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Numerical Analysis

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preprint2016arXiv

Parallel Tensor Compression for Large-Scale Scientific Data

As parallel computing trends towards the exascale, scientific data produced by high-fidelity simulations are growing increasingly massive. For instance, a simulation on a three-dimensional spatial grid with 512 points per dimension that tracks 64 variables per grid point for 128 time steps yields 8~TB of data, assuming double precision. By viewing the data as a dense five-way tensor, we can compute a Tucker decomposition to find inherent low-dimensional multilinear structure, achieving compression ratios of up to 5000 on real-world data sets with negligible loss in accuracy. So that we can operate on such massive data, we present the first-ever distributed-memory parallel implementation for the Tucker decomposition, whose key computations correspond to parallel linear algebra operations, albeit with nonstandard data layouts. Our approach specifies a data distribution for tensors that avoids any tensor data redistribution, either locally or in parallel. We provide accompanying analysis of the computation and communication costs of the algorithms. To demonstrate the compression and accuracy of the method, we apply our approach to real-world data sets from combustion science simulatio

preprint2017arXiv

Mixed one-bit compressive sensing with applications to overexposure correction for CT reconstruction

When a measurement falls outside the quantization or measurable range, it becomes saturated and cannot be used in classical reconstruction methods. For example, in C-arm angiography systems, which provide projection radiography, fluoroscopy, digital subtraction angiography, and are widely used for medical diagnoses and interventions, the limited dynamic range of C-arm flat detectors leads to overexposure in some projections during an acquisition, such as imaging relatively thin body parts (e.g., the knee). Aiming at overexposure correction for computed tomography (CT) reconstruction, we in this paper propose a mixed one-bit compressive sensing (M1bit-CS) to acquire information from both regular and saturated measurements. This method is inspired by the recent progress on one-bit compressive sensing, which deals with only sign observations. Its successful applications imply that information carried by saturated measurements is useful to improve recovery quality. For the proposed M1bit-CS model, alternating direction methods of multipliers is developed and an iterative saturation detection scheme is established. Then we evaluate M1bit-CS on one-dimensional signal recovery tasks. In s

preprint2017arXiv

The Unum Number Format: Mathematical Foundations, Implementation and Comparison to IEEE 754 Floating-Point Numbers

This thesis examines a modern concept for machine numbers based on interval arithmetic called 'Unums' and compares it to IEEE 754 floating-point arithmetic, evaluating possible uses of this format where floating-point numbers are inadequate. In the course of this examination, this thesis builds theoretical foundations for IEEE 754 floating-point numbers, interval arithmetic based on the projectively extended real numbers and Unums.

preprint2016arXiv

Fast symmetric factorization of hierarchical matrices with applications

We present a fast direct algorithm for computing symmetric factorizations, i.e. $A = WW^T$, of symmetric positive-definite hierarchical matrices with weak-admissibility conditions. The computational cost for the symmetric factorization scales as $\mathcal{O}(n \log^2 n)$ for hierarchically off-diagonal low-rank matrices. Once this factorization is obtained, the cost for inversion, application, and determinant computation scales as $\mathcal{O}(n \log n)$. In particular, this allows for the near optimal generation of correlated random variates in the case where $A$ is a covariance matrix. This symmetric factorization algorithm depends on two key ingredients. First, we present a novel symmetric factorization formula for low-rank updates to the identity of the form $I+UKU^T$. This factorization can be computed in $\mathcal{O}(n)$ time if the rank of the perturbation is sufficiently small. Second, combining this formula with a recursive divide-and-conquer strategy, near linear complexity symmetric factorizations for hierarchically structured matrices can be obtained. We present numerical results for matrices relevant to problems in probability \& statistics (Gaussian processes), interp

preprint2016arXiv

A Field Guide to Forward-Backward Splitting with a FASTA Implementation

Non-differentiable and constrained optimization play a key role in machine learning, signal and image processing, communications, and beyond. For high-dimensional minimization problems involving large datasets or many unknowns, the forward-backward splitting method provides a simple, practical solver. Despite its apparently simplicity, the performance of the forward-backward splitting is highly sensitive to implementation details. This article is an introductory review of forward-backward splitting with a special emphasis on practical implementation concerns. Issues like stepsize selection, acceleration, stopping conditions, and initialization are considered. Numerical experiments are used to compare the effectiveness of different approaches. Many variations of forward-backward splitting are implemented in the solver FASTA (short for Fast Adaptive Shrinkage/Thresholding Algorithm). FASTA provides a simple interface for applying forward-backward splitting to a broad range of problems.

preprint2016arXiv

Power Series Method applied to Inverse Analysis in Chemical Kinetics Problem

Power Series Solution Method has been traditionally used to solve Ordinary and Partial Linear Differential Equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-Type differential equations in order to demonstrate its scope and applicability.

preprint2016arXiv

A multigrid scheme for 3D Monge-Ampère equations

The elliptic Monge-Ampère equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fast three dimensional solvers are needed, for example in medical image registration but are not yet available. We build fast solvers for smooth solutions in three dimensions using a nonlinear full-approximation storage multigrid method. Starting from a second-order accurate centered finite difference approximation, we present a nonlinear Gauss-Seidel iterative method which has a mechanism for selecting the convex solution of the equation. The iterative method is used as an effective smoother, combined with the full-approximation storage multigrid method. Numerical experiments are provided to validate the accuracy of the finite difference scheme and illustrate the computational efficiency of the proposed multigrid solver.

preprint2012arXiv

The Modified Direct Method: an Approach for Smoothing Planar and Surface Meshes

The Modified Direct Method (MDM) is an iterative mesh smoothing method for smoothing planar and surface meshes, which is developed from the non-iterative smoothing method originated by Balendran [1]. When smooth planar meshes, the performance of the MDM is effectively identical to that of Laplacian smoothing, for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian smoothing for tri-quad meshes. When smooth surface meshes, for trian-gular, quadrilateral and quad-dominant mixed meshes, the mean quality(MQ) of all mesh elements always increases and the mean square error (MSE) decreases during smoothing; For tri-dominant mixed mesh, the quality of triangles always descends while that of quads ascends. Test examples show that the MDM is convergent for both planar and surface triangular, quadrilateral and tri-quad meshes.

preprint2016arXiv

Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, linearity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton's method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers' equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this se

preprint2016arXiv

Nonuniformly weighted Schwarz smoothers for spectral element multigrid

A hybrid Schwarz/multigrid method for spectral element solvers to the Poisson equation in $\mathbb R^2$ is presented. It extends the additive Schwarz method studied by J. Lottes and P. Fischer (J. Sci. Comput. 24:45--78, 2005) by introducing nonuniform weight distributions based on the smoothed sign function. Using a V-cycle with only one pre-smoothing, the new method attains logarithmic convergence rates in the range from 1.2 to 1.9, which corresponds to residual reductions of almost two orders of magnitude. Compared to the original method, it reduces the iteration count by a factor of 1.5 to 3, leading to runtime savings of about 50 percent. In numerical experiments the method proved robust with respect to the mesh size and polynomial orders up to 32. Used as a preconditioner for the (inexact) CG method it is also suited for anisotropic meshes and easily extended to diffusion problems with variable coefficients.

preprint2016arXiv

Convergence Rates for Greedy Kaczmarz Algorithms, and Faster Randomized Kaczmarz Rules Using the Orthogonality Graph

The Kaczmarz method is an iterative algorithm for solving systems of linear equalities and inequalities, that iteratively projects onto these constraints. Recently, Strohmer and Vershynin [J. Fourier Anal. Appl., 15(2):262-278, 2009] gave a non-asymptotic convergence rate analysis for this algorithm, spurring numerous extensions and generalizations of the Kaczmarz method. Rather than the randomized selection rule analyzed in that work, in this paper we instead discuss greedy and approximate greedy selection rules. We show that in some applications the computational costs of greedy and random selection are comparable, and that in many cases greedy selection rules give faster convergence rates than random selection rules. Further, we give the first multi-step analysis of Kaczmarz methods for a particular greedy rule, and propose a provably-faster randomized selection rule for matrices with many pairwise-orthogonal rows.

preprint2016arXiv

Streaming GPU Singular Value and Dynamic Mode Decompositions

This work develops a parallelized algorithm to compute the dynamic mode decomposition (DMD) on a graphics processing unit using the streaming method of snapshots singular value decomposition. This allows the algorithm to operate efficiently on streaming data by avoiding redundant inner-products as new data becomes available. In addition, it is possible to leverage the native compressed format of many data streams, such as HD video and computational physics codes that are represented sparsely in the Fourier domain, to massively reduce data transfer from CPU to GPU and to enable sparse matrix multiplications. Taken together, these algorithms facilitate real-time streaming DMD on high-dimensional data streams. We demonstrate the proposed method on numerous high-dimensional data sets ranging from video background modeling to scientific computing applications, where DMD is becoming a mainstay algorithm. The computational framework is developed as an open-source library written in C++ with CUDA, and the algorithms may be generalized to include other DMD advances, such as compressed sensing DMD, multi resolution DMD, or DMD with control. Keywords: Singular value decomposition, dynamic mod

preprint2016arXiv

Online and stochastic Douglas-Rachford splitting method for large scale machine learning

Online and stochastic learning has emerged as powerful tool in large scale optimization. In this work, we generalize the Douglas-Rachford splitting (DRs) method for minimizing composite functions to online and stochastic settings (to our best knowledge this is the first time DRs been generalized to sequential version). We first establish an $O(1/\sqrt{T})$ regret bound for batch DRs method. Then we proved that the online DRs splitting method enjoy an $O(1)$ regret bound and stochastic DRs splitting has a convergence rate of $O(1/\sqrt{T})$. The proof is simple and intuitive, and the results and technique can be served as a initiate for the research on the large scale machine learning employ the DRs method. Numerical experiments of the proposed method demonstrate the effectiveness of the online and stochastic update rule, and further confirm our regret and convergence analysis.

preprint2016arXiv

An efficient hybrid tridiagonal divide-and-conquer algorithm on distributed memory architectures

In this paper, an efficient divide-and-conquer (DC) algorithm is proposed for the symmetric tridiagonal matrices based on ScaLAPACK and the hierarchically semiseparable (HSS) matrices. HSS is an important type of rank-structured matrices.Most time of the DC algorithm is cost by computing the eigenvectors via the matrix-matrix multiplications (MMM). In our parallel hybrid DC (PHDC) algorithm, MMM is accelerated by using the HSS matrix techniques when the intermediate matrix is large. All the HSS algorithms are done via the package STRUMPACK. PHDC has been tested by using many different matrices. Compared with the DC implementation in MKL, PHDC can be faster for some matrices with few deflations when using hundreds of processes. However, the gains decrease as the number of processes increases. The comparisons of PHDC with ELPA (the Eigenvalue soLvers for Petascale Applications library) are similar. PHDC is usually slower than MKL and ELPA when using 300 or more processes on Tianhe-2 supercomputer.

preprint2016arXiv

Uniform Penalty inversion of two-dimensional NMR Relaxation data

The inversion of two-dimensional NMR data is an ill-posed problem related to the numerical computation of the inverse Laplace transform. In this paper we present the 2DUPEN algorithm that extends the Uniform Penalty (UPEN) algorithm [Borgia, Brown, Fantazzini, {\em Journal of Magnetic Resonance}, 1998] to two-dimensional data. The UPEN algorithm, defined for the inversion of one-dimensional NMR relaxation data, uses Tikhonov-like regularization and optionally non-negativity constraints in order to implement locally adapted regularization. In this paper, we analyze the regularization properties of this approach. Moreover, we extend the one-dimensional UPEN algorithm to the two-dimensional case and present an efficient implementation based on the Newton Projection method. Without any a-priori information on the noise norm, 2DUPEN automatically computes the locally adapted regularization parameters and the distribution of the unknown NMR parameters by using variable smoothing. Results of numerical experiments on simulated and real data are presented in order to illustrate the potential of the proposed method in reconstructing peaks and flat regions with the same accuracy.

preprint2016arXiv

The Reduced-Order Hybrid Monte Carlo Sampling Smoother

Hybrid Monte-Carlo (HMC) sampling smoother is a fully non-Gaussian four-dimensional data assimilation algorithm that works by directly sampling the posterior distribution formulated in the Bayesian framework. The smoother in its original formulation is computationally expensive due to the intrinsic requirement of running the forward and adjoint models repeatedly. Here we present computationally efficient versions of the HMC sampling smoother based on reduced-order approximations of the underlying model dynamics. The schemes developed herein are tested numerically using the shallow-water equations model on Cartesian coordinates. The results reveal that the reduced-order versions of the smoother are capable of accurately capturing the posterior probability density, while being significantly faster than the original full order formulation.

preprint2016arXiv

Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D

We present a polynomial multigrid method for the nodal interior penalty formulation of the Poisson equation on three-dimensional Cartesian grids. Its key ingredient is a weighted overlapping Schwarz smoother operating on element-centered subdomains. The MG method reaches superior convergence rates corresponding to residual reductions of about two orders of magnitude within a single V(1,1) cycle. It is robust with respect to the mesh size and the ansatz order, at least up to ${P=32}$. Rigorous exploitation of tensor-product factorization yields a computational complexity of $O(PN)$ for $N$ unknowns, whereas numerical experiments indicate even linear runtime scaling. Moreover, by allowing adjustable subdomain overlaps and adding Krylov acceleration, the method proved feasible for anisotropic grids with element aspect ratios up to 48.

preprint2016arXiv

Well-balanced nodal discontinuous Galerkin method for Euler equations with gravity

We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss-Lobatto-Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.

preprint2016arXiv

Parameter Estimation of a Nonlinear Burgers Model using Nanoindentation and Finite Element-based Inverse Analysis

Nanoindentation involves probing a hard diamond tip into a material, where the load and the displacement experienced by the tip is recorded continuously. This load-displacement data is a direct function of material's innate stress-strain behavior. Thus, theoretically it is possible to extract mechanical properties of a material through nanoindentation. However, due to various nonlinearities associated with nanoindentation the process of interpreting load-displacement data into material properties is difficult. Although, simple elastic behavior can be characterized easily, a method to characterize complicated material behavior such as nonlinear viscoelasticity is still lacking. In this study, a nanoindentation-based material characterization technique is developed to characterize soft materials exhibiting nonlinear viscoelasticity. Nanoindentation experiment was modeled in finite element analysis software (ABAQUS), where a nonlinear viscoelastic behavior was incorporated using user-defined subroutine (UMAT). The model parameters were calibrated using a process called inverse analysis. In this study, a surrogate model-based approach was used for the inverse analysis. The differen

preprint2015arXiv

Learning-based Reduced Order Model Stabilization for Partial Differential Equations: Application to the Coupled Burgers Equation

We present results on stabilization for reduced order models (ROM) of partial differential equations using learning. Stabilization is achieved via closure models for ROMs, where we use a model-free extremum seeking (ES) dither-based algorithm to learn the best closure models' parameters, for optimal ROM stabilization. We first propose to auto-tune linear closure models using ES, and then extend the results to a closure model combining linear and nonlinear terms, for better stabilization performance. The coupled Burgers' equation is employed as a test-bed for the proposed tuning method.

preprint2016arXiv

A search for extensible low-WAFOM point sets

Matsumoto, Saito, and Matoba recently proposed the Walsh figure of merit (WAFOM), which is a computable criterion for quasi-Monte Carlo point sets using digital nets. Several algorithms have been proposed for finding low-WAFOM point sets. In the existing algorithms, the number of points is fixed in advance, but extensible point sets are preferred in some applications. In this paper, we propose a random search algorithm for extensible low-WAFOM point sets. For this, we introduce a method that uses lookup tables to compute WAFOM faster. Numerical results show that our extensible low-WAFOM point sets are comparable with Niederreiter--Xing sequences for some low-dimensional and smooth test functions.

preprint2016arXiv

Implementation and evaluation of data-compression algorithms for irregular-grid iterative methods on the PEZY-SC processor

Iterative methods on irregular grids have been used widely in all areas of comptational science and engineering for solving partial differential equations with complex geometry. They provide the flexibility to express complex shapes with relatively low computational cost. However, the direction of the evolution of high-performance processors in the last two decades have caused serious degradation of the computational efficiency of iterative methods on irregular grids, because of relatively low memory bandwidth. Data compression can in principle reduce the necessary memory memory bandwidth of iterative methods and thus improve the efficiency. We have implemented several data compression algorithms on the PEZY-SC processor, using the matrix generated for the HPCG benchmark as an example. For the SpMV (Sparse Matrix-Vector multiplication) part of the HPCG benchmark, the best implementation without data compression achieved 11.6Gflops/chip, close to the theoretical limit due to the memory bandwidth. Our implementation with data compression has achieved 32.4Gflops. This is of course rather extreme case, since the grid used in HPCG is geometrically regular and thus its compression effici

preprint2016arXiv

Towards a Global Controller Design for Guaranteed Synchronization of Switched Chaotic Systems

In this paper, synchronization of identical switched chaotic systems is explored based on Lyapunov theory of guaranteed stability. Concepts from robust control principles and switched linear systems are merged together to derive a sufficient condition for synchronization of identical master-slave switched nonlinear chaotic systems and are expressed in the form of bilinear matrix inequalities (BMIs). The nonlinear controller design problem is then recast in the form of linear matrix inequalities (LMIs) to facilitate numerical computation by standard LMI solvers and is illustrated by appropriate examples.

preprint2016arXiv

Fast Multipole Method based filtering of non-uniformly sampled data

Non-uniform fast Fourier Transform (NUFFT) and inverse NUFFT (INUFFT) algorithms, based on the Fast Multipole Method (FMM) are developed and tested. Our algorithms are based on a novel factorization of the FFT kernel, and are implemented with attention to data structures and error analysis. Note: This unpublished manuscript was available on our web pages and has been referred to by others in the literature. To provide a proper archival reference we are placing it on arXiv.

360 works