A short note on a generalization of the Givens transformation

A new transformation, a generalization of the Givens rotation, is introduced here. Its properties are studied. This transformation has some free parameters, which can be chosen to attain pre-established conditions. Some special choices of those parameters are discussed, mainly to improve numerical properties of the transformation. © 2013 Elsevier Ltd. All rights reserved.A new transformation, a generalization of the Givens rotation, is introduced here. Its properties are studied. This transformation has some free parameters, which can be chosen to attain pre-established conditions. Some special choices of those parameters6615661CAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOSEM INFORMAÇÃOSEM INFORMAÇÃOBjörck, A., (1996) Numerical Methods for Least Squares Problems, , SIAM PhiladelphiaGolub, G.H., Loan, C.V., (1996) Matrix Computation, , 3rd Edition The Johns Hopkins University Press Baltimore and LondonStewart, G.W., (1973) Introduction to Matrix Computations, , Academic Press New YorkStewart, G.W., (1998) Matrix Algorithms I: Basic Decompositions, , SIAM PhiladelphiaBai, Z., Demmel, J., Dongarra, J., Ruhe, A., Van Der Vorst, H., (2000) Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, , SIAM PhiladelphiaGolub, G.H., (1999) Numerical Methods for Large Scale Eigenvalue Problems, Teaching Note, , Stanford UniversityParlette, B., (1997) The Symmetric Eigenvalue Problem, 20. , Reprinted as Classics in Applied Mathematics SIAM PhiladelphiaStewart, G.W., (2001) Matrix Algorithms II: Eigensystems, , SIAM PhiladelphiaGerck, E., D'Oliveira, A.B., Continued fraction calculation of the eigenvalues of tridiagonal matrices arising from the Schrödinger equation (1980) Journal of Computational and Applied Mathematics, 6, pp. 81-82Golub, G.H., Robertson, N.T., A generalized Bairstow algorithm (1967) Communication on Applied and Computational Mathematics, 10, pp. 371-373Im, Y., Ri, S., An algorithm for the calculation of eigenvalues of tridiagonal matrices using QD-transformations and the LR (RL) method (1995) Su-hak: Academy of Science of the People's Democratic Republic of Korea, 2, pp. 12-15Kulkarni, D., Schmidt, D., Tsui, S.K., Eigenvalues of tridiagonal pseudo-toeplitz matrices (1999) Linear Algebra and Its Applications, 297Pasquini, L., Pavani, R., Computing the eigenvalues of non-normal tridiagonal matrices (1995) Rendiconti Del Seminario Matematico e Fisico di Milano, 65, pp. 109-138Veselic, K., On real eigenvalues of real tridiagonal matrices (1979) Linear Algebra and Its Applications, 27, pp. 167-171Golub, G.H., Yuan, J.Y., Biloti, R., Ramos, J., Optimal generalized Householder transformation with application (2005) Tech. Rep., Universidade Federal Do Paraná, , BrazilLabudde, C.D., The reduction of an arbitrary real square matrix to tridiagonal form using similarity transformations (1963) Mathematics of Computation, 17, pp. 433-437Stathopolous, A., Saad, Y., Wu, K., Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods (1998) SIAM Journal on Scientific Computing, 19, pp. 227-24

Inexact Newton Methods for Solving Generalized Equations on Riemannian Manifolds

The local convergence of an inexact Newton method is studied for solving generalized equations on Riemannian manifolds by using the metric regularity property which is explored as well. Under suitable conditions and without any additional geometric assumptions, local convergence results with linear and quadratic rate and a semi-local convergence result are obtained for the proposed method. Finally, the theory can be applied to problems of finding a singularity of the sum of two vector fields.Comment: 34 page

Improved optimization methods for image registration problems

In this paper, we propose new multilevel optimization methods for minimizing continuously differentiable functions obtained by discretizing models for image registration problems. These multilevel schemes rely on a novel two-step Gauss-Newton method, in which a second step is computed within each iteration by minimizing a quadratic approximation of the objective function over a certain two-dimensional subspace. Numerical results on image registration problems show that the proposed methods can outperform the standard multilevel Gauss-Newton method

An Adaptive Cubic Regularization quasi-Newton Method on Riemannian Manifolds

A quasi-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most ${\cal O} (\epsilon_g^{-3/2})$ iterations to achieve a gradient smaller than $\epsilon_g$ for given $\epsilon_g$, and at most $\mathcal O(\max\{ \epsilon_g^{-\frac{3}{2}}, \epsilon_H^{-3} \})$ iterations to reach a second-order stationary point respectively. Notably, the proposed algorithm remains applicable even in cases of the gradient and Hessian of the objective function unknown. Numerical experiments are performed with gradient and Hessian being approximated by forward finite-differences to illustrate the theoretical results and numerical comparison

Focused ultrasound-enhanced delivery of intranasally administered anti-programmed cell death-ligand 1 antibody to an intracranial murine glioma model

Immune checkpoint inhibitors have great potential for the treatment of gliomas; however, their therapeutic efficacy has been partially limited by their inability to efficiently cross the blood-brain barrier (BBB). The objective of this study was to evaluate the capability of focused-ultrasound-mediated intranasal brain drug delivery (FUSIN) in achieving the locally enhanced delivery of anti-programmed cell death-ligand 1 antibody (aPD-L1) to the brain. Both non-tumor mice and mice transcranially implanted with GL261 glioma cells at the brainstem were used in this study. aPD-L1 was labeled with a near-infrared fluorescence dye (IRDye 800CW) and administered to mice through the nasal route to the brain, followed by focused ultrasound sonication in the presence of systemically injected microbubbles. FUSIN enhanced the accumulation of aPD-L1 at the FUS-targeted brainstem by an average of 4.03- and 3.74-fold compared with intranasal (IN) administration alone in the non-tumor mice and glioma mice, respectively. Immunohistochemistry staining found that aPD-L1 was mainly located within the perivascular spaces after IN delivery, while FUSIN further enhanced the penetration depth and delivery efficiency of aPD-L1 to the brain parenchyma. The delivered aPD-L1 was found to be colocalized with the tumor cells after FUSIN delivery to the brainstem glioma. These findings suggest that FUSIN is a promising technique to enhance the delivery of immune checkpoint inhibitors to gliomas

Mechanically manipulating glymphatic transport by ultrasound combined with microbubbles

The glymphatic system is a perivascular fluid transport system for waste clearance. Glymphatic transport is believed to be driven by the perivascular pumping effect created by the pulsation of the arterial wall caused by the cardiac cycle. Ultrasound sonication of circulating microbubbles (MBs) in the cerebral vasculature induces volumetric expansion and contraction of MBs that push and pull on the vessel wall to generate a MB pumping effect. The objective of this study was to evaluate whether glymphatic transport can be mechanically manipulated by focused ultrasound (FUS) sonication of MBs. The glymphatic pathway in intact mouse brains was studied using intranasal administration of fluorescently labeled albumin as fluid tracers, followed by FUS sonication at a deep brain target (thalamus) in the presence of intravenously injected MBs. Intracisternal magna injection, the conventional technique used in studying glymphatic transport, was employed to provide a comparative reference. Three-dimensional confocal microscopy imaging of optically cleared brain tissue revealed that FUS sonication enhanced the transport of fluorescently labeled albumin tracer in the perivascular space (PVS) along microvessels, primarily the arterioles. We also obtained evidence of FUS-enhanced penetration of the albumin tracer from the PVS into the interstitial space. This study revealed that ultrasound combined with circulating MBs could mechanically enhance glymphatic transport in the brain