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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]

In this paper we study the convergence rate and inverse theorem for spherical multiscale interpolation in L p and Sobolev norms. The multiscale interpolation is constructed using a sequence of scaled, compactly supported radial basis functions restricted to the unit sphere S n . For the interpolation scheme the problem called “native space barrier” is considered. In addition, a Bernstein type inequality is established to derive an inverse theorem for the multiscale interpolation, and some numerical experiments to illustrate the theoretical results are given. [ABSTRACT FROM AUTHOR]