A numerical comparison of discrete Kalman filtering algorithms: An orbit determination case study

The numerical stability and accuracy of various Kalman filter algorithms are thoroughly studied. Numerical results and conclusions are based on a realistic planetary approach orbit determination study. The case study results of this report highlight the numerical instability of the conventional and stabilized Kalman algorithms. Numerical errors associated with these algorithms can be so large as to obscure important mismodeling effects and thus give misleading estimates of filter accuracy. The positive result of this study is that the Bierman-Thornton U-D covariance factorization algorithm is computationally efficient, with CPU costs that differ negligibly from the conventional Kalman costs. In addition, accuracy of the U-D filter using single-precision arithmetic consistently matches the double-precision reference results. Numerical stability of the U-D filter is further demonstrated by its insensitivity of variations in the a priori statistics

Non-negative matrix factorization for self-calibration of photometric redshift scatter in weak lensing surveys

Photo-z error is one of the major sources of systematics degrading the accuracy of weak lensing cosmological inferences. Zhang et al. (2010) proposed a self-calibration method combining galaxy-galaxy correlations and galaxy-shear correlations between different photo-z bins. Fisher matrix analysis shows that it can determine the rate of photo-z outliers at a level of 0.01-1% merely using photometric data and do not rely on any prior knowledge. In this paper, we develop a new algorithm to implement this method by solving a constrained nonlinear optimization problem arising in the self-calibration process. Based on the techniques of fixed-point iteration and non-negative matrix factorization, the proposed algorithm can efficiently and robustly reconstruct the scattering probabilities between the true-z and photo-z bins. The algorithm has been tested extensively by applying it to mock data from simulated stage IV weak lensing projects. We find that the algorithm provides a successful recovery of the scatter rates at the level of 0.01-1%, and the true mean redshifts of photo-z bins at the level of 0.001, which may satisfy the requirements in future lensing surveys.Comment: 12 pages, 6 figures. Accepted for publication in ApJ. Updated to match the published versio

Implication of the B→(ρ,ω)γB \to (\rho, \omega) \gamma Branching Ratios for the CKM Phenomenology

We study the implication of the recent measurement by the BELLE collaboration of the averaged branching fraction $\bar B_{exp} [B \to (\rho, \omega) \gamma] = (1.8^{+0.6}_{-0.5} \pm 0.1) \times 10^{-6}$ for the CKM phenomenology. Combined with the averaged branching fraction $\bar B_{exp} (B \to K^* \gamma) = (4.06 \pm 0.26) \times 10^{-5}$ measured earlier, this yields $\bar R_{exp} [(\rho, \omega) \gamma/K^* \gamma] = (4.2 \pm 1.3)$ for the ratio of the two branching fractions. Updating earlier theoretical analysis of these decays based on the QCD factorization framework, and constraining the CKM-Wolfenstein parameters from the unitarity fits, our results yield $\bar B_{th} [B \to (\rho, \omega) \gamma] = (1.38 \pm 0.42) \times 10^{-6}$ and $\bar R_{th} [(\rho, \omega) \gamma/K^* \gamma] = (3.3 \pm 1.0)$, in agreement with the BELLE data. Leaving instead the CKM-Wolfenstein parameters free, our analysis gives (at 68% C.L.) $0.16\leq |V_{td}/V_{ts}| \leq 0.29$, which is in agreement with but less precise than the indirect CKM-unitarity fit of the same, $0.18 \leq |V_{td}/V_{ts}| \leq 0.22$. The isospin-violating ratio in the $B \to \rho \gamma$ decays and the SU(3)-violating ratio in the $B_d^0 \to (\rho^0, \omega) \gamma$ decays are presented together with estimates of the direct and mixing-induced CP-asymmetries in the $B \to (\rho,\omega) \gamma$ decays within the SM. Their measurements will overconstrain the angle $\alpha$ of the CKM-unitarity triangle.Comment: 21 pages, 3 figures. Included a discussion of model-dependent estimates of the long-distance/rescattering contributions in radiative B-decays; added a reference. Version accepted for publication in Physics Letters

Relaxed Majorization-Minimization for Non-smooth and Non-convex Optimization

We propose a new majorization-minimization (MM) method for non-smooth and non-convex programs, which is general enough to include the existing MM methods. Besides the local majorization condition, we only require that the difference between the directional derivatives of the objective function and its surrogate function vanishes when the number of iterations approaches infinity, which is a very weak condition. So our method can use a surrogate function that directly approximates the non-smooth objective function. In comparison, all the existing MM methods construct the surrogate function by approximating the smooth component of the objective function. We apply our relaxed MM methods to the robust matrix factorization (RMF) problem with different regularizations, where our locally majorant algorithm shows advantages over the state-of-the-art approaches for RMF. This is the first algorithm for RMF ensuring, without extra assumptions, that any limit point of the iterates is a stationary point.Comment: AAAI1