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Abstract: Abstract The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive specific results for the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented, and their applications are demonstrated on the Hénon–Heiles system and a Lotka–Volterra system, and on both the training and integration of a pendulum system learned from data by a neural network. PubDate: 2022-12-01

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Abstract: Abstract We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection ‘flow’ vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in space and time variables, and utilizing variants of existing fixed-point methods as well as a recently developed Newton–Krylov scheme. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters, in both two and three dimensions. A key contribution is the provision of software which allows the discretization and solution of a range of optimization problems constrained by differential equations describing particle dynamics. PubDate: 2022-12-01

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Abstract: Abstract We consider the problem of solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. While some classical approaches are theoretically well founded, they can face difficulties when the matrix of constraints contains dense rows or if an algorithmic transformation used in the solution process results in a modified problem that is much denser than the original one. We propose modifications with an emphasis on requiring that the constraints be satisfied with a small residual. We examine combining the null-space method with our recently developed algorithm for computing a null-space basis matrix for a “wide” matrix. We further show that a direct elimination approach enhanced by careful pivoting can be effective in transforming the problem to an unconstrained sparse-dense least squares problem that can be solved with existing direct or iterative methods. We also present a number of solution variants that employ an augmented system formulation, which can be attractive for solving a sequence of related problems. Numerical experiments on problems coming from practical applications are used throughout to demonstrate the effectiveness of the different approaches. PubDate: 2022-12-01

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Abstract: Abstract In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and diffusion terms in the exponential operator. We prove that the midpoint Lawson schemes preserve quadratic invariants and discuss this property as well for the trapezoidal Lawson scheme. Numerical experiments demonstrate that the integration error for highly oscillatory problems is smaller than that of some standard methods. PubDate: 2022-12-01

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Abstract: Abstract In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker–Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations. PubDate: 2022-12-01

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Abstract: Abstract Very recently, a novel class of parallelizable high-order time discretization schemes has been introduced in Schütz et al. (J Sci Comput 90(54):1–33, 2022). In this current work, we analyze the stability properties of those schemes and introduce a small but effective modification which only necessitates minor modifications of existing implementations. It is shown how this modification leads to A( \(\alpha \) )-stable schemes with \(\alpha \) being close to \(90^{\circ }\) . Numerical examples illustrate an additional favorable influence of this modification on the accuracy of those schemes. PubDate: 2022-12-01

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Abstract: Abstract We identify a relationship between the solutions of a nonsymmetric algebraic \(T\) -Riccati equation ( \(T\) -NARE) and the deflating subspaces of a palindromic matrix pencil, obtained by arranging the coefficients of the \(T\) -NARE. The interplay between \(T\) -NAREs and palindromic pencils allows one to derive both theoretical properties of the solutions of the equation, and new methods for its numerical solution. In particular, we propose methods based on the (palindromic) QZ algorithm and the doubling algorithm, whose effectiveness is demonstrated by several numerical tests. PubDate: 2022-12-01

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Abstract: Abstract We consider a system of non-linear eikonal equations in one space dimension that describes the evolution of interfaces moving with non-signed strongly coupled velocities. We have recently proven the global existence and uniqueness of viscosity solutions for this system, under a BV estimate. In this paper, we propose a semi-explicit scheme that satisfies the same BV estimate proven in the continuous case, at the discrete level, and we show that a certain linear interpolation of the discrete solution to the scheme converges to a viscosity solution of the main system considered. We also provide some numerical simulations in the case of dislocation dynamics. PubDate: 2022-12-01

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Abstract: Abstract In this article, a new class of stochastic exponential Runge-Kutta (SERK) methods is developed for solving stochastic differential equations. The proposed SERK methods can preserve conformal quadratic invariants and conformal symplectic structure automatically under certain coefficient conditions. Stochastic B-series theory is generalized, which allows the study of the mean-square convergence order conditions of the SERK methods. Some low stage stochastic exponential integrators with 1 order mean-square convergence and structure-preserving properties are given. For damped Hamiltonian systems with additive noise terms, a class of stochastic exponential integrators with 1.5 order mean-square convergence and conformal symplectic structure preservation is constructed. Numerical tests show the efficacy of the stochastic exponential integrators. PubDate: 2022-12-01

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Abstract: Abstract General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. In the present work, we consider a random elliptic partial differential equation (PDE) where the discontinuous subordinated Gaussian random fields occur in the diffusion coefficient. Problem specific multilevel Monte Carlo (MLMC) Finite Element methods are constructed to approximate the mean of the solution to the random elliptic PDE. We prove a-priori convergence of a standard MLMC estimator and a modified MLMC—control variate estimator and validate our results in various numerical examples. PubDate: 2022-12-01

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Abstract: Abstract This paper presents a sequence of variable time step deferred correction (DC) methods constructed recursively from the second-order backward differentiation formula (BDF2) applied to the numerical solution of initial value problems for first-order ordinary differential equations (ODE). The sequence of corrections starts with the BDF2 then considered as DC2. We prove that this improvement from a p-order solution (DCp) results in a \(p+1\) -order accurate solution (DC \(p+1\) ). This one-order increment in accuracy holds for the least stringent BDF2 0-stability conditions. If we introduce additional requirements for the ratio of consecutive variable time step sizes, then the order increment is 2, allowing a direct transition from DCp to DC \(p+2\) . These requirements include the constant time step DCp methods. We also prove that all these DCp methods are A-stable. We briefly discuss two other DC variants to illustrate how a proper transition from DCp to DC \(p+1\) is critical to maintaining A-stability at all orders. Numerical experiments based on two manufactured (closed-form) solutions confirmed the accuracy orders of the DCp – for DCp, \(p=2,3,4,5\) – both with constant or alternating time step sizes. We showed that the theoretical conditions required to obtain an increment of orders 1 and 2 are satisfied in practice. Finally, a test case shows that we can estimate the error on the DCp solution with the DC \(p+1\) solution, and a last test case that our new methods maintain their order of accuracy for a stiff system. PubDate: 2022-12-01

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Abstract: Abstract We present a low-rank greedily adapted hp-finite element algorithm for computing an approximation to the solution of the Lyapunov operator equation. We show that there is a hidden regularity in eigenfunctions of the solution of the Lyapunov equation which can be utilized to justify the use of high order finite element spaces. Our numerical experiments indicate that we achieve eight figures of accuracy for computing the trace of the solution of the Lyapunov equation posed in a dumbbell-domain using a finite element space of dimension of only \(10^4\) degrees of freedom. Even more surprising is the observation that hp-refinement has an effect of reducing the rank of the approximation of the solution. PubDate: 2022-12-01

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Abstract: Abstract In this paper, we consider iterative solution of certain large scale block two-by-two linear systems arising from numerical solution process of some PDE-constrained optimal control problems. Based upon skillful rotating technique, a new fast and robust stationary iteration method is constructed from the idea of classical block successive over relaxation (BSOR) iteration. Equipped with a practical problem independent parameter choice strategy, the proposed method can result in a sharp parameter independent convergence rate close to 0.17. Moreover, a robust preconditioner is developed from an equivalent form of the new iteration method, which is suitable for inexact variable right preconditioning within Krylov subspace acceleration. Numerical examples from both distributed steady control problem and unsteady control problem which leads to complex Kronecker structured linear system are tested to show that the new solution methods are competitive to some existing ones. PubDate: 2022-12-01

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Abstract: Abstract A rank-adaptive integrator for the dynamical low-rank approximation of matrix and tensor differential equations is presented. The fixed-rank integrator recently proposed by two of the authors is extended to allow for an adaptive choice of the rank, using subspaces that are generated by the integrator itself. The integrator first updates the evolving bases and then does a Galerkin step in the subspace generated by both the new and old bases, which is followed by rank truncation to a given tolerance. It is shown that the adaptive low-rank integrator retains the exactness, robustness and symmetry-preserving properties of the previously proposed fixed-rank integrator. Beyond that, up to the truncation tolerance, the rank-adaptive integrator preserves the norm when the differential equation does, it preserves the energy for Schrödinger equations and Hamiltonian systems, and it preserves the monotonic decrease of the functional in gradient flows. Numerical experiments illustrate the behaviour of the rank-adaptive integrator. PubDate: 2022-12-01

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Abstract: Abstract In this paper we deal with construction and analysis of a multiwavelet spectral element scheme for a generalized Cauchy type problem with Caputo fractional derivative. Numerical schemes for this type of problems, often suffer from the draw-back of spurious oscillations. A common remedy is to render the problem to an equivalent integral equation. For the generalized Cauchy type problem, a corresponding integral equation is of nonlinear Volterra type. In this paper we investigate wellposedness and convergence of a stabilizing multiwavelet scheme for a, one-dimensional case (in [a, b] or [0, 1]), of this problem. Based on multiwavelets, we construct an approximation procedure for the fractional integral operator that yields a linear system of equations with sparse coefficient matrix. In this setting, choosing an appropriate threshold, the number of non-zero coefficients in the system is substantially reduced. A severe obstacle in the convergence analysis is the lack of continuous derivatives in the vicinity of the inflow/ starting boundary point. We overcome this issue through separating a J (mesh)-dependent, small, neighborhood of a (or origin) from the interval, where we only take \(L_2\) -norm. The estimate in this part relies on Chebyshev polynomials, viz. As reported by Richardson( Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms, Oxford University, UK, 2012) and decreases, almost, exponentially by raising J. At the remaining part of the domain the solution is sufficiently regular to derive the desired optimal error bound. We construct such a modified scheme and analyze its wellposedness, efficiency and accuracy. The robustness of the proposed scheme is confirmed implementing numerical examples. PubDate: 2022-12-01

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Abstract: Abstract We present a new refinement strategy for locally refined B-splines which ensures the local linear independence of the basis functions. The strategy also guarantees the spanning of the full spline space on the underlying locally refined mesh. The resulting mesh has nice grading properties which grant the preservation of shape regularity and local quasi uniformity of the elements in the refining process. PubDate: 2022-12-01

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Abstract: Abstract We present an algorithm for the numerical solution of ordinary differential equations by random enumeration of the Butcher trees used in the implementation of the Runge–Kutta method. Our Monte Carlo scheme allows for the direct numerical evaluation of an ODE solution at any given time within a certain interval, without iteration through multiple time steps. In particular, this approach does not involve a discretization step size, and it does not require the truncation of Taylor series. PubDate: 2022-10-17

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Abstract: Abstract In this paper, we present a new block preconditioner for solving the saddle point linear systems. The proposed method is developed from an augmented reformulation of the saddle point problem into a new linear system with an almost block triangular coefficient matrix. Theoretical results are derived on the eigenvalue distribution of the preconditioned matrix, and an efficient algorithmic implementation is developed and presented. Several numerical examples are reported to support the theoretical findings and to illustrate the favourable convergence properties of the proposed preconditioner, also compared to other popular solvers for saddle point problems. PubDate: 2022-09-14 DOI: 10.1007/s10543-022-00938-8

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Abstract: Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to \(n+k\) with \(0<k\le n\) . Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the \(L^2\) norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as \(n\rightarrow \infty \) if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. PubDate: 2022-09-05 DOI: 10.1007/s10543-022-00935-x