Application of Nonsmooth Optimization in Ensemble and Variational Data Assimilation Problems
Abstract: Cost functions formulated in four-dimensional variational data assimilation (4DVAR) are nonsmooth in the presence of discontinuous physical processes (i.e., the presence of on-off switches in NWP models). The adjoint model integration produces values of subgradients, instead of gradients, of these cost functions with respect to the models control variables at discontinuous points. Minimization of these cost functions using conventional optimization algorithms may encounter diffculties. Nondifferentiable optimization algorithms were able to find the true minima in cases where the differentiable minimization failed. Optimal control of the 1-D Riemann problem of Euler equations was studied, with the initial values for pressure and density as control parameters. The cost functional employs either distributed observations in time or observations calculated at the end of the assimilation window. Smooth and nonsmooth optimization methods employ the numerical gradient (respectively, a subgradient) of the cost functional, obtained from adjoint of the discrete forward model. The numerical flow obtained with the optimal initial conditions obtained from application of nonsmooth minimization matches very well with the observations. The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. Derivation reveals that a new non-differentiable minimization method can be defined as a generalization of the gradient-based unconstrained methods, such as the preconditioned conjugate-gradient and quasi-Newton methods. In the new minimization algorithm the vector of first order increments of the cost function is defined as a generalized gradient, while the symmetric matrix of second order increments of the cost function is defined as a generalized Hessian matrix. For differentiable observation operators, the minimization algorithm reduces to the standard gradient-based form. The non-differentiable aspect of the MLEF algorithm is illustrated with one-dimensional Burgers model. The MLEF algorithm has a robust performance, producing satisfactory results for tested non-differentiable observation operators.
High-order Methods for Centroidal voronoi Tessellations on a Sphere
Abstract: It is well known that voronoi tessellations and their dual Delaunay tessellations are useful tools in the construction of unstructured mesh. Centroidal voronoi tessellations and their dual centroidal voronoi Delaunay triangulations can help with grid generation and optimization and nodal distribution in meshfree computing. These could be very useful in atmosphere and ocean modeling. In this talk, we will discuss high-order methods on these grids.
Efficient and Stable Spectral Methods for Dynamics of Bose-Einstein Condensation
Abstract: In this talk, an efficient and accurate spectral method will be presented to simulate the dynamics of non-rotating Bose-Einstein condensates (BECs). This method has higher order accuracy in both space and time. Then this method is also extended to study the dynamics of rotating BECs based on the Gross-Pitaevskii equation (GPE) with an angular momentum rotation term. Finally, some numerical results are presented to show the efficiency of these methods.
Adaptive Anisotropic Meshes for Steady-State Convection Dominated Problems
Abstract: Obtaining accurate solutions for the convection-diffusion equation is challenging due to the presence of layers when convection dominates diffusion. To solve this problem we have designed an adaptive algorithm which optimizes the alignment of anisotropic meshes with a numerical solution. Three main ingredients are used. First, a streamline upwind Petrov Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor. Our algorithm has been tested on a variety of 2-dimensional examples. The results show better accuracy and convergence rates at lower computational cost when compared with those of methods using isotropic meshes.