We consider a stochastic linear elliptic boundary value problem whose stochastic coefficient a(x, omega) is expressed by a finite number N-KL of mutually independent random variables, and transform this problem into a deterministic one. We show how to choose a suitable N-KL which should be as low as possible for practical reasons, and we give the a priori estimates for modeling error when a(x, omega) is completely known. When a random function a(x, omega) is selected to fit the experimental data, we address the estimation of the error in this selection due to insufficient experimental data. We present a simple model problem, simulate the experiments, and give the numerical results and error estimates.

We prove an a posteriori error estimate for an inverse acoustic scattering problem, where the objective is to reconstruct an unknown wave speed coefficient inside a body from measured wave reflection data in time on parts of the surface of the body. The inverse problem is formulated as a problem of finding a zero of a Jacobian of a Lagrangian. The a posterori error estimate couples residuals of the computed solution to weights the reconstruction reflecting the sensitivity of the reconstruction obtained by solving an associated linaerized problem for the Hessian of the Lagrangian. We show concrete examples of reconstrution including a posteriori error estimation.

3.

Henning, Patrick

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.

This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in L∞(L2) under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.

In this paper we study a subgrid model based on extrapolation of a modeling residual, in the case of a linear convection-diffusion-reaction problem Lu=f in two dimensions. The solution u to the exact problem satisfies an equation L_{h}u=[f]^{h}+F_{h}(u), where L_{h} is the operator used in the computation on the finest computational scale h, [f]^{h} is the approximation of f on the scale h, and F_{h}(u) is a modeling residual, which needs to be modeled. The subgrid modeling problem is to compute approximations of F_{h}(u) without using finer scales than h. In this study we model F_{h}(u) by extrapolation from coarser scales than h, where F_{h}(u) is directly computed with the finest scale h as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a mesh of size less than h/4.

In this paper, we describe an incompressible Unified Continuum(UC) model in Euler (laboratory) coordinates with a moving mesh for tracking the fluid-structure interface as part of the discretization, allowing simple and general formulation and efficient computation. The model consists of conservation equations for mass and momentum, a phase convection equation and a Cauchy stress and phase variable theta as data for defining material properties and constitutive laws. We target realistic 3D turbulent fluid-structure interaction (FSI) applications, where we show simulation results of a flexible flag mounted in the turbulent wake behind a cube as a qualitative test of the method, and quantitative results for 2D benchmarks, leaving adaptive error control for future work. We compute piecewise linear continuous discrete solutions in space and time by a general Galerkin (G2) finite element method (FEM). We introduce and compensate for mesh motion by defining a local arbitrary Euler-Lagrange (ALE) map on each space-time slab as part of the discretization, allowing a sharp phase interface given by theta on cell facets. The Unicorn implementation is published as part of the FEniCS Free Software system for automation of computational mathematical modeling. Simulation results are given for a 2D stationary convergence test, indicating quadratic convergence of the displacement, a simple 2D Poiseuille test for verifying correct treatment of the fluid-structure interface, showing quadratic convergence to the exact drag value, an established 2D dynamic flag benchmark test, showing a good match to published reference solutions and a 3D turbulent flag test as indicated above.

We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q greater than or equal to 0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.

Computational modeling of dynamical systems2005In: Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, Vol. 15, no 3, p. 471-481Article in journal (Refereed)

Abstract [en]

In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10(-9) computed over the time interval [0, 100], and a lattice consisting of large and small point masses.

8.

Szepessy, Anders

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a KacZwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio M of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M^{-1/2}) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small O(M ^{-1/2}) friction and o(M^{-1/2}) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperature, motivated by a stability and consistency argument. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuationdissipation relation.