A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

We present a numerical method for the variable coefficient Poisson equation in three-dimensional irregular domains and with interfacial discontinuities. The discretization embeds the domain and interface into a uniform Cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions. The matrix associated with the discretization is symmetric positive definite and equal to the standard 7-point finite difference Poisson stencil away from embedded interfaces and boundaries. Numerical evidence suggests second order accuracy in the L^∞-norm. Our approach improves the treatment of Dirichlet and jump constraints in the recent work of Bedrossian et al. [1] and introduces innovations necessary in three dimensions. Specifically, we construct new constraint-based Lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations; we provide a method for cell-local polyhedral approximation to the zero isocontour surface of a level set needed for three-dimensional embedding; and we show that the new Lagrange multiplier spaces naturally lead to a class of easy-to-implement multigrid methods that achieve near optimal efficiency, as shown by numerical examples. For the specific case of a continuous Poisson coefficient in interface problems, we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints. As in [1], this is used in a discontinuity removal technique that yields the standard 7-point stencil across the interface and only requires a modification to the right-hand side of the linear system.     

A second order virtual node method for elliptic problems with interfaces and irregular domains

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities or on irregular domains, handling both cases with the same approach. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in L^∞.     

Fast Kalman Filtering and Forward–Backward Smoothing via a Low-Rank Perturbative Approach

Kalman filtering-smoothing is a fundamental tool in statistical time-series analysis. However, standard implementations of the Kalman filter-smoother require O(d³) time and O(d²) space per time step, where d is the dimension of the state variable, and are therefore impractical in high-dimensional problems. In this article we note that if a relatively small number of observations are available per time step, the Kalman equations may be approximated in terms of a low-rank perturbation of the prior state covariance matrix in the absence of any observations. In many cases this approximation may be computed and updated very efficiently (often in just O(k²d) or O(k²d + kdlog?d) time and space per time step, where k is the rank of the perturbation and in general k ? d), using fast methods from numerical linear algebra. We justify our approach and give bounds on the rank of the perturbation as a function of the desired accuracy. For the case of smoothing, we also quantify the error of our algorithm because of the low-rank approximation and show that it can be made arbitrarily low at the expense of a moderate computational cost. We describe applications involving smoothing of spatiotemporal neuroscience data. This article has online supplementary material.     

Performance of interventions with manipulator-driven real-time MR guidance: implementation and initial in vitro tests

The purpose of this work was to implement and assess the performance of interventions inside a cylindrical magnetic resonance imaging (MRI) scanner with an MR-compatible manipulator system and manipulator-driven real-time MR guidance. The interventional system is based on a seven degree-of-freedom MR-compatible manipulator, which offers man-in-the-loop control either with a graphical user interface or with a master/slave device. The position and the orientation of the interventional tool are sent to an MR scanner for a manipulator-driven dynamic update of the imaging plane to track, visualize and guide the motion of an end-effector. Studies on phantoms were performed with a cylindrical 1.5-T scanner using an operator-managed triggered gradient-recalled echo (GRE) or a computer-managed dynamic True Fast Imaging with Steady Precession (TrueFISP). Targets were acquired with an accuracy of 3.2 mm and with an in-plane path orientation of 2.5 degrees relative to the prescribed one. Path planning, including negotiation of obstacles and needle bending, was successfully performed. The signal-to-noise ratio (SNR) of TrueFISP was 25.3+/-2.1 when the manipulator was idle and was 18.6+/-2.4 during its operation. The SNR of triggered GRE (acquired only when the manipulator was idle) was 61.3+/-1.8. In conclusion, this study shows the feasibility of performing manually directed interventions inside cylindrical MR scanners with real-time MRI.