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Gautschi proposed a method for computing incomplete beta functions I x (a , b) using Miller's algorithm with a three-term recurrence relation and showed a computation program in ALGOL. In this paper, first, Miller's algorithm using the recurrence relation satisfied by f k (x) = I x (a + k , b) is described. Next, another solution that is first-order independent of f k (x) of the recurrence relation is given, and its general solution can be expressed as a linear sum of these. Using this general solution, an error analysis for the function I x (a , b) is performed for the first time. The relative error of the function values is then expressed in a new formula to a trend of the error behavior. Also, Miller's algorithm with a normalizing sum is explained and its error analysis is performed. Since Miller's algorithm requires a predefined number of iterations of the recurrence relation, it is necessary to repeat the computation of the recurrence relation with increasing number of iterations until the required accuracy will be met. Therefore, in this paper, we apply Deuflhard's algorithm, which can automatically obtain the function value with the required accuracy. This algorithm requires far fewer iterations than Gautschi's algorithm to obtain the same accuracy. [ABSTRACT FROM AUTHOR]

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is "disjoint" in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake. [ABSTRACT FROM AUTHOR]

Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal boundaries. In this paper, alternative approaches are developed for some interface and internal layer problems based on non-matching grids, in which two mesh sizes are used to obtain comparable high order accuracy near and away from an interface or an internal layer. For one-dimensional, or two-dimensional problems with straight interfaces or boundary layers that are parallel to one of the axes, the discussion is relatively easy. One of the challenges is how to construct a fourth order compact finite difference scheme at border grid points that connect two meshes. The idea is to employ a second order discretization near the interface in the fine mesh and a fourth order discretization away from the interface in the coarse and border grid points. For two-dimensional problems with a curved interface or an internal layer, a level set representation is utilized for which we can build a fine mesh within a tube | φ (x) | ≤ δ h of the interface. A new super-third seven-point discretization that can guarantee the discrete maximum principle has been developed at hanging nodes. The coefficient matrices of the finite difference equations developed in this paper are M-matrices, which leads to the convergence of the finite difference schemes. Non-trivial numerical examples have confirmed the desired accuracy and convergence of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study a variation of the HSS iteration method which was first formulated by Bai et al. in 2003. This variation is similar to that of Kellogg's variation of the Peaceman-Radford method. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate Sobolev orthogonality and families of orthogonal polynomials on the triangle as a generalization of the results in Xu, Y. Constr. Approx. 46, 349-434 (2017). We define inner products or pseudo-inner products that contain derivatives up to third-order in the Sobolev space on a triangle and study associated orthogonal polynomials that are analogs of the Jacobi polynomials J k , n α , β , γ but with α , β , γ being - 1 , - 2 , or - 3 , and two other families derived under simultaneous permutations of x , y , 1 - x - y and α , β , γ . [ABSTRACT FROM AUTHOR]

Least squares models appear frequently in many fields, such as data fitting, signal processing, machine learning, and especially artificial intelligence. Nowadays, the model is a popular and sophisticated way to make predictions about real-world problems. Meanwhile, conjugate gradient methods are traditionally known as efficient tools to solve unconstrained optimization problems, especially in high-dimensional cases. This paper presents a new structured spectral conjugate gradient method based on a modification of the modified structured secant equation of Zhang, Xue, and Zhang. The proposed method uses a novel appropriate spectral parameter. It is proved that the new direction satisfies the sufficient descent condition regardless of the line search. The global convergence of the proposed method is demonstrated under some standard assumptions. Numerical experiments show that our proposed method is efficient and can compete with other existing algorithms in this area. [ABSTRACT FROM AUTHOR]

In this paper, we propose the idea for combining several iteration functions in order to construct new families of iterative methods with both high-order of convergence and high computational efficiency. As an illustration, combining the most famous iteration functions with two or three arbitrary iteration functions, we obtain three new families of high-order iterative methods for the simultaneous computation of simple or multiple polynomial zeros. To demonstrate the influence of the different corrections and to show how one can manage the convergence order and the efficiency when constructing particular methods, we prove a local convergence theorem with a priori and a posteriori error estimates and an estimate of the asymptotic error constant about one of the proposed general families called Schröder-like methods with two corrections. [ABSTRACT FROM AUTHOR]

By introducing two parameters in the splittings of the matrices A and B, this paper presents a parameterized accelerated iteration (PAI) method for solving the matrix equation A X B = C . The convergence property of the PAI method and the choices of the parameters are thoroughly investigated. Additionally, based on some special splittings of the matrices A and B, several variants of the PAI method are established. Furthermore, for some certain cases, the optimal parameters can be determined, and it is demonstrated that the PAI method is more efficient than the gradient-based iteration (GBI) method (Ding et al. Appl. Math. Comput. 197, 41–50 2008). Finally, by comparing it with several existing iteration methods, the effectiveness of the PAI method is verified through four numerical examples. [ABSTRACT FROM AUTHOR]

In 2015, Mao (J. Comput. Appl. Math., 290, 370–384, 2015) proposed the truncated Euler-Maruyama (EM) method for stochastic differential equations (SDEs) under the local Lipschitz condition plus the Khasminskii-type condition. Adapting the truncation idea from Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016), lots of modified truncated EM methods are proposed (see, e.g., Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) and the references therein). These truncated-type EM methods Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016) and Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) construct the numerical solutions by defining an appropriate truncation projection, then applying the truncation projection to the numerical solutions before substituting them into the coefficients in each iteration. In this paper, we develop a new class of explicit schemes for superlinear stochastic differential equations with piecewise continuous arguments (SDEPCAs), which are defined by directly truncating the coefficients. Our method has a more simple structure and is easier to implement. We not only show the explicit schemes converge strongly to SDEPCAs but also demonstrate the convergence rate is optimal 1/2. A numerical example is provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a numerical strategy is introduced for solving a class of linear fractional-order Darboux problem that includes the Caputo derivative. The presented strategy is grounded on the Newton-Cotes quadrature rule and is established by implementing Lagrange interpolation polynomial. A formula is developed to evaluate the two-dimensional fractional integrals by using the multiplicative integration strategy. Then, this formula is employed to derive a scheme for solving a class of fractional-order Darboux problem. The accuracy and efficiency of all the presented schemes are illustrated through numerical experiments. Furthermore, the estimation of the upper bounds of error in the numerical approximation for the proposed multiplicative integration strategy is established. [ABSTRACT FROM AUTHOR]

In this paper, the behavior of derivative-free techniques to approximate solutions of nonlinear Hammerstein-type integral equations in Banach space C ([ α , β ]) as alternatives against the well-known Newton's method is examined. In particular, from the well-known Kurchatov method, we construct a uniparametric family of derivative-free iterative schemes in order to achieve an approximation of a solution of a Hammerstein-type integral equation with a non-differentiable Nemyskii operator. We compare the uniparametric family built and the Kurchatov method, obtaining improvements in the precision and also in the accessibility through studying its dynamic behavior. [ABSTRACT FROM AUTHOR]

In this paper, efficient algorithms will be studied for solving nonlinear saddle point problems from the elliptic optimal control problems. Based on the preconditioned inexact Uzawa (PIU) algorithmic framework, an improved preconditioned inexact Uzawa (I-PIU) algorithmic framework will first be constructed for solving a class of block-structured nonlinear systems, which possess a more general form of the nonlinear saddle point systems. When the I-PIU algorithmic framework is used for solving the block-structured nonlinear systems, a detailed theoretical analysis of the global convergence properties is proposed under the case that the nonlinear operator is single-valued. Besides, a special I-PIU method is established based on a new matrix splitting technique at the first iterative step of the I-PIU algorithmic framework, and then we establish the detailed theoretical results of the convergence. Finally, some numerical experiments are given to demonstrate the efficiency of the I-PIU method. [ABSTRACT FROM AUTHOR]

The tensor equation A x m - 1 = b with the tensor A of order m and dimension n and the vector b , has practical applications in several fields including signal processing, high-dimensional PDEs, high-order statistics, and so on. In this paper, a class of exponential accelerated iterative methods is proposed for solving the tensor equation mentioned above in the sense that the coefficient tensor A is a symmetric and nonsingular or singular M -tensor. The obtained iterative schemes involve the classical Newton's method as a special case. It is shown that the proposed method for nonsingular case is superlinearly convergent, while for singular cases, it is linearly convergent. The performed numerical experiments demonstrate that our methods outperform some existing ones. [ABSTRACT FROM AUTHOR]

In this paper, a two-level Newton iterative method is proposed for the stationary micropolar fluid equations. Firstly, the original equations are solved on a coarse grid based on Newton-type linearization. Then, the simplified linearized equations are solved on a fine grid. The stability and error estimates of the method are given in the theoretical part. The results of the theoretical analysis show that when the coarse mesh size H and fine mesh size h satisfy the relation h = O (H 2) , the two-level Newton iterative method can achieve an optimal convergence rate. Finally, the effectiveness and applicability of the method are verified by some numerical experiments. [ABSTRACT FROM AUTHOR]

Due to their several attractive properties, BDF-type multistep methods are usually the method-of-choice for solving stiff initial value problems (IVPs) of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs). Recently, a class of BDF-type methods based on linear barycentric rational interpolants (LBRIs), referred to as RBDF methods, was introduced for solving ODEs. In the present paper, we are going to introduce a new family of LBRIs-based BDF-type formulas for the numerical solution of ODEs and DAEs with desirable stability properties and smaller error constants than those of the RBDF methods. Numerical experiments of the proposed methods on some well-known IVPs for ODEs and DAEs with index ≤ 3 illustrate the efficiency and capability of the methods in solving such problems. [ABSTRACT FROM AUTHOR]

The classical parareal algorithm for time-periodic problems, solving a periodic-like coarse problem, called the periodic parareal algorithm with periodic coarse problem (PP-PC), usually converges slowly. In this paper, we present a new parallel-in-time algorithm for time-periodic problems based on the classical PP-PC algorithm and the Krylov subspace method. In this new algorithm, a new propagator derived by the Krylov subspace is chosen as the coarse propagator instead of the classical coarse propagator in the PP-PC algorithm. And because of the special characteristic of time-periodic problems, the Krylov subspace enhanced PP-PC algorithm needs to solve a periodic coarse problem on the coarse time grid on each iteration. We provide two different kinds of theoretical bounds under different assumptions for the proposed algorithm. Numerical results illustrate our analysis with two effective theoretical bounds for the heat equation, the wave equation, and the viscous Burgers equation, where we could also find that the new proposed algorithm converges faster than the classical PP-PC algorithm. [ABSTRACT FROM AUTHOR]

The quaternionic Schrödinger equation ∂ ∂ t | f ⟩ = - A | f ⟩ is the crucial part of the study of quaternionic quantum mechanics and plays indispensable roles in related fields. One of the practical and special cases that has received more attention from mathematicians and physicists is that A is a Hermitian quaternion matrix. The problem can be equivalent to a Hermitian quaternion right eigenvalue problem A α = α λ by discretization. This paper, by means of a complex representation method, studies the Hermitian quaternion Schrödinger equation problem, and proposes a novel algebraic method (complex structure-preserving method) for right eigenvalue problems of Hermitian quaternion matrices. Moreover, the complex structure-preserving method is superior and formally simple compared to previous methods, and numerical experiments also demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, we construct a reduced order model (ROM) to solve the advection–diffusion-reaction (ADR) equation with variable coefficients. In order to get the desired fidelity solution, we introduce the interior penalty discontinuous Galerkin (IPDG) and implicit–explicit Runge–Kutta (IMEXRK) schemes to construct the full-order model (FOM). The IPDG scheme can achieve higher-order accuracy in spatial discretization. We split the model into the advection-reaction and diffusion terms so as to ensure the stability of the full-discrete scheme. The IMEXRK scheme discrete the advection-reaction term explicitly and the diffusion term implicitly to obtain a higher-order scheme in time discretization. Then, the POD method and Galerkin projection are introduced to construct the ROM. The resulting ROM can maintain the higher-order accuracy of the original IPDG method and greatly reduce the computation cost. Some numerical examples are used to demonstrate the accuracy and effectiveness of the ROM. [ABSTRACT FROM AUTHOR]

This paper focuses on algorithms for the truncated Takagi factorization of complex symmetric matrices. The problem is formulated as a Riemannian optimization problem on a complex Stiefel manifold and then is converted into a real Riemannian optimization problem on the intersection of the real Stiefel manifold and the quasi-symplectic set. The steepest descent, the Riemannian nonmonotone conjugate gradient, Newton, and hybrid methods are used for solving the problem and they are compared in their performance for the optimization task. Numerical experiments are provided to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a two-metric variable scaled forward-backward algorithm is proposed to solve the nonconvex noncontinuous composite optimization problems with ℓ 0 regularization. We first explore using a class of locally Lipschitz scale folded concave functions to approach the ℓ 0 . Then, a convex half-absolute method is proposed to precisely approximate these nonconvex nonsmooth functions. A double iterative algorithm is considered to solve the convex-relaxed composite optimization problems, where the inner iterative subproblem is solved using a well-designed two-metric variable scaled forward-backward algorithm. Specifically, while the variable metric in the forward step is derived by quasi-Newton methods, in the backward implicit step, however, the scaled matrix is chosen naturally derived by the inherent nonconvex properties of the established folded concave regularization function. The convergence of the proposed algorithm is analyzed, and the effectiveness of the proposed approach is validated by extensive numerical experiments, including sparse signal recovery, breast cancer data classification, and image restoration. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time stability-preserving properties of the two-step backward differentiation formula (BDF2) fully discrete scheme for the fractional complex Ginzburg-Landau equation. More precisely, we consider the BDF2 time discretization together with a general spatial spectral discretization and show that the numerical scheme can unconditionally preserve the long-time stability in the L 2 , H 1 , H 1 + α and H 2 norms with the aid of the discrete uniform Gronwall lemma. As a special case, we obtain the long-time stability-preserving properties of the fully discrete BDF2 spectral method for the standard complex Ginzburg-Landau equation for the first time. Numerical examples are presented to support our theoretical results. [ABSTRACT FROM AUTHOR]

In the present paper, the regularity and finite difference methods for Caputo-Hadamard fractional differential equations with initial value singularity are taken into consideration. To overcome the weak singularity and enhance convergence precision, a fitted scheme on nonuniform meshes is applied to such problems. Firstly, based on L log , 2 - 1 σ approximation, the temporal convergence accuracy of fitted scheme for the sub-diffusion equations is O (N - min { 2 r α , 2 }) , where N denotes the number of time steps, α is the fractional order and r is the mesh grading parameter. It is indicated that the performance of the fitted scheme is better than that of the standard L log , 2 - 1 σ scheme on (i) exponent meshes (i.e., r = 1 ) and (ii) graded meshes with the optimal choice of the mesh grading. Secondly, a second-order fitted scheme on exponent meshes for the diffusion-wave equations is obtained. Furthermore, for the sake of improving the computational efficiency and demonstrating the effectiveness of the decomposition of the solution, the fast algorithm and further decomposition of the solution for the sub-diffusion equations are investigated. Ultimately, some examples are presented to verify the availability of our theoretical results. [ABSTRACT FROM AUTHOR]

Nonsymmetric algebraic Riccati equations are a class of nonlinear matrix equations with wide applications. In this paper, numerical solution of nonsymmetric algebraic Riccati equations is studied. A generalized ALI iteration method is proposed to solve the equation, which is an extension of the ALI iteration method and the MALI iteration method. Theoretical analysis and numerical experiments show that the proposed method is feasible and is more effective than the ALI iteration method and the MALI iteration method. [ABSTRACT FROM AUTHOR]

In this paper, we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient than other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices. [ABSTRACT FROM AUTHOR]

In this paper, for solving a broad class of large-scale nonconvex and nonsmooth optimization problems, we propose a stochastic two-step inertial Bregman proximal alternating linearized minimization (STiBPALM) algorithm with variance-reduced stochastic gradient estimators. And we show that SAGA and SARAH are variance-reduced gradient estimators. Under expectation conditions with the Kurdyka–Łojasiewicz property and some suitable conditions on the parameters, we obtain that the sequence generated by the proposed algorithm converges to a critical point. And the general convergence rate is also provided. Numerical experiments on sparse nonnegative matrix factorization and blind image-deblurring are presented to demonstrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In the present paper, we study the approximation of solutions (if any) of a system of equations, where the only condition required to the involved functions is to be continuous. To be more precise, by using the so called α -dense curves, if the given system has some solution in a densifiable set we propose a method, which only use evaluations of single variable functions, to approximate at least one solution of the system in such set. The feasibility and reliability of the proposed method is illustrated by several examples, and its drawbacks are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we develop a branch and bound algorithm for globally solving linear multiplicative programs (LMP). The problem LMP is firstly converted to an equivalent problem (EP2), and then a novel linear relaxation technique is constructed for (EP2) to get the lower bound of the optimum of LMP. Subsequently, the presented region pruning technique aims to eliminate as many areas as possible where the optimal solution does not exist. The convergence analysis of the algorithm is provided, and the number of worst-case iterations required is estimated to obtain an ε -optimal solution. Finally, the numerical experiments show the effectiveness of our presented algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a new conjugate gradient method on Riemannian manifolds and establishes its global convergence under the standard Wolfe line search. The proposed algorithm is a generalization of a Wei-Yao-Liu-type Hestenes-Stiefel method from Euclidean space to the Riemannian setting. We prove that the new algorithm is well-defined, generates a descent direction at each iteration, and globally converges when the step lengths satisfy the standard Wolfe conditions. Numerical experiments on the matrix completion problem demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies a stabilized virtual element method for the unsteady Navier-Stokes problems on polygonal meshes. Using "equal-order" virtual elements in space and the Crank-Nicolson scheme in time, we give a fully discrete formula. By introducing the local-projection type stabilizations, the method can not only circumvent the discrete inf-sup condition but also control spurious oscillations caused by high Reynolds numbers. Stability and error estimates for velocity and pressure are analyzed. Particularly, error estimates are derived in which the constants are independent of the negative powers of the viscosity. Several numerical experiments are simulated to support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents three tensor randomized extended Kaczmarz methods for solving a tensor inconsistent linear system of equations under the t-product between tensors. The randomized extended Kaczmarz, the randomized extended block Kaczmarz, and the randomized extended greedy block Kaczmarz methods in tensor form are proposed to solve inconsistent tensor linear system of equations, respectively. The convergence of each method is proved. Several numerical examples are given to show the efficiency and effectiveness of our methods. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the global approximation of solutions of stochastic differential equations (SDEs) driven by countably dimensional Wiener process. Under certain regularity conditions imposed on the coefficients, we show lower bounds for exact asymptotic error behaviour. For that reason, we analyse separately two classes of admissible algorithms: based on equidistant, and possibly not equidistant meshes. Our results indicate that in both cases, decrease of any method error requires significant increase of the cost term, which is illustrated by the product of cost and error diverging to infinity. This is, however, not visible in the finite-dimensional case. In addition, we propose an implementable, path-independent Euler algorithm with adaptive step-size control, which is asymptotically optimal among algorithms using specified truncation levels of the underlying Wiener process. Our theoretical findings are supported by numerical simulation in Python language. [ABSTRACT FROM AUTHOR]

Solving large-scale systems of nonlinear equations (SoNE) is a central task in mathematics that traverses different areas of applications. There are several derivative-free methods for finding SoNE solutions. However, most of the methods contributed to find SoNE solutions involve a monotone cost function. Methods dealing with pseudomonotone cost function remain rare. In this paper, we introduce two classes of derivative-free spectral three-term methods to solve large-scale continuous pseudomonotone SoNE. We combine the projection method of Solodov and Svaiter with the structure of the recently developed spectral three-term conjugate gradient method for unconstrained optimization by Amini and Faramarzi. We prove that the proposed methods possess sufficient descent property, trust region property, and global convergence without relying on Lipschitz continuity. Numerical experiments show that the proposed methods are efficient and competitive with existing methods. Finally, the proposed methods have been successfully applied to recover a sparse signal from incomplete and contaminated sampling measurements, yielding promising results. [ABSTRACT FROM AUTHOR]

In this paper, we consider solutions for discrete systems arising from multi-term time-fractional diffusion equations. Using discrete sine transform techniques, we find that all-at-once systems of such equations have a structure similar to that of diagonal-plus-Toeplitz matrices. We establish a generalized circulant approximate inverse preconditioner for the all-at-once systems. Through a detailed analysis of the preconditioned matrices, we show that the spectrum of the obtained preconditioned matrices is clustered around one. We give some numerical examples to demonstrate the effectiveness of the proposed preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for efficiently obtaining an approximate solution for constrained nonlinear monotone operator equations. The search direction of the proposed method closely aligns with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) direction, known for its low storage requirement. Notably, the search direction is shown to be sufficiently descent and bounded without using the line search condition. Furthermore, under some standard assumptions, the proposed method converges globally. As an application, the proposed method is applied to solve image restoration problems. The efficiency and robustness of the method in comparison to other methods are tested by numerical experiments using some test problems. [ABSTRACT FROM AUTHOR]

In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs). A unifying theme of this paper is our focus on software packages for these problem classes that compute error-controlled, continuous numerical solutions. A continuous numerical solution can be accessed by the user at any point in the domain. We focus on error-control software; this means that the software adapts the computation until it obtains a continuous approximate solution with a corresponding error estimate that satisfies the user tolerance. The second section of the paper will provide an overview of recent work on the development of COLNEWSC, an updated version of the widely used collocation BVODE solver, COLNEW, that returns an error-controlled continuous approximate solution based on the use of a superconvergent interpolant to the underlying collocation solution. The third section of the paper gives a brief review of recent work on the development of a new 1DPDE solver, BACOLIKR, that provides time- and space-dependent event detection for an error-controlled continuous numerical solution. In the fourth section of the paper, we briefly review the state of the art in IVODE software for the computation of error-controlled continuous numerical solutions. [ABSTRACT FROM AUTHOR]

The numerical solution of initial-value problems (IVP) for ordinary differential equations (ODE) is at this time a mature subject, with many high-quality codes freely available. Second-order linear equations without singularities are an especially simple class of problems to solve, even more so if only a single scalar equation such as the Mathieu equation y ′ ′ + (a - 2 q cos 2 x) y = 0 is being considered. Nonetheless, the topic is not yet exhausted, and this paper considers the case of writing an efficient arbitrary-precision code for the solution of such equations. For this purpose, an implicit Hermite–Obreshkov method attains nearly spectral accuracy at a cost only polynomial in the number of bits of accuracy requested. This is interesting for the Mathieu equation in particular because the solutions can be highly oscillatory of variable frequency and be highly ill-conditioned. This paper reports on the details of the prototype Maple implementation of the method and summarizes the approximation theoretic results justifying the choice of a balanced Hermite–Obreshkov method including its backward stability and decent Lebesgue constants. This method may be of especial interest for the solution of so-called D-finite equations, for which Taylor series coefficients up to degree m are available at cost only O(m), instead of the more usual O (m 2) . This paper celebrates the happy occasion of the 90th birthday of John C. Butcher. [ABSTRACT FROM AUTHOR]

Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices are connected by both inter-layer and intra-layer edges. In this paper, we investigate multiplex networks, in which vertices in different layers are identified with each other, and the only inter-layer edges are those that connect a vertex with its copy in other layers. Let the third-order adjacency tensor A ∈ R N × N × L and the parameter γ ≥ 0 , which is associated with the ease of communication between layers, represent a multiplex network with N vertices and L layers. To measure the ease of communication in a multiplex network, we focus on the average inverse geodesic length, which we refer to as the multiplex global efficiency e A (γ) by means of the multiplex path length matrix P ∈ R N × N . This paper generalizes the approach proposed in [15] for single-layer networks. We describe an algorithm based on min-plus matrix multiplication to construct P, as well as variants P K that only take into account multiplex paths made up of at most K intra-layer edges. These matrices are applied to detect redundant edges and to determine non-decreasing lower bounds e A K (γ) for e A (γ) , for K = 1 , 2 , ⋯ , N - 2 . Finally, the sensitivity of e A K (γ) to changes of the entries of the adjacency tensor A is investigated to determine edges that should be strengthened to enhance the multiplex global efficiency the most. [ABSTRACT FROM AUTHOR]

In this paper we analyze the stability of the problem of performing a rational QZ step with a shift that is an eigenvalue of a given regular pencil H - λ K in unreduced Hessenberg–Hessenberg form. In exact arithmetic, the backward rational QZ step moves the eigenvalue to the top of the pencil, while the rest of the pencil is maintained in Hessenberg–Hessenberg form, which then yields a deflation of the given shift. But in finite-precision the rational QZ step gets "blurred" and precludes the deflation of the given shift at the top of the pencil. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the rational QZ step can be constructed using this eigenvector, so that the exact deflation is also obtained in finite-precision. [ABSTRACT FROM AUTHOR]

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long-time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lions et al. in 2001, for multiscale Hamiltonian systems. The first method involves constructing a correction operator to improve a given inaccurate coarse solver through solving a Procrustes problem using data collected online along parareal trajectories. The second method involves constructing an efficient, high-fidelity solver by a neural network trained with offline generated data. For the second method, we address the issues of effective data generation and proper loss function design based on the Hamiltonian function. We show proof-of-concept by applying the proposed methods to a Fermi-Pasta-Ulam (FPU) problem. The numerical results demonstrate that the Procrustes parareal method is able to produce solutions that are more stable in energy compared to the standard parareal. The neural network solver can achieve comparable or better runtime performance compared to numerical solvers of similar accuracy. When combined with the standard parareal algorithm, the improved neural network solutions are slightly more stable in energy than the improved numerical coarse solutions. [ABSTRACT FROM AUTHOR]

This article presents the discovery of an explicit 16-stage Runge–Kutta method that numerically satisfies the Runge–Kutta order conditions (in Butcher form) to 10th order, conjecturally improving the best-known number of stages in an explicit 10th-order Runge–Kutta method from 17 to 16. Unlike the vast majority of published Runge–Kutta methods, the method presented in this paper was not constructed via symbolic analysis or the use of simplifying assumptions, but instead by directly applying numerical optimization and root-finding algorithms to the order conditions. A naïve implementation of these algorithms would be made computationally infeasible by the considerable size and complex structure of the Butcher equations. However, we present a collection of software optimizations that greatly accelerate the evaluation of the Butcher equations and their derivatives while mitigating the effects of destructive cancellation, pushing these techniques within the realm of feasibility. While we do not have a formal proof of order, we present the results of numerical experiments to demonstrate that our method satisfies the Butcher equations to an accuracy of over 3000 decimal digits. [ABSTRACT FROM AUTHOR]

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new method based on a library of chemical reactions for constructing a system of ordinary differential equations from stochastic simulations arising from an agent-based model. The advantage of this approach is that this library respects any coupling between systems components, whereas the SINDy algorithm (introduced by Brunton et al.) treats the individual components as decoupled from one another. Another advantage of our approach is that we can use a non-negative least squares algorithm to find the non-negative rate constants in a very robust, stable and simple manner. We illustrate our ideas on an agent-based model of tumour growth on a 2D lattice. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new framework for deriving partitioned implicit-exponential integrators for stiff systems of ordinary differential equations and construct several time integrators of this type. The new approach is suited for solving systems of equations where the forcing term is comprised of several additive nonlinear terms. We analyze the stability, convergence, and efficiency of the new integrators and compare their performance with existing schemes for such systems using several numerical examples. We also propose a novel approach to visualizing the linear stability of the partitioned schemes, which provides a more intuitive way to understand and compare the stability properties of various schemes. Our new integrators are A-stable, second-order methods that require only one call to the linear system solver and one exponential-like matrix function evaluation per time step. [ABSTRACT FROM AUTHOR]

Considering validated Runge-Kutta schemes requires the computation of local truncation error. Computation of such error is expensive in term of complexity, thus few techniques have been proposed to compute or at least estimate it. For example, Runge-Kutta pairs provide an error estimation by computing the difference of two schemes of different orders while sharing the same stages. With the goal of proposing a validated scheme, an approximation is not sufficient, but could be helpful after some adjustments. In this paper, we compute a new pair and show its performance on a non-trivial problem. Two new techniques for the local truncation error computation are then presented: a symbolic based approach dedicated to a specific problem and a preliminary study of pair's approximation. [ABSTRACT FROM AUTHOR]

In this paper, we develop a convex splitting algorithm for the time-dependent Ginzburg-Landau equation, which can preserve both the energy stability and maximum bound principle. The basic idea of the convex splitting method is to decompose the energy functional into the convex part and the concave part. The term corresponding to the convex part of the equation is implicitly treated, and the concave part is explicitly processed. The backward Euler time discretizing method is chosen for the time-dependent Ginzburg-Landau equation. The theoretical analysis proves that the convex splitting method can preserve the maximum bound principle and energy stability. The numerical results show that the numerical algorithm is stable. [ABSTRACT FROM AUTHOR]

Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections, and mappings. Projective geometry identifies a line with a single point, like the perspective on the horizon line and, due to this fact, it requires a restructuring of the real mathematical and numerical analysis. In particular, the problem of interpolating data must be refocused. In this paper, we define a linear structure along with a metric on a projective space, and prove that the space thus constructed is complete. Then, we consider an iterated function system giving rise to a fractal interpolation function of a set of data. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop and analyze an adaptive collocation method for Fredholm integral equations of the second kind, even if the equation exhibits localized rapid variations, steep gradients, or steep front. The strategy of the adaptive procedure is to transform the given equation into an equivalent one with a sufficiently smooth behavior in order to ensure the convergence of the Legendre spectral collocation method without dividing the domain of the integral equation, as usual, into the sub intervals. Existence and uniqueness of solutions are discussed. Convergence rates and error estimates are given for the numerical scheme in both L ∞ -norm and L 2 -norm. Finally, several numerical examples are provided to show that the proposed method is preferable to its classical predecessor and some other existing approaches. [ABSTRACT FROM AUTHOR]

This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. For the spatial discretization of the state and co-state variables, the lowest-order Raviart-Thomas spaces are utilized, although for the control variable, variational discretization technique is used. The backward-Euler implicit method is applied for temporal discretization. To provide a posteriori error estimates for the state and control variables in the L ∞ (L 2) -norm, an elliptic reconstruction approach paired with an energy strategy is utilized. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of singularly perturbed semilinear parabolic convection-diffusion initial-boundary-value problems exhibiting a boundary layer. This type of model problem often appears in modeling various physical phenomena, particularly, in mathematical biology; and thus, it requires effective numerical techniques for analyzing them computationally. For this purpose, we approximate the considered nonlinear problem by developing two efficient fitted mesh methods followed by the extrapolation technique. The first one is the fully implicit fitted mesh method which utilizes the implicit-Euler method for the temporal discretization; and the other one is the implicit-explicit (IMEX) fitted mesh method which utilizes the IMEX-Euler method for the temporal discretization. The spatial discretization for both the numerical methods is based on a new hybrid finite difference scheme. To accomplish this, the spatial domain is discretized by an appropriate layer-adapted mesh and the time domain by an equidistant mesh. At first, we analyze stability and study the asymptotic behavior of the analytical solution of the governing nonlinear problem. Then, we perform stability analysis and establish the parameter-uniform convergence of both the newly proposed methods in the discrete supremum norm. Thereafter, we analyze the Richardson extrapolation technique solely for the time variable to improve the order of convergence in the temporal direction. Hereby, we provide a comparative error analysis to achieve parameter-robust higher-order numerical approximations (concerning both space and time) for the considered nonlinear problem utilizing two new algorithms on a nonuniform grid. The theoretical outcomes are finally supported by the extensive numerical experiments, which also include comparison of the proposed numerical methods along with the fully-implicit upwind method in terms of the order of accuracy and the computational cost. [ABSTRACT FROM AUTHOR]

In this paper, a reliable Jacobi Galerkin method is developed and analyzed to solve a particular class of cordial Volterra integral equations. Existence and uniqueness theorems approve that these kinds of equations possess smooth solutions versus smooth data, so representing their Galerkin solutions based on suitable polynomial basis produces spectrally accurate approximations. Moreover, in order to control condition number growth, we designed a reliable approach that calculates the approximate solutions recursively without solving any high-conditioning systems. Indeed, this approach deals with solving equations in a long integration domain with even high oscillatory solutions. The convergence analysis of the presented approach is established, and the familiar spectral accuracy is justified in L ∞ -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method. Finally, we provide an application of this method to approximate solution of a two-dimensional case. [ABSTRACT FROM AUTHOR]