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20
Random projections for the nonnegative leastsquares problem
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 2009
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Subspace fluid resimulation
 ACM Trans. Graph
, 2013
"... Figure 1: An efficient subspace resimulation of novel fluid dynamics. This scene was generated an order of magnitude faster than the original. The solver itself, without velocity reconstruction (§5), runs three orders of magnitude faster. We present a new subspace integration method that is capable ..."
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Cited by 11 (2 self)
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Figure 1: An efficient subspace resimulation of novel fluid dynamics. This scene was generated an order of magnitude faster than the original. The solver itself, without velocity reconstruction (§5), runs three orders of magnitude faster. We present a new subspace integration method that is capable of efficiently adding and subtracting dynamics from an existing highresolution fluid simulation. We show how to analyze the results of an existing highresolution simulation, discover an efficient reduced approximation, and use it to quickly “resimulate ” novel variations of the original dynamics. Prior subspace methods have had difficulty resimulating the original input dynamics because they lack efficient means of handling semiLagrangian advection methods. We show that multidimensional cubature schemes can be applied to this and other advection methods, such as MacCormack advection. The remaining pressure and diffusion stages can be written as a single matrixvector multiply, so as with previous subspace methods, no matrix inversion is needed at runtime. We additionally propose a novel importance samplingbased fitting algorithm that asymptotically accelerates the precomputation stage, and show that the Iterated Orthogonal Projection method can be used to elegantly incorporate moving internal boundaries into a subspace simulation. In addition to efficiently producing variations of the original input, our method can produce novel, abstract fluid motions that we have not seen from any other solver.
A Reduced Newton Method for Constrained Linear LeastSquares Problems
"... We propose an iterative method that solves constrained linear leastsquares problems by formulating them as nonlinear systems of equations and applying the Newton scheme. The method reduces the size of the linear system to be solved at each iteration by considering only a subset of the unknown varia ..."
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Cited by 9 (3 self)
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We propose an iterative method that solves constrained linear leastsquares problems by formulating them as nonlinear systems of equations and applying the Newton scheme. The method reduces the size of the linear system to be solved at each iteration by considering only a subset of the unknown variables. Hence the linear system can be solved more efficiently. We prove that the method is locally quadratic convergent. Applications to image deblurring problems show that our method gives better restored images than those obtained by projecting or scaling the solution into the dynamic range.
Exact Post Model Selection Inference for Marginal Screening
, 2014
"... We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response y, conditional on the model being selected (“condition on selection ” fram ..."
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We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response y, conditional on the model being selected (“condition on selection ” framework). This allows us to construct valid confidence intervals and hypothesis tests for regression coefficients that account for the selection procedure. In contrast to recent work in highdimensional statistics, our results are exact (nonasymptotic) and require no eigenvaluelike assumptions on the design matrix X. Furthermore, the computational cost of marginal regression, constructing confidence intervals and hypothesis testing is negligible compared to the cost of linear regression, thus making our methods particularly suitable for extremely large datasets. Although we focus on marginal screening to illustrate the applicability of the condition on selection framework, this framework is much more broadly applicable. We show how to apply the proposed framework to several other selection procedures including orthogonal matching pursuit, nonnegative least squares, and marginal screening+Lasso. 1
Efficient parallel nonnegative least squares on the GPU. submitted to the
 SIAM Journal on Scientific Computing
"... Abstract. We parallelize a version of the activeset iterative algorithm derived from the original works of Lawson and Hanson (1974) on multicore architectures. This algorithm requires the solution of an unconstrained least squares problem in every step of the iteration for a matrix composed of the ..."
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Abstract. We parallelize a version of the activeset iterative algorithm derived from the original works of Lawson and Hanson (1974) on multicore architectures. This algorithm requires the solution of an unconstrained least squares problem in every step of the iteration for a matrix composed of the passive columns of the original system matrix. To achieve improved performance, we use parallelizable procedures to efficiently update and downdate the QR factorization of the matrix at each iteration, to account for inserted and removed columns. We use a reordering strategy of the columns in the decomposition to reduce computation and memory access costs. We consider graphics processing units (GPUs) as a new mode for efficient parallel computations and compare our implementations to that of multicore CPUs. Both synthetic and nonsynthetic data are used in the experiments. Key words. Nonnegative least squares, activeset, QR updating, parallelism, multicore, GPU, deconvolution
Simulating Articulated Subspace SelfContact
"... Figure 1: A hand mesh composed of 458K tetrahedra, running at 5.8 FPS (171 ms), including both selfcontact detection and resolution. Our algorithm accelerates the computation of complex selfcontacts by a factor of 5 × to 52 × over other subspace methods and 166 × to 391× over fullrank simulations ..."
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Cited by 3 (1 self)
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Figure 1: A hand mesh composed of 458K tetrahedra, running at 5.8 FPS (171 ms), including both selfcontact detection and resolution. Our algorithm accelerates the computation of complex selfcontacts by a factor of 5 × to 52 × over other subspace methods and 166 × to 391× over fullrank simulations. Our selfcontact computation never dominates the total time, and takes up at most 46 % of a single frame. We present an efficient new subspace method for simulating the selfcontact of articulated deformable bodies, such as characters. Selfcontact is highly structured in this setting, as the limited space of possible articulations produces a predictable set of coherent collisions. Subspace methods can leverage this coherence, and have been used in the past to accelerate the collision detection stage of contact simulation. We show that these methods can be used to accelerate the entire contact computation, and allow selfcontact to be resolved without looking at all of the contact points. Our analysis of the problem yields a broader insight into the types of nonlinearities that subspace methods can efficiently approximate, and leads us to design a posespace cubature scheme. Our algorithm accelerates selfcontact by up to an order of magnitude over other subspace simulations, and accelerates the overall simulation by two orders of magnitude over fullrank simulations. We demonstrate the simulation of high resolution (100K – 400K elements) meshes in selfcontact at interactive rates (5.8 – 50 FPS).
Linesource based Xray Tomography
"... Abstract: Current computed tomography (CT) scanners, including microCT scanners, utilize a point xray source. As we target higher and higher spatial resolutions, the reduced xray focal spot size limits the temporal and contrast resolutions achievable. To overcome this limitation, in this paper we ..."
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Abstract: Current computed tomography (CT) scanners, including microCT scanners, utilize a point xray source. As we target higher and higher spatial resolutions, the reduced xray focal spot size limits the temporal and contrast resolutions achievable. To overcome this limitation, in this paper we propose to use a lineshaped xray source so that many more photons can be generated, given a data acquisition interval. In reference to the simultaneous algebraic reconstruction technique (SART) algorithm for image reconstruction from projection data generated by an xray point source, here we develop a generalized SART algorithm for image reconstruction from projection data generated by an xray line source. Our numerical simulation results demonstrate the feasibility of our novel linesourcebased xray CT approach and the proposed generalized SART algorithm.
A HYBRID MULTILEVELACTIVE SET METHOD FOR LARGE BOXCONSTRAINED DISCRETE ILLPOSED INVERSE PROBLEMS
"... Abstract. Many questions in science and engineering give rise to illposed inverse problems whose solution is known to satisfy box constrains, such as nonnegativity. The solution of discretized versions of these problems is highly sensitive to perturbations in the data, discretization errors, and ro ..."
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Abstract. Many questions in science and engineering give rise to illposed inverse problems whose solution is known to satisfy box constrains, such as nonnegativity. The solution of discretized versions of these problems is highly sensitive to perturbations in the data, discretization errors, and roundoff errors introduced during the computations. It is therefore often beneficial to impose known constraints during the solution process. This paper paper describes a twophase algorithm for the solution of largescale boxconstrained discrete illposed problems. The first phase applies a cascadic multilevel method and imposes the constraints on each level by orthogonal projection. The second phase improves the computed approximate solution on the finest level by an active set method. The latter allows several indices of the active set to be updated simultaneously. This reduces the computational effort significantly, when compared to standard active set methods that update one index at a time. Applications to image restoration are presented.