Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space

In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. As shown here, reconstruction from samples at arbitrary locations can be understood as approximation of vector-valued functions from the acquired samples and formulated using a Reproducing Kernel Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between approximation theory and parallel imaging. Theoretical tools from approximation theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional g-factor noise analysis to both noise amplification and approximation errors. This is demonstrated with numerical examples.Comment: 28 pages, 7 figure

Numerical solution of third order three-point boundary value problems of ordinary differential equations with variable coefficients using variational-composite hybrid fixed point iterative method

This paper explores variational–composite hybrid fixed point iterative scheme for the solution of third order three-point boundary value problems. The method shows a strong convergence which makes it an efficient and reliable technique for finding approximate analytical solutions for third order three-point boundary value problems of ordinary differential equations with variable coefficients. From the numerical experiments carried out, the accuracy of the method was confirmed through the order of convergence obtained

A numerical algorithm for nonlinear multi-point boundary value problems

AbstractIn this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method

Reproducing Kernel Functions

In this chapter, we obtain some reproducing kernel spaces. We obtain reproducing kernel functions in these spaces. These reproducing kernel functions are very important for solving ordinary and partial differential equations