1,630 research outputs found
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing
and computing with multivariate functions using the functional tensor-train
(FT), a continuous extension of the tensor-train (TT) decomposition. The FT
represents functions using a tensor-train ansatz by replacing the
three-dimensional TT cores with univariate matrix-valued functions. The main
contribution of this paper is a framework to compute the FT that employs
adaptive approximations of univariate fibers, and that is not tied to any
tensorized discretization. The algorithm can be coupled with any univariate
linear or nonlinear approximation procedure. We demonstrate that this approach
can generate multivariate function approximations that are several orders of
magnitude more accurate, for the same cost, than those based on the
conventional approach of compressing the coefficient tensor of a tensor-product
basis. Our approach is in the spirit of other continuous computation packages
such as Chebfun, and yields an algorithm which requires the computation of
"continuous" matrix factorizations such as the LU and QR decompositions of
vector-valued functions. To support these developments, we describe continuous
versions of an approximate maximum-volume cross approximation algorithm and of
a rounding algorithm that re-approximates an FT by one of lower ranks. We
demonstrate that our technique improves accuracy and robustness, compared to TT
and quantics-TT approaches with fixed parameterizations, of high-dimensional
integration, differentiation, and approximation of functions with local
features such as discontinuities and other nonlinearities
MGARD: A multigrid framework for high-performance, error-controlled data compression and refactoring
We describe MGARD, a software providing MultiGrid Adaptive Reduction for
floating-point scientific data on structured and unstructured grids. With
exceptional data compression capability and precise error control, MGARD
addresses a wide range of requirements, including storage reduction,
high-performance I/O, and in-situ data analysis. It features a unified
application programming interface (API) that seamlessly operates across diverse
computing architectures. MGARD has been optimized with highly-tuned GPU kernels
and efficient memory and device management mechanisms, ensuring scalable and
rapid operations.Comment: 20 pages, 8 figure
Deep Hierarchical Super-Resolution for Scientific Data Reduction and Visualization
We present an approach for hierarchical super resolution (SR) using neural
networks on an octree data representation. We train a hierarchy of neural
networks, each capable of 2x upscaling in each spatial dimension between two
levels of detail, and use these networks in tandem to facilitate large scale
factor super resolution, scaling with the number of trained networks. We
utilize these networks in a hierarchical super resolution algorithm that
upscales multiresolution data to a uniform high resolution without introducing
seam artifacts on octree node boundaries. We evaluate application of this
algorithm in a data reduction framework by dynamically downscaling input data
to an octree-based data structure to represent the multiresolution data before
compressing for additional storage reduction. We demonstrate that our approach
avoids seam artifacts common to multiresolution data formats, and show how
neural network super resolution assisted data reduction can preserve global
features better than compressors alone at the same compression ratios
Region-Adaptive, Error-Controlled Scientific Data Compression using Multilevel Decomposition
The increase of computer processing speed is significantly outpacing improvements in network and storage bandwidth, leading to the big data challenge in modern science, where scientific applications can quickly generate much more data than that can be transferred and stored. As a result, big scientific data must be reduced by a few orders of magnitude while the accuracy of the reduced data needs to be guaranteed for further scientific explorations. Moreover, scientists are often interested in some specific spatial/temporal regions in their data, where higher accuracy is required. The locations of the regions requiring high accuracy can sometimes be prescribed based on application knowledge, while other times they must be estimated based on general spatial/temporal variation. In this paper, we develop a novel multilevel approach which allows users to impose region-wise compression error bounds. Our method utilizes the byproduct of a multilevel compressor to detect regions where details are rich and we provide the theoretical underpinning for region-wise error control. With spatially varying precision preservation, our approach can achieve significantly higher compression ratios than single-error bounded compression approaches and control errors in the regions of interest. We conduct the evaluations on two climate use cases-one targeting small-scale, node features and the other focusing on long, areal features. For both use cases, the locations of the features were unknown ahead of the compression. By selecting approximately 16% of the data based on multi-scale spatial variations and compressing those regions with smaller error tolerances than the rest, our approach improves the accuracy of post-analysis by approximately 2 x compared to single-error-bounded compression at the same compression ratio. Using the same error bound for the region of interest, our approach can achieve an increase of more than 50% in overall compression ratio
Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
Stochastic Galerkin methods for non-affine coefficient representations are
known to cause major difficulties from theoretical and numerical points of
view. In this work, an adaptive Galerkin FE method for linear parametric PDEs
with lognormal coefficients discretized in Hermite chaos polynomials is
derived. It employs problem-adapted function spaces to ensure solvability of
the variational formulation. The inherently high computational complexity of
the parametric operator is made tractable by using hierarchical tensor
representations. For this, a new tensor train format of the lognormal
coefficient is derived and verified numerically. The central novelty is the
derivation of a reliable residual-based a posteriori error estimator. This can
be regarded as a unique feature of stochastic Galerkin methods. It allows for
an adaptive algorithm to steer the refinements of the physical mesh and the
anisotropic Wiener chaos polynomial degrees. For the evaluation of the error
estimator to become feasible, a numerically efficient tensor format
discretization is developed. Benchmark examples with unbounded lognormal
coefficient fields illustrate the performance of the proposed Galerkin
discretization and the fully adaptive algorithm
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