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In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg--Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called $\mu$-mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg--Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge--Kutta integrator, for stringent accuracies.

Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial Differential Equations containing such operators and integrated in time with exponential integrators, it is then of paramount importance to efficiently approximate the actions of $\varphi$-functions of the arising matrices. In this work, we show how to produce directional split approximations of third order with respect to the time step size. They conveniently employ tensor-matrix products (the so-called $\mu$-mode product and related Tucker operator, realized in practice with high performance level 3 BLAS), and allow for the effective usage of exponential Runge--Kutta integrators up to order three. The technique can also be efficiently implemented on modern computer hardware such as Graphic Processing Units. The approach has been successfully tested against state-of-the-art techniques on two well-known physical models that lead to Turing patterns, namely the 2D Schnakenberg and the 3D FitzHugh--Nagumo systems, on different hardware and software architectures.

We propose a second order exponential scheme suitable for two-component coupled systems of stiff evolutionary advection--diffusion--reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small-sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension. Several numerical examples in 2D and 3D with physically relevant (advective) Schnakenberg, FitzHugh--Nagumo, DIB, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques.

In this manuscript, we propose an efficient, practical and easy-to-implement way to approximate actions of $\varphi$-functions for matrices with $d$-dimensional Kronecker sum structure in the context of exponential integrators up to second order. The method is based on a direction splitting of the involved matrix functions, which lets us exploit the highly efficient level 3 BLAS for the actual computation of the required actions in a $\mu$-mode fashion. The approach has been successfully tested on two- and three-dimensional problems with various exponential integrators, resulting in a consistent speedup with respect to a technique designed to compute actions of $\varphi$-functions for Kronecker sums.

In this paper we consider an approach to improve the performance of exponential Runge--Kutta integrators and Lawson schemes} in cases where the solution of a related, but usually much simpler, problem can be computed efficiently. While for implicit methods such an approach is common (e.g. by using preconditioners), for exponential integrators this has proven more challenging. Here we propose to extract a constant coefficient differential operator from the semilinear advection-diffusion-reaction equation for which, in many situations, efficient methods are known to compute the required matrix functions. Both a linear stability analysis and {\color{black} extensive} numerical experiments show that the resulting schemes can be unconditionally stable. In fact, we find that exponential integrators of Runge--Kutta type and Lawson schemes can have better stability properties than similarly constructed implicit-explicit schemes. We also derive two new Lawson type integrators that further improve on these stability properties. The overall effectiveness of the approach is highlighted by a number of performance comparisons on examples in two and three space dimensions.

In this paper we consider mean-field optimal control problems with selective action of the control, where the constraint is a continuity equation involving a non-local term and diffusion. First order optimality conditions are formally derived in a general framework, accounting for boundary conditions. Hence, the optimality system is used to construct a reduced gradient method, where we introduce a novel algorithm for the numerical realization of the forward and the backward equations, based on exponential integrators. We illustrate extensive numerical experiments on different control problems for collective motion in the context of opinion formation and pedestrian dynamics.

We present a method for computing actions of the exponential-like $\varphi$-functions for a Kronecker sum $K$ of $d$ arbitrary matrices $A_\mu$. It is based on the approximation of the integral representation of the $\varphi$-functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call PHIKS, evaluates the required actions by means of $\mu$-mode products involving exponentials of the small sized matrices $A_\mu$, without forming the large sized matrix $K$ itself. PHIKS, which profits from the highly efficient level 3 BLAS, is designed to compute different $\varphi$-functions applied on the same vector or a linear combination of actions of $\varphi$-functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of PHIKS in the exponential integration context. In fact, our newly designed method has been tested on popular exponential Runge--Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of $\varphi$-functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection--diffusion--reaction, Allen--Cahn, and Brusselator equations show the superiority of the proposed $\mu$-mode approach.

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension $d$ by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the $\mu$-mode product (also known as tensor-matrix product or mode- $n$ product) and related operations, such as the Tucker operator. Their MathWorks MATLAB/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach.

Running kinetic simulations using grid-based methods is extremely expensive due to the up to six-dimensional phase space. Recently, it has been shown that dynamical low-rank algorithms can drastically reduce the required computational effort, while still accurately resolving important physical features such as filamentation and Landau damping. In this paper, we propose a new second order projector-splitting dynamical low-rank algorithm for the full six-dimensional Vlasov--Poisson equations. An exponential integrator based Fourier spectral method is employed to obtain a numerical scheme that is CFL condition free but still fully explicit. The resulting method is implemented with the aid of Ensign, a software framework which facilitates the efficient implementation of dynamical low-rank algorithms on modern multi-core CPU as well as GPU based systems. Its usage and features are briefly described in the paper as well. The presented numerical results demonstrate that 6D simulations can be run on a single workstation and highlight the significant speedup that can be obtained using GPUs.

In this paper, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a $d$-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $\mu$-mode products can be used to compute spectral transforms efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving, among the others, three-dimensional linear and nonlinear Schr\"odinger equations, and we show that the $\mu$-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of $10$ and $20$, depending on the problem.