An economical representation of PDE solution by using compressive sensing approach

Kang, Hongmei; Lai, Ming-Jun; Li, Xin

doi:10.1016/j.cad.2019.05.021

Cited by 6 publications

(12 citation statements)

References 27 publications

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“…The intuition is the following: all (or most of) the splines in the lowest levels are activated to approximate the coarse component of the solution and only a few splines in the high-resolution levels are activated to capture local features or sharp transitions. The dictionary Ψ p,l0,L has been also considered in [24].…”

Section: Multilevel Dictionary Of B-splines and The Sparsity Assumptionmentioning

confidence: 99%

“…A second disclaimer is that upon completion of the manuscript we became aware of the work [24], which bears some similarities with CossIGA. Our work shares with [24] the idea that using a spline dictionary instead of a basis could promote sparsity/compressibility of the PDE solution, and that a sparse version of the solution can then be recovered by suitable 1 minimization algorithms. However, in [24] randomized selection of the rows is not present and the compressive sensing paradigm is therefore not fully exploited.…”

Section: Introductionmentioning

confidence: 99%

“…Our work shares with [24] the idea that using a spline dictionary instead of a basis could promote sparsity/compressibility of the PDE solution, and that a sparse version of the solution can then be recovered by suitable 1 minimization algorithms. However, in [24] randomized selection of the rows is not present and the compressive sensing paradigm is therefore not fully exploited. In fact, the entire Galerkin matrix is assembled, and not just a fraction of it as proposed here (in our opinion, this is where most of the computational gain can be potentially obtained).…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the entire Galerkin matrix is assembled, and not just a fraction of it as proposed here (in our opinion, this is where most of the computational gain can be potentially obtained). Furthermore, the numerical tests discussed in [24] only include 2D square domains, while here we take into account more general (2D and 3D) geometries. Finally, in [24] sparse recovery is performed via 1 minimization, whereas we employ the greedy algorithm orthogonal matching pursuit.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the numerical tests discussed in [24] only include 2D square domains, while here we take into account more general (2D and 3D) geometries. Finally, in [24] sparse recovery is performed via 1 minimization, whereas we employ the greedy algorithm orthogonal matching pursuit.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Compressive Isogeometric Analysis

Brugiapaglia¹,

Tamellini

Tani

2020

Preprint

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This work is motivated by the difficulty in assembling the Galerkin matrix when solving Partial Differential Equations (PDEs) with Isogeometric Analysis (IGA) using B-splines of moderate-to-high polynomial degree. To mitigate this problem, we propose a novel methodology named CossIGA (COmpreSSive IsoGeometric Analysis), which combines the IGA principle with CORSING, a recently introduced sparse recovery approach for PDEs based on compressive sensing. CossIGA assembles only a small portion of a suitable IGA Petrov-Galerkin discretization and is effective whenever the PDE solution is sufficiently sparse or compressible, i.e., when most of its coefficients are zero or negligible. The sparsity of the solution is promoted by employing a multilevel dictionary of B-splines as opposed to a basis. Thanks to sparsity and the fact that only a fraction of the full discretization matrix is assembled, the proposed technique has the potential to lead to significant computational savings. We show the effectiveness of CossIGA for the solution of the 2D and 3D Poisson equation over nontrivial geometries by means of an extensive numerical investigation.

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Section: Multilevel Dictionary Of B-splines and The Sparsity Assumptionmentioning

confidence: 99%