Abstract: We show how to simulate numerically the evolution of 1D quantum systems under dissipation as well as in thermal equilibrium. The method applies to both finite and inhomogeneous systems, and it is based on two ideas: (a) a representation for density operators which extends that of matrix product states to mixed states; (b) an algorithm to approximate the evolution (in real or imaginary time) of matrix product states which is variational.

Abstract: In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of O(N) for N grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.

Abstract: We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.

Abstract: We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the com- putations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method gener- ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The sec- ond method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h -o rp-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.

Abstract: We investigate a method for the numerical solution of the nonlinear fractional differential equation D * α y(t)=f(t,y(t)), equipped with initial conditions y (k)(0)=y 0 (k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.

Abstract: 1 Mean-square approximation for stochastic differential equations.- 2 Weak approximation for stochastic differential equations.- 3 Numerical methods for SDEs with small noise.- 4 Stochastic Hamiltonian systems and Langevin-type equations.- 5 Simulation of space and space-time bounded diffusions.- 6 Random walks for linear boundary value problems.- 7 Probabilistic approach to numerical solution of the Cauchy problem for nonlinear parabolic equations.- 8 Numerical solution of the nonlinear Dirichlet and Neumann problems based on the probabilistic approach.- 9 Application of stochastic numerics to models with stochastic resonance and to Brownian ratchets.- A Appendix: Practical guidance to implementation of the stochastic numerical methods.- A.1 Mean-square methods.- A.2 Weak methods and the Monte Carlo technique.- A.3 Algorithms for bounded diffusions.- A.4 Random walks for linear boundary value problems.- A.5 Nonlinear PDEs.- A.6 Miscellaneous.- References.

Abstract: Preface 1. Introduction to MATLAB 2. Linear equations 3. Interpolation 4. Zeros and roots 5. Least squares 6. Quadrature 7. Ordinary differential equations 8. Fourier analysis 9. Random numbers 10. Eigenvalues and singular values 11. Partial differential equations Bibliography Index.

Abstract: Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies. The present paper modifies the adaptive method of Morin, Nochetto, and Siebert [17] for solving the Laplace equation with piecewise linear elements on domains in ℝ2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the H1 norm). Namely, it is shown that whenever s>0 and the solution u is such that for each n≥1, it can be approximated to accuracy O(n−s) in the energy norm by a continuous, piecewise linear function on a triangulation with n cells (using complete knowledge of u), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order O(n) for each n≥1. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation.

Abstract: In this paper, we present a continuous normalized gradient flow (CNGF) and prove its energy diminishing property, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the CNGF. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD) method, the other is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for the linear case and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g., Crank--Nicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving the energy diminishing property of the CNGF. Numerical results in one, two, and three dimensions with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a far-blue detuned Gaussian laser beam, are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the CNGF and its BEFD discretization can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.

Abstract: Part One: Modeling, Computers, and Error Analysis Chapter 1: Mathematical Modeling, Numerical Methods and Problem Solving Chapter 2: MATLAB Fundamentals Chapter 3: Programming with MATLAB Chapter 4: Roundoff and Truncation Errors Part Two: Roots and Optimization Chapter 5: Roots: Bracketing Methods Chapter 6: Roots: Open Methods Chapter 7: Optimization Part Three: Linear Systems Chapter 8: Linear Algebraic Equations and Matrices Chapter 9: Gauss Elimination Chapter 10: LU Factorization Chapter 11: Matrix Inverse and Condition Chapter 12: Iterative Methods Chapter 13: Eigenvalues Part Four: Curve Fitting Chapter 14: Linear Regression Chapter 15: General Linear Least-Squares and Nonlinear Regression Chapter 16: Fourier Analysis Chapter 17: Polynomial Interpolation Chapter 18: Splines and Piecewise Interpolation Part Five: Integration and Differentiation Chapter 19: Numerical Integration Formulas Chapter 20: Numerical Integration of Functions Chapter 21: Numerical Differentiation Part Six: Ordinary Differential Equations Chapter 22: Initial-Value Problems Chapter 23: Adaptive Methods and Stiff Systems Chapter 24: Boundary-Value Problems Appendix A: MATLAB Built-in Functions Appendix B: MATLAB M-file Functions Bibliography Index

Abstract: For the positive semidefinite system of linear equations of a block two-by-two structure, by making use of the Hermitian/skew-Hermitian splitting iteration technique we establish a class of preconditioned Hermitian/skew-Hermitian splitting iteration methods. Theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly. Numerical examples further confirm the correctness of the theory and the effectiveness of the method.

Abstract: The Lagrange representation of the interpolating polynomial can be rewritten in two more computationally attractive forms: a modified Lagrange form and a barycentric form. We give an error analysis of the evaluation of the interpolating polynomial using these two forms. The modified Lagrange formula is shown to be backward stable. The barycentric formula has a less favourable error analysis, but is forward stable for any set of interpolating points with a small Lebesgue constant. Therefore the barycentric formula can be significantly less accurate than the modified Lagrange formula only for a poor choice of interpolating points. This analysis provides further weight to the argument of Berrut and Trefethen that barycentric Lagrange interpolation should be the polynomial interpolation method of choice.

Abstract: A natural mixed-element approach for the Stokes equations in the velocity-pressure formulation would approximate the velocity by continuous piecewise-polynomials and would approximate the pressure by discontinuous piecewise-polynomials of one degree lower. However; many such elements are unstable in 2D and 3D. This paper is devoted to proving that the mixed finite elements of this P k -P k-1 type when k > 3 satisfy the stability condition-the Babuska-Brezzi inequality on macro-tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra. This type of mesh simplifies the implementation since it has no restrictions on the initial mesh. The new element also suits the multigrid method.

Abstract: Population balance equations have been used to model a wide range of processes including polymerization, crystallization, cloud formation, and cell dynamics. Rather than developing new algorithms specific to population balance equations, it is proposed to adapt the high-resolution finite volume methods developed for compressible gas dynamics, which have been applied to aerodynamics, astrophysics, detonation waves, and related fields where shock waves occur. High-resolution algorithms are presented for simulating multidimensional population balance equations with nucleation and size-dependent growth rates. For sharp distributions, these high-resolution algorithms can achieve improved numerical accuracy with orders-of-magnitude lower computational cost than other finite difference and finite volume algorithms. The algorithms are implemented in the ParticleSolver software package, which is applied to batch and continuous processes with one and multiple internal coordinates. © 2004 American Institute of Chemical Engineers AIChE J, 50: 2738 –2749, 2004

Abstract: A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and L 2 estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.

Abstract: Introduction and Overview Review of Basic Concepts and Results from Theoretical Linear Algebra Fundamental Tools and Concepts from Numerical Linear Algebra Canonical Forms Obtained via Orthogonal Transformations Linear State Space Models and Solutions of the State Equations Contollability, Observability and Distance to Uncontrollability Stability, Inertia and Robust Stability Numerical Solutions and Conditioning of Lyapunov and Sylvester Equations Realization and Subspace Identification Feedback Stabilization, Eigenvalue Assignment and Optimal Control Numerical Methods and Conditioning of the Eigenvalue Assignment Problems State Estimation Numerical Solutions and Conditioning of Algebraic Riccati Equations Internal Balancing and Model Reduction Large-Scale Matrix Computations in Control: Krylov Subspace Methods Numerical Methods for Matrix-Second-Order Control Systems Existing Software for Control Systems Design and Analysis

Abstract: Fundamental Physical and Model Equations.- The Fluid Flow Equations.- The Viscous Fluid Flow Equations.- Curvilinear Coordinates and Transformed Equations.- Overview of Various Formulations and Model Equations.- Basic Principles in Numerical Analysis.- Time Integration Methods.- Numerical Linear Algebra.- Solution Approaches.- Compressible and Preconditioned-Compressible Solvers.- The Artificial Compressibility Method.- Projection Methods: The Basic Theory and the Exact Projection Method.- Approximate Projection Methods.- Modern High-Resolution Methods.- to Modern High-Resolution Methods.- High-Resolution Godunov-Type Methods for Projection Methods.- Centered High-Resolution Methods.- Riemann Solvers and TVD Methods in Strict Conservation Form.- Beyond Second-Order Methods.- Applications.- Variable Density Flows and Volume Tracking Methods.- High-Resolution Methods and Turbulent Flow Computation.