Domain Decomposition Research Articles

Adaptive Meta-Learning Probabilistic Inference Framework for Long Sequence Prediction

Long sequence prediction has broad and significant application value in fields such as finance, wind power, and weather. However, the complex long-term dependencies of long sequence data and the potential domain shift problems limit the effectiveness of traditional models in practical scenarios. To this end, we propose an Adaptive Meta-Learning Probabilistic Inference Framework (AMPIF) based on sequence decomposition, which can effectively enhance the long sequence prediction ability of various basic models. Specifically, first, we decouple complex sequences into seasonal and trend components through a frequency domain decomposition module. Then, we design an adaptive meta-learning task construction strategy, which divides the seasonal and trend components into different tasks through a clustering-matching approach. Finally, we design a dual-stream amortized network (ST-DAN) to capture shared information between seasonal-trend tasks and use the support set to generate task-specific parameters for rapid generalization learning on the query set. We conducted extensive experiments on six datasets, including wind power and finance scenarios, and the results show that our method significantly outperforms baseline methods in prediction accuracy, interpretability, and algorithm stability and can effectively enhance the long sequence prediction capabilities of base models. The source code is publicly available at https://github.com/Zhu-JP/AMPIF.

Physics-informed shape optimization using coordinate projection

The rapid growth of artificial intelligence is revolutionizing classical engineering society, offering novel approaches to material and structural design and analysis. Among various scientific machine learning techniques, physics-informed neural network (PINN) has been one of the most researched subjects, for its ability to incorporate physics prior knowledge into model training. However, the intrinsic continuity requirement of PINN demands the adoption of domain decomposition when multiple materials with distinct properties exist. This greatly complicates the gradient computation of design features, restricting the application of PINN to structural shape optimization. To address this, we present a novel framework that employs neural network coordinate projection for shape optimization within PINN. This technique allows for direct mapping from a standard shape to its optimal counterpart, optimizing the design objective without the need for traditional transition functions or the definition of intermediate material properties. Our method demonstrates a high degree of adaptability, allowing the incorporation of diverse constraints and objectives directly as training penalties. The proposed approach is tested on magnetostatic problems for iron core shape optimization, a scenario typically plagued by the high permeability contrast between materials. Validation with finite-element analysis confirms the accuracy and efficiency of our approach. The results highlight the framework’s capability as a viable tool for shape optimization in complex material design tasks.

Optimal time decay of the compressible micropolar fluids with discontinuous initial data in 3D

The global existence of weak solutions to Cauchy problem for compressible micropolar fluids with discontinuous initial data has been established in [4]. However, the optimal decay rates of these weak solutions remain an open problem. The current work is to show the optimal decay rates of these weak solutions in Lr-norm with 2<r≤∞ and the optimal decay rates of the first order derivative of the velocity and micro-rotational velocity in L2-norm which are the same as the rates of classical solutions in [20]. In this process, we combine electromagnetic and fluid decomposition, frequency domain decomposition, and construct a series of temporal energy functional to solve complex nonlinear problems with low regularity.

Structural Health Assessment of Kretek II Bridge using Enhanced Frequency Domain Decomposition

The development of infrastructure is growing rapidly in Indonesia. Kretek II bridge is one of the infrastructures built in the country. Dynamics aspects are one of the very important aspects used to validate structural analysis model or being linked to Structural Health Monitoring (SHM) of the bridge. Dynamics properties such as natural frequency, damping ratio, and mode shape also referred as modal parameters. The objective of this research is to determine the modal parameters of Kretek II bridge using Enhanced Frequency Domain Decomposition (EFDD) which is commonly used to extract modal parameters from the acceleration data recorded. Compared to the conventional method, EFDD is very practical and robust in structural health assessment because of its user-friendly uses in the ARTeMIS Modal software. To make sure that the results from EFDD are accurate, numerical modeling is necessary to validate it. This research was conducted using dynamics load test results as data for the modal extraction with EFDD method assisted by ARTeMIS Modal software. Dynamics load test was performed by placing 8 accelerometers in the span and exciting it with impact load from dropped truck on ramp. Modal parameters from the EFDD results are then compared to the numerical modeling results. The first two modes of the EFDD and numerical modeling results consecutively are 3.09 Hz, 3.745 Hz and 3.06 Hz, 3.49 Hz. The EFDD and numerical modeling results are both in agreement with both results having similar mode shape on their first two mode and low error percentage with only 0.89% and 7.17% respectively.

D 3 PBO: D ynamic D omain D ecomposition-based P arallel B ayesian O ptimization for Large-scale Analog Circuit Sizing

Bayesian optimization (BO) is an efficient global optimization method for expensive black-box functions, but the expansion for high-dimensional problems and large sample budgets still remains a severe challenge. In order to extend BO for large-scale analog circuit synthesis, a novel computationally efficient parallel BO method, D 3 PBO, is proposed for high-dimensional problems in this work. We introduce the dynamic domain decomposition method based on maximum variance between clusters. The search space is decomposed into subdomains progressively to limit the maximal number of observations in each domain. The promising domain is explored by multi-trust region-based batch BO with the local Gaussian process (GP) model. As the domain decomposition progresses, the basin-shaped domain is identified using a GP-assisted quadratic regression method and exploited by the local search method BOBYQA to achieve a faster convergence rate. The time complexity of D 3 PBO is constant for each iteration. Experiments demonstrate that D 3 PBO obtains better results with significantly less runtime consumption compared to state-of-the-art methods. For the circuit optimization experiments, D 3 PBO achieves up to 10× runtime speedup compared to TuRBO with better solutions.

Improving Deep Learning-Based Digital Image Correlation with Domain Decomposition Method

Improving Deep Learning-Based Digital Image Correlation with Domain Decomposition Method

SPASTC: a Spatial Partitioning Algorithm for Scalable Travel-time Computation

Travel-time computation with large transportation networks is often computationally intensive for two main reasons: 1) large computer memory is required to handle large networks; and 2) calculating shortest-distance paths over large networks is computing intensive. Therefore, previous research tends to limit their spatial extent to reduce computational intensity or resolve computational intensity with advanced cyberinfrastructure. In this context, this article describes a new Spatial Partitioning Algorithm for Scalable Travel-time Computation (SPASTC) that is designed based on spatial domain decomposition with computer memory limit explicitly considered. SPASTC preserves spatial relationships required for travel-time computation and respects a user-specified memory limit, which allows efficient and large-scale travel-time computation within the given memory limit. We demonstrate SPASTC by computing spatial accessibility to hospital beds across the conterminous United States. Our case study shows that SPASTC achieves significant efficiency and scalability making the travel-time computation tens of times faster.

Enhanced parallel computation for time-fractional fluid dynamics: A fast time-stepping method with Newton-Krylov-Schwarz solver

This paper presents a sum-of-exponentials domain decomposition method for the numerical simulation of two-dimensional unsteady fluid flow and heat transfer using a time-fractional fluid model. We employ a fast time-stepping approach to discretize the time-fractional derivatives, followed by the application of a parallel Newton-Krylov-Schwarz algorithm to solve the resulting discrete nonlinear system. The numerical experiments demonstrate a superior performance of the fast time-stepping method compared to the traditional L1 scheme, particularly in the context of parallel computation, as it substantially reduces the computational complexity and memory usage. The algorithm exhibits robust performance across a wide range of model parameters and solver settings. Notably, it achieves 60% parallel efficiency with the use of 768 processor cores. This study underscores the efficacy of parallel processing as a potent computational strategy for addressing the challenges in solving time-fractional partial differential equations.

A Reduced Order Model for Domain Decompositions with Non-conforming Interfaces

A Reduced Order Model for Domain Decompositions with Non-conforming Interfaces

Robust domain decomposition methods for high-contrast multiscale problems on irregular domains with virtual element discretizations

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains.The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.

Wave interaction with multiple adjacent floating solar panels with arbitrary constraints

The problem of wave interaction with multiple adjacent floating solar panels with arbitrary types and numbers of constraints is considered. All the solar panels are assumed to be homogeneous, with the same physical properties, as well as modeled by using the Kirchhoff-Love plate theory. The motion of the fluid is described by the linear velocity potential theory. The domain decomposition method is employed to obtain the solutions. In particular, the entire fluid domain is divided into two types, the one below the free surface, and the other below elastic plates. The velocity potential in the free surface domain is expressed into a series of eigenfunctions. By contrast, the boundary integral equation and the Green function are employed to construct the velocity potential of fluid beneath the entire elastic cover, with unknowns distributed along two interfaces and jumps of physical parameters of the plates. All these unknowns are solved from the system of linear equations, which is established from the matching conditions of velocity potentials and edge conditions. This approach is confirmed with much higher computational efficiency compared with the one only involving eigenfunction expansion for the fluid beneath each plate. Extensive results and discussions are provided for the reflection and transmission coefficients of water waves, maximum deflection, and principal strain of the elastic plates; especially, the influence of different types and numbers of edge constraints are investigated in detail.

Domain Decomposition for 3-D Nonlinear Magnetostatic Problems: Newton–Krylov–Schur Versus Schur–Newton–Krylov Methods

Domain Decomposition for 3-D Nonlinear Magnetostatic Problems: Newton–Krylov–Schur Versus Schur–Newton–Krylov Methods

The improved backward compatible physics-informed neural networks for reducing error accumulation and applications in data-driven higher-order rogue waves.

Due to the dynamic characteristics of instantaneity and steepness, employing domain decomposition techniques for simulating rogue wave solutions is highly appropriate. Wherein, the backward compatible physics-informed neural network (bc-PINN) is a temporally sequential scheme to solve PDEs over successive time segments while satisfying all previously obtained solutions. In this work, we propose improvements to the original bc-PINN algorithm in two aspects based on the characteristics of error propagation. One is to modify the loss term for ensuring backward compatibility by selecting the earliest learned solution for each sub-domain as pseudo-reference solution. The other is to adopt the concatenation of solutions obtained from individual subnetworks as the final form of the predicted solution. The improved backward compatible PINN (Ibc-PINN) is applied to study data-driven higher-order rogue waves for the nonlinear Schrödinger (NLS) equation and the AB system to demonstrate the effectiveness and advantages. Transfer learning and initial condition guided learning (ICGL) techniques are also utilized to accelerate the training. Moreover, the error analysis is conducted on each sub-domain, and it turns out that the slowdown of Ibc-PINN in error accumulation speed can yield greater advantages in accuracy. In short, numerical results fully indicate that Ibc-PINN significantly outperforms bc-PINN in terms of accuracy and stability without sacrificing efficiency.

Solving Surface-volume Integral Equations for PEC and Inhomogeneous/Anisotropic Materials with Multibranch Basis Functions

Multibranch basis functions have been confirmed to be effective for local refinement of domain decomposition methods in the application of solving surface and volume integral equations. Surface-volume integral equations (SVIEs) are applied for solving the hybrid electromagnetic scattering problems involving perfect electric conductors (PEC) and dielectrics, especially inhomogeneous and anisotropic media. In this paper, multibranch Rao-Wilton-Glisson basis functions (MB-RWGs) are applied in conjunction with multibranch Schaubert-Wilton-Glisson basis functions (MB-SWGs) for solving the SVIEs. Block diagonal preconditioners (BDPs) are used to accelerate the iteration convergence based on generalized minimum residual (GMRES) algorithms. The numerical results demonstrate the accuracy of the multibranch basis functions in solving SVIEs, and also show that proper BDPs can accelerate the iteration convergency.

BROM: An extension of beam theory through model order reduction

Beam analysis played a crucial role in the design of structures throughout modern history. Classical beam theories rely on analytical derivations under certain a priori kinematic assumptions (and are accurate with low computational cost), but they are not amenable to, for example, orthotropic materials and arbitrary cross sections. On the other hand, full 3D linear elasticity theory solved via the Finite Element (FE) method provides solutions for a wide range of engineering problems at the expense of a high computational cost. Reduced-order models try to provide a theory as general as 3D linear elasticity but as efficient as classical beam theories by assuming the solution to be represented in a low-dimensional basis. Among the several options available in the literature, the domain decomposition and hyperreduction via Empirical Cubature Method (ECM), which we name ddROM [1], exhibits this compelling feature. However, ddROM applied to beam structures, may provide more Degrees of Freedom (DoF) per node than most standard commercial FE codes which only consider rigid body motion of the cross section. This paper presents a methodology, called beam Reduced Order Model (bROM), to integrate ddROM in standard FE codes by means of a condensation process followed by a regression procedure. Condensation gives the stiffness matrix, required by FE commercial codes, of a super-element of a given length. This is achieved by assembling several elements and finding the relationship between applied displacements and reactions (Lagrange multipliers). The condensation process is then used to create a database of stiffness matrices as a function of the beam length over which a regression is performed. Some numerical examples are first presented to validate the proposed methodology against the classical beam theory for the case of isotropic beams, and finally, two orthotropic beam showing the utility of the method in comparison with FE.

A randomized operator splitting scheme inspired by stochastic optimization methods

In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an abstract setting and provide an error analysis where we prove that the temporal convergence order of the scheme is at least 1/2. We illustrate the theory by numerical experiments on both linear and quasilinear diffusion problems, using a randomized domain decomposition approach. We conclude that choosing the randomization in certain ways may improve the order to 1. This is as accurate as applying e.g. backward (implicit) Euler to the full problem, without splitting.

Parallel acoustic analysis based on the domain decomposition method with higher-order element

Parallel acoustic analysis based on the domain decomposition method with higher-order element

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

Quasi-optimal domain decomposition method for neural network-based computation of the time-dependent Schrödinger equation

In this paper, we derive and analyze the performance of optimal/quasi-optimal Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for the time-dependent Schrödinger equation when implement with neural network-based Partial Differential Equations (PDE) solvers. Optimal SWR methods, which are based on Dirichlet-to-Neumann operators, are known to have a higher convergence rate than classical or optimized SWR methods. However, they are usually considered prohibitive due to their computational costs with standard PDE solvers. Thanks to Physics Informed Neural Network acceleration within the Schwarz waveform relaxation process and an efficient computation of Dirichlet-to-Neumann transmission operators, we demonstrate that optimal and quasi-optimal SWR methods can be performed almost as efficiently as classical or optimized SWR methods while maintaining a faster convergence rate. We present a few numerical examples to illustrate the performance and convergence of the proposed method.

Trans-Net: A transferable pretrained neural networks based on temporal domain decomposition for solving partial differential equations

Physics-Informed Neural Networks (PINNs) has provided a novel direction for solving partial differential equations (PDEs) and has achieved significant advancements in the field of scientific computing. PINNs effectively incorporate the physical constraints of equations into the loss function, enabling neural networks to learn and approximate the behavior of physical systems by optimizing the loss function. According to the existing research, we propose a transferable pretrained neural networks framework based on temporal domain decomposition to solve partial differential equations. Specifically, we divide the domain into multiple subdomains according to time, each of which is solved using an individual neural network. Subsequently, the trained subnetwork and the predicted values at the common boundary are used as the pretrained model and initial values for the next subdomain, respectively. This not only improves the prediction accuracy of the subnetwork, but also reduces the training time. Finally, we present a series of classical numerical experiments including one-dimensional, two-dimensional, and three-dimensional partial differential equations. The experimental results indicate that the proposed method outperforms existing approaches in terms of accuracy and efficiency. Relevant data and code can be accessed on GitHub via https://github.com/karry-Zhang/Trans-Net.