Manifold interpolation and model reduction

One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of model reduction. Moreover, the principal approaches to data interpolation and Taylor-like extrapolation on matrix manifolds are outlined and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model Order Reduction

Grid sensitivity for aerodynamic optimization and flow analysis

After reviewing relevant literature, it is apparent that one aspect of aerodynamic sensitivity analysis, namely grid sensitivity, has not been investigated extensively. The grid sensitivity algorithms in most of these studies are based on structural design models. Such models, although sufficient for preliminary or conceptional design, are not acceptable for detailed design analysis. Careless grid sensitivity evaluations, would introduce gradient errors within the sensitivity module, therefore, infecting the overall optimization process. Development of an efficient and reliable grid sensitivity module with special emphasis on aerodynamic applications appear essential. The organization of this study is as follows. The physical and geometric representations of a typical model are derived in chapter 2. The grid generation algorithm and boundary grid distribution are developed in chapter 3. Chapter 4 discusses the theoretical formulation and aerodynamic sensitivity equation. The method of solution is provided in chapter 5. The results are presented and discussed in chapter 6. Finally, some concluding remarks are provided in chapter 7

Gaussian integration with rescaling of abscissas and weights

An algorithm for integration of polynomial functions with variable weight is considered. It provides extension of the Gaussian integration, with appropriate scaling of the abscissas and weights. Method is a good alternative to usually adopted interval splitting.Comment: 14 pages, 5 figure

Stochastic collocation on unstructured multivariate meshes

Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification (UQ) community. Techniques for least-squares regularization, compressive sampling recovery, and interpolatory reconstruction are becoming standard tools used in a variety of applications. Selection of a collocation mesh is frequently a challenge, but methods that construct geometrically "unstructured" collocation meshes have shown great potential due to attractive theoretical properties and direct, simple generation and implementation. We investigate properties of these meshes, presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure

STiC -- A multi-atom non-LTE PRD inversion code for full-Stokes solar observations

The inference of the underlying state of the plasma in the solar chromosphere remains extremely challenging because of the nonlocal character of the observed radiation and plasma conditions in this layer. Inversion methods allow us to derive a model atmosphere that can reproduce the observed spectra by undertaking several physical assumptions. The most advanced approaches involve a depth-stratified model atmosphere described by temperature, line-of-sight velocity, turbulent velocity, the three components of the magnetic field vector, and gas and electron pressure. The parameters of the radiative transfer equation are computed from a solid ground of physical principles. To apply these techniques to spectral lines that sample the chromosphere, NLTE effects must be included in the calculations. We developed a new inversion code STiC to study spectral lines that sample the upper chromosphere. The code is based the RH synthetis code, which we modified to make the inversions faster and more stable. For the first time, STiC facilitates the processing of lines from multiple atoms in non-LTE, also including partial redistribution effects. Furthermore, we include a regularization strategy that allows for model atmospheres with a complex stratification, without introducing artifacts in the reconstructed physical parameters, which are usually manifested in the form of oscillatory behavior. This approach takes steps toward a node-less inversion, in which the value of the physical parameters at each grid point can be considered a free parameter. In this paper we discuss the implementation of the aforementioned techniques, the description of the model atmosphere, and the optimizations that we applied to the code. We carry out some numerical experiments to show the performance of the code and the regularization techniques that we implemented. We made STiC publicly available to the community.Comment: Accepted for publication in Astronomy & Astrophysic

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion

In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a series expansion of the deviation of its logarithm from that of a Gaussian distribution. Through this procedure, dubbed {\em exponent expansion}, the transition probability is obtained as a power series in $\Delta t$. This becomes asymptotically exact if an increasing number of terms is included, and provides remarkably accurate results even when truncated to the first few (say 3) terms. The coefficients of such expansion can be determined straightforwardly through a recursion, and involve simple one-dimensional integrals. We present several examples of financial interest, and we compare our results with the state-of-the-art approximation of discretely sampled diffusions [A\"it-Sahalia, {\it Journal of Finance} {\bf 54}, 1361 (1999)]. We find that the exponent expansion provides a similar accuracy in most of the cases, but a better behavior in the low-volatility regime. Furthermore the implementation of the present approach turns out to be simpler. Within the functional integration framework the exponent expansion allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. This is illustrated with the application to simple path-dependent interest rate derivatives. Finally we discuss how these results can also be used to increase the efficiency of numerical (both deterministic and stochastic) approaches to derivative pricing.Comment: 28 pages, 7 figure

A diffusion-driven Characteristic Mapping method for particle management

We present a novel particle management method using the Characteristic Mapping framework. In the context of explicit evolution of parametrized curves and surfaces, the surface distribution of marker points created from sampling the parametric space is controlled by the area element of the parametrization function. As the surface evolves, the area element becomes uneven and the sampling, suboptimal. In this method we maintain the quality of the sampling by pre-composition of the parametrization with a deformation map of the parametric space. This deformation is generated by the velocity field associated to the diffusion process on the space of probability distributions and induces a uniform redistribution of the marker points. We also exploit the semigroup property of the heat equation to generate a submap decomposition of the deformation map which provides an efficient way of maintaining evenly distributed marker points on curves and surfaces undergoing extensive deformations