Two-constraint Domain Decomposition with Space Filling Curves

In scientific computing, Space Filling Curves are a widely used tool for one-constraint domain decomposition. They provide a mechanism to sort multi-dimensional data in a locality preserving way, and, in this way, a (one dimensional) list of mesh elements is established which is subsequently split into partitions under consideration of the constraint. This procedure has a runtime of O(N logN) (N is the number of mesh elements) while nearly perfect load balancing can be established with reasonable partition surface sizes. Inthiswork, wediscuss theextensibilityofthisproceduretotwo-constraintsettings whichis desirable, since the methodology is extremely fast. Here, the splitting operation is subject to two constraints, and, unlike to the one-constraint case, obtaining near perfect balancing is often hard to establish, and is, even more as in the one-constraint case, in conflict with the induced surface sizes (or edge-cuts). We discuss multiple strategies to tackle the splitting, and we present a fast, O(N logN) splitting heuristic algorithm which provides an integer σ that allows to trade off between balancing and surface sizes which results in a O(N logN) two-constraint decomposition method. Results are compared to the multi-constraint domain decomposition abilities implemented in the Metis software package, and show that the method produces higher surface sizes, but is orders of magnitudes faster which makes the method superior for certain applications

Frequency-splitting Dynamic MRI Reconstruction using Multi-scale 3D Convolutional Sparse Coding and Automatic Parameter Selection

Department of Computer Science and EngineeringIn this thesis, we propose a novel image reconstruction algorithm using multi-scale 3D con- volutional sparse coding and a spectral decomposition technique for highly undersampled dy- namic Magnetic Resonance Imaging (MRI) data. The proposed method recovers high-frequency information using a shared 3D convolution-based dictionary built progressively during the re- construction process in an unsupervised manner, while low-frequency information is recovered using a total variation-based energy minimization method that leverages temporal coherence in dynamic MRI. Additionally, the proposed 3D dictionary is built across three different scales to more efficiently adapt to various feature sizes, and elastic net regularization is employed to promote a better approximation to the sparse input data. Furthermore, the computational com- plexity of each component in our iterative method is analyzed. We also propose an automatic parameter selection technique based on a genetic algorithm to find optimal parameters for our numerical solver which is a variant of the alternating direction method of multipliers (ADMM). We demonstrate the performance of our method by comparing it with state-of-the-art methods on 15 single-coil cardiac, 7 single-coil DCE, and a multi-coil brain MRI datasets at different sampling rates (12.5%, 25% and 50%). The results show that our method significantly outper- forms the other state-of-the-art methods in reconstruction quality with a comparable running time and is resilient to noise.ope

On domain decomposition with space filling curves for the parallel solution of the coupled Maxwell/Vlasov equations

Space filling Curves (SFCs) are increasingly used for combinatorial scientific computing and in particular for designing fast domain decomposition (partitioning) methods. In the context of parallel particle simulations for solving the system of Maxwell/Vlasov equations with a coupled FE/PIC (Finite Element/Particle-In-Cell) unstructured mesh based solver, one has to deal with a two-constraint partitioning problem. Moreover, this problem has to be solved several times during the simulation. Therefore, a fast and scalable partitioning problem is required. For this purpose, we propose here a new SFC based method which is well adapted to multi-constraint partitioning problems. This method is compared to graph based partitioning methods from the widely used MeTiS tool. Experimental results show that the proposed SFC based method is at least 100 times faster than MeTiS to the disadvantage of edge-cuts that are between 2 to 4 times worse than those achieved by the MeTiS methods

Holomorphic matrix models

This is a study of holomorphic matrix models, the matrix models which underlie the conjecture of Dijkgraaf and Vafa. I first give a systematic description of the holomorphic one-matrix model. After discussing its convergence sectors, I show that certain puzzles related to its perturbative expansion admit a simple resolution in the holomorphic set-up. Constructing a `complex' microcanonical ensemble, I check that the basic requirements of the conjecture (in particular, the special geometry relations involving chemical potentials) hold in the absence of the hermicity constraint. I also show that planar solutions of the holomorphic model probe the entire moduli space of the associated algebraic curve. Finally, I give a brief discussion of holomorphic $ADE$ models, focusing on the example of the $A_2$ quiver, for which I extract explicitly the relevant Riemann surface. In this case, use of the holomorphic model is crucial, since the Hermitian approach and its attending regularization would lead to a singular algebraic curve, thus contradicting the requirements of the conjecture. In particular, I show how an appropriate regularization of the holomorphic $A_2$ model produces the desired smooth Riemann surface in the limit when the regulator is removed, and that this limit can be described as a statistical ensemble of `reduced' holomorphic models.Comment: 45 pages, reference adde

Initial Conditions for Large Cosmological Simulations

This technical paper describes a software package that was designed to produce initial conditions for large cosmological simulations in the context of the Horizon collaboration. These tools generalize E. Bertschinger's Grafic1 software to distributed parallel architectures and offer a flexible alternative to the Grafic2 software for ``zoom'' initial conditions, at the price of large cumulated cpu and memory usage. The codes have been validated up to resolutions of 4096^3 and were used to generate the initial conditions of large hydrodynamical and dark matter simulations. They also provide means to generate constrained realisations for the purpose of generating initial conditions compatible with, e.g. the local group, or the SDSS catalog.Comment: 12 pages, 11 figures, submitted to ApJ

Binary black holes in circular orbits. II. Numerical methods and first results

We present the first results from a new method for computing spacetimes representing corotating binary black holes in circular orbits. The method is based on the assumption of exact equilibrium. It uses the standard 3+1 decomposition of Einstein equations and conformal flatness approximation for the 3-metric. Contrary to previous numerical approaches to this problem, we do not solve only the constraint equations but rather a set of five equations for the lapse function, the conformal factor and the shift vector. The orbital velocity is unambiguously determined by imposing that, at infinity, the metric behaves like the Schwarzschild one, a requirement which is equivalent to the virial theorem. The numerical scheme has been implemented using multi-domain spectral methods and passed numerous tests. A sequence of corotating black holes of equal mass is calculated. Defining the sequence by requiring that the ADM mass decrease is equal to the angular momentum decrease multiplied by the orbital angular velocity, it is found that the area of the apparent horizons is constant along the sequence. We also find a turning point in the ADM mass and angular momentum curves, which may be interpreted as an innermost stable circular orbit (ISCO). The values of the global quantities at the ISCO, especially the orbital velocity, are in much better agreement with those from third post-Newtonian calculations than with those resulting from previous numerical approaches.Comment: 27 pages, 20 PostScript figures, improved presentation of the regularization procedure for the shift vector, new section devoted to the check of the momentum constraint, references added + minor corrections, accepted for publication in Phys. Rev.

Spin-Charge Separation in the t−Jt-J Model: Magnetic and Transport Anomalies

A real spin-charge separation scheme is found based on a saddle-point state of the $t-J$ model. In the one-dimensional (1D) case, such a saddle-point reproduces the correct asymptotic correlations at the strong-coupling fixed-point of the model. In the two-dimensional (2D) case, the transverse gauge field confining spinon and holon is shown to be gapped at {\em finite doping} so that a spin-charge deconfinement is obtained for its first time in 2D. The gap in the gauge fluctuation disappears at half-filling limit, where a long-range antiferromagnetic order is recovered at zero temperature and spinons become confined. The most interesting features of spin dynamics and transport are exhibited at finite doping where exotic {\em residual} couplings between spin and charge degrees of freedom lead to systematic anomalies with regard to a Fermi-liquid system. In spin dynamics, a commensurate antiferromagnetic fluctuation with a small, doping-dependent energy scale is found, which is characterized in momentum space by a Gaussian peak at ($\pi/a$, $\pi/a$) with a doping-dependent width ($\propto \sqrt{\delta}$, $\delta$ is the doping concentration). This commensurate magnetic fluctuation contributes a non-Korringa behavior for the NMR spin-lattice relaxation rate. There also exits a characteristic temperature scale below which a pseudogap behavior appears in the spin dynamics. Furthermore, an incommensurate magnetic fluctuation is also obtained at a {\em finite} energy regime. In transport, a strong short-range phase interference leads to an effective holon Lagrangian which can give rise to a series of interesting phenomena including linear-$T$ resistivity and $T^2$ Hall-angle. We discuss the striking similarities of these theoretical features with those found in the high-$T_c$ cuprates and give aComment: 70 pages, RevTex, hard copies of 7 figures available upon request; minor revisions in the text and references have been made; To be published in July 1 issue of Phys. Rev. B52, (1995