Ole H. Hald

Convergence of Vortex methods for Euler's equations, III

We prove the convergence of a large class of vortex methods for two-dimensional incompressible, inviscid flows with Holder continuous initial data. We present several infinite order methods and est...

Optimal prediction with memory

Abstract Optimal prediction methods estimate the solution of nonlinear time-dependent problems when that solution is too complex to be fully resolved or when data are missing. The initial conditions for the unresolved components of the solution are drawn from a probability distribution, and their effect on a small set of variables that are actually computed is evaluated via statistical projection. The formalism resembles the projection methods of irreversible statistical mechanics, supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation, the orthogonal dynamics equation, needed to evaluate a non-Markovian memory term. The result of the computations is close to the best possible estimate that can be obtained given the partial data. We present the constructions in detail together with several useful variants, provide simple examples, and point out the relation to the fluctuation–dissipation formulas of statistical physics.

Inverse eigenvalue problems for Jacobi matrices

Abstract It is proved that a real symmetric tridiagonal matrix with positive codiagonal elements is uniquely determined by its eigenvalues and the eigenvalues of the largest leading principal submatrix, and can be constructed from these data. The matrix depends continuously on the data, but because of rounding errors the investigated algorithm might in practice break down for large matrices.

Solution of inverse nodal problems

The authors show that the coefficients in a second-order differential equation can be determined from the positions of the nodes for the eigenfunctions. They prove uniqueness results, derive approximate solutions, give error bounds and present numerical experiments.

On a Newton-Moser type method

We prove that a variant of Mosers iterative method for solving nonlinear equations is quadratically convergent and give error bounds. We estimate the amount of arithmetic for the method and compare it to Newtons method. Finally we use the method to solve a problem with small divisors.

Inverse problems: recovery of BV coefficients from nodes

We consider the Sturm-Liouville problem on a finite interval with Dirichlet boundary conditions. Let the elastic modulus and the density be of bounded variation. Results for both the forward problem and the inverse problem are established. For the forward problem, new bounds are established for the eigenfrequencies. The bounds are sharp. For the inverse problem, it is shown that the elastic modulus is uniquely determined, up to one arbitrary constant, by a dense subset of the nodes of the eigenfunctions when the density is known. Similarly it is shown that the density is uniquely determined, up to one arbitrary constant, by a dense subset of the nodes of the eigenfunctions when the elastic modulus is known. Algorithms for finding piecewise constant approximations to the unknown elastic modulus or density are established and are shown to converge to the unknown function at every point of continuity. Results from numerical calculations are presented.

Convergence of fourier methods for Navier-Stokes equations

Abstract The Fourier-Galerkin method is used to simulate fluid flows in two and three dimensions, on domains with periodic boundary conditions. It is proved that the numerical solution converges towards the solution of Navier-Stokes equations. The rate of convergence depends on the smoothness of the mathematical solution. Finally, it is shown that the Fourier-Galerkin method can be interpreted as a projection method. This observation may lead to more sophisticated convergence proofs.

Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions

The “t-model” for dimensional reduction is applied to the estimation of the rate of decay of solutions of the Burgers equation and of the Euler equations in two and three space dimensions. The model was first derived in a statistical mechanics context, but here we analyze it purely as a numerical tool and prove its convergence. In the Burgers case, the model captures the rate of decay exactly, as was previously shown. For the Euler equations in two space dimensions, the model preserves energy as it should. In three dimensions, we find a power law decay in time and observe a temporal intermittency.

Inverse nodal problems : finding the potential from nodal lines

Introduction Separation of eigenvalues for the Laplacian Eigenvalues for the finite dimensional problem Eigenfunctions for the finite dimensional problem Eigenvalues for

Constants of motion in models of two‐dimensional turbulence

- \Delta + q