Non-overlapping Domain Research Articles

Convergence analysis and applicability of a domain decomposition method with nonlocal interface boundary conditions

In the past, the domain decomposition method was developed successfully for solving large-scale linear systems. However, the problems with significant nonlocal effect remain a major challenger for applying the method efficiently. In order to sort out the problem, a non-overlapping domain decomposition method with nonlocal interface boundary conditions was recently proposed and studied both theoretically and numerically. This paper is the report on the further development of the method, aiming to provide a comprehensive convergence analysis of the method, with supplementary numerical tests to support the theoretical result. The nonlocal effect of the problem is found to be reflected in both the governing equation and boundary conditions, and the effect of the latter was never taken into account, although playing a significant role in affecting the convergence. In addition, the paper extends the applicability of the analysis result drawn from the Poisson’s equation to more complicated problems by examining the symbols of the Steklov–Poincaré operators. The extended application includes a model equation arising from fluid dynamics and the high performance of the domain decomposition method in solving this equation is better elaborated.

A Chimera method for thermal part-scale metal additive manufacturing simulation

This paper presents a Chimera approach for the thermal problems in welding and metallic Additive Manufacturing (AM). In particular, a moving mesh is attached to the moving heat source while a fixed background mesh covers the rest of the computational domain. The thermal field of the moving mesh is solved in the heat source reference frame. The chosen framework to couple the solutions on both meshes is a non-overlapping Domain Decomposition (DD) with Neumann–Dirichlet transmission conditions.Increased steadiness and accuracy within the vicinity of the Heat Affected Zone (HAZ) are the main advantages of this approach. The steadiness gain allows for the use of larger time steps, which is vital in AM applications and, in particular, Laser Powder Bed Fusion (LPBF), where the disparity of time scales represents a major hurdle. Moreover, enhanced accuracy can be observed in the resulting morphology of the melt pool. It will be shown that the method addresses classical shortcomings pointed out by Goldak without requiring the use of an asymmetrical heat source profile.

A Curved Surface Integral Method for Reliability Analysis of Multiple Failure Modes System with Non-Overlapping Failure Domains

Abstract In the study of reliability of systems with multiple failure modes, approximations can be obtained by calculating the probability of failure for each state function. The first-order reliability method and the second-order reliability method are effective, but they may introduce significant errors when dealing with certain nonlinear situations. Simulation methods such as line sampling method and response surface method can solve implicit function problems, but the large amount of calculation results in low efficiency. The curved surface integral method (CSI) has good accuracy in dealing with nonlinear problems. Therefore, a system reliability analysis method (CSIMMS) is proposed on the basis of CSI for solving multiple failure modes system reliability problems with non-overlapping failure domains. The order of magnitude of the failure probability is evaluated based on the reliability index and the degree of nonlinearity, ignoring the influence of low order of magnitude failure modes, and reducing the calculation of the system failure probability. Finally, CSIMMS and other methods are compared by three numerical examples, and the results show the stability and accuracy of the proposed method.

Information Density Enhancement Using Lossy Compression in DNA Data Storage.

This study develops two deoxyribonucleic acid (DNA)lossy compression models, Models A and B, to encode grayscale images into DNA sequences, enhance information density, and enable high-fidelity image recovery. These models, distinguished by their handling of pixel domains and interpolation methods, offer a novel approach to data storage for DNA. Model A processes pixels in overlapped domains using linear interpolation (LI), whereas Model B uses non-overlapped domains with nearest-neighbor interpolation (NNI). Through a comparative analysis with Joint Photographic Experts Group (JPEG)compression, the DNA lossy compression models demonstrate competitive advantages in terms of information density and image quality restoration. The application of these models to the Modified National Institute of Standards and Technology (MNIST) dataset reveals their efficiency and the recognizability of decompressed images, which is validated by convolutional neural network (CNN) performance. In particular, Model B2, a version of Model B, emerges as an effective method for balancing high information density (surpassing over 20 times the typical densities of two bits per nucleotide) with reasonably good image quality. These findings highlight the potential of DNA-based data storage systems for high-density and efficient compression, indicating a promising future for biological data storage solutions.

Zanidatamab in previously-treated HER2-positive (HER2+) biliary tract cancer (BTC): Overall survival (OS) and longer follow-up from the phase 2b HERIZON-BTC-01 study.

4091 Background: For patients with BTC that has progressed after first-line therapy, prognosis is poor with a median OS of 6-9 months with subsequent chemotherapy. Zanidatamab is a HER2-targeted bispecific antibody that binds to two non-overlapping HER2 domains and crosslinks neighboring HER2 proteins. In the primary analysis of the phase 2b HERIZON-BTC-01 trial, after a median follow-up of 12.4 months (data cutoff: October 10, 2022), zanidatamab showed encouraging antitumor activity (41% confirmed objective response rate [cORR]) with rapid and durable responses and a manageable safety profile in patients with previously treated HER2+ BTC. OS data were immature at that time. Here, we report updated analyses, including OS. Methods: HERIZON-BTC-01 (NCT04466891) is an ongoing phase 2b trial assessing zanidatamab (20 mg/kg IV Q2W) in patients with HER2/ERBB2 gene amplification and immunohistochemistry (IHC) 2+ or 3+ (Cohort 1; HER2+); or 0 or 1+ (Cohort 2) locally advanced, unresectable, or metastatic BTC (gallbladder cancer, intra/extrahepatic cholangiocarcinoma) who received prior gemcitabine-containing treatment. The primary endpoint was cORR. Select secondary endpoints included duration of response (DoR), OS, and frequency and severity of adverse events (AEs). Updated efficacy analyses include only Cohort 1; safety analyses include Cohorts 1 and 2. Results: As of the data cutoff (July 28, 2023), median (range) follow-up was 21.9 (16-34) months. In the 80 patients in Cohort 1 (HER2+), treatment was ongoing for 9 (11%) patients and 20 (25%) patients remained on the study. cORR was unchanged from the primary analysis (n = 33; 41%) with 1 additional complete response (n = 2; 2.5%). The median DoR (95% CI) increased approximately 2 months to 14.9 (7.4, not reached) months. Median OS (95% CI) was 15.5 (10.4, 18.5) months; 12-month OS (95% CI) was 56.2% (44.3, 66.5). Among all 87 patients (Cohort 1 and Cohort 2), 18 (21%) experienced grade ≥3 treatment-related AEs (TRAEs). The most common grade 3 TRAEs (occurring in > 2 patients) were diarrhea (4 [5%]); anemia (3 [3%]), and ejection fraction decreased (3 [3%]). Only one patient had a grade 4 TRAE (aspartate aminotransferase increased). There were no deaths due to TRAEs. Serious AEs occurred in 8 (9%) patients; 2 (2%) patients discontinued treatment due to an AE (pneumonitis and ejection fraction decreased; 1 patient each). Conclusions: With close to 2 years of median follow-up, zanidatamab demonstrated a median OS of 15.5 months and a median duration of response of 14.9 months (an increase compared with the initial report) in previously-treated patients with HER2+ BTC; a patient population with significant unmet need. With additional follow-up, safety remained manageable. A phase 3 trial of zanidatamab in the first-line setting for patients with metastatic BTC is planned. Clinical trial information: NCT04466891 .

Acceleration of computations in modelling of processes in complex objects and systems

The development of methods of parallelization of computing processes, which involve the decomposition of the computational domain, is an urgent task in the modeling of complex objects and systems. Complex objects and systems can contain a large number of elements and interactions. Decomposition allows you to break down a system into simpler subsystems, which simplifies the analysis and management of complexity. By dividing the calculation area of the part, it is possible to perform parallel calculations, which increases the efficiency of calculations and reduces simulation time. Domain decomposition makes it easy to scale the model to work with larger or more detailed systems. With the right choice of decomposition methods, the accuracy of the simulation can be improved, since different parts of the system may have different levels of detail and require appropriate methods of additional analysis. Decomposition allows the simulation to be distributed between different participants or devices, which is relevant for distributed systems or collaborative work on a project. In this work, mathematical models are built, which consist in the construction of iterative procedures for "stitching" several areas into a single whole. The models provide for different complexity of calculation domains, which makes it possible to perform different decomposition approaches, in particular, both overlapping and non-overlapping domain decomposition. The obtained mathematical models of subject domain decomposition can be applied to objects and systems that have different geometric complexity. Domain decomposition models that do not use overlap contain different iterative methods of "stitching" on a common boundary depending on the types of boundary conditions (a condition of the first kind is a Dirichlet condition, or a condition of the second year is a Neumann condition), and domain decomposition models with an overlap of two or more areas consist of the minimization problem for constructing the iterative condition of "stitching" areas. It should be noted that the obtained models will work effectively on all applied tasks that describe the dynamic behavior of objects and their systems, but the high degree of efficiency of one model may be lower than the corresponding the degree of effectiveness of another model, since each task is individual. Keywords: mathematical modelling, decomposition of the computational domain, parallelization, optimization, complex objects and systems.

An Efficient Cross-Domain Recommendation System Based on Clustering Approach and User-Score

Growing consumers and online commercial products have put a great challenge on Recommender systems. Abundant browsing and online shopping across the world have provided immense opportunities to E-commercial giants to effectively learn the behaviour and attract users. However, due to the variety of available products and distinct consumers, crucial issues of data sparsity and cold start problems have been introduced. Single-domain recommendation systems are unable to handle such issues and research has been concentrated on developing more sophisticated cross-domain recommendation systems. Concerning other systems, non-overlapping domains form the most difficult part to solve using the cross-domain approach. This article focuses on partially overlapping domains and suggests a simple and efficient approach based on mapping users' interests by clustering books and movies. The cluster information from the auxiliary domain is transferred to the target domain using the user score and mapped properly to recommend books. The user and item scores in both domains are computed using the transfer frequency-inverse document frequency approach. The proposed recommender system offers low computational complexity and requires less time.

A non-overlapping optimization-based domain decomposition approach to component-based model reduction of incompressible flows

We present a component-based model order reduction procedure to efficiently and accurately solve parameterized incompressible flows governed by the Navier-Stokes equations. Our approach leverages a non-overlapping optimization-based domain decomposition technique to determine the control variable that minimizes jumps across the interfaces between sub-domains. To solve the resulting constrained optimization problem, we propose both Gauss-Newton and sequential quadratic programming methods, which effectively transform the constrained problem into an unconstrained formulation. Furthermore, we integrate model order reduction techniques into the optimization framework, to speed up computations. In particular, we incorporate localized training and adaptive enrichment to reduce the burden associated with the training of the local reduced-order models. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology.

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media

Abstract The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.

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.

A family of Nonoverlapping Spectral Additive Schwarz methods (NOSAS) and their economic versions

NOSAS (Nonoverlapping Spectral Additive Schwarz) methods are nonoverlapping iterative domain decomposition methods for efficiently solving large sparse linear systems arising from elliptic problems with heterogeneous coefficients. NOSAS methods are of additive Schwarz types with local Dirichlet solver on each nonoverlapping subdomain Ωi, plus a coarse problem on the interface of the subdomains. The coarse problem has global and local interaction components which are associated respectively to low-frequency and high-frequency modes obtained locally from a generalized eigenvalue problem defined on each subdomain’s interface. In this paper, we propose several variants of the NOSAS methods by selecting different generalized eigenvalue problems and different coarse space extensions. Additionally, we propose an economic variant to reduce the complexity of these generalized eigenvalue problems. The analysis of all the variants is established, and numerical tests are provided.

Investigation of combustion model via the local collocation technique based on moving Taylor polynomial (MTP) approximation/domain decomposition method with error analysis

In this paper, we develop a new meshless numerical procedure for simulating the combustion model. To that end, we employ a local meshless collocation method according to the moving Taylor polynomial (MTP) approximation. The space derivative is approximated by using the local approach and then the Crank–Nicolson algorithm is utilized to approximate the time derivative. The stability and convergence of the time-discrete formulation are discussed, analytically and numerically. The Broyden method is applied to solve this nonlinear system. Since the size of the physical domain is large, we employ the non-overlapping domain decomposition method (DDM) to obtain a faster numerical algorithm. The local meshless approaches are efficient numerical techniques to simulate models in the fluid flow. The obtained results show that the proposed numerical formulation has efficient results for solving this mathematical model.

Adaptive fail-safe topology optimization using a hierarchical parallelization scheme

This work presents an efficient, flexible, and scalable strategy to implement density-based topology optimization formulation for fail-safe structural design. Such optimized designs can operate after faults modeled as loss of stiffness. However, the need for assessing the physical behavior in all the fault cases can make this problem unfeasible computationally. We combine many ingredients to exploit the different levels of parallelism of the formulation to deal successfully with this problem. We use a non-overlapping domain decomposition method to solve the failure cases using multi-core computing with distributed memory computation. These subdomains use adaptive mesh refinement (AMR) techniques to focus computing efforts on the regions of interest. A key point to evaluate the fault cases is the organization of the computing threads, especially in clusters of computers. We group the computing threads by physical computing nodes to reduce the inter-node communications, maximizing intra-node communications to mitigate bandwidth problems. Another ingredient of paramount importance is the use of computing buffers to adapt the evaluation of the fault cases to the computing resources. We test the scalability of the computational framework using a large cluster of computers.

Multiscale sampling for the inverse modeling of partial differential equations

We are concerned with a novel Bayesian statistical framework for the characterization of natural subsurface formations, a very challenging task. Because of the large dimension of the stochastic space of the prior distribution in the framework, typically a dimensional reduction method, such as a Karhunen-Leove expansion (KLE), needs to be applied to the prior distribution to make the characterization computationally tractable. Due to the large variability of properties of subsurface formations (such as permeability and porosity) it may be of value to localize the sampling strategy so that it can better adapt to large local variability of rock properties.In this paper, we introduce the concept of multiscale sampling to localize the search in the stochastic space. We combine the simplicity of a preconditioned Markov Chain Monte Carlo method with a new algorithm to decompose the stochastic space into orthogonal subspaces, through a one-to-one mapping of the subspaces to subdomains of a non-overlapping domain decomposition of the region of interest. The localization of the search is performed by a multiscale blocking strategy within Gibbs sampling: we apply a KL expansion locally, at the subdomain level. Within each subdomain, blocking is applied again, for the sampling of the KLE random coefficients.The effectiveness of the proposed framework is tested in the solution of inverse problems related to elliptic partial differential equations arising in porous media flows. We use multi-chain studies in a multi-GPU cluster to show that the new algorithm clearly improves the convergence rate of the preconditioned MCMC method. Moreover, we illustrate the importance of a few conditioning points to further improve the convergence of the proposed method.

Fast Non-overlapping Domain Decomposition Methods for Continuous Multi-phase Labeling Problem

Fast Non-overlapping Domain Decomposition Methods for Continuous Multi-phase Labeling Problem

Non-overlapping domain decomposition algorithms with only primal velocity unknowns for the discontinuous viscosity Stokes problem

In this paper, non-overlapping domain decomposition algorithms for the Stokes problem with a discontinuous viscosity coefficient are proposed and analyzed. A pair of inf–sup stable finite element spaces with discontinuous pressure finite element functions is employed to obtain a discrete Stokes system. By introducing a non-overlapping subdomain partition for the given finite element spaces, the discrete Stokes system is reformulated into an algebraic system on dual unknowns, where the continuity on the velocity unknowns at the subdomain vertices is enforced strongly and the continuity on the remaining interface velocity unknowns is enforced weakly using the dual unknowns. The resulting algebraic system on the dual unknowns is then solved by an iterative method. The coarse problem in the resulting algebraic system is only related to the strongly coupled primal velocity unknowns and it is thus obtained as a symmetric positive definite matrix equation. To accelerate the iteration convergence, practical versions of lumped and Dirichlet preconditioners are proposed and their condition numbers are also analyzed for the resulting dual algebraic system. Numerical results are also included to confirm our condition number estimates.

Spatially non-overlapping Ca2+ signals drive distinct forms of neurotransmission

Calcium (Ca2+) signaling is tightly regulated within a presynaptic bouton. Here, we visualize Ca2+ signals within hippocampal presynaptic boutons using GCaMP8s tagged to synaptobrevin, a synaptic vesicle protein. We identify evoked presynaptic Ca2+ transients (ePreCTs) that derive from synchronized voltage-gated Ca2+ channel openings, spontaneous presynaptic Ca2+ transients (sPreCTs) that originate from ryanodine sensitive Ca2+ stores, and a baseline Ca2+ signal that arises from stochastic voltage-gated Ca2+ channel openings. We find that baseline Ca2+, but not sPreCTs, contributes to spontaneous glutamate release. We employ photobleaching as a use-dependent tool to probe nano-organization of Ca2+ signals and observe that all three occur in non-overlapping domains within the synapse at near-resting conditions. However, increased depolarization induces intermixing of these Ca2+ domains via both local and non-local synaptic vesicle turnover. Our findings reveal nanosegregation of Ca2+ signals within a presynaptic terminal that derive from multiple sources and in turn drive specific modes of neurotransmission.

Navigating Faith and Science

Navigating Faith and Science

A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problems

A locally optimal preconditioned Newton-Schur method is proposed for solving symmetric elliptic eigenvalue problems. Firstly, the Steklov-Poincaré operator is used to project the eigenvalue problem on the domain Ω \Omega onto the nonlinear eigenvalue subproblem on Γ \Gamma , which is the union of subdomain boundaries. Then, the direction of correction is obtained via applying a non-overlapping domain decomposition method on Γ \Gamma . Four different strategies are proposed to build the hierarchical subspace U k + 1 U_{k+1} over the boundaries, which are based on the combination of the coarse-subspace with the directions of correction. Finally, the approximation of eigenpair is updated by solving a local optimization problem on the subspace U k + 1 U_{k+1} . The convergence rate of the locally optimal preconditioned Newton-Schur method is proved to be γ = 1 − c 0 T h , H − 1 \gamma =1-c_{0}T_{h,H}^{-1} , where c 0 c_{0} is a constant independent of the fine mesh size h h , the coarse mesh size H H and jumps of the coefficients; whereas T h , H T_{h,H} is the constant depending on stability of the decomposition. Numerical results confirm our theoretical analysis.