is vanishing potential. By using some suitable conditions on K and f, we obtain a Palais–Smale sequence by using Pohožaev equality and prove the ground-state solution for this system by employing variational methods. Our result improves the related one in the literature. 相似文献
11.
Tobias Danczul Clemens Hofreither Joachim Schöberl 《Numerical Linear Algebra with Applications》2023,30(5):e2488
We present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix-vector product of the form , where is the discretization matrix of the spatial operator, a prescribed vector, and a parametric function, such as a fractional power or the Mittag-Leffler function. In the abstract framework of Stieltjes and complete Bernstein functions, to which the functions we are interested in belong to, we apply a rational Krylov method and prove uniform convergence when using poles based on Zolotarëv's minimal deviation problem. The latter are particularly suited for fractional diffusion as they allow for an efficient query of the map and do not degenerate as the fractional parameters approach zero. We also present a variety of both novel and existing pole selection strategies for which we develop a computable error certificate. Our numerical experiments comprise a detailed parameter study of space-time fractional diffusion problems and compare the performance of the poles with the ones predicted by our certificate. 相似文献
12.
Nathan Heavner Chao Chen Abinand Gopal Per-Gunnar Martinsson 《Numerical Linear Algebra with Applications》2023,30(6):e2515
Standard rank-revealing factorizations such as the singular value decomposition (SVD) and column pivoted QR factorization are challenging to implement efficiently on a GPU. A major difficulty in this regard is the inability of standard algorithms to cast most operations in terms of the Level-3 BLAS. This article presents two alternative algorithms for computing a rank-revealing factorization of the form , where and are orthogonal and is trapezoidal (or triangular if is square). Both algorithms use randomized projection techniques to cast most of the flops in terms of matrix-matrix multiplication, which is exceptionally efficient on the GPU. Numerical experiments illustrate that these algorithms achieve significant acceleration over finely tuned GPU implementations of the SVD while providing low rank approximation errors close to that of the SVD. 相似文献
13.
Tahir Boudjeriou 《Mathematische Nachrichten》2023,296(3):938-956
In this paper, we consider the following class of wave equation involving fractional p-Laplacian with logarithmic nonlinearity where is a bounded domain with Lipschitz boundary, , , and is the critical exponent in the Sobolev inequality. First, via the Galerkin approximations, the existence of local solutions are obtained when . Next, by combining the potential well theory with the Nehari manifold, we establish the existence of global solutions when . Then, via the Pohozaev manifold, the existence of global solutions are obtained when . By virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we discuss the asymptotic behavior of solutions as time tends to infinity. Here, we point out that the main difficulty is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian. 相似文献
14.
Anh Tuan Duong Tran Thi Loan Dao Trong Quyet Dao Manh Thang 《Mathematische Nachrichten》2023,296(6):2321-2331
In this paper, we are concerned with the fractional Choquard equation on the whole space with , and . We first prove that the equation does not possess any positive solution for . When , we establish a Liouville type theorem saying that if then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation. 相似文献
15.
Paulo Cesar Carrião Olímpio Hiroshi Miyagaki André Vicente 《Mathematische Nachrichten》2023,296(1):130-151
In this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space . The space is the unit disc of endowed with the Riemannian metric g given by , where and , if and , if . Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis–Marcus, which allows us to overcome the difficulty involving the singularities. 相似文献
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17.
In this paper, we study the existence and properties of normalized solutions for the following Sobolev critical Schrödinger equation involving Hardy term: with prescribed mass where 2* is the Sobolev critical exponent. For a L2-subcritical, L2-critical, or L2-supercritical perturbation , we prove several existence results of normalized ground state when and non-existence results when . Furthermore, we also consider the asymptotic behavior of the normalized solutions u as or . 相似文献
18.
Andreas Chatziafratis Spyridon Kamvissis Ioannis G. Stratis 《Studies in Applied Mathematics》2023,150(2):339-379
In this paper, we consider the solution to the linear Korteweg-De Vries (KdV) equation, both homogeneous and forced, on the quadrant via the unified transform method of Fokas and we provide a complete rigorous study of the integrals of the formula provided by the method, especially focusing on the explicit verification of the considered initial-boundary-value problems (IBVPs), with generic data, as well as on the uniform convergence of all its derivatives, as approaches the boundary of the quadrant, and their rapid decay as . 相似文献
19.
By Aguglia et al., new quasi-Hermitian varieties in depending on a pair of parameters from the underlying field have been constructed. In the present paper we study the structure of the lines contained in and consequently determine the projective equivalence classes of such varieties for odd and . As a byproduct, we also prove that the collinearity graph of is connected with diameter 3 for . 相似文献