This paper describes inverse eigenvalue problems that arise in studying qualitative dynamics in systems biology models. An algorithm based on lift-and-project iterations is proposed, where the lifting step entails solving a constrained matrix inverse eigenvalue problem. In particular, prior to carrying out the iterative steps, $a$-$priori$ bounds on the entries of the Jacobian matrix are computed by relying on the reaction network structure as well as the form of the rate law expressions for the model under consideration. Numerical results on a number of models show that the proposed algorithm can be used to computationally explore the possible dynamical scenarios while identifying the important mechanisms via the use of sparsity-promoting regularization.

In this article, a level-set approach for solving nonlinear elliptic Cauchy problems with piecewise constant solutions is proposed, which allows the definition of a Tikhonov functional on a space of level-set functions. We provide convergence analysis for the Tikhonov approach, including stability and convergence results. Moreover, a numerical investigation of the proposed Tikhonov regularization method is presented. Newton-type methods are used for the solution of the optimality systems, which can be interpreted as stabilized versions of algorithms in a previous work and yield a substantial improvement in performance. The whole approach is focused on three dimensional models, better suited for real life applications.

In this paper, we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆² ± λ². Similar to the derivation of fundamental solutions, it is non-trivial to derive particular solutions for higher order differential operators. In this paper, we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D. The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration. Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the $δ$-function. For the approximation of the $δ$-function, the direct projection method is used that was proposed in [6]. The $δ$-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The $δ$-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the non-stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.

Ever since its introduction by Kane Yee over forty years ago, the finite-difference time-domain (FDTD) method has been a widely-used technique for solving the time-dependent Maxwell's equations that has also inspired many other methods. This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability, based on Krylov subspace spectral (KSS) methods. These methods have previously been applied to the variable-coefficient heat equation and wave equation, and have demonstrated high-order accuracy, as well as stability characteristic of implicit time-stepping schemes, even though KSS methods are explicit.  KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. We show how they can be generalized to coupled systems of equations, such as Maxwell's equations, by choosing appropriate basis functions that, while induced by this coupling, still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields.  We also discuss the application of block KSS methods to problems involving non-self-adjoint spatial differential operators, which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices.

Symmetric Taylor-Görtler-like vortices at $Re$=3200 and 5000 in 3D rectangular cavities with a moving top lid are studied numerically and tested with a spanwise aspect ratio of 1 : 1 : $L$, where $L$=1, 2, 3. Solutions are obtained by solving the momentum equations and the continuity equations using the consistent splitting scheme. The code presented here was ported to the Parallel Interoperable Computational Mechanics System Simulator (PICMSS). Stable solutions are obtained as limit cases of the transients.

In this paper, we are presenting a proposal for new modified algorithms for RRGMRES and AGMRES. It is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed problems. In this paper we have focused on the residual norm and have come up with two improvements where successive updates and the stabilization of decreases for the residual norm improve performance respectively. Our numerical experiments confirm that our improved algorithms are effective for linear discrete ill-posed problems.

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieves asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.

We consider pricing options in a jump-diffusion model which requires solving a partial integro-differential equation. Discretizing the spatial direction with a fourth order compact scheme leads to a linear system of ordinary differential equations. For the temporal direction, we utilize the favorable boundary value methods owing to their advantageous stability properties. In addition, the resulting large sparse system can be solved rapidly by the GMRES method with a circulant Strang-type preconditioner. Numerical results demonstrate the high order accuracy of our scheme and the efficiency of the preconditioned GMRES method.

We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties. An important feature of the methods is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations. Although the stability of the method has been established, for the homogeneous Stokes equations, the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space. This makes it much simpler and more attractive. The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations.