Advances in Computational Mathematics最新文献

A standard task in solid state physics and quantum chemistry is the computation of localized molecular orbitals known as Wannier functions. In this manuscript, we propose a new procedure for computing Wannier functions in one-dimensional crystalline materials. Our approach proceeds by first performing parallel transport of the Bloch functions using numerical integration. Then, a simple analytically computable correction is introduced to yield the optimally localized Wannier function. The resulting scheme is rapidly convergent and is proven to yield real-valued Wannier functions that achieve global optimality. The analysis in this manuscript can also be viewed as a proof of the existence of exponentially localized Wannier functions in one dimension. We illustrate the performance of the scheme by a number of numerical experiments.

This paper introduces a new method for multiple image fusion that minimizes a nonlocal isotropic osmosis regularizer combined with a similarity term applied to a specific subregion. The proposed model captures nonlocal pixel interactions through nonlocal differential operators while also accounting for contrast variations. Using the semi-group theory, we demonstrate the existence and uniqueness of a solution for the corresponding evolution partial differential equation, and establish several properties that make the model well-suited for image processing. Experimental results show that this new method outperforms other existing state-of-the-art techniques in both visual quality and quantitative evaluation for two and multiple-image fusion, including multi-focus image fusion.

We show that much of the theory of finite tight frames can be generalised to vector spaces over the quaternions. This includes the variational characterisation, group frames and the characterisations of projective and unitary equivalence. We are particularly interested in sets of equiangular lines (equi-isoclinic subspaces) and the groups associated with them, and how to move them between the spaces (mathbb {R}^d), (mathbb {C}^d) and (mathbb {H}^d). We discuss what the analogue of Zauner’s conjecture for equiangular lines in (mathbb {H}^d) might be.

The original compactly supported radial basis functions of Wendland (Adv. Comput. Math., 4, 389–396, 1995) and Wu (Adv. Comput. Math., 4, 283–292, 1995) have a polynomial form and are constructed using a two-step dimension walk strategy. Focussing on the Wendland functions, Schaback (Adv. Comput. Math., 34(1), 67–81, 2011) proposed a one-step dimension walk which is shown to recover the original Wendland functions at every second step but also introduces new examples, the so-called missing Wendland functions at the intermediate steps. In a recent paper (Science China Mathematics Published online, 2025), the analogue of Schaback’s work is presented for the Wu functions and so delivers the so-called missing Wu functions. The original and missing Wendland functions belong to a much wider class proposed by Buhmann (Math. Comput., 70(233), 307–318, 2001). The classical Buhmann functions, which are related to thin-plate spline radial basis functions, also belong to this much wider class. The theme uniting the classical Buhmann functions and the missing Wendland/Wu functions is that they are non-polynomial, and closed-form expressions are not known for all of them. In this paper, we revisit these functions and show how closed-form representations can be given using direct techniques. The results for the classical Buhmann and Wu functions are new, and the resulting expressions for the missing Wendland functions improve on those given in Hubbert (Adv. Comput. Math., 36, 115–136, 2012) and so their implementation should be more straightforward.

This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial approximation. A pathwise uniform convergence rate is derived for general spatial ( L^q )-norms, by using the discrete deterministic and stochastic maximal ( L^p )-regularity estimates. Additionally, the theoretical convergence rate is validated through numerical experiments. The primary contribution of this work is the introduction of a technique to establish the pathwise uniform convergence of fully discrete finite element approximations for nonlinear stochastic parabolic equations within the framework of general spatial ( L^q )-norms.

The primary goal of this article is to propose an efficient virtual element method formulation for solving a two-dimensional time-fractional Emden-Fowler model. The virtual element technique is a generalization of the finite element approach to polygonal and polyhedral meshes in the Galerkin approximation framework. A fully discrete virtual element scheme is obtained by using a fractional version of the Grünwald-Letnikov approximation for the temporal discretization and the virtual element method for the spatial discretization. We establish the existence and uniqueness of the discrete solution, that is, the well-posedness of the approach. The error analysis and optimal convergence order with respect to the (L^2-)norm and the (H^1-)seminorm are presented. The numerical experiments validated the theoretical analysis and demonstrated the technique’s efficacy on convex and non-convex polygonal meshes.

The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here, we present new scalable MFS formulations for the corresponding elastance and mobility problems. The elastance problem computes the potentials of conductors with given net charges, while the mobility problem—crucial to rheology and complex fluid applications—computes rigid body velocities given net forces and torques on the particles. The key idea is orthogonal projection of the net charge (or forces and torques) in a rectangular variant of a “completion flow.” The proposal is compatible with one-body preconditioning, resulting in well-conditioned square linear systems amenable to fast multipole accelerated iterative solution, thus a cost linear in the particle number. For large suspensions with moderate lubrication forces, MFS sources on inner proxy-surfaces give accuracy on par with a well-resolved boundary integral formulation. Our several numerical tests include a suspension of 10,000 nearby ellipsoids, using (2.6times 10^7) total preconditioned degrees of freedom, where GMRES converges to five digits of accuracy in under two hours on one workstation.

A new iterative projection method is proposed to solve the unsteady Navier–Stokes equations with high Reynolds numbers. The convectional projection method attempts to project the intermediate velocity to the divergence-free space only once per time step. However, such a velocity is not genuinely divergence-free in general practice, which can yield large errors when the Reynolds number is high. The new method has several important features: the BDF2 time discretization, the skew-symmetric convection in a semi-implicit form, two modulating parameters, and the iterative projections in each time step. A major difficulty in the proof of iteration convergence is the nonlinear convection. We solve this problem by first analyzing the non-convective scheme with a focus on the spectral properties of the iterative matrix and then employing a delicate perturbation analysis for the convective scheme. The work achieves the weakly divergence-free velocity (strongly divergence-free for divergence-free finite element spaces) and the rigorous stability and error analysis when the iterations converge The three-dimensional numerical tests confirm that this new method can effectively treat high Reynolds numbers with only a few iterations per time, where the convectional projection method and the iterative projection method with the explicit convection would fail.

This work is about a new two-level solver for Helmholtz equations discretized by finite elements. The method is inspired by two-grid methods for finite-difference Helmholtz problems as well as by previous work on two-level domain-decomposition methods. For the coarse-level discretization, a compact-stencil finite-difference method is used that minimizes dispersion errors. The smoother involves a domain-decomposition solver applied to a complex-shifted Helmholtz operator. Local Fourier analysis shows the method is convergent if the number of degrees of freedom per wavelength is larger than some lower bound that depends on the order, e.g., more than 8 for order 4. In numerical tests, with problem sizes up to 80 wavelenghts, convergence was fast, and almost independent of problem size unlike what is observed for conventional methods. Analysis and comparison with dispersion-error data shows that, for good convergence of a two-grid method for Helmholtz problems, it is essential that fine- and coarse-level dispersion relations closely match.

Based on previous work, we extend a primal-dual semi-smooth Newton method for minimizing a general (varvec{L^1})-(varvec{L^2})-(varvec{TV}) functional over the space of functions of bounded variations by adaptivity in a finite element setting. For automatically generating an adaptive grid, we introduce indicators based on a-posteriori error estimates. Further, we discuss data interpolation methods on unstructured grids in the context of image processing and present a pixel-based interpolation method. The efficiency of our derived adaptive finite element scheme is demonstrated on image inpainting and the task of computing the optical flow in image sequences. In particular, for optical flow estimation, we derive an adaptive finite element coarse-to-fine scheme which allows resolving large displacements and speeds up the computing time significantly.