GMRES-Accelerated ADMM for Quadratic Objectives

We consider the sequence acceleration problem for the alternating direction method-of-multipliers (ADMM) applied to a class of equality-constrained problems with strongly convex quadratic objectives, which frequently arise as the Newton subproblem of interior-point methods. Within this context, the ADMM update equations are linear, the iterates are confined within a Krylov subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its ability to accelerate convergence. The basic ADMM method solves a $\kappa$-conditioned problem in $O(\sqrt{\kappa})$ iterations. We give theoretical justification and numerical evidence that the GMRES-accelerated variant consistently solves the same problem in $O(\kappa^{1/4})$ iterations for an order-of-magnitude reduction in iterations, despite a worst-case bound of $O(\sqrt{\kappa})$ iterations. The method is shown to be competitive against standard preconditioned Krylov subspace methods for saddle-point problems. The method is embedded within SeDuMi, a popular open-source solver for conic optimization written in MATLAB, and used to solve many large-scale semidefinite programs with error that decreases like $O(1/k^{2})$, instead of $O(1/k)$, where $k$ is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on Optimization (SIOPT

Automatic Generation of Geometrically Parameterized Reduced Order Models for Integrated Spiral RF-Inductors

In this paper we describe an approach to generating low-order models of spiral inductors that accurately capture the dependence on both frequency and geometry (width and spacing) parameters. The approach is based on adapting a multiparameter Krylov-subspace based moment matching method to reducing an integral equation for the three dimensional electromagnetic behavior of the spiral inductor. The approach is demonstrated on a typical on-chip rectangular inductor.Singapore-MIT Alliance (SMA

Substrate Resistance Extraction Using a Multi-Domain Surface Integral Formulation

In order to assess and optimize layout strategies for minimizing substrate noise, it is necessary to have fast and accurate techniques for computing contact coupling resistances associated with the substrate. In this talk, we describe an extraction method capable of full-chip analysis which combines modest geometric approximations, a novel integral formulation, and an FFT-accelerated preconditioned iterative method.Singapore-MIT Alliance (SMA

On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems

A computational scheme for solving 2D Laplace boundary-value problems using rational functions as the basis functions is described. The scheme belongs to the class of desingularized methods, for which the location of singularities and testing points is a major issue that is addressed by the proposed scheme, in the context of the 2D Laplace equation. Well-established rational-function fitting techniques are used to set the poles, while residues are determined by enforcing the boundary conditions in the least-squares sense at the nodes of rational Gauss-Chebyshev quadrature rules. Numerical results show that errors approaching the machine epsilon can be obtained for sharp and almost sharp corners, nearly-touching boundaries, and almost-singular boundary data. We show various examples of these cases in which the method yields compact solutions, requiring fewer basis functions than the Nystr\"{o}m method, for the same accuracy. A scheme for solving fairly large-scale problems is also presented

A novel boundary element method using surface conductive absorbers for full-wave analysis of 3-D nanophotonics

Fast surface integral equation (SIE) solvers seem to be ideal approaches for simulating 3-D nanophotonic devices, as these devices generate fields both in an interior channel and in the infinite exterior domain. However, many devices of interest, such as optical couplers, have channels that can not be terminated without generating reflections. Generating absorbers for these channels is a new problem for SIE methods, as the methods were initially developed for problems with finite surfaces. In this paper we show that the obvious approach for eliminating reflections, making the channel mildly conductive outside the domain of interest, is inaccurate. We describe a new method, in which the absorber has a gradually increasing surface conductivity; such an absorber can be easily incorporated in fast integral equation solvers. Numerical experiments from a surface-conductivity modified FFT-accelerated PMCHW-based solver are correlated with analytic results, demonstrating that this new method is orders of magnitude more effective than a volume absorber, and that the smoothness of the surface conductivity function determines the performance of the absorber. In particular, we show that the magnitude of the transition reflection is proportional to 1/L^(2d+2), where L is the absorber length and d is the order of the differentiability of the surface conductivity function.Comment: 10 page

The Use of Bone Morphogenetic Protein in the Intervertebral Disk Space in Minimally Invasive Transforaminal Lumbar Interbody Fusion

Study Design: Retrospective Cohort. Objective: The objective of this study was to characterize one surgeon’s experience over a 10-year period using rhBMP-2 in the disk space for minimally invasive transforaminal lumbar interbody fusion (MIS TLIF). Summary of Background Data: MIS TLIF has been utilized as a technique for decreasing patients’ immediate postoperative pain, decreasing blood loss, and shortened hospital stays. Effectiveness and complications of rhBMP-2’s use in the disk space is limited because of its off-label status. Methods: Retrospective analysis of consecutive MIS TLIFs performed by senior author between 2004 and 2014. rhBMP-2 was used in the disk space in all cases. Patients were stratified based on the dose of rhBMP-2 utilized. Patients had 9 to 12 month computerized tomography scan to evaluate for bony fusion and continued follow-up for 18 months. Results: A total of 688 patients underwent a MIS TLIF. A medium kit of rhBMP-2 was utilized in 97 patients, and small kit was used in 591 patients. Fusion rate was 97.9% and this was not different between the 2 groups with 96/97 patients fusing in the medium kit group and 577/591 patients fusing in the small kit group. Five patients taken back to the operating room for symptomatic pseudoarthrosis, 4 reoperated for bony hyperostosis, and 10 radiographic pseudoarthroses that did not require reoperation. A statistically significant difference in the rate of foraminal hyperostosis was found when using a medium sized kit of rhBMP-2 was 4.12% (4/97 patients), compared with a small kit (0/591 patients, P=0.0004). Conclusions: Utilization of rhBMP-2 in an MIS TLIF leads to high fusion rate (97.9%), with an acceptable complication profile. The development of foraminal hyperostosis is a rare complication that only affected 0.6% of patients, and seems to be a dose related complication, as this complication was eliminated when a lower dose of rhBMP-2 was utilized

A Precorrected-FFT Method for Coupled Electrostatic-Stokes Flow Problem

We present the application of the boundary integral equation method for solving the motion of biological cell or particle under Stokes flow in the presence of electrostatic field. The huge dense matrix-vector product from the boundary integral method poses a computationally challenging problem for solving the large system of equations generated. In our work, we used the precorrected-FFT (pFFT) method to reduce the computational time and memory usage drastically, so that large scale simulations can be performed quickly on a personal computer. Results on the force field acting on the particle, as well as the behavior of the particle through cell trap are presented.Singapore-MIT Alliance (SMA

Numerical Study of the Poisson-Boltzmann Equation for Biomolecular Electrostatics

Electrostatics interaction plays a very important role in almost all biomolecular systems. The Poisson-Boltzmann equation is widely used to treat this electrostatic effect in an ionic solution. In this work, a simple mixed discrete-continuum model is considered and boundary element method is used to solve for the solution.Singapore-MIT Alliance (SMA