This paper describes how to specify probability models for data analysis via a backward induction procedure. The new approach yields coherent, priorfree uncertainty assessment. The backward induction approach is first demonstrated on two familiar models — the Bernoulli distribution and the Gaussian distribution — to compare the resulting specifications to their standard counterparts arising as ...

We present an overview of recent work together with J. Hoffman, A. Logg and R. Scott developing a new approach towards resolving the classical paradox of irreversibility in formally reversible Hamiltonian systems such as inviscid fluid models and particle models. We base our solution on finite precision computation in the form of General Galerkin G2, instead of statistical mechanics, which is t...

We study the backward Euler fully discrete mixed finite element method for time-dependent Schrödinger equation; error result of solution is obtained in L2-norm with order O(τ+hk+1). Then, a two-grid presented scheme. Using this method, we solve original problem on much coarser grid and elliptic equations fine grid. In addition, also O(τ+hk+1+Hk+2). The numerical experiment provided to demonstra...

We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin (IPDG) methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l(H) and l(L) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin (SIPG) scheme with the implicit θ-schemes in time...

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order erro...

The long time numerical approximation of the parabolic p-Laplacian problem with a time-independent forcing term and sufficiently smooth initial data is studied. Convergence and stability results which are uniform for t ∈ [0,∞) are established in the L2, W 1,p norms for the backward Euler and the Crank–Nicholson schemes with the finite element method (FEM). This result extends the existing unifo...

he nonFickian flow of fluid in porous media is complicated by the history effect which characterizes various mixing length growth of the flow, and can be modeled by an integro-differential equation. This paper studies a backward Euler scheme for the mixed finite element approximate solution of such problems with non-smooth initial data. A new regularity result is derived for the model problem, ...

We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e., ut = νuxx−unux+mu+h(x)). We show existence of an absorbing ball in L2[0,1] and uniqueness of steady state solutions for all integer n ≥ 1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to...

Tasks. Implement the implicit finite difference scheme (i.e., backward Euler scheme) and the Crank-Nicolson scheme and compute the option prices in this way. Moreover, study the empirical convergence rates. You need to choose the following numerical parameters: • The truncation values for the infinite domain xmin < 0 < xmax. To simplify the analysis, you should use the same values for all the r...

This paper considers the stability of both continuous and discrete time-varying linear systems. Stability estimates are obtained in either case in terms of the Lipschitz constant for the governing matrices and the assumed uniform decay rate of the corresponding frozen time linear systems. The main techniques used in the analysis are comparison methods, scaling and the application of continuous ...