Evolution PDEs Research Articles

Optimal block circulant preconditioners for block Toeplitz systems with application to evolutionary PDEs

In this work, we propose a preconditioned minimal residual (MINRES) method for a class of non-Hermitian block Toeplitz systems. Namely, considering an mn-by-mn non-Hermitian block Toeplitz matrix T(n,m) with m-by-m commuting Hermitian blocks, we first premultiply it by a simple permutation matrix to obtain a Hermitian matrix and then construct a Hermitian positive definite block circulant preconditioner for the modified matrix. Under certain conditions, we show that the eigenvalues of the preconditioned matrix are clustered around ±1 when n is sufficiently large. Due to the Hermitian nature of the modified matrix, MINRES with our proposed preconditioner can achieve theoretically guaranteed superlinear convergence under suitable conditions. In addition, we provide several useful properties of block circulant matrices with commuting Hermitian blocks, including diagonalizability and symmetrization. A generalization of our result to the multilevel block case is also provided. We in particular indicate that our work can be applied to the all-at-once systems arising from solving evolutionary partial differential equations. Numerical examples are given to illustrate the effectiveness of our preconditioning strategy.

Nonlinear Evolutionary PDE-Based Refinement of Optical Flow

The goal of this paper is to propose two nonlinear variational models for obtaining a refined motion estimation from an image sequence. Both the proposed models can be considered as a part of a generalized framework for an accurate estimation of physics-based flow fields such as rotational and fluid flow. The first model is novel in the sense that it is divided into two phases: the first phase obtains a crude estimate of the optical flow and then the second phase refines this estimate using additional constraints. The correctness of this model is proved using an evolutionary PDE approach. The second model achieves the same refinement as the first model, but in a standard manner, using a single functional. A special feature of our models is that they permit us to provide efficient numerical implementations through the first-order primal-dual Chambolle-Pock scheme. Both the models are compared in the context of accurate estimation of angle by performing an anisotropic regularization of the divergence and curl of the flow respectively. We observe that, although both the models obtain the same level of accuracy, the two-phase model is more efficient. In fact, we empirically demonstrate that the single-phase and the two-phase models have convergence rates of order O(1/N2) and O(1/N) respectively.

A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems in unsteady convection-diffusion PDEs

In this paper we propose a model reduction technique to speed up the diagonalization-based parallel-in-time (ParaDIAG) preconditioner, for iteratively solving all-at-once systems from evolutionary PDEs. In particular, we use the reduced basis method to seek a low-dimensional approximation to the sequence of complex-shifted systems arising from Step-(b) of the ParaDIAG preconditioning procedure. Different from the standard reduced order modeling that uses the separation of offline and online stages, we have to build the reduced order model (ROM) online for the considered systems at each iteration. Therefore, several heuristic acceleration techniques are introduced in the greedy basis generation algorithm, that is built upon a residual-based error indicator, to further boost up its computational efficiency. Several numerical experiments are conducted, which illustrate the favorable computational efficiency of our proposed ROM-accelerated ParaDIAG preconditioner, in comparison with the multigrid-based one.

The unified transform for evolution equations on the half‐line with time‐periodic boundary conditions*

AbstractThis paper elaborates on a new approach for solving the generalized Dirichlet‐to‐Neumann map, in the large time limit, for linear evolution PDEs formulated on the half‐line with time‐periodic boundary conditions. First, by employing the unified transform (also known as the Fokas method) it can be shown that the solution becomes time‐periodic for large . Second, it is shown that the coefficients of the Fourier series of the unknown boundary values can be determined explicitly in terms of the coefficients of the Fourier series of the given boundary data in a very simple, algebraic way. This approach is illustrated for second‐order linear evolution equations and also for linear evolution equations containing spatial derivatives of arbitrary order. The simple and explicit determination of the unknown boundary values is based on the “‐equation”, which for the linearized nonlinear Schrödinger equation is the linear limit of the quadratic ‐equation introduced by Lenells and Fokas [Proc. R. Soc. A, 471, 2015]. Regarding the latter equation, it is also shown here that it provides a very simple, algebraic way for rederiving the remarkable results of Boutet de Monvel, Kotlyarov, and Shepelsky [Int. Math. Res. Not. issue 3, 2009] for the particular boundary condition of a single exponential.

Long-time behavior of partially damped systems modeling degenerate plates with piers

We consider a partially damped nonlinear beam-wave system of evolution PDE’s modeling the dynamics of a degenerate plate. The plate can move both vertically and torsionally and, consequently, the solution has two components. We show that the component from the damped beam equation always vanishes asymptotically while the component from the (undamped) wave equation does not. In case of small energies we show that the first component vanishes at exponential rate. Our results highlight that partial damping is not enough to steer the whole solution to rest and that the partially damped system can be less stable than the undamped system. Hence, the model and the behavior of the solution enter in the framework of the so-called indirect damping and destabilization paradox. These phenomena are valorized by the physical interpretation in the final section, leading to possible new explanations of the Tacoma Narrows Bridge collapse. Several natural problems are left open.

The unique continuation property for second order evolution PDEs

The unique continuation property for second order evolution PDEs

Computing Spectral Measures and Spectral Types

Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous and singular continuous parts) often characterise relevant physical properties such as the long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts have focused on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as the linear Schrödinger equation on L^2({mathbb {R}}^d). Computational spectral problems in infinite dimensions have led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. We classify the computation of measures, measure decompositions, types of spectra, functional calculus, and Radon–Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from orthogonal polynomials on the real line and the unit circle (giving, for example, computational realisations of Favard’s theorem and Verblunsky’s theorem, respectively), and are applied to evolution equations on a two-dimensional quasicrystal.

Open topological recursion relations in genus 1 and integrable systems

The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov-Witten invariants in genus 1. We find an explicit formula for any solution analogous to the Dijkgraaf-Witten formula for a descendent Gromov-Witten potential in genus 1. We then prove that at the approximation up to genus 1 the exponent of an open descendent potential satisfies a system of explicitly constructed linear evolutionary PDEs with one spatial variable.

On Incompressible Heat-Conducting Viscoelastic Rate-Type Fluids with Stress-Diffusion and Purely Spherical Elastic Response

We prove the existence of large-data global-in-time weak solutions to an evolutionary PDE system describing flows of incompressible heat-conducting viscoelastic rate-type fluids with stress-diffusi...

ERROR CONTROL IN THE NUMERICAL POSTERIOR DISTRIBUTION IN THE BAYESIAN UQ ANALYSIS OF A SEMILINEAR EVOLUTION PDE

We elaborate on results obtained in \cite{christen2018} for controlling the numerical posterior error for Bayesian UQ problems, now considering forward maps arising from the solution of a semilinear evolution partial differential equation. Results in \cite{christen2018} demand an estimate for the absolute global error (AGE) of the numeric forward map. Our contribution is a numerical method for computing the AGE for semilinear evolution PDEs and shows the potential applicability of \cite{christen2018} in this important wide range family of PDEs. Numerical examples are given to illustrate the efficiency of the proposed method, obtaining numerical posterior distributions for unknown parameters that are nearly identical to the corresponding theoretical posterior, by keeping their Bayes factor close to 1.

Sobolev space of functions valued in a monotone Banach family

We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a nested family of Banach spaces. In this case we adapt the definition of Newtonian spaces. For a monotone family, we show the existence of weak derivative, obtain an isomorphism to the standard Sobolev space, and provide some scalar characteristics.

The geometry of differential constraints for a class of evolution PDEs

The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in order to obtain explicit differential constraints for PDEs belonging to this family. Several examples, with applications in non-linear stochastic filtering theory, stochastic perturbation of soliton equations and non-isospectral integrable systems, are discussed in detail to verify the effectiveness of the method.

Long-time behaviour of solutions to an evolution PDE with nonstandard growth

Abstract In this paper, we prove time estimates for solutions to a general nonhomogeneous parabolic problem whose operator satisfies nonstandard growth conditions. We also study the asymptotic behaviour of solutions to an anisotropic problem.

On the Drinfeld-Sokolov hierarchy of type $$E_6^{(1)}$$ and its topological solution

In this paper we report some explicit evolutionary PDEs of the Drinfeld-Sokolov hierarchy of type $$E_6^{(1)}$$, and show how the unknown functions in these PDEs are related to the tau function. Moreover, for this hierarchy we compute its topological solution of formal series up to a certain degree, whose coefficients of monomials give the Fan-Jarvis-Ruan-Witten invariants for the E6 simple singularity. Based on such results we also derive several explicit evolutionary PDEs and some low-degree terms of the topological solution for the Drinfeld-Sokolov hierarchy of type $$F_4^{(1)}$$.

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In [13], for any (1+1)-dimensional scalar evolution equation E, we defined a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. In [12] we showed that, using these algebras, one obtains necessary conditions for existence of a Bäcklund transformation between two given equations.In this paper we show that, using the algebras F(E), one obtains some necessary conditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order 5. Also, we prove a result announced in [13] on the structure of the algebras F(E) for certain classes of equations of orders 3, 5, 7, which include KdV, mKdV, Kaup–Kupershmidt, Sawada–Kotera type equations. Among the obtained algebras for equations considered in this paper and in [12], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus 1 and 0.

On the spectral stability of soliton-like solutions to a non-local hydrodynamic-type model

A model of nonlinear elastic medium with internal structure is considered. The medium is assumed to contain cavities, microcracks or blotches of substances that differ sharply in physical properties from the base material. To describe the wave processes in such a medium, the averaged values of physical fields are used. This leads to nonlinear evolutionary PDEs, differing from the classical balance equations. The system under consideration possesses a family of invariant soliton-like solutions. These solutions are shown to be spectrally stable under certain restrictions on the parameters.

The gradient flow of the potential energy on the space of arcs

We study the gradient flow of the potential energy on the infinite-dimensional Riemannian manifold of spatial curves parametrized by the arc length, which models overdamped motion of a falling inextensible string. We prove existence of generalized solutions to the corresponding nonlinear evolutionary PDE and their exponential decay to the equilibrium. We also observe that the system admits solutions backwards in time, which leads to non-uniqueness of trajectories.

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn‐Hilliard equation in one‐space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an “energy approach," already proposed for various evolution PDEs, including the Allen‐Cahn and the Cahn‐Hilliard equations. In particular, we shall prove that certain solutions maintain a N‐transition layer structure for a very long time, thus proving their metastable dynamics. More precisely, we will show that, for an exponentially long time, such solutions are very close to piecewise constant functions assuming only the minimal points of the potential, with a finitely number of transition layers, which move with an exponentially small velocity.

A note on parallel preconditioning for all-at-once evolutionary PDEs

McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40 (2018), pp. A1012–A1033] present a method for preconditioning time-dependent PDEs via an approximation by a nearby time-periodic problem, that is, they employ circulant-related matrices as preconditioners for the non-symmetric block Toeplitz matrices which arise from an all-at-once formulation. They suggest that such an approach might be efficiently implemented in parallel. In this short article, we present parallel numerical results for their preconditioner which exhibit strong scaling. We also extend their preconditioner via a Neumann series approach which also allows for efficient parallel execution. Results are shown for both parabolic and hyperbolic PDEs. Our simple implementation (in C++ and MPI) is available at the Git repository PARALAAOMPI.

An automatic stopping criterion for nonlinear anisotropic diffusion

Nonlinear anisotropic diffusion (NAD) filtering is a procedure based on nonlinear evolution PDEs which seeks to improve images qualitatively by removing noise while preserving details and even enhancing edges. However, well-known implementations are sensitive to parameters which are necessarily tuned to sharpen a narrow range of edge slopes; otherwise, edges are either blurred or staircased [Kee2002]. One important parameter is the iterations number, for that reason, in this paper a stopping criterion to halt the diffusion process is proposed. To meet this goal, two stopping criteria were compared. The first is the stopping criterion proposed by Joao et. al. [Joa2016], which is based on the Mean Squared Error (MSE). The second is our proposed method based on the CIRR contrast measure. To this end, a comparative analysis of five diffusion methods is performed. Four of them are nonlinear anisotropic diffusion methods and the fifth is the Perona-Malik method. According to the tests performed, the number of iterations required by the smoothing algorithms using the proposed stopping criterion is lower.