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In this paper, we consider an iterative scheme for solving nonlinear ill-posed operator equations of monotone types under minimal and weaker assumptions. The convergence analysis of the scheme is carried out with both a priori and a posteriori parameter choice strategies and the error estimate is derived accordingly. Finally, we supply numerical examples to illustrate the proposed scheme by considering the model of nonlinear integral equations in Wiener-type filtering theory, and the efficiency of the proposed scheme is compared with other schemes. [ABSTRACT FROM AUTHOR]

Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting points and complex roots, it is possible to examine the basins of attraction in the complex plane in order to compare the performance of various iterative techniques. In this paper, a one-parameter family of third-order root-finding methods is studied by varying its parameter A within −2.0 and 2.4 and applying it to a polynomial equation of high degree (degree 25). This family includes the Euler–Chebyshev's (A = 0), Halley's (A = 1) and BSC (A = 2) techniques. According to the results, the one-parameter family provides the best performance for values near A = 1, which equals to the Halley's method. [ABSTRACT FROM AUTHOR]

The present paper studies a new class of problems of optimal control theory with special differential inclusions described by higher-order linear differential operators (HLDOs). There arises a rather complicated problem with simultaneous determination of the HLDOs and a Mayer functional depending of high-order derivatives of searched functions. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and "transversality" conditions at the endpoints t = − 1, 0 and t = 1 are derived. One of the key features in the proof of sufficient conditions is the notion of locally adjoint mappings. Then, we demonstrate how these conditions can be transformed into Pontryagin's maximum principle in some particular cases. [ABSTRACT FROM AUTHOR]

A usual way to characterize the quality of different a posteriori parameter choices is to prove their order-optimality on the different sets of solutions. In paper by Raus and Hämarik (J Inverse Ill-Posed Probl 15(4):419-439, ) we introduced the property of the quasi-optimality to characterize the quality of the rule of the a posteriori choice of the regularization parameter for concrete problem Au = f in case of exact operator and discussed the quasi-optimality of different well-known rules for the a posteriori parameter choice as the discrepancy principle, the modification of the discrepancy principle, balancing principle and monotone error rule. In this paper we generalize the concept of the quasi-optimality for the case of a noisy operator and concretize results for the mentioned parameter choice rules. [ABSTRACT FROM AUTHOR]

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect gaps in essential spectra and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these long-standing problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the solvability complexity index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale's comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications. [ABSTRACT FROM AUTHOR]

The paper investigates the Galerkin method for an initial boundary value problem for heat convection equations. New error estimates for the approximate solutions and their derivatives in strong norm are obtained. [ABSTRACT FROM AUTHOR]

In this paper we discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and geometric proof based on the preconditioned inverse iteration (PINVIT) for matrices (Knyazev and Neymeyr, SIAM J Matrix Anal 31:621-628, ) is extended to the case of operators. We show that convergence is retained up to any tolerance if one only uses approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). We then analyze the Besov regularity of the eigenfunctions of the Poisson eigenvalue problem on a polygonal domain, showing the advantage of an adaptive solver to uniform refinement when using a stable wavelet base. A numerical example for PPINVIT, applied to the model problem on the L-shaped domain, is shown to reproduce the predicted behaviour. [ABSTRACT FROM AUTHOR]

In the present paper we develop a representation of a norm frequently used in the analysis of multilevel methods. This allows us to examine the convergence of additive Schwarz schemes also in the case of non-nested subspaces. We demonstrate the usefulness of the given norm representation by studying in detail the stability of sparse grid splittings due to Griebel and Oswald, which turns out to be a special case of our unified theory. Further applications concerning approximation spaces and non-nested finite element spaces are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider discontinuous Galerkin (DG) finite element together with finite difference (FD) scheme for solving Merton’s jump–diffusion model, which is given by a partial integro-differential equations (PIDEs). Spatial differential operators are discretized using FD on a uniform grid, and time stepping is performed using the DG finite element method. The treatment of the integral term associated with jumps in models is more challenging. The discretization of this integral term will lead to full matrices for the non-locality of the integral operator. To fast solve this model, multigrid method is used for solving such linear algebraical system. Numerical comparison of multigrid method and GMRES method shows that multigrid method is superior to and more effective than GMRES method in solving the dense algebraic systems resulting from the FD approximations of the PIDEs. [ABSTRACT FROM AUTHOR]

If a spectral numerical method for solving ordinary or partial differential equations is written as a biinfinite linear system b = Z a with a map Z : ℓ 2 → ℓ 2 that has a continuous inverse, this paper shows that one can discretize the biinfinite system in such a way that the resulting finite linear system b ̃ = Z ̃ a ̃ is uniquely solvable and is unconditionally stable, i.e. the stability can be made to depend on Z only, not on the discretization. Convergence rates of finite approximations b ̃ of b then carry over to convergence rates of finite approximations a ̃ of a . Spectral convergence is a special case. Some examples are added for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we present a priori error analysis of a hybridizable discontinuous Galerkin (HDG) method for a distributed optimal control problem governed by diffusion equations. The error estimates are established based on the projection-based approach recently used to analyze these methods for the diffusion equation. We proved that for approximations of degree k on conforming meshes, the orders of convergence of the approximation to fluxes and scalar variables are k + 1 when the local stabilization parameter is suitably chosen. [ABSTRACT FROM AUTHOR]

Over any field F every square matrix A can be factored into the product of two symmetric matrices as A = S 1 ⋅ S 2 with S i = S i T ∈ F n , n and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910. Frobenius' symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such that SA is symmetric, was inspired by an iterative linear systems algorithm of Huang and Nong (2010) in 2013 [29,30] . The resulting iterative algorithm has solved this computational problem over R and C , but at high computational cost. This paper develops and tests another linear equations solver, as well as eigen- and principal vector or Schur Normal Form based algorithms for solving the matrix symmetrizer problem numerically. Four new eigendata based algorithms use, respectively, SVD based principal vector chain constructions, Gram–Schmidt orthogonalization techniques, the Arnoldi method, or the Schur Normal Form of A in their formulations. They are helped by Datta's 1973 method that symmetrizes unreduced Hessenberg matrices directly. The eigendata based methods work well and quickly for generic matrices A and create well conditioned matrix symmetrizers through eigenvector dyad accumulation. But all of the eigen based methods have differing deficiencies with matrices A that have ill-conditioned or complicated eigen structures with nontrivial Jordan normal forms. Our symmetrizer studies for matrices with ill-conditioned eigensystems lead to two open problems of matrix optimization. [ABSTRACT FROM AUTHOR]

In this paper we introduce a modified spectral method for solving the linear operator equation L u = f , L : D ( L ) ⊆ H 1 → H 2 , where H 1 and H 2 are normed vector spaces with norms ‖ . ‖ 1 and ‖ . ‖ , respectively and D ( L ) is the domain of L . Also for each h ∈ H 2 , ‖ h ‖ 2 = ( h , h ) where ( . , . ) is an inner product on H 2 . In this method we make a new set { ψ n } n = 0 ∞ for H 1 using L and two sets in H 1 and H 2 . Then using the new set { ψ n } n = 0 ∞ we solve this linear operator equation. We show that this method does not have some shortcomings of spectral method, also we prove the stability and convergence of the new method. After introducing the method we give some conditions that under them the nonlinear operator equation L u + N u = f can be solved. Some examples are considered to show the efficiency of method. [ABSTRACT FROM AUTHOR]

In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. To analyze the behavior of the different schemes considered, a 1D wave propagation problem is solved. [ABSTRACT FROM AUTHOR]

We consider a convolution-type operator on vector bundles over metric-measure spaces. This operator extends the analogous convolution Laplacian on functions in our earlier work to vector bundles, and is a natural extension of the graph connection Laplacian. We prove that for Euclidean or Hermitian connections on closed Riemannian manifolds, the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D. [Copyright &y& Elsevier]

By using the pseudo-metric introduced in Golse and Paul (Arch Ration Mech Anal 223:57–94, 2017), which is an analogue of the Wasserstein distance of exponent 2 between a quantum density operator and a classical (phase-space) density, we prove that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant ħ . We obtain explicit uniform in ħ error estimates for the first-order Lie–Trotter, and the second-order Strang splitting methods. [ABSTRACT FROM AUTHOR]

The main objective of this paper is finding new types of discrete transforms with tuning factor t. This is not only analogy to the continuous Laplace transform but gives discrete Laplace-Fibonacci transform (LFt). This type of Laplace-Fibonacci transform is not available in the continuous case. The LFt generates uncountably many outcomes when the parameter t varies on (0,∞). This possibility is not available in the existing Laplace transform. All the formulae and results derived are verified by MATLAB.

The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed. [ABSTRACT FROM AUTHOR]

This work presents a new approach to numerically solve the general linear two-point boundary value problems with Dirichlet boundary conditions. Multilevel bases from the anti-derivatives of the Daubechies wavelets are constructed in conjunction with the augmentation method. The accuracy of numerical solutions can be improved by increasing the number of basis levels, but the computational cost also increases drastically. The multilevel augmentation method can be applied to reduce the computational time by splitting the coefficient matrix into smaller submatrices. Then the unknown coefficients in the higher level can be solved separately. The convergent rate of this method is $2^{s}$, where $1 \leq s \leq p+1$, when the anti-derivatives of the Daubechies wavelets order p are applied. Some numerical examples are also presented to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we conduct a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy. [ABSTRACT FROM AUTHOR]

A functional type a posteriori error estimator for the finite element discretization of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimization problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings. [ABSTRACT FROM AUTHOR]

We apply fast multiscale methods for solving ill-posed integral equations via the Tikhonov regularization. A modified a posteriori parameter choice strategy is presented, which leads to optimal convergence rates. Numerical experiments are given to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

We consider numerical approximation to the solution of non-autonomous evolution equations. The order of convergence of the simplest possible Magnus method is investigated. [ABSTRACT FROM AUTHOR]

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points. [ABSTRACT FROM AUTHOR]

We develop a stability and convergence analysis of Galerkin–Petrov schemes based on a general setting of multiresolution generated by several refinable functions for the numerical solution of pseudodifferential equations on smooth closed curves. Particular realizations of such a multiresolution analysis are trial spaces generated by biorthogonal wavelets or by splines with multiple knots. The main result presents necessary and sufficient conditions for the stability of the numerical method in terms of the principal symbol of the pseudodifferential operator and the Fourier transforms of the generating multiscaling functions as well as of the test functionals. Moreover, optimal convergence rates for the approximate solutions in a range of Sobolev spaces are established. [ABSTRACT FROM AUTHOR]