Search

The periodic nonuniform sampling series, involving periodic samples of both the function and its first r derivatives, was initially introduced by Nathan (Inform Control 22: 172–182, 1973). Since then, various authors have extended this sampling series in different contexts over the past decades. However, the application of the periodic nonuniform derivative sampling series in approximation theory has been limited due to its slow convergence. In this article, we introduce a modification to the periodic nonuniform sampling involving derivatives by incorporating a Gaussian multiplier. This modification results in a significantly improved convergence rate, which now follows an exponential order. This is a significant improvement compared to the original series, which had a convergence rate of O (N - 1 / p) where p > 1 . The introduced modification relies on a complex-analytic technique and is applicable to a wide range of functions. Specifically, it is suitable for the class of entire functions of exponential type that satisfy a decay condition, as well as for the class of analytic functions defined on a horizontal strip. To validate the presented theoretical analysis, the paper includes rigorous numerical experiments. [ABSTRACT FROM AUTHOR]

We consider a subdiffusive fractional differential problem characterized by an equation that incorporates a time Riemann-Liouville fractional derivative of order 1 − α , α ∈ (0 , 1) , on its right-hand side, while the diffusive coefficient is allowed to vary with both space and time. An high order numerical method for the subdiffusion problem is derived based on the fractional splines of degree β ∈ (1 , 2 ]. The main purpose of this work is to apply fractional splines for approximating the fractional integral in the definition of the Riemann-Liouville fractional derivative, and hence explain how to solve the subdiffusion problem using this approach. It is discussed how the rate of convergence of the numerical method depends on the solution, the degree of the spline and the order of the fractional derivative. [ABSTRACT FROM AUTHOR]

The primary objective of this research is to develop and analyze a robust computational method based on exponential B splines for solving fractional sub-diffusion equations. The fractional operator includes the Mittag-Leffler function of one parameter in the form of a kernel that is non-local and non-singular in nature. The current approach is based on an effective finite difference method for discretizing in time, and the exponential B-spline functions for discretizing in space. The proposed scheme is proven to be unconditionally stable and convergent. Also, the unique solvability of the method is established. Numerical simulations conducted for multiple test examples validate the agreement between the obtained theoretical results and the corresponding numerical outcomes. [ABSTRACT FROM AUTHOR]

This paper derives the sharp bounds for Hermite–Hadamard inequalities in the context of Riemann–Liouville fractional integrals. A generalization of Jensen's inequality called the Jensen–Mercer inequality is used for general points to find the new and refined bounds of fractional Hermite–Hadamard inequalities. The existing Hermite–Hadamard inequalities in classical or fractional calculus have been proved for convex functions, typically involving only two points as in Jensen's inequality. By applying the general points in Jensen–Mercer inequalities, we extend the scope of the existing results, which were previously proved for two points in the Jensen's inequality or the Jensen–Mercer inequality. The use of left and right Riemann–Liouville fractional integrals in inequalities is challenging because of the general values involved in the Jensen–Mercer inequality, which we overcame by considering different cases. The use of the Jensen–Mercer inequality for general points to prove the refined bounds is a very interesting finding of this work, because it simultaneously generalizes many existing results in fractional and classical calculus. The application of these new results is demonstrated through error analysis of numerical integration formulas. To show the validity and significance of the findings, various numerical examples are tested. The numerical examples clearly demonstrate the significance of this new approach, as using more points in the Jensen–Mercer inequality leads to sharper bounds. [ABSTRACT FROM AUTHOR]

Utility Preference Robust Optimization with Moment-Type Information Structure In some decision-making problems, information on the true utility function of the decision maker may be incomplete, which may bring potential modeling risk. In "Utility Preference Robust Optimization with Moment-Type Information Structure," Guo, Xu, and Zhang propose a maximin utility preference robust optimization model where information about the DM's preference is constructed by moment-type conditions. The authors propose a piecewise linear approximation approach to tackle the maximin problem, reformulate the approximate problem as a single mixed integer linear program, and derive error bounds for the approximate ambiguity set, the optimal value, and the optimal solutions. To examine the performance of the model and the computational schemes, they carry out extensive numerical tests and demonstrate the effectiveness of the model and efficiency of the computational methods. Utility preference robust optimization (PRO) models have recently been proposed to deal with decision-making problems where the decision-maker's true utility function is unknown and the optimal decision is based on the worst-case utility function in an ambiguity set of utility functions. In this paper, we consider the case where the ambiguity set is constructed using some moment-type conditions. We propose piecewise linear approximation of the utility functions in the ambiguity set. The approximate maximin problem can be solved directly by derivative-free methods when the utility functions are nonconcave. Alternatively, we can reformulate the approximate problem as a single mixed integer linear program (MILP) and solve the MILP by existing solvers such as Gurobi. To justify the approximation scheme, we derive error bounds for the approximate ambiguity set, the optimal value and optimal solutions of the approximate maximin problem. To address the data perturbation/contamination issues arising from the construction of the ambiguity set, we derive some stability results which quantify the variation of the ambiguity set against perturbations in the elicitation data and its propagation to the optimal value and optimal solutions of the PRO model. Moreover, we extend the PRO models to allow some inconsistencies in the process of eliciting the decision-maker's preferences. Finally, we carry out numerical tests to evaluate the performances of the proposed numerical schemes and show that the computational schemes work fairly efficiently, the PRO model is resilient against small perturbations in data (with respect to both exogenous uncertainty data and preference elicitation data), and there is a potential to improve the efficiency of the preference elicitation by incorporating an optimal selection strategy. Funding: This project is supported by Hong Kong RGC [Grant 14500620] and The Chinese University of Hong Kong start-up grant. The research of S. Guo is supported by the National Key R&D Program of China [Grant 2022YFA1004000] and the Natural Science Foundation of China [Grant 12271077]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2023.2464. [ABSTRACT FROM AUTHOR]

A new subclass of H-matrices named generalized S D D 2 (for shortly, G S D D 2 ) matrices is introduced and some properties of G S D D 2 matrices are presented. The relationship between G S D D 2 matrices and other subclasses of H-matrices is studied. Moreover, the infinity norm bounds for the inverse of G S D D 2 matrices are provided. Using the proposed infinity norm bounds, error bounds of the linear complementarity problems for G S D D 2 matrices are given, which improve some existing results. Numerical examples are given to illustrate the validity of new results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a mapped Jacobi spectral Galerkin method for solving the multi-term Fredholm integral equations (MFIEs) with two-sided weakly singularities. We introduce a new family of mapped Jacobi functions (MJFs) and establish the corresponding spectral approximation results on these MJFs in weighted Sobolev spaces involving the mapped Jacobi weight function. These MJFs serve as the basis functions in our algorithm design and are tailored to the two-sided end-points singularities of the solution by using suitable mapping. Moreover, we derive the error estimates of the proposed method for MFIEs. Finally, the numerical examples are provided to demonstrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

A new subclass of nonsingular $ H $-matrix named $ SDD_1^{+} $ matrices is studied in this paper. The relationships between $ SDD_1^{+} $ matrices and other subclasses of nonsingular $ H $-matrices are analyzed by numerical examples. Moreover, the infinity norm bounds of the inverse for $ SDD_1^{+} $ matrices are derived in two different methods. As applications, two error bounds of the linear complementarity problems ($ LCP $) for $ SDD_1^{+} $ matrices are given. Finally, the effectiveness of corresponding results is illustrated by numerical examples.

A new subclass of nonsingular H-matrix named S DD+1 matrices is studied in this paper. The relationships between S DD+1 matrices and other subclasses of nonsingular H-matrices are analyzed by numerical examples. Moreover, the infinity norm bounds of the inverse for S DD+1 matrices are derived in two different methods. As applications, two error bounds of the linear complementarity problems (LCP) for S DD+1 matrices are given. Finally, the effectiveness of corresponding results is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

This paper delves into the numerical analysis of the extended Fisher-Kolmogorov (EFK) equation within open bounded convex domains R ⊂ R d , where d ≤ 3. Two distinct finite element schemes are introduced, namely the semi-discrete and fully-discrete finite element approximations. The existence and uniqueness of solutions are established for both the semi-discrete and fully-discrete finite element approximations. Error bounds are investigated across different scenarios, including comparisons between the semi-discrete and exact solutions, as well as between the semi-discrete and fully-discrete solutions, along with the fully-discrete and exact solutions. An effective algorithm has been proposed to solve the nonlinear system resulting from the fully-discrete finite element approximation at each time step. The paper also provides computed numerical error results and showcases a variety of numerical experiments to further illustrate and support the findings. [ABSTRACT FROM AUTHOR]

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, to guarantee well-posedness, are known [(see, e.g., Exercise 2.14 of Ern and Guermond (Theory and practice of finite elements, Applied mathematical, sciences, Springer, 2004)]. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately. [ABSTRACT FROM AUTHOR]

The upper bounds for the powers of the iteration matrix derived via a numerical method are intimately related to the stability analysis of numerical processes. In this paper, we establish upper bounds for the norm of the nth power of the iteration matrix derived via a fourth-order compact $ \theta $-method to obtain the numerical solutions of delay parabolic equations, and thus present conclusions about the stability properties. We prove that, under certain conditions, the numerical process behaves in a stable manner within its stability region. Finally, we illustrate the theoretical results through the use of several numerical experiments.

We revisit our prior work on clock skew compensation immune to floating-point precision loss and provide practical as well as theoretical bounds on the initial value of the skew-compensated clock based on a systematic analysis of the errors of floating-point operations; by practical bounds, we mean the actual values of the theoretical bounds calculated at resource-constrained computing platforms like WSN nodes equipped with 32-bit single-precision floating-point format. Numerical examples demonstrate that the proposed practical bounds on single-precision floating-point format do not violate the theoretical bounds and thereby can guarantee the correctness of the clock skew compensation on resource-constrained computing platforms. [ABSTRACT FROM AUTHOR]

We investigate error bounds for numerical solutions of divergence structure linear elliptic partial differential equations (PDEs) on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of stability that guarantees the existence of a bounded solution. In many settings including the Dirichlet problem, it is easy to show that the resulting solution error is proportional to the formal consistency error of the scheme. We make the surprising observation that this need not be true for PDEs posed on compact manifolds without boundary. We propose a particular class of approximation schemes built around an underlying monotone scheme with consistency error |$O(h^{\alpha })$|⁠. By carefully constructing barrier functions, we prove that the solution error is bounded by |$O(h^{\alpha /(d+1)})$| in dimension |$d$|⁠. We also provide a specific example where this predicted convergence rate is observed numerically. Using these error bounds, we further design a family of provably convergent approximations to the solution gradient. [ABSTRACT FROM AUTHOR]

We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and Hölderian error bounds and includes logarithmic and entropic error bounds found in the exponential cone. It also includes the error bounds obtainable under the theory of amenable cones. Our main result is that the convergence rate of several projection algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function. Another feature is the usage of Karamata theory and functions of regular variations which allows us to reason about convergence rates while bypassing certain complicated expressions. Finally, applications to conic feasibility problems are given and we show that a number of algorithms have convergence rates depending explicitly on the singularity degree of the problem. [ABSTRACT FROM AUTHOR]

This work investigates novel fractional Hadamard integral inequalities by utilizing extended convex functions and generalized Riemann-Liouville operators. By carefully using extended integral formulations, we not only find novel inequalities but also improve the accuracy of error bounds related to fractional Hadamard integrals. Our study broadens the applicability of these inequalities and shows that they are useful for a variety of convexity cases. Our results contribute to the advancement of mathematical analysis and provide useful information for theoretical comprehension as well as practical applications across several scientific directions. [ABSTRACT FROM AUTHOR]

We provide state-dependent error bounds for strongly continuous unitary representations of connected Lie groups. That is, we bound the difference of two unitaries applied to a state in terms of the energy with respect to a reference Hamiltonian associated with the representation and a left-invariant metric distance on the group. Our method works for any connected Lie group, and the metric is independent of the chosen representation. The approach also applies to projective representations and allows us to provide bounds on the energy-constrained diamond norm distance of any suitably continuous channel representation of the group. [ABSTRACT FROM AUTHOR]

This paper studies the interplay between the concepts of error bounds and the Kurdyka–Łojasiewicz (KL) inequality on Hadamard manifolds. To this end, we extend some properties and existence results of a solution for differential inclusions on Hadamard manifolds. As a second contribution, we show how the KL inequality can be used to obtain the convergence of the gradient method for solving convex feasibility problems on Hadamard manifolds. The convergence results of the alternating projection method are also established for cyclic and random projections on Hadamard manifolds and, more generally, CAT(0) spaces. [ABSTRACT FROM AUTHOR]

The present paper deals with the lacunary interpolation problem called the mixed values problem or (0, 3; 0, 2) problem for which known data points are function values at all the points, third derivatives at even knots, and second derivatives at odd knots of the unit interval I = [0,1]. For this problem, we obtained an interpolating function. The paper is divided into two parts, where we have shown that the spline function exists and is convergent.

This paper derives the sharp bounds for Hermite–Hadamard inequalities in the context of Riemann–Liouville fractional integrals. A generalization of Jensen’s inequality called the Jensen–Mercer inequality is used for general points to find the new and refined bounds of fractional Hermite–Hadamard inequalities. The existing Hermite–Hadamard inequalities in classical or fractional calculus have been proved for convex functions, typically involving only two points as in Jensen’s inequality. By applying the general points in Jensen–Mercer inequalities, we extend the scope of the existing results, which were previously proved for two points in the Jensen’s inequality or the Jensen–Mercer inequality. The use of left and right Riemann–Liouville fractional integrals in inequalities is challenging because of the general values involved in the Jensen–Mercer inequality, which we overcame by considering different cases. The use of the Jensen–Mercer inequality for general points to prove the refined bounds is a very interesting finding of this work, because it simultaneously generalizes many existing results in fractional and classical calculus. The application of these new results is demonstrated through error analysis of numerical integration formulas. To show the validity and significance of the findings, various numerical examples are tested. The numerical examples clearly demonstrate the significance of this new approach, as using more points in the Jensen–Mercer inequality leads to sharper bounds.

Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and dynamic mode decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models. [ABSTRACT FROM AUTHOR]

This article is devoted to constructing and analyzing two new approximations (CPL2‐1 σ$$ {}_{\sigma } $$ and CPL‐2 formulas) for the Caputo–Prabhakar fractional derivative. The error bounds for the CPL2‐1 σ$$ {}_{\sigma } $$ and CPL‐2 formulas are proved to be of order 2−α$$ 2-\alpha $$ and 3−α$$ 3-\alpha $$, respectively, where α$$ \alpha $$ is the order of time‐fractional derivative. The newly developed approximations are then used in the numerical treatment of a reaction–diffusion problem with variable coefficients defined in the Caputo–Prabhakar sense. Moreover, the space variable in the developed numerical schemes, CFD1 and CFD2, is discretized using a fourth‐order compact difference operator. Both schemes' stability and convergence analysis are demonstrated thoroughly using the discrete energy method. It is shown that the convergence orders of CFD1 and CFD2 schemes are O(Δt2−α,Δt2,h4)$$ \mathcal{O}\left(\Delta {t}^{2-\alpha },\Delta {t}^2,{h}^4\right) $$ and O(Δt3−α,h4)$$ \mathcal{O}\left(\Delta {t}^{3-\alpha },{h}^4\right) $$, respectively, where Δt$$ \Delta t $$ and h$$ h $$ represent the mesh spacing in time and space directions, respectively. In addition, numerical results are obtained for three test problems to confirm the theory and demonstrate the efficiency and superiority of the proposed schemes. [ABSTRACT FROM AUTHOR]

This article deals with the singularly perturbed nonlinear Volterra-Fredholm integrodifferential equations. Firstly, some priori bounds are presented. Then, the finite difference scheme is constructed on non-uniform mesh by using interpolating quadrature rules [5] and composite numerical integration formulas. The error estimates are derived in the discrete maximum norm. Finally, theoretical results are performed on two examples and they are compared for both Bakhvalov (B-type) and Shishkin (S-type) meshes. [ABSTRACT FROM AUTHOR]

We study a stochastically perturbed version of the well-known Krasnoselskii--Mann iteration for computing fixed points of nonexpansive maps in finite dimensional normed spaces. We discuss sufficient conditions on the stochastic noise and stepsizes that guarantee almost sure convergence of the iterates towards a fixed point and derive nonasymptotic error bounds and convergence rates for the fixed-point residuals. Our main results concern the case of a martingale difference noise with variances that can possibly grow unbounded. This supports an application to reinforcement learning for average reward Markov decision processes, for which we establish convergence and asymptotic rates. We also analyze in depth the case where the noise has uniformly bounded variance, obtaining error bounds with explicit computable constants. [ABSTRACT FROM AUTHOR]

This paper purposes to present a new discrete scheme for the singularly perturbed semilinear Volterra–Fredholm integro-differential equation including two integral boundary conditions. Initially, some analytical properties of the solution are given. Then, using the composite numerical integration formulas and implicit difference rules, the finite difference scheme is established on a uniform mesh. Error approximations for the approximate solution and stability bounds are investigated in the discrete maximum norm. Finally, a numerical example is solved to show -uniform convergence of the suggested difference scheme. [ABSTRACT FROM AUTHOR]

In this paper, we develop Stein’s method for binomial approximation using the stop-loss metric that allows one to obtain a bound on the error term between the expectation of call functions. We obtain the results for a locally dependent collateralized debt obligation (CDO), under certain conditions on moments. The results are also exemplified for an independent CDO. Finally, it is shown that our bounds are sharper than the existing bounds. [ABSTRACT FROM AUTHOR]

In this paper, a new error bound for the linear complementarity problems of $ B- $matrices which is a subclass of the $ P- $matrices is presented. Theoretical analysis and numerical example illustrate that the new error bound improves some existing results.

In this paper, we develop Stein's method for negative binomial distribution using call function defined by fz(k) = (k - z)+ = max{k - z, 0}, for k ≥ 0 and z ≥ 0. We obtain error bounds between E [ fz(Nr,p)] and E [ fz(V )], where Nr,p follows negative binomial distribution and V is the sum of locally dependent random variables, using certain conditions on moments. We demonstrate our results through an interesting application, namely, collateralized debt obligation (CDO), and compare the bounds with the existing bounds.

In this paper, we study a nonlocal approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the nonlocal problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and nonlocal problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the nonlocal problem and that of the local one, both in continuous time and forward Euler time discretization. We then turn to studying continuum limits of nonlocal problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows then, almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large. [ABSTRACT FROM AUTHOR]

We present a methodology for the definition of the matrices involved in stress based finite element approximations in three‐dimensional problems. In order to handle the increased complexity of the 3D case, a rotation of the global reference frame is used, which greatly simplifies the expressions involved. This rotation is also applicable to the 2D case. The main focus of this work concerns the matrices used in the implementation of the hybrid equilibrium approach. However, a similar complementary methodology (displacement based) is also presented, allowing for an efficient definition of the matrices required for dual analysis, wherein bounds of the solution error are obtained from pairs of complementary solutions, one compatible and the other equilibrated. [ABSTRACT FROM AUTHOR]

Consider the problem of solving systems of linear algebraic equations A x = b with a real symmetric positive definite matrix A using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the A -norm of the error. This quantity cannot be easily computed; however, it can be estimated. In this paper we discuss and analyze the behavior of the Gauss-Radau upper bound on the A -norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate μ to the smallest eigenvalue of A . We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of μ . We construct a model problem that is used to demonstrate and study the behavior of the upper bound in dependence of μ , and developed formulas that are helpful in understanding this behavior. We show that the above-mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of A . It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate μ . We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations. [ABSTRACT FROM AUTHOR]

The divergence-free generalized moving least squares (Div-Free GMLS) approximation has recently been utilized to solve some incompressible fluid flows problems. In our recent work (Mohammadi and Dehghan (2021) [28]), we have presented its formulation more precisely, and also the error estimates of derivatives have been carried out in L ∞ (Ω) , where Ω ⊂ R d is a bounded set satisfying an interior cone condition. However, the error estimates of this vector-valued approximation in Sobolev spaces are not done. So, in this paper we make the error estimates of Div-Free GMLS derivatives approximations in L q (Ω) , where 1 ≤ q ≤ ∞ , using a stable local divergence-free polynomial reproduction property. Note that, the method is a direct approximants of exact derivatives of a divergence-free vector field, which possesses the optimal rates of convergence. This vector-valued technique can also be developed to find the numerical solution of the incompressible fluid flows problems easier than the other available mesh-dependent methods. Finally, we have shown how the proposed approximation can recover the velocity field variable of the well-known Darcy's problem in a two-dimensional space. • The error analysis of the div-free GMLS derivatives approximations has been released on the Sobolev spaces. • The method can approximate the vector field satisfying the div-free property with the order of O (h m + 1 − | α |). • The developed vector-valued approximation satisfies analytically the div-free property for all polynomials belong to P m. • We have shown how the div-free GMLS approximation can recover the velocity field variable related to the Darcy's problem. [ABSTRACT FROM AUTHOR]

In this paper, a new error bound for the linear complementarity problems of B-matrices which is a subclass of the P-matrices is presented. Theoretical analysis and numerical example illustrate that the new error bound improves some existing results. [ABSTRACT FROM AUTHOR]

Stochastic Parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as Parareal. Similarly to Parareal, it combines fine- and coarse-grained solutions to an ODE using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics. [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence properties of a randomized block-coordinate descent algorithm for the minimization of a composite convex objective function, where the block-coordinates are updated asynchronously and randomly according to an arbitrary probability distribution. We prove that the iterates generated by the algorithm form a stochastic quasi-Fejér sequence and thus converge almost surely to a minimizer of the objective function. Moreover, we prove a general sublinear rate of convergence in expectation for the function values and a linear rate of convergence in expectation under an error bound condition of Tseng type. Under the same condition strong convergence of the iterates is provided as well as their linear convergence rate. [ABSTRACT FROM AUTHOR]

The disjunctive system is a system involving a disjunctive set which is the union of finitely many polyhedral convex sets. In this paper, we introduce a notion of the relaxed constant positive linear dependence constraint qualification (RCPLD) for the disjunctive system. For a disjunctive system, our notion is weaker than the one we introduced for a more general system recently (J. Glob. Optim. 2020) and is still a constraint qualification. To obtain the local error bound for the disjunctive system, we introduce the piecewise RCPLD under which the error bound property holds if all inequality constraint functions are subdifferentially regular and the rest of the constraint functions are smooth. We then specialize our results to the ortho-disjunctive program, which includes the mathematical program with equilibrium constraints (MPEC), the mathematical program with vanishing constraints (MPVC) and the mathematical program with switching constraints (MPSC) as special cases. For MPEC, we recover MPEC-RCPLD, an MPEC variant of RCPLD and propose the MPEC piecewise RCPLD to obtain the error bound property. For MPVC, we introduce new constraint qualifications MPVC-RCPLD and the piecewise RCPLD, which also implies the local error bound. For MPSC, we show that both RCPLD and the piecewise RCPLD coincide and hence it leads to the local error bound. [ABSTRACT FROM AUTHOR]

This work investigates novel fractional Hadamard integral inequalities by utilizing extended convex functions and generalized Riemann-Liouville operators. By carefully using extended integral formulations, we not only find novel inequalities but also improve the accuracy of error bounds related to fractional Hadamard integrals. Our study broadens the applicability of these inequalities and shows that they are useful for a variety of convexity cases. Our results contribute to the advancement of mathematical analysis and provide useful information for theoretical comprehension as well as practical applications across several scientific directions.

In this study, we examine the Generalized Equations' subregularity in Asplund spaces utilizing a novel approach. We obtain sufficient conditions for a family of multifunctions to be metrically subregular which are stronger than the known sufficient conditions thanks to a modification of the well-known coderivative concept and of the partial sequential normal compactness.

We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method. Numerical examples are provided for the equation - Δ M u + u = f on the 2- and 3-spheres, where Δ M is the Laplace-Beltrami operator. [ABSTRACT FROM AUTHOR]

In the present paper, we prove convergence rates for the pressure of the local discontinuous Galerkin approximation, proposed in Part I of the paper [A. Kaltenbach and M. Růžička, SIAM J. Numer. Anal., 61 (2023), pp. 1613--1640], of systems of p-Navier-Stokes type and p-Stokes type with p ∈ (2,∞). The results are supported by numerical experiments. [ABSTRACT FROM AUTHOR]

In the present paper, we prove convergence rates for the velocity of the local discontinuous Galerkin approximation, proposed in Part I of the paper [A. Kaltenbach and M. Růžička, SIAM J. Numer. Anal., to appear], of systems of p-Navier-Stokes type and p-Stokes type with p ∈ (2,∞). The convergence rates are optimal for linear ansatz functions. The results are supported by numerical experiments. [ABSTRACT FROM AUTHOR]

To solve a parabolic variational inequality with a nonlinear and nonlocal spatial operator depending on the Dirichlet integral and a one-sided constraint on the solution, a numerical method based on the penalty method, finite elements, and the implicit Euler scheme is proposed and studied. Optimal estimates for the accuracy of the approximate solution in the energy norm are obtained. Numerical results which confirm theoretical findings are discussed. [ABSTRACT FROM AUTHOR]

Some inequalities for elements relation of strictly diagonally dominant M-matrix and its inverse are given, and new upper bounds of the infinity norm for the inverse of strictly diagonally dominant M-matrix are presented. We apply these new bounds to linear complementarity problems (LCPs) and obtain an alternative error bound for LCPs of B S -matrices. A lower bound for the smallest singular value is also provided. Theoretical analysis and numerical examples show the validity of the results. [ABSTRACT FROM AUTHOR]

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of O ε - 2 , where 0 < ε ≪ 1 is inversely proportional to the speed of light. Yongyong Cai and Yan Wang have shown, however, that such solutions can be approximated up to an error of O ε 2 by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the explicit exponential midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy significantly. [ABSTRACT FROM AUTHOR]