Conservation laws and symmetries of difference equations

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Probabilistic state estimation in regimes of nonlinear error growth

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences, 2005.Includes bibliographical references (p. 273-286).State estimation, or data assimilation as it is often called, is a key component of numerical weather prediction (NWP). Nearly all implementable methods of state estimation suitable for NWP are forced to assume that errors remain in regimes of linear error growth and retain distributions of Gaussian uncertainty, yet nonlinear systems like the atmosphere can readily allow regimes of nonlinear error growth and, in turn, produce distributions of non- Gaussian uncertainty. State-of-the-art, ensemble-based methods of state estimation suitable for NWP are examined to gauge the consequences and relevance of violating the linear error growth assumption. For quite generic sources of non-Gaussian uncertainty, the methods are observed to fail, as they must, and the obtained analyses become probabilistically unreliable before becoming inaccurate. The mispositioning of coherent features is identified as a specific, geophysically relevant source of non-Gaussian uncertainty that can easily cause the state-of-the-art methods of state estimation to fail. However, an understanding of relevant phenomenology sometimes allows these same methods to remain successful owing to an available redefinition of the involved errors. The redefinition is phrased as an alternative error model. It is recognized and exploited that non-Gaussian additive Eulerian errors can come from Gaussian Lagrangian position errors. A two-step, augmented state vector approach is developed that is suitable for use with coherent features and that relies only on implementable methods of state estimation.(cont.) By combining the dual Eulerian and Lagrangian state information into one vector, an ensemble can approximate their covariance, thus allowing each component's uncertainty to be reduced. The first step of the two-step approach reduces the feature position errors in an effort to render the residual additive errors Gaussian, thereby allowing the second step of an implementable state estimation method to proceed successfully. Philosophically, the two-step approach uses physical knowledge of the problem (as phrased by the error model) to compensate for neglected important non-Gaussian uncertainty structure in the state estimation process. The proposed two-step approach successfully allows use of implementable methods of state estimation to obtain probabilistically reliable analyses in regimes of nonlinear error growth, something unavailable using current standards.by W. Gregory Lawson.Ph.D

Geometry and dynamics in Hamiltonian lattices

E. Fermi, J. Pasta and S. Ulam introduced the Fermi-Pasta-Ulam lattice in the 1950s as a classical mechanical model for a mono-atomic crystal or a one-dimensional continuum. The model consisted of a discrete number of equal point masses that interact with their nearest neighbours only. On the basis of statistical mechanics, they expected that when the interparticle forces were anharmonic, the lattice would reach a thermal equilibrium. This means that averaged over time, its initial energy would be equipartitioned among all its Fourier modes. The famous computer experiment that they performed in Los Alamos in 1955, was intended to investigate in what manner and at what time-scale the equipartitioning would take place. The result was astonishing: the lattice did not come close to thermal equilibrium, but behaved more or less quasi-periodically. Only when the initial energy was larger than a certain threshold, did the lattice seem to `thermalise'. This paradox is nowadays known as the `Fermi-Pasta-Ulam problem'. The observations of Fermi, Pasta and Ulam were a great impulse for nonlinear dynamics. One possible explanation for the quasi-periodic behaviour of the FPU system, is based on the Kolmogorov-Arnol'd-Moser theorem, which states that most of the invariant Lagrangean tori of a Liouville integrable Hamiltonian system survive under small perturbations. It is required though for the KAM theorem that the integrable system satisfies a nondegeneracy condition. Unfortunately, it has for a long time been unclear how the Fermi-Pasta-Ulam lattice can be viewed as a perturbation of a nondegenerate integrable system. Nishida in 1971 and Sanders in 1977, investigated a Birkhoff normal form for the FPU lattice. Under the assumption of a rather strong nonresonance condition on the linear frequencies $\omega_k = 2 \sin (k\pi/n)$ of the lattice, they showed that this Birkhoff normal form is integrable and nondegenerate. This means that the KAM theorem can indeed be applied. The problem is of course that their required nonresonance condition is usually not met. This leaves a large gap in the proofs. The new idea in this thesis is to incorporate the discrete symmetries of the lattice in the argument. It turns out that this enables us to show that the nonresonance condition of Nishida and Sanders is not needed: every resonance is overruled by a symmetry. Hence the Birkhoff normal form is integrable and this proves the applicability of the KAM theorem. Moreover, much attention is paid to the analysis of exact and approximate solutions of the lattice equations. New exact solutions and invariant manifolds are found in the fixed point sets of the symmetries of the lattice. Also, the Birkhoff normal reveals approximate integrals and solutions in the low energy domain of the phase space. The analysis makes use of invariant theory and singular reduction. One of the conclusions is that the lattice with an even number of particles contains travelling wave solutions that change their direction. Moreover the integrable normal form contains nontrivial monodromy, meaning that it does not allow global action-angle variable