| 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. |
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| 5. |
- Gyllingberg, Linnéa, et al.
(författare)
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Finding analytical approximations for discrete, stochastic, individual-based models of ecology
- 2023
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Ingår i: Mathematical Biosciences. - : Elsevier. - 0025-5564 .- 1879-3134. ; 365
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Tidskriftsartikel (refereegranskat)abstract
- Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.
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| 6. |
- Liu, Hong, et al.
(författare)
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On the maximum number of integer colourings with forbidden monochromatic sums
- 2021
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Ingår i: The Electronic Journal of Combinatorics. - : The Electronic Journal of Combinatorics. - 1097-1440 .- 1077-8926. ; 28:1
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Tidskriftsartikel (refereegranskat)abstract
- Let f(n, r) denote the maximum number of colourings of A ⊆ {1, …, n} with r colours such that each colour class is sum-free. Here, a sum is a subset {x, y, z} such that x + y = z. We show that f(n, 2) = 2⌈n/2⌉, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of f(n, r) for r ≤ 5. Similar results were obtained by Hán and Jiménez in the setting of finite abelian groups.
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| 7. |
- Prodinger, Helmut
(författare)
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Counting ternary trees according to the number of middle edges and factorizing into (3 / 2) -ary trees
- 2022
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Ingår i: Asian-European Journal of Mathematics. - : World Scientific Publishing. - 1793-5571 .- 1793-7183. ; 15:11
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Tidskriftsartikel (refereegranskat)abstract
- The sequence A120986 in the Encyclopedia of Integer Sequences counts ternary trees according to the number of edges (equivalently nodes) and the number of middle edges. Using a certain substitution, the underlying cubic equation can be factored. This leads to an extension of the concept of (3/2)-ary trees, introduced by Knuth in his christmas lecture from 2014.
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| 8. |
- 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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| 9. |
- 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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| 10. |
- 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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