共查询到20条相似文献,搜索用时 62 毫秒
1.
Britta Schmitt Boris N. Khoromskij Venera Khoromskaia Volker Schulz 《Numerical Linear Algebra with Applications》2022,29(1):e2404
We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two- (2D) and three-dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D spacial grids. Using the diagonalization of the arising matrix-valued functions in the eigenbasis of the one-dimensional Sturm–Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right-hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low-rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an -threshold. This method reduces the numerical cost for solving the control problem to (plus the quadratic term with a small weight), which outperforms traditional approaches with complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by . This essentially outperforms the traditional methods operating with fully populated matrices and vectors in . Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n. 相似文献
2.
For studying spectral properties of a nonnormal matrix , information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε‐pseudospectra, i.e. the level sets of the resolvent norm function . The computation of ε‐pseudospectra requires determining the smallest singular values for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation, which provides highly accurate estimates of pseudospectra in the region of interest, in particular, for pseudospectra estimates in isolated parts of the spectrum containing few eigenvalues of A. It incorporates the sampled singular vectors of zI ? A for different values of z, and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. In addition, we propose a domain splitting technique for tackling numerically more challenging examples. We present a comparison of our algorithms to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time. 相似文献
3.
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
4.
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
5.
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
6.
Zhao Yang 《Numerical Linear Algebra with Applications》2023,30(5):e2483
A class of negative matrices including Vandermonde-like matrices tends to be extremely ill-conditioned, and linear systems associated with this class of matrices appear in the polynomial interpolation problems. In this article, we present a fast and accurate algorithm with complexity to solve the linear systems whose coefficient matrices belong to the class of negative matrix. We show that the inverse of any such matrix is generated in a subtraction-free manner. Consequently, the solutions of linear systems associated with the class of negative matrix are accurately determined by parameterization matrices of coefficient matrices, and a pleasantly componentwise forward error is provided to illustrate that each component of the solution is computed to high accuracy. Numerical experiments are performed to confirm the claimed high accuracy. 相似文献
7.
This article is devoted to the efficient numerical solution of the Helmholtz equation in a two‐ or three‐dimensional (2D or 3D) rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT‐based direct solver is operations. Numerical results for both 2D and 3D problems are presented confirming the efficiency of the method discussed. 相似文献
8.
Hiroyuki Tsurumi 《Mathematische Nachrichten》2023,296(4):1651-1668
We consider the stationary Navier–Stokes equations in the two-dimensional torus . For any , we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces for given small external forces in when . These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well-posedness is proved by using the embedding property and the para-product estimate in homogeneous Besov spaces. In addition, for the case , we can show the ill-posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill-posedness by showing the discontinuity of a certain solution map from to . 相似文献
9.
Petr N. Vabishchevich 《Numerical Linear Algebra with Applications》2023,30(6):e2522
In computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required parameters. The approximating function is linear in the first parameter; these parameters are assumed to be positive. The individual terms of the approximating function represent a fixed function that depends nonlinearly on the second parameter. A numerical approximation minimizes the residual functional by approximating function values at individual points. The second parameter's value is set on a more extensive set of points of the interval of permissible values. The proposed approach's key feature consists in determining the first parameter on each separate iteration of the classical nonnegative least squares method. The computational algorithm is used to rational approximate the function . The second example concerns the approximation of the stretching exponential function at by the sum of exponents. 相似文献
10.
In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: with small initial data. For and μ ∈ (0, μ?) , where μ? > 0 is depending on the nonlinearties' powers and the space dimension (μ? satisfies ), we prove that the wave equation, in this case, behaves like the one without dissipation (μ = 0 ). Our result completes the previous studies in the case where the dissipation is given by , where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term is important. 相似文献
11.
12.
Paola Ferrari Nikos Barakitis Stefano Serra‐Capizzano 《Numerical Linear Algebra with Applications》2021,28(1)
The singular value distribution of the matrix‐sequence {YnTn[f]}n , with Tn[f] generated by , was shown in [J. Pestana and A.J. Wathen, SIAM J Matrix Anal Appl. 2015;36(1):273‐288]. The results on the spectral distribution of {YnTn[f]}n were obtained independently in [M. Mazza and J. Pestana, BIT, 59(2):463‐482, 2019] and [P. Ferrari, I. Furci, S. Hon, M.A. Mursaleen, and S. Serra‐Capizzano, SIAM J. Matrix Anal. Appl., 40(3):1066‐1086, 2019]. In the latter reference, the authors prove that {YnTn[f]}n is distributed in the eigenvalue sense as under the assumptions that f belongs to and has real Fourier coefficients. The purpose of this paper is to extend the latter result to matrix‐sequences of the form {h(Tn[f])}n , where h is an analytic function. In particular, we provide the singular value distribution of the sequence {h(Tn[f])}n , the eigenvalue distribution of the sequence {Ynh(Tn[f])}n , and the conditions on f and h for these distributions to hold. Finally, the implications of our findings are discussed, in terms of preconditioning and of fast solution methods for the related linear systems. 相似文献
13.
We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space for . It is clarified that if w is small in for and , then for every small initial disturbance with and (), there exists a unique solution of the nonstationary Navier–Stokes equations on (0, ∞) with such that and as , for , , and small . 相似文献
14.
15.
Mohammad Assem Mahmoud 《Mathematical Logic Quarterly》2019,65(3):293-304
In this paper, we show that for any computable ordinal α, there exists a computable tree of rank with strong degree of categoricity if α is finite, and with strong degree of categoricity if α is infinite. In fact, these are the greatest possible degrees of categoricity for such trees. For a computable limit ordinal α, we show that there is a computable tree of rank α with strong degree of categoricity (which equals ). It follows from our proofs that, for every computable ordinal , the isomorphism problem for trees of rank α is ‐complete. 相似文献
16.
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
17.
In this paper, we study the existence of ground state solutions for the modified fractional Schrödinger equations where , , , and are positive parameters, , denotes the fractional Laplacian of order . For the case and the case , the existence results of ground state solutions are given, respectively. 相似文献
18.
Nelson Faustino 《Mathematische Nachrichten》2023,296(7):2758-2779
In this paper, we introduce a wide class of space-fractional and time-fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space-time lattice (), resembling to fractional semidiscrete counterparts of the so-called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup , carrying the parameter constraints and . The results obtained involve the study of Cauchy problems on . 相似文献
19.
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
20.
For and variable exponents and with values in [1, ∞], let the variable exponents be defined by The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space to the variable Lebesgue space for , then where C is an interpolation constant independent of T. We consider two different modulars and generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that and , as well as, lead to sufficient conditions for and . We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that , are Lipschitz continuous and bounded away from one and infinity (in this case, ). 相似文献