We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in generic Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general the theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems -- including thin film epitaxy with slope selection and the square phase field crystal model -- are carried out to verify the efficiency of the scheme.
Gradient-type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD-id) is one of such methods. The convergence behavior of the PSD-id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non-asymptotic estimates indicate a superlinear convergence of the PSD-id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis…
Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N-GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, while the second employs a predefined small step. A simple global convergence proof is provided for the N-GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioni…
A method of estimating all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in T. Gergelits, K.A. Mardal, B.F. Nielsen, Z. Strakoš: Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator, SIAM Journal on Numerical Analysis 57(3) (2019), 1369-1394. Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elastici…
In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time-space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded $L1$ formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the $M$-matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use t…
In this work, we discuss the task of finding a direction of optimal descent for problems in Shape Optimisation and its relation to the dual problem in Optimal Transport. This link was first observed in a previous work which sought minimisers of a shape derivative over the space of Lipschitz functions which may be closely related to the $\infty$-Laplacian. We provide some results of shape optimisation using this novel Lipschitz approach, highlighting the difference between the Lipschitz and $W^{1,\infty}$ semi-norms. After this, we provide an overview of the necessary results from Optimal trans…
Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number $μ$
We present a simple discretization scheme for the hypersingular integral representation of the fractional Laplace operator and solver for the corresponding fractional Laplacian problem. Through singularity subtraction, we obtain a regularized integrand that is amenable to the trapezoidal rule with equispaced nodes, assuming a high degree of regularity in the underlying function (i.e., $u\in C^6(R^d)$). The resulting quadrature scheme gives a discrete operator on a regular grid that is translation-invariant and thus can be applied quickly with the fast Fourier transform. For discretizations of …
Convergence theory for preconditioned eigenvalue solvers in a nutshell
University Colorado Denver（科罗拉多大学丹佛分校）；Mitsubishi Electric Research Laboratories（三菱电机研究实验室）；Universität Rostock（罗斯托克大学）；STFC Rutherford Appleton Laboratory（STFC鲁瑟福阿普尔顿实验室）
We show that the mass matrix derived from finite elements can be effectively used as a preconditioner for iteratively solving the linear system arising from finite-difference discretization of the Poisson equation, using the conjugate gradient method. We derive analytically the condition number of the preconditioned operator. Theoretical analysis shows that the ratio of the condition number of the Laplacian to the preconditioned operator is $8/3$ in one dimension, $9/2$ in two dimensions, and $2^9/3^4 \approx 6.3$ in three dimensions. From this it follows that the expected iteration count for …
Diagonal and normal with Toeplitz-block splitting iteration method for space fractional coupled nonlinear Schrödinger equations with repulsive nonlinearities
Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator
Institute of Computer Sciences, Czech Academy of Science（捷克科学院计算机科学研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）；Department of Mathematics, University of Oslo（奥斯陆大学数学系）；Department of Numerical Analysis and Scientific Computing, Simula Research Laboratory（Simula研究实验室数值分析与科学计算部）；Faculty of Science and Technology, Norwegian University of Life Sciences（挪威生命科技大学理学院）
Due to its significance in terms of wave phenomena a considerable effort has been put into the design of preconditioners for the Helmholtz equation. One option to derive a preconditioner is to apply a multigrid method on a shifted operator. In such an approach, the wavenumber is shifted by some imaginary value. This step is motivated by the observation that the shifted problem can be more efficiently handled by iterative solvers when compared to the standard Helmholtz equation. However, up to now, it is not obvious what the best strategy for the choice of the shift parameter is. It is well kno…
An Energy Stable Finite-Difference Scheme for Functionalized Cahn-Hilliard Equation and its Convergence Analysis
The University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）；The University of California, Irvine（加州大学尔湾分校）；The University of Massachusetts, North Dartmouth（麻省大学北达特茅斯分校）
In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. To overcome the difficulties associated with this highly nonlinear operator, we design numerical algorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral app…
Continuous meets discrete: geometry dependence of the performance of iterative solutions for nonsymmetric PDEs in terms of Laplace eigenvalues
Chalmers University and the University of Gothenburg（查尔姆斯理工大学和哥德堡大学）；ELTE Eötvös Loránd University（罗兰·厄特沃什大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
We introduce tensor numerical techniques for solving optimal control problems constrained by elliptic operators in $\mathbb{R}^d$, $d=2,3$, with variable coefficients, which can be represented in a low rank separable form. We construct a preconditioned iterative method with an adaptive rank truncation for solving the equation for the control function, governed by a sum of the elliptic operator and its inverse $M=A + A^{-1}$, both discretized over large $n^{\otimes d}$, $d=2,3$, spatial grids. Two basic solution schemes are proposed and analyzed. In the first approach, one solves iteratively th…
