Fast Parallel Solution Research Articles

Fast uncertainty quantification of activation sequences in patient-specific cardiac electrophysiology meeting clinical time constraints.

We present a fast, patient-specific methodology for uncertainty quantification in electrophysiology, aimed at meeting the time constraints of clinical practitioners. We focus on computing the statistics of the activation map, given the uncertainties associated with the conductivity tensor modeling the fiber orientation in the heart. We use a fast parallel solution method implemented on a graphics processing unit for the eikonal approximation, in order to compute the activation map and to sample the random fiber field with correlation on the basis of geodesic distances. While this enables to perform uncertainty quantification studies with a manageable computational effort, the required time frame still exceeds clinically suitable time expectations. In order to reduce it further by 2 orders of magnitude, we rely on Bayesian multifidelity methods. In particular, we propose a low-fidelity model that is patient-specific and free from the additional training cost associated with reduced models. This is achieved by a sound physics-based simplification of the full eikonal model. The low-fidelity output is then corrected by the standard multifidelity framework. In practice, the complete procedure only requires approximately 100 new runs of our eikonal graphics processing unit solver for producing the sought estimates and their associated credible intervals, enabling a full online analysis in less than 5minutes.

Fast parallel solution of boundary integral equations and related problems

This article is concerned with the efficient numerical solution of Fredholm integral equations on a parallel computer with shared or distributed memory. Parallel algorithms for both, the approximation of the discrete operator by hierarchical matrices using adaptive cross approximation (ACA) and the parallel matrix-vector multiplication of such matrices by a vector, are presented. The first algorithm has a complexity of order p -1 N log2d-1 N, while the latter is of order p -1 N log d N, where N, d and p are the number of unknowns, the spatial dimension and the number of processors, respectively. The approximant needs Ω(p -1 N log d N) units of storage on each processor.

Fast parallel molecular solutions for DNA-based supercomputing: the subset-product problem

In this paper our main purpose is to give molecular solutions for the subset-product problem. In order to achieve this, we propose three DNA-based algorithms – parallel adder, parallel multiplier and parallel comparator – that formally verify our designed molecular solutions for the subset-product problem. We also show that Boolean circuits are not needed to perform mathematical operations on a molecular computer. Furthermore, this work indicates that the subset-product problem is solved and also presents clear evidence of the ability of molecular computing to perform complicated mathematical operations.

Fast parallel molecular solution to the dominating-set problem on massively parallel bio-computing

This paper shows how to use DNA strands to construct solution space of molecules for the dominating-set problem and how to apply biological operations to solve the problem from the solution space of molecules. In order to achieve this, we have proposed some DNA based parallel algorithms using the operations in Adleman–Lipton model, together with the analysis of the computational complexity for DNA parallel algorithms.

Quantum Neural Networks

This paper initiates the study of quantum computing within the constraints of using a polylogarithmic (O(logkn), k⩾1) number of qubits and a polylogarithmic number of computation steps. The current research in the literature has focussed on using a polynomial number of qubits. A new mathematical model of computation called Quantum Neural Networks (QNNs) is defined, building on Deutsch's model of quantum computational network. The model introduces a nonlinear and irreversible gate, similar to the speculative operator defined by Abrams and Lloyd. The precise dynamics of this operator are defined and while giving examples in which nonlinear Schrödinger's equations are applied, we speculate on its possible implementation. The many practical problems associated with the current model of quantum computing are alleviated in the new model. It is shown that QNNs of logarithmic size and constant depth have the same computational power as threshold circuits, which are used for modeling neural networks. QNNs of polylogarithmic size and polylogarithmic depth can solve the problems in NC, the class of problems with theoretically fast parallel solutions. Thus, the new model may indeed provide an approach for building scalable parallel computers.

Fast parallel solvers for elliptic problems

In an earlier work [1], we have considered the use of τ [2] preconditioners for the solution of Dirichlet boundary value problems by means of the preconditioned conjugate gradients methods. In this paper we propose the use of Toeplitz matrices for the fast parallel solution of differential problems. We propose some algorithms to solve preconditioned systems which are especially suitable in parallel environments. We also present several numerical results to prove the effectiveness of our technique

Fast parallel solution of the Poisson equation on irregular domains

We present a fast parallel solution method for the Poisson equation on irregular domains. Due to a simple embedding method using harmonic polynomial approximation, a dominant part of the computation becomes solving one Poisson problem on a disk.

Fast parallel solution of fixed point equations for the performance evaluation of circuit-switched networks

Massively parallel algorithms are presented for solving systems of fixed point equations, modeling state-dependent routing in large asymmetric circuit-switched networks. Our focus is on the Aggregated Least Busy Alternative (ALBA) routing policy of Mitra, Gibbens and Huang. On a 16384 processor MasPar parallel computer, about a minute is required to compute estimates of the call blocking probabilities for every node-pair, for realistic networks of over 100 nodes. A few hours are required on a high speed workstation.

Fast and Efficient Parallel Solution of Sparse Linear Systems

This paper presents a parallel algorithm for the solution of a linear system $A{\bf x} = {\bf b}$ with a sparse $n \times n$ symmetric positive definite matrix A, associated with the graph $G(A)$ that has n vertices and has an edge for each nonzero entry of A. If $G(A)$ has an $s(n)$-separator family and a known $s(n)$-separator tree, then the algorithm requires only $O(\log ^3 n)$ time and $(|E| + {{M(s(n)))} / {\log n}}$ processors for the evaluation of the solution vector ${\bf x} = A^{ - 1} {\bf b}$, where $|E|$ is the number of edges in $G(A)$ and $M(n)$ is the number of processors sufficient for multiplying two $n \times n$ rational matrices in time $O(\log n)$. Furthermore, for this computational cost the algorithm computes a recursive factorization of A such that the solution of any other linear system $A{\bf x} = {\bf b}'$ with the same matrix A requires only $O(\log ^2 n)$ time and $({{|E|} / {\log n}}) + s(n)^2 $ processors.

BREADTH-DEPTH SEARCH IS ${\mathcal P}$-COMPLETE

The parallel complexity of a search strategy that combines attributes of both breadth-first search and depth-first search is studied. The search called breadth-depth search was defined by Horowitz and Sahni. The search technique has applications in branch-and-bound strategies. Kindervater and Lenstra posed the complexity of this type of search strategy as an open problem. We resolve their question by showing that a natural decision problem based on breadth-depth search is [Formula: see text]-complete. Specifically, we prove that if given a graph G=(V, E) either directed or undirected, a start vertex s∈V, and two designated vertices u and v in V, then the problem of deciding whether u is visited before v by a breadth-depth search originating from s is [Formula: see text]-complete. The search can be based either on vertex numbers or fixed ordered adjacency lists. Our reductions differ for directed/undirected graphs and depending on whether vertex numbers/fixed ordered adjacency lists are used. These results indicate breadth-depth search is highly sequential in nature and probably will not adapt to a fast parallel solution, unless [Formula: see text] equals [Formula: see text].

Space station static and dynamic analyses using parallel methods

Algorithms for high-performance parallel computers are applied to perform static analyses of large-scale Space Station finite-element models (FEMs). Several parallel-vector algorithms under development at NASA Langley are assessed. Sparse matrix solvers were found to be more efficient than banded symmetric or iterative solvers for the static analysis of large-scale applications. In addition, new sparse and “out-of-core” solvers were found superior to substructure (superelement) techniques which require significant additional cost and time to perform static condensation during global FEM matrix generation as well as the subsequent recovery and expansion. A method to extend the fast parallel static solution techniques to reduce the computation time for dynamic analysis is also described. The resulting static and dynamic algorithms offer design economy for preliminary multidisciplinary design optimization and FEM validation against test modes. The algorithms are being optimized for parallel computers to solve one-million degrees-of-freedom (DOF) FEMs. The high-performance computers at NASA afforded effective software development, testing, efficient and accurate solution with timely system response and graphical interpretation of results rarely found in industry. Based on the author's experience, similar cooperation between industry and government should be encouraged for similar large-scale projects in the future.

Fast and efficient parallel solution of dense linear systems

The most efficient previously known parallel algorithms for the inversion of a nonsingular n × n matrix A or solving a linear system Ax = b over the rational numbers require O(log 2 n) time and M(n) n processors [provided that M( n) processors suffice in order to multiply two n × n rational matrices in time O(log n)]. Furthermore, the known polylog arithmetic time algorithms for those problems are numerically unstable. In this paper we apply Newton's iteration and initially choose an approximate inverse matrix by following Ben-Israel. This quadratically convergent and numerically stable iterative method takes O(log 2 n) parallel time using M( n) processors to compute the inverse (within the relative precision 2 − n c for a positive constant c) of an n × n rational matrix A with the condition number at most n d for a constant d. This is the optimum processor bound and by a factor of n improvement of the previously known processor bounds for polylogarithmic time matrix inversion. The algorithm does not require to precompute the condition number of the input matrix, but it just converges slower for ill-conditioned input matrices.

The Depth Efficacy of Unbounded: Characteristic Finite Field Arithmetic

We introduce an arithmetic model of parallel computation. The basic operations are ½ and Š gates over finite fields. Functions computed are unary and increasing input size is modelled by shifting the arithmetic base to a larger field. When only finite fields of bounded characteristic are used, then the above model is fully general for parallel computations in that size and depth of optimal arithmetic solutions are polynomially related to size and depth of general (boolean) solutions. In the case of finite fields of unbounded characteristic, we prove that the existence of a fast parallel (boolean) solution to the problem of powering an integer modulo a prime (and powering a polynomial modulo an irreducible polynomial) in combination with the existence of a fast parallel (arithmetic) solution for the problem of computing a single canonical function, f<em>(x)</em>, in the prime fields, guarantees the full generality of the finite field model of computation. We prove that the function f<em>(x)</em>, has a fast parallel arithmetic solution for any ''shallow'' class of primes, i.e. primes <em>p</em> such that any prime power divisor <em>q</em> of <em>p</em> -1 is bounded in value by a polynomial in log <em>p</em>.

Parallel algorithms for solvable permutation groups

A number of basic problems involving solvable and nilpotent permutation groups are shown to have fast parallel solutions. Testing solvability is in NC as well as, for solvable groups, finding order, testing membership, finding centralizers, finding centers, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in NC, find a central composition series, and find pointwise stabilizers of sets. The latter is applied to an instance of graph isomorphism. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to NC.

An elimination approach for fast parallel solution of the model fourth-order elliptic problem

A fast algorithm is developed for the parallel numerical solution of the first biharmonic boundary value problem on a rectangular region with N 2 interior grid points. The parallel computer considered is of SIMD type. The iterative procedure where one iteration consists in solving two transformed Poisson equations with relaxation is used. This approach allows one to apply the direct block-elimination method with parallel algorithm for linear recurrence relations efficiently to the evaluation of one iteration. For our algorithm the time per iteration does not exceed 9 log N time units, on N 2 processors. Thus, the technique presented brings a reduction in the arithmetic steps required for the solution of the problem considered.