Featured Cover Hao, Shourong; Shen, Yongxing
International journal for numerical methods in engineering,
03/2024, Letnik:
125, Številka:
6
Journal Article
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The cover image is based on the Research Article An efficient parallel solution scheme for the phase field approach to dynamic fracture based on a domain decomposition method by Shourong Hao et al., ...https://doi.org/10.1002/nme.7405.
It is well-known that the convergence rate of non-overlapping domain decomposition methods (DDMs) applied to the parallel finite-element solution of large-scale time-harmonic wave problems strongly ...depends on the transmission condition enforced at the interfaces between the subdomains. Transmission operators based on perfectly matched layers (PMLs) have proved to be well-suited for configurations with layered domain partitions. They are shown to be a good compromise between basic impedance conditions, which can lead to slow convergence, and computational expensive conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain. Unfortunately, the extension of the PML-based DDM for more general partitions with cross-points (where more than two subdomains meet) is rather tricky and requires some care.
In this work, we present a non-overlapping substructured DDM with PML transmission conditions for checkerboard (Cartesian) decompositions that takes cross-points into account. In such decompositions, each subdomain is surrounded by PMLs associated to edges and corners. The continuity of Dirichlet traces at the interfaces between a subdomain and PMLs is enforced with Lagrange multipliers. This coupling strategy offers the benefit of naturally computing Neumann traces, which allows to use the PMLs as discrete operators approximating the exact Dirichlet-to-Neumann maps. Two possible Lagrange multiplier finite element spaces are presented, and the behavior of the corresponding DDM is analyzed on several numerical examples.
•A novel PML-based cross-point treatment is proposed for a non-overlapping DDM.•Two discretizations with several stabilization strategies are presented.•2D finite element results with acoustic scattering benchmarks are presented.•The influence of relevant parameters is studied.•Settings with smoothly varying heterogeneous media are discussed.
Two parallel, non-iterative, multi-physics, domain decomposition methods are proposed to solve a coupled time-dependent Stokes-Darcy system with the Beavers-Joseph-Saffman-Jones interface condition. ...For both methods, spatial discretization is effected using finite element methods. The backward Euler method and a three-step backward differentiation method are used for the temporal discretization. Results obtained at previous time steps are used to approximate the coupling information on the interface between the Darcy and Stokes subdomains at the current time step. Hence, at each time step, only a single Stokes and a single Darcy problem need be solved; as these are uncoupled, they can be solved in parallel. The unconditional stability and convergence of the first method is proved and also illustrated through numerical experiments. The improved temporal convergence and unconditional stability of the second method is also illustrated through numerical experiments.
We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-refinement via domain ...decomposition and projection onto the space of high-order polynomials. The trial space is the space of neural network, which is defined globally over the entire computational domain, while the test space contains piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with a local learning algorithm that can efficiently localize the network parameter optimization. We demonstrate the advantages of hp-VPINNs in both accuracy and training cost for several numerical examples of function approximation and in solving differential equations.
•Development of a general framework for hp-variational physics-informed neural networks•Nonlinear approximation of neural networks, projection onto space of high-order polynomials.•Domain decomposition•Comparison with other methods that use neural networks•Local and global approximations with locally/globally defined test functions.•Different loss functions based on the variational form and integration by parts.•Detailed derivation of the hp-VPINN formulation.
Decision trees are widely-used classification and regression models because of their interpretability and good accuracy. Classical methods such as CART are based on greedy approaches but a growing ...attention has recently been devoted to optimal decision trees. We investigate the nonlinear continuous optimization formulation proposed in Blanquero et al. (2020) for training sparse optimal randomized classification trees. Sparsity is important not only for feature selection but also to improve interpretability. We first consider alternative methods to sparsify such trees based on concave approximations of the l0 “norm”. Promising results are obtained on 24 datasets in comparison with the original l1 and l∞ regularizations. Then, we derive bounds on the VC dimension of multivariate randomized classification trees. Finally, since training is computationally challenging for large datasets, we propose a general node-based decomposition scheme and a practical version of it. Experiments on larger datasets show that the proposed decomposition method is able to significantly reduce the training times without compromising the testing accuracy.
•l0-based regularization terms induce sparsity in multivariate randomized classification trees.•new lower and upper bounds on the VC dimension of multivariate randomized classification trees.•node-based decomposition methods for multivariate randomized classification trees.
This study presents a hybrid technique for polarimetric synthetic aperture radar (PolSAR) data decomposition. The proposed technique aims to overcome the negative power problem of model-based ...decomposition methods. To achieve this, matrix rotation theory is used along with hybrid scattering models. The matrix rotation theory is utilised on the basis of underlying dominant scatterer to remove maximal of the cross-polarisation power generated by the coupling between orthogonal states of polarisation. Removing the coupling energy between orthogonal states not only optimised the PolSAR coherency matrix but also transform it more close to reflection symmetry condition. The applicability of the proposed approach is shown through the implementation of hybrid three- and four-component decomposition methods. The proposed hybrid methods are experimentally verified on C-band Radarsat-2 San Francisco and L-band UAVSAR Hayward datasets. For further analysis, different land-cover patches are selected. Moreover, the variations in percentage of negative power pixels are investigated by employing different volume scattering models. Comparative analyses are presented with existing PolSAR decomposition techniques in terms of normalised scattering power means and amount of negative power pixels. All experimental analyses clearly report the superiority of the proposed hybrid approach through improvements over existing PolSAR decomposition techniques along with non-negative scattering powers.
•Construction and implementation of new domain-decomposition based parallel algorithm is proposed for cPINNs and XPINNs methods.•The proposed algorithm adds another dimension of parallelism in SciML ...primarily driven by data and model parallelism.•The proposed algorithm is shown its scaling for CPU and CPU+GPU architectures.
We develop a distributed framework for the physics-informed neural networks (PINNs) based on two recent extensions, namely conservative PINNs (cPINNs) and extended PINNs (XPINNs), which employ domain decomposition in space and in time-space, respectively. This domain decomposition endows cPINNs and XPINNs with several advantages over the vanilla PINNs, such as parallelization capacity, large representation capacity, efficient hyperparameter tuning, and is particularly effective for multi-scale and multi-physics problems. Here, we present a parallel algorithm for cPINNs and XPINNs constructed with a hybrid programming model described by MPI + X, where X ∈{CPUs,GPUs}. The main advantage of cPINN and XPINN over the more classical data and model parallel approaches is the flexibility of optimizing all hyperparameters of each neural network separately in each subdomain. We compare the performance of distributed cPINNs and XPINNs for various forward problems, using both weak and strong scalings. Our results indicate that for space domain decomposition, cPINNs are more efficient in terms of communication cost but XPINNs provide greater flexibility as they can also handle time-domain decomposition for any differential equations, and can deal with any arbitrarily shaped complex subdomains. To this end, we also present an application of the parallel XPINN method for solving an inverse diffusion problem with variable conductivity on the United States map, using ten regions as subdomains.
We present a coupling framework for Stokes-Darcy systems valid for arbitrary flow direction at low Reynolds numbers and for isotropic porous media. The proposed method is based on an overlapping ...domain decomposition concept to represent the transition region between the free-fluid and the porous-medium regimes. Matching conditions at the interfaces of the decomposition impose the continuity of velocity (on one interface) and pressure (on the other one) and the resulting algorithm can be easily implemented in a non-intrusive way. The numerical approximations of the fluid velocity and pressure obtained by the studied method converge to the corresponding counterparts computed by direct numerical simulation at the microscale, with convergence rates equal to suitable powers of the scale separation parameter ε in agreement with classical results in homogenization.
•Robust and efficient overlapping domain decomposition method to model filtration of fluids in porous media.•No auxiliary closure problems needed to determine coupling parameters.•Simple Dirichlet interface conditions on velocity and pressure.•Validation versus direct numerical simulations.•Numerical convergence rates in agreement with homogenization theory for filtration problems.
In this article, we propose a nonconformal multidomain and multisolver method (MDMSM) to solve the problem of time-consuming repetitive modeling of radiation and scattering in complex multiscale ...models. The MDMSM is a hybrid method based on the domain decomposition framework, which is compatible with the embedded finite element domain decomposition method (FEM-DDM), non-overlapping FEM-DDM, and the boundary element domain decomposition method (BEM-DDM). This method can flexibly select the appropriate partition and calculation methods for various solution scenarios. The subdomain solution is calculated by choosing the most suitable decomposition methods (DDMs) for sub-coupling calculations, and different transmission conditions are used for calculations between different subdomains. In multicase scene calculations, we propose a superimposed inheritance calculation method to save calculation time further. We apply this method to simulate the embedded antenna array structure in complex multiscale aircraft targets and prove its practicability and effectiveness.
A multibranch Rao-Wilton-Glisson (MB-RWG) basis function is proposed in this article, which is composed of one positive triangle with a larger edge length and several negative triangles with smaller ...edge lengths. All negative triangles are aggregated to replace the negative triangle in the traditional RWG basis function, with exactly the same function expression defined on the triangles. The number of negative triangles can be changed in different mesh structures. Similar to traditional RWG basis functions, the proposed MB-RWG basis functions guarantee normal current continuity across the common edges. No line charges exist and the total charge on one MB-RWG is zero. MB-RWG basis functions can be used very conveniently to connect one surface with a coarse mesh scheme to another with a fine mesh scheme and can be flexibly applied in the domain decomposition method (DDM). Numerical results validate the accuracy and demonstrate the versatility of the proposed basis function in modeling multiscale perfect electrically conducting (PEC) objects.
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