| 1. |
- Elfverson, Daniel, et al.
(författare)
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Multiscale methods for problems with complex geometry
- 2017
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 321, s. 103-123
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Tidskriftsartikel (refereegranskat)abstract
- We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in the energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.
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| 2. |
- Gerken, Jan, 1991, et al.
(författare)
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Geometric deep learning and equivariant neural networks
- 2023
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Ingår i: Artificial Intelligence Review. - : Springer Nature. - 1573-7462 .- 0269-2821. ; 56:12, s. 14605-14662
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Tidskriftsartikel (refereegranskat)abstract
- We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M= G/ K , which are instead equivariant with respect to the global symmetry G on M . Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M= S2= SO (3) / SO (2) . Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G= SO (3) , illustrating the power of representation theory for deep learning.
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| 3. |
- Galindo-Prieto, Beatriz, et al.
(författare)
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A new approach for variable influence on projection (VIP) in O2PLS models
- 2017
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Ingår i: Chemometrics and Intelligent Laboratory Systems. - : Elsevier. - 0169-7439 .- 1873-3239. ; 160, s. 110-124
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Tidskriftsartikel (refereegranskat)abstract
- A novel variable influence on projection approach for O2PLS® models, named VIPO2PLS, is presented in this paper. VIPO2PLS is a model-based method for judging the importance of variables. Its cornerstone is the 2-way formalism of the O2PLS models; i.e. the use of both predictive and orthogonal normalized loadings of the two modelled data matrices, and also a new weighting system based on the sum of squares of both data blocks (X, Y). The VIPO2PLS algorithm has been tested in one synthetic data set and two real cases, and the outcomes have been compared to the PLS-VIP, VIPOPLS, and i-PLS methods. The purpose is to achieve a sharper and enhanced model interpretation of O2PLS models by using the new VIPO2PLS method for assessing the importance of both X- and Y- variables.
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| 4. |
- Berg, André, et al.
(författare)
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APPROXIMATED EXPONENTIAL INTEGRATORS FOR THE STOCHASTIC MANAKOV EQUATION
- 2023
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Ingår i: Journal of Computational Dynamics. - : American Institute of Mathematical Sciences (AIMS). - 2158-2491 .- 2158-2505. ; 10:2, s. 323-344
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Tidskriftsartikel (refereegranskat)abstract
- . This article presents and analyzes an approximated exponential integrator for the (inhomogeneous) stochastic Manakov system. This system of SPDE occurs in the study of pulse propagation in randomly birefringent optical fibers. For a globally Lipschitz-continuous nonlinearity, we prove that the strong order of the time integrator is 1/2. This is then used to prove that the approximated exponential integrator has convergence order 1/2 in probability and almost sure order 1/2-, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the approximated exponential integrator as well as a modified version of it.
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| 5. |
- Berg, André, et al.
(författare)
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Lie-Trotter Splitting for the Nonlinear Stochastic Manakov System
- 2021
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 88:1
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Tidskriftsartikel (refereegranskat)abstract
- This article analyses the convergence of the Lie-Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 1/2- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie-Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities.
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| 6. |
- Burman, Erik, et al.
(författare)
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CutFEM based on extended finite element spaces
- 2022
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 152, s. 331-369
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Tidskriftsartikel (refereegranskat)abstract
- We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More, precisely the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.
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| 7. |
- Burman, Erik, et al.
(författare)
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Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions
- 2024
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 93:345, s. 35-54
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Tidskriftsartikel (refereegranskat)abstract
- We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution u ∈ Hs with s ∈ (1, 3/2]. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.
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| 8. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
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Miniversal deformations of matrices of bilinear forms
- 2012
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 436:7, s. 2670-2700
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Tidskriftsartikel (refereegranskat)abstract
- Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct miniversal deformations of matrices under congruence.
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| 9. |
- Hansbo, Peter, et al.
(författare)
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STABILIZED FINITE ELEMENT APPROXIMATION OF THE MEAN CURVATURE VECTOR ON CLOSED SURFACES
- 2015
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 53:4, s. 1806-1832
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Tidskriftsartikel (refereegranskat)abstract
- The mean curvature vector of a surface is obtained by letting the Laplace-Beltrami operator act on the embedding of the surface in R-3. In this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in L-2. The stabilization involves the jump in the tangent gradient in the direction of the outer co-normals at each edge in the surface mesh. We consider both standard meshed surfaces and so-called cut surfaces that are level sets of piecewise linear distance functions. We prove a priori error estimates and verify the theoretical results numerically.
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| 10. |
- Gerken, Jan, 1991, et al.
(författare)
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Equivariance versus augmentation for spherical images
- 2022
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Ingår i: Proceedings of Machine Learning Resaerch. ; , s. 7404-7421
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Konferensbidrag (refereegranskat)abstract
- We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST- or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we find that by considerably increasing the amount of data augmentation and the size of the networks, it is possible for the standard CNNs to reach at least the same performance as the equivariant network. In contrast, for the inherently equivariant task of semantic segmentation, the non-equivariant networks are consistently outperformed by the equivariant networks with significantly fewer parameters. We also analyze and compare the inference latency and training times of the different networks, enabling detailed tradeoff considerations between equivariant architectures and data augmentation for practical problems.
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