Sparse grid

About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.

Experimental validation of sparse sensing technique in subsurface microwave holography

Abstract: In this paper the implementation of the Sparse or Compressed Sensing (CS) technique to subsurface holography allowing data acquisition on a sparse grid is considered, which raises the scanning speed, reduces the amount of stored and transmitted data. To test various aspects of applying CS technique, a custom-built experimental setup acquiring data at several programmable sampling intervals was used. The setup consists of a vector network analyzer (VNA) with connected antenna and a two-coordinate mechanical scanner that moves the sample in the vicinity of the antenna. This technique allows simulating sparse sampling and testing various scanning and sounding parameters. As the test object a sample of cryogenic fuel tank thermal insulation was selected for the purpose of nondestructive testing (NDT). Upon acquisition, a complex data processing algorithm on the basis of windowing functions, interpolation, and filtering was applied. Then the Fourier-based back propagation algorithm was used for obtaining radar images by complex holograms reconstruction. Further development of the considered sensing technique is suggested, including the adaptive scanning on a non-equidistant grid.

Sparse tensors and discrete-time nonlinear filtering

Abstract: In many applications it is desired that discrete-discrete filtering problem can be solved in a reliable and computationally efficient manner. In particular, the signal and measurement models often include nonlinearity and/or non-Gaussian characteristics. In this paper, it is pointed out that this can be done efficiently by noting two key observations. Firstly, the bulk of the computations associated with the ldquopredictionrdquo step can be done off-line. The second key point is that the transition probability tensor and the conditional probability density are effectively sparse and so can be efficiently stored and manipulated using sparse tensors. These ideas are crucial for efficiently solving the higher dimensional filtering problems. The resulting technique, termed sparse grid filtering, is demonstrated by some examples, where it is shown that it works very well.

Sparse Grid Adaptive Interpolation in Problems of Modeling Dynamic Systems with Interval Parameters

Abstract: The paper is concerned with the issues of modeling dynamic systems with interval parameters In previous works, the authors proposed an adaptive interpolation algorithm for solving interval problems; the essence of the algorithm is the dynamic construction of a piecewise polynomial function that interpolates the solution of the problem with a given accuracy The main problem of applying the algorithm is related to the curse of dimension, ie, exponential complexity relative to the number of interval uncertainties in parameters The main objective of this work is to apply the previously proposed adaptive interpolation algorithm to dynamic systems with a large number of interval parameters In order to reduce the computational complexity of the algorithm, the authors propose using adaptive sparse grids This article introduces a novelty approach of applying sparse grids to problems with interval uncertainties The efficiency of the proposed approach has been demonstrated on representative interval problems of nonlinear dynamics and computational materials science

Maximum a posteriori density estimation and the sparse grid combination technique

Abstract: We study a novel method for maximum a posteriori (MAP) estimation of the probability density function of an arbitrary, independent and identically distributed \(d\)-dimensional data set. We give an interpretation of the MAP algorithm in terms of regularised maximum likelihood. We also present numerical experiments using a sparse grid combination technique and the `opticom' method. The numerical results demonstrate the viability of parallelisation for the combination technique. References H. J. Bungartz, M. Griebel, D. Roschke and C. Zenger. Pointwise convergence of the combination technique for the Laplace equation. East-West J. Numer. Math , 2:21--45 (1994). http://zbmath.org/?q=an:00653220 J. Garcke. Regression with the optimised combination technique. In Proceedings of the 23rd international conference on Machine learning , ICML '06, pages 321--328 (2006). doi:10.1145/1143844.1143885 J. Garcke. Sparse grid tutorial. Technical report (2011). http://page.math.tu-berlin.de/ garcke/paper/sparseGridTutorial.pdf M. Griebel and M. Hegland. A finite element method for density estimation with Gaussian process priors. SIAM J. Numer. Anal. , 47:4759--4792 (2010). doi:10.1137/080736478 M. Griebel, M. Schneider and C. Zenger. A combination technique for the solution of sparse grid problems. In Iterative methods in linear algebra (Brussels, 1991) , pages 263--281. North-Holland, Amsterdam (1992). M. Hegland. Adaptive sparse grids. ANZIAM J. , 44:C335--C353 (2003). http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/685 M. Hegland. Approximate maximum a posteriori with Gaussian process priors. Constr. Approx. , 26:205--224 (2007). doi:10.1007/s00365-006-0661-4 M. Hegland, J. Garcke, and V. Challis. The combination technique and some generalisations. Linear Algebra Appl. , 420:249--275 (2007). doi:10.1016/j.laa.2006.07.014 C. T. Kelley. Solving nonlinear equations with Newton's method . Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003). H. Kobayashi, B.L. Mark, and W. Turin. Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance . Cambridge University Press (2012). C. Pflaum and A. Zhou. Error analysis of the combination technique. Numerische Mathematik , 84:327--350 (1999). doi:10.1007/s002110050474 D. W. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization . John Wiley and Sons (2004). C. Zenger. Sparse grids. Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar , 31 (1990).