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Recently, fractional partial differential equations have been widely applied in option pricing problems, which better explains many important empirical facts of financial markets, but rare paper considers the multi-state options pricing problem based on fractional diffusion models. Thus, multi-state European option pricing problem under regime-switching tempered fractional partial differential equation is considered in this paper. Due to the expensive computational cost caused by the implicit finite difference scheme, a novel implicit-explicit finite difference scheme has been developed with consistency, stability, and convergence guarantee. Since the resulting coefficient matrix equals to the direct sum of several Toeplitz matrices, a preconditioned direct method has been proposed with O (S ̄ N log N + S ̄ 2 N) operation cost on each time level with adaptability analysis, where S ̄ is the number of states and N is the number of grid points. Related numerical experiments including an empirical example have been presented to demonstrate the effectiveness and accuracy of the proposed numerical method. [ABSTRACT FROM AUTHOR]

This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular M -tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative componentwise nonincreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method. [ABSTRACT FROM AUTHOR]

Previous papers have shown the impact of partial convergence of discretized PDEs on the accuracy of tangent and adjoint linearizations. A series of papers suggested linearization of the fixed-point iteration used in the solution process as a means of computing the sensitivities rather than linearizing the discretized PDE, as the lack of convergence of the nonlinear problem indicates that the discretized form of the governing equations has not been satisfied. These works showed that the accuracy of an approximate linearization depends in part on the convergence of the nonlinear system. This work shows an error analysis of the impact of the approximate linearization and the convergence of the nonlinear problem for both the tangent and adjoint modes and provides a series of results for an exact Newton solver, an inexact Newton solver, and a low-storage explicit Runge-Kutta scheme to confirm the error analyses. [ABSTRACT FROM AUTHOR]

The indirect solution of optimal control problems (OCPs) with inequality constraints and parameters is obtained by solving the two-point boundary value problem (BVP) involving index-1 differential-algebraic equations (DAEs) associated with its first-order optimality conditions. This paper introduces an adaptive mesh refinement method based on a collocation method for solving the index-1 BVP-DAEs. The paper first derives a method to estimate the relative error between the numerical solution and the exact solution. The relative error estimate is then used to guide the mesh refinement process. The mesh size is increased when the estimated error within a mesh interval is beyond the numerical tolerance by either increasing the order of the approximating polynomial or dividing the interval into multiple subintervals. In the mesh interval where the error tolerance has been met, the mesh size is reduced by either decreasing the degree of the approximating polynomial or merging adjacent mesh intervals. An efficient parallel implementation of the method is implemented using Python and CUDA. The paper presents three examples which show that the approach is more computationally efficient and robust when compared with fixed-order methods. [ABSTRACT FROM AUTHOR]

Approximation of a multivariate function is an important theme in the field of numerical analysis and its applications, which continues to receive a constant attention. In this paper, we provide a parameterized family of self-referential (fractal) approximants for a given multivariate smooth function defined on an axis-aligned hyper-rectangle. Each element of this class preserves the smoothness of the original function and interpolates the original function at a prefixed gridded data set. As an application of this construction, we deduce a fractal methodology to approach a multivariate Hermite interpolation problem. This part of our paper extends the classical bivariate Hermite's interpolation formula by Ahlin (Math. Comp. 18:264–273, 1964) in a twofold sense: (i) records, in particular, a multivariate generalization of this bivariate interpolation theory; (ii) replaces the unicity of the Hermite interpolant with a parameterized family of self-referential Hermite interpolants which contains the multivariate analogue of Ahlin's interpolant as a particular case. Some related aspects including the approximation by multivariate self-referential functions preserving Popoviciu convexity are given, too. [ABSTRACT FROM AUTHOR]

In many applications, it is useful to use piecewise polynomials that satisfy certain regularity conditions at the joint points. Cubic spline functions emerge as good candidates having C2 regularity. On the other hand, if the data points present discontinuities, the classical spline approximations produce Gibbs oscillations. In a recent paper, we have introduced a new nonlinear spline approximation avoiding the presence of these oscillations. Unfortunately, this new reconstruction loses the C2 regularity. This paper introduces a new nonlinear spline that preserves the regularity at all the joint points except at the end points of an interval containing a discontinuity, and that avoids the Gibbs oscillations. [ABSTRACT FROM AUTHOR]

In this paper, a one-dimensional two-phase inverse Stefan problem is studied. The free surface is considered unknown here, which is more realistic from the practical point of view. The problem is ill-posed since small errors in the input data can lead to large deviations from the desired solution. To obtain a stable numerical solution, we propose a method based on discrete mollification combined with space marching. We use the integration matrix in Bernstein polynomial basis for the discrete mollification method. Through this method, our numerical integration does not involve any direct function integration and only contains algebraic calculations. Furthermore, the stability and convergence of the process are proved. Finally, the results of this paper are illustrated and examined through some numerical examples. The numerical examples supporting our theoretical analysis are provided and discussed. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic optimization problem over the fixed point sets of quasinonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian hierarchical optimization problems over complicated sets, such as the intersection of many closed convex sets, the set of all minimizers of a nonsmooth convex function, and the intersection of sublevel sets of nonsmooth convex functions. We focus on adaptive learning rate optimization algorithms, which adapt step-sizes (referred to as learning rates in the machine learning field) to find optimal solutions quickly. We then propose a Riemannian stochastic fixed point optimization algorithm, which combines fixed point approximation methods on Riemannian manifolds with the adaptive learning rate optimization algorithms. We also give convergence analyses of the proposed algorithm for nonsmooth convex and smooth nonconvex optimization. The analysis results indicate that, with small constant step-sizes, the proposed algorithm approximates a solution to the problem. Consideration of the case in which step-size sequences are diminishing demonstrates that the proposed algorithm solves the problem with a guaranteed convergence rate. This paper also provides numerical comparisons that demonstrate the effectiveness of the proposed algorithms with formulas based on the adaptive learning rate optimization algorithms, such as Adam and AMSGrad. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to analyze nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix-Raviart–type finite element is used to approximate the velocity and the conforming piecewise linear element P1 is used for the pressure. Since the finite element method for the velocity field and the pressure is unstable, a simple locally stabilization term is added to satisfy the weak inf-sup condition. Then, the well-posedness and the optimal error estimates of the continuous and discrete problems are analyzed with the nonlinear terms being iteratively updated. Three effective iterative methods are proposed and their stability and convergence analyses are carried out. Finally, the theoretical analysis presented in this paper is verified by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we trace back the genesis of Aitken's Δ2 process and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis. [ABSTRACT FROM AUTHOR]

This paper develops an improved tri-coloured rooted-tree theory for the order conditions for ERKN methods solving general multi-frequency and multidimensional second-order oscillatory systems. The bottleneck of the original tricoloured rooted-tree theory is the existence of numerous redundant trees. In light of the fact that the sum of the products of the symmetries and the elementary differentials is meaningful, this paper naturally introduces the so-called extended elementary differential mappings. Then, the new improved tri-coloured rooted tree theory is established based on a subset of the original tri-coloured rooted-tree set. This new theory makes all redundant trees disappear, and thus, the order conditions of ERKN methods for general multi-frequency and multidimensional second-order oscillatory systems are reduced greatly. Furthermore, with this new theory, we present some new ERKN methods of order up to four. Numerical experiments are implemented and the results show that ERKN methods can be competitive with other existing methods in the scientific literature, especially when comparatively large stepsizes are used. [ABSTRACT FROM AUTHOR]

In this paper, we consider a decoupled and mass-conservative numerical scheme for the incompressible miscible displacement problem in porous media by employing multipoint flux mixed finite element method for velocity-pressure equation and mass-conservation characteristic finite element method for concentration equation. This feature is designed to avoid handling saddle-point system and keep mass globally. Convergence and L2 error estimates of full discrete scheme are obtained. Numerical experiments are also presented to verify the correctness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a high order multiquadric trigonometric quasi-interpolation method for function approximation and derivative approximation based on periodic sampling data. Moreover, we apply it to solve time-dependent nonlinear partial differential equations (PDEs) with periodic solutions. Learning from the construction of the trigonometric B-spline quasi-interpolation, we derive the new quasi-interpolation scheme by replacing the truncated trigonometric polynomial with a high degree, infinitely smooth multiquadric trigonometric kernel. By appropriately choosing the shape parameter in the kernel, we prove that the proposed method achieves higher order of convergence than the existing multiquadric trigonometric quasi-interpolation method. Finally, we conduct extensive numerical experiments to demonstrate the accuracy and efficiency of the proposed method for approximating unknown function and solving different types of PDEs including the one-dimensional KdV equation and two-dimensional Allen-Cahn equation. [ABSTRACT FROM AUTHOR]

It is difficult to convert the PDEs defined on the whole space with higher order space derivatives into initial boundary value problems by proposing boundary conditions. In this paper, we use an absorbing boundary condition method to solve the Cauchy problem for one-dimensional Euler-Bernoulli beam with fast convolution boundary condition which is derived through the Padé approximation for the square root function ⋅ . We also introduce a constant damping term to control the error between the resulting approximation of the Euler-Bernoulli system and the original one. Numerical examples verify the theoretical results and demonstrate the performance for the fast numerical method. [ABSTRACT FROM AUTHOR]

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli (Numer. Algo., 27, 119–130 2021). In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a general framework that provides a unified convergence analysis of the derivative-free projection-based method (DFPM) for solving large-scale constrained nonlinear monotone equations. The new results provide a complete picture on the convergence guarantees of DFPM and cover the existing relevant convergence results as special cases. Preliminary numerical experiment results are also reported to show the numerical performance of six line search schemes used in the existing DFPM. [ABSTRACT FROM AUTHOR]

When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z2 − c has the same form regardless of which algorithm has been used. In this paper, we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree d polynomials p(z) = zd − c. Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algorithms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods. [ABSTRACT FROM AUTHOR]

In this paper, we present a collocation method for the approximate solution of two-dimensional Hammerstein integral equations over polygonal regions in ℝ 2 . For a reformulation of the equation, we triangulate the polygonal region and use a special type of quadratic interpolation on each triangle. This produces a quadrature formula with a higher degree of precision than expected, leading to a numerical method that is superconvergent at the collocation nodes. We prove the convergence of the method and give error estimates. The implementation is also discussed and the applicability of the proposed scheme is presented on numerical examples. The paper concludes with a discussion on the efficiency of the proposed method, also highlighting some ideas for future research in this area. [ABSTRACT FROM AUTHOR]

In this paper, we consider the sum of linear ratios problem (SLR) that is known to be NP-hard and often arises in various practical applications such as data envelopment analysis and financial investment. We first introduce an equivalent problem (EP) of SLR that involves differences of square terms in inequality constraints. Subsequently, the concave parts of the non-convex constraints in problem (EP) are replaced with the piecewise linear functions. Using the resulting second-order cone program (SOCP), we design a spatial branch and bound algorithm, which iteratively refines the piecewise linear approximations by dividing rectangles and solving a series of problems (SOCP) to obtain the solution of the original problem. Also, a region compression technique is proposed to accelerate the convergence of the algorithm. Furthermore, we demonstrate that the bound on the optimality gap is a function of approximation errors at the iteration and estimate that the worst-case number of iterations is in the order of O (ε) to attain an ε-optimal solution. Numerical results illustrate that the proposed algorithm scales better than both the existing LP-based algorithms and the off-the-shelf solvers SCIP to solve the problem (SLR). It is worth mentioning that the proposed algorithm takes significantly less time to reach four-digit accuracy than the time required by the known algorithms on small to medium problem instances. [ABSTRACT FROM AUTHOR]

In a recent paper, a conformable fractional Newton-type method was proposed for solving nonlinear equations. This method involves a lower computational cost compared to other fractional iterative methods. Indeed, the theoretical order of convergence is held in practice, and it presents a better numerical behaviour than fractional Newton-type methods formerly proposed, even compared to classical Newton-Raphson method. In this work, we design a generalization of this method for solving nonlinear systems by using a new conformable fractional Jacobian matrix, and a suitable conformable Taylor power series; and it is compared with classical Newton's scheme. The necessary concepts and results are stated in order to design this method. Convergence analysis is made and a quadratic order of convergence is obtained, as in classical Newton's method. Numerical tests are made, and the Approximated Computational Order of Convergence (ACOC) supports the theory. Also, the proposed scheme shows good stability properties observed by means of convergence planes. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm for finding a zero of a pseudomonotone vector field with a convex constraint on a Hadamard manifold. This new method is the combination of the hyperplane projection method with specially constructed search directions. The global convergence property of this algorithm is established under the assumptions that the constructed halfspace is closed and convex, the tangent vector field is continuous, and the solution set is nonempty. Numerical experiments show the efficiency of this new derivative-free iterative method. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a tempered fractional integrodifferential equation in three-dimensional (3D) space. The Crank-Nicolson (CN) method and trapezoidal convolution quadrature rule are used to approximate the time derivative and tempered fractional integral term respectively, and finite difference/compact difference approaches combined with corresponding alternating direction implicit (ADI) algorithms are employed for spatial discretizations, which obtains the fully discrete CN ADI finite difference/compact difference schemes. Then, convergence analysis of two kinds of ADI schemes is derived via the energy argument and the generating function of Toeplitz matrices. Provided numerical examples confirm our theoretical estimates. [ABSTRACT FROM AUTHOR]

In this paper, we construct a stable finite difference scheme for the generalized non-Fickian time-fractional reaction-diffusion equations with time delay. The proposed difference scheme has second-order accuracy in both space and time directions. The stability and convergence of the difference solutions are proved rigorously in the maximum norm. Three representative models with delay are carried out to verify the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In the fields of artificial intelligence and control engineering, generalized-Sylvester matrix equation is considered as an important mathematic problem, and its solving process is usually viewed as a challenge that deserves particular attention. In this paper, a creative discrete-form recurrent neural network (RNN) model is developed, analyzed and investigated for solving discrete-form time-variant generalized-Sylvester matrix equation (DF-TV-GSME), which is derived by a direct discretization numerical method. Specifically, first of all, DF-TV-GSME, which includes the well-known Lyapunov matrix equation and Sylvester matrix equation, is presented as the target problem of this research. Secondly, for solving such problem, different from the traditional discrete-form RNN design philosophy, second-order Taylor expansion is applied to derive the discrete-form RNN model. This creative process avoids involving the continuous time-variant environment and continuous-form model. Then, theoretical properties analyses are presented, which present the convergence and precision of the discrete-form RNN model. Abundant numerical experiments are further carried out with different perspectives of DF-TV-GSME, which further confirm the excellent properties of discrete-form RNN model. [ABSTRACT FROM AUTHOR]

In this paper, first we propose a general modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems. In order to improve the computing efficiency, we further propose a preconditioned general modulus-based matrix splitting iteration method. We establish the convergence theorems when the coefficient matrices are (symmetric) positive definite matrices and H+-matrices, respectively. Numerical results show that the proposed preconditioned general modulus-based matrix splitting iteration method is superior than some existing methods and the general modulus-based matrix splitting iteration method. [ABSTRACT FROM AUTHOR]

Theta functions and theta constants in low genus, especially genus 1 and 2, can be evaluated at any given point in quasi-linear time in the required precision using Newton schemes based on Borchardt sequences. Our goal in this paper is to provide the necessary tools to implement these algorithms in a provably correct way. In particular, we obtain uniform and explicit convergence results in the case of theta constants in genus 1 and 2, and theta functions in genus 1: the associated Newton schemes will converge starting from approximations to N bits of precision for N = 60, 300, and 1600 respectively, for all suitably reduced arguments. We also describe a uniform quasi-linear time algorithm to evaluate genus 2 theta constants on the Siegel fundamental domain. Our main tool is a detailed study of Borchardt means as multivariate analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, by combining a partition of unity with backtracking technique, a modified local and parallel finite element method (MLPFEM) is proposed and investigated for the coupled Stokes-Darcy problem with the Beavers-Joseph (BJ) interface condition. The well-posedness of the coupled Stokes-Darcy model with BJ interface condition is established when the parameter αBJ is small enough in Cao et al. (Commun. Math. Sci.8(1), 1–25 2010). The MLPFEM is adopted based on its significant advantages that we only need to solve a series of local subproblems once the coarse approximation is derived. Compared with the global discontinuous solution by the algorithm in Du and Zuo (2017), the main features of our algorithm are as follows: (1) partition of unity functions are utilized to gather local approximations computed on a fine grid by local and parallel procedures to generate a global continuous solution; (2) a further global coarse correction, namely the backtracking technique, is considered to obtain optimal error bounds of the velocity field and the piezometric head in L2 norm. Optimal error bounds are derived and numerical tests are carried out to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

As a generalization of k-tridiagonal matrices, many variations of (p,q)-tridiagonal matrices have attracted much attention over the years. In this paper, we present an efficient algorithm for numerically computing the inverses of n-square (p,q)-tridiagonal matrices under a certain condition. The algorithm is based on the combination of a bidiagonalization approach which preserves the banded structure and sparsity of the original matrix, and a recursive algorithm for inverting general lower bidiagonal matrices. Some numerical results with simulations in MATLAB implementation are provided in order to illustrate the accuracy and efficiency of the proposed algorithms, and its competitiveness with MATLAB built-in function. [ABSTRACT FROM AUTHOR]

In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular, a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show near-optimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the special case of unknown sparsity. [ABSTRACT FROM AUTHOR]

The modified phase field crystal (MPFC) model is a sixth-order evolutive nonlinear partial differential equation which can describe many crystal phenomena. In this paper, we propose a highly efficient and accurate numerical method to construct linear and unconditionally energy stable schemes for the MPFC model. In recent years, the scalar auxiliary variable (SAV) and SAV-based methods have attracted much attention in numerical solution for dissipative systems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. The considered numerical schemes are based on a new SAV-type approach which named new scalar auxiliary variable (nSAV) approach. We first give a first-order energy stable numerical scheme by introducing a new SAV R(t). Then, the high-order nSAV schemes based on the k-step backward differentiation formula (BDFk) are constructed. The considered nSAV schemes allow us to construct high-order schemes for both the phase variable ϕ and ψ while only using a first-order approximation of the energy balance equation. To our knowledge, there is no careful research to give high-order accurate but energy stable schemes for the MPFC model. Meanwhile, the proposed approach only needs to solve linear equation with constant coefficients in one time step which is easy to use fast Fourier transform (FFT) to save more CPU time in calculation. Some numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, for solving large sparse vertical linear complementarity problems, the modulus-based synchronous multisplitting iteration method is established. Convergence theorems of the proposed method are given, which are shown to generalize and improve the existing results of the serial method. Numerical tests by OpenACC technique are given to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem of nonlinear systems with mixed control variables and probability constraints. To obtain a numerical solution of this optimal control problem, our target is to formulate this problem as a constrained nonlinear parameter optimization problem (CNPOP), which can be solved by using any gradient-based numerical computation method. Firstly, some binary functions are introduced for each value of the discrete-valued control variable (DCV). Following that, we relax these binary functions and impose a penalty term on the relaxation such that the solution of the resulting relaxed problem (RP) can converge to the solution of the original problem as the penalty parameter increases. Secondly, we introduce a simple initial transformation for the probability constraints. Following that, an adaptive sample approximation method (ASAM) and a novel smooth approximation technique (NSAT) are adopted to formulate the probability constraints as some deterministic constraints. Thirdly, a control parameterization approach (CPA) is used to transform the deterministic problem (i.e., an infinite dimensional problem) into a finite dimensional CNPOP. Fourthly, in order to combine the advantages of limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithms and differential evolution (DE) algorithms, a L-BFGS-based DE (L-BFGS-DE) algorithm is proposed for solving the resulting approximation problem based on an improvied L-BFGS (IL-BFGS) method and an improved DE (IDE) algorithm. Following that, we establish the convergence result of the L-BFGS-DE algorithm. The L-BFGS-DE algorithm consists of two stages. The objectives of the first and second stages are to obtain a probable position of the global solution and to accelerate the convergence rate, respectively. In the IL-BFGS method, we propose a novel updating rule (NUR), which uses not only the gradient information of the objective function but also the value of the objective function. This will improved the performance of the IL-BFGS method. In the IDE algorithm, a novel adaptive parameter adjustment (NAPA) method, a novel population size decrease (NPSD) strategy, and an improved mutation (IM) scheme are proposed to improve its performance. Finally, an anti-cancer drug therapy problem (ADTP) is further extended to illustrate the effectiveness of the L-BFGS-DE algorithm by taking into account some probability constraints. Numerical results show that the L-BFGS-DE algorithm has good performance and can obtain a stable and robust performance when considering the small noise perturbations in initial state. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary λ-Hölder continuous process, λ ∈ (0,1). We prove that, under some mild moment assumptions on the Hölder constant of the noise, the L r (Ω ; L ∞ ([ 0 , T ])) -approximation error converges to 0 as O(Δλ), Δ → 0. To exemplify, we consider numerical schemes for the generalized Cox–Ingersoll–Ross and Tsallis–Stariolo–Borland models. The results are illustrated by simulations. [ABSTRACT FROM AUTHOR]

This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem are derived. Computable expressions and upper bounds for these condition numbers are also given to avoid the costly Kronecker product-based operations. The results unify the ones for the TLS problem. For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition number-based estimates are much tighter. [ABSTRACT FROM AUTHOR]

Recently, a number of efficient iteration methods for the solution of a special class of block 3 × 3 saddle point systems have been proposed by some authors. In order to easily evaluate the strong stability of these numerical algorithms and provide a practical and reliable termination criterion, in this paper, we perform the structured backward error analysis for this type of block 3 × 3 saddle point system and present an explicit and computable formula of the normwise structured backward error. Some numerical experiments are performed to demonstrate that our results can be used to easily test the stability of running algorithms, and the new stopping criterion based on the derived structured backward error is more suitable and efficient than the commonly used residue-based one. [ABSTRACT FROM AUTHOR]

This paper studies an implicit-explicit (IMEX) finite difference scheme for solving a system of moving boundary partial integro-differential equations (PIDEs) which arises in Asian option pricing under regime-switching jump-diffusion models. First, the moving boundary PIDEs are recast into a fixed boundary problem of the PIDEs. Then the IMEX scheme is proposed to solve the problem and the second-order convergence rates are proved. Numerical examples are carried out to validate the theoretical results. [ABSTRACT FROM AUTHOR]