arXiv subjects

Zhiwen Zhang

Publications and source records attributed to Zhiwen Zhang.

At least 55 records · Page 3Linked to original sources

arXiv2024

A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals with application to a UQ problem for the Helmholtz equation

In this paper, we combine the Smolyak technique for multi-dimensional interpolation with the Filon-Clenshaw-Curtis (FCC) rule for one-dimensional oscillatory integration, to obtain a new Filon-Clenshaw-Curtis-Smolyak (FCCS) rule for oscillatory integrals with linear phase over the $d-$dimensional cube $[-1,1]^d$. By combining stability and convergence estimates for the FCC rule with error estimates for the Smolyak interpolation operator, we obtain an error estimate for the FCCS rule, consisting of the product of a Smolyak-type error estimate multiplied by a term that decreases with $\mathcal{O}(k^{-\tilde{d}})$, where $k$ is the wavenumber and $\tilde{d}$ is the number of oscillatory dimensions. If all dimensions are oscillatory, a higher negative power of $k$ appears in the estimate. As an application, we consider the forward problem of uncertainty quantification (UQ) for a one-space-dimensional Helmholtz problem with wavenumber $k$ and a random heterogeneous refractive index, depending in an affine way on $d$ i.i.d. uniform random variables. After applying a classical hybrid numerical-asymptotic approximation, expectations of functionals of the solution of this problem can be formulated as a sum of oscillatory integrals over $[-1,1]^d$, which we compute using the FCCS rule. We give numerical results for the FCCS rule and the UQ algorithm showing that accuracy improves when both $k$ and the order of the rule increase. We also give results for dimension-adaptive sparse grid FCCS quadrature showing its efficiency as dimension increases.

math.NA↗

arXiv2024

A DeepParticle method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systems

We study a regularized interacting particle method for computing aggregation patterns and near singular solutions of a Keller-Segal (KS) chemotaxis system in two and three space dimensions, then further develop DeepParticle (DP) method to learn and generate solutions under variations of physical parameters. The KS solutions are approximated as empirical measures of particles which self-adapt to the high gradient part of solutions. We utilize the expressiveness of deep neural networks (DNNs) to represent the transform of samples from a given initial (source) distribution to a target distribution at finite time T prior to blowup without assuming invertibility of the transforms. In the training stage, we update the network weights by minimizing a discrete 2-Wasserstein distance between the input and target empirical measures. To reduce computational cost, we develop an iterative divide-and-conquer algorithm to find the optimal transition matrix in the Wasserstein distance. We present numerical results of DP framework for successful learning and generation of KS dynamics in the presence of laminar and chaotic flows. The physical parameter in this work is either the small diffusivity of chemo-attractant or the reciprocal of the flow amplitude in the advection-dominated regime.

physics.comp-ph↗

arXiv2023

Randomized methods for computing optimal transport without regularization and their convergence analysis

The optimal transport (OT) problem can be reduced to a linear programming (LP) problem through discretization. In this paper, we introduced the random block coordinate descent (RBCD) methods to directly solve this LP problem. Our approach involves restricting the potentially large-scale optimization problem to small LP subproblems constructed via randomly chosen working sets. By using a random Gauss-Southwell-$q$ rule to select these working sets, we equip the vanilla version of (${\bf{\rm RBCD}_0}$) with almost sure convergence and a linear convergence rate to solve general standard LP problems. To further improve the efficiency of the (${\bf{\rm RBCD}_0}$) method, we explore the special structure of constraints in the OT problems and leverage the theory of linear systems to propose several approaches for refining the random working set selection and accelerating the vanilla method. Inexact versions of the RBCD methods are also discussed. Our preliminary numerical experiments demonstrate that the accelerated random block coordinate descent (${\bf {\rm ARBCD}}$) method compares well with other solvers including Sinkhorn's algorithm when seeking solutions with relatively high accuracy, and offers the advantage of saving memory.

math.OC↗

arXiv2023

Error estimate of a quasi-Monte Carlo time-splitting pseudospectral method for nonlinear Schrodinger equation with random potentials

In this paper, we consider the numerical solution of a nonlinear Schrodinger equation with spatial random potential. The randomly shifted quasi-Monte Carlo (QMC) lattice rule combined with the time-splitting pseudospectral discretization is applied and analyzed. The nonlinearity in the equation induces difficulties in estimating the regularity of the solution in random space. By the technique of weighted Sobolev space, we identify the possible weights and show the existence of QMC that converges optimally at the almost-linear rate without dependence on dimensions. The full error estimate of the scheme is established. We present numerical results to verify the accuracy and investigate the wave propagation.

math.NA↗

arXiv2023

Stochastic convergence of regularized solutions for backward heat conduction problems

In this paper, we study the stochastic convergence of regularized solutions for backward heat conduction problems. These problems are recognized as ill-posed due to the exponential decay of eigenvalues associated with the forward problems. We derive an error estimate for the least-squares regularized minimization problem within the framework of stochastic convergence. Our analysis reveals that the optimal error of the Tikhonov-type least-squares optimization problem depends on the noise level, the number of sensors, and the underlying ground truth. Moreover, we propose a self-adaptive algorithm to identify the optimal regularization parameter for the optimization problem without requiring knowledge of the noise level or any other prior information, which will be very practical in applications. We present numerical examples to demonstrate the accuracy and efficiency of our proposed method. These numerical results show that our method is efficient in solving backward heat conduction problems.

math.NA↗

arXiv2023

Stochastic Domain Decomposition Based on Variable-Separation Method

In this work, we propose a new stochastic domain decomposition method for solving steady-state partial differential equations (PDEs) with random inputs. Based on the efficiency of the Variable-separation (VS) method in simulating stochastic partial differential equations (SPDEs), we extend it to stochastic algebraic systems and apply it to stochastic domain decomposition. The resulting Stochastic Domain Decomposition based on the Variable-separation method (SDD-VS) effectively addresses the ``curse of dimensionality" by leveraging the explicit representation of stochastic functions derived from physical systems. The SDD-VS method aims to obtain a separated representation of the solution for the stochastic interface problem. To enhance efficiency, an offline-online computational decomposition is introduced. In the offline phase, the affine representation of stochastic algebraic systems is obtained through the successive application of the VS method. This serves as a crucial foundation for the SDD-VS method. In the online phase, the interface unknowns of SPDEs are estimated using a quasi-optimal separated representation, enabling the construction of efficient surrogate models for subproblems. The effectiveness of the proposed method is demonstrated via the numerical results of three concrete examples.

math.NA↗

arXiv2023

Multitask Weakly Supervised Learning for Origin Destination Travel Time Estimation

Travel time estimation from GPS trips is of great importance to order duration, ridesharing, taxi dispatching, etc. However, the dense trajectory is not always available due to the limitation of data privacy and acquisition, while the origin destination (OD) type of data, such as NYC taxi data, NYC bike data, and Capital Bikeshare data, is more accessible. To address this issue, this paper starts to estimate the OD trips travel time combined with the road network. Subsequently, a Multitask Weakly Supervised Learning Framework for Travel Time Estimation (MWSL TTE) has been proposed to infer transition probability between roads segments, and the travel time on road segments and intersection simultaneously. Technically, given an OD pair, the transition probability intends to recover the most possible route. And then, the output of travel time is equal to the summation of all segments' and intersections' travel time in this route. A novel route recovery function has been proposed to iteratively maximize the current route's co occurrence probability, and minimize the discrepancy between routes' probability distribution and the inverse distribution of routes' estimation loss. Moreover, the expected log likelihood function based on a weakly supervised framework has been deployed in optimizing the travel time from road segments and intersections concurrently. We conduct experiments on a wide range of real world taxi datasets in Xi'an and Chengdu and demonstrate our method's effectiveness on route recovery and travel time estimation.

cs.AI↗

arXiv2022

A Deep Learning Approach to Nonconvex Energy Minimization for Martensitic Phase Transitions

We propose a mesh-free method to solve nonconvex energy minimization problems for martensitic phase transitions and twinning in crystals, using the deep learning approach. These problems pose multiple challenges to both analysis and computation, as they involve multiwell gradient energies with large numbers of local minima, each involving a topologically complex microstructure of free boundaries with gradient jumps. We use the Deep Ritz method, whereby candidates for minimizers are represented by parameter-dependent deep neural networks, and the energy is minimized with respect to network parameters. The new essential ingredient is a novel activation function proposed here, which is a smoothened rectified linear unit we call SmReLU; this captures the structure of minimizers where usual activation functions fail. The method is mesh-free and thus can approximate free boundaries essential to this problem without any special treatment, and is extremely simple to implement. We show the results of many numerical computations demonstrating the success of our method.

cs.CE↗

arXiv2022

Deterministic-Statistical Approach for an Inverse Acoustic Source Problem using Multiple Frequency Limited Aperture Data

We propose a deterministic-statistical method for an inverse source problem using multiple frequency limited aperture far field data. The direct sampling method is used to obtain a disc such that it contains the compact support of the source. The Dirichlet eigenfunctions of the disc are used to expand the source function. Then the inverse problem is recast as a statistical inference problem for the expansion coefficients and the Bayesian inversion is employed to reconstruct the coefficients. The stability of the statistical inverse problem with respect to the measured data is justified in the sense of Hellinger distance. A preconditioned Crank-Nicolson (pCN) Metropolis-Hastings (MH) algorithm is implemented to explore the posterior density function of the unknowns. Numerical examples show that the proposed method is effective for both smooth and non-smooth sources given limited-aperture data.

math.NA↗

arXiv2022

A variational neural network approach for glacier modelling with nonlinear rheology

In this paper, we propose a mesh-free method to solve full stokes equation which models the glacier movement with nonlinear rheology. Our approach is inspired by the Deep-Ritz method proposed in [12]. We first formulate the solution of non-Newtonian ice flow model into the minimizer of a variational integral with boundary constraints. The solution is then approximated by a deep neural network whose loss function is the variational integral plus soft constraint from the mixed boundary conditions. Instead of introducing mesh grids or basis functions to evaluate the loss function, our method only requires uniform samplers of the domain and boundaries. To address instability in real-world scaling, we re-normalize the input of the network at the first layer and balance the regularizing factors for each individual boundary. Finally, we illustrate the performance of our method by several numerical experiments, including a 2D model with analytical solution, Arolla glacier model with real scaling and a 3D model with periodic boundary conditions. Numerical results show that our proposed method is efficient in solving the non-Newtonian mechanics arising from glacier modeling with nonlinear rheology.

physics.comp-ph↗

arXiv2022

GOF-TTE: Generative Online Federated Learning Framework for Travel Time Estimation

Estimating the travel time of a path is an essential topic for intelligent transportation systems. It serves as the foundation for real-world applications, such as traffic monitoring, route planning, and taxi dispatching. However, building a model for such a data-driven task requires a large amount of users' travel information, which directly relates to their privacy and thus is less likely to be shared. The non-Independent and Identically Distributed (non-IID) trajectory data across data owners also make a predictive model extremely challenging to be personalized if we directly apply federated learning. Finally, previous work on travel time estimation does not consider the real-time traffic state of roads, which we argue can significantly influence the prediction. To address the above challenges, we introduce GOF-TTE for the mobile user group, Generative Online Federated Learning Framework for Travel Time Estimation, which I) utilizes the federated learning approach, allowing private data to be kept on client devices while training, and designs the global model as an online generative model shared by all clients to infer the real-time road traffic state. II) apart from sharing a base model at the server, adapts a fine-tuned personalized model for every client to study their personal driving habits, making up for the residual error made by localized global model prediction. % III) designs the global model as an online generative model shared by all clients to infer the real-time road traffic state. We also employ a simple privacy attack to our framework and implement the differential privacy mechanism to further guarantee privacy safety. Finally, we conduct experiments on two real-world public taxi datasets of DiDi Chengdu and Xi'an. The experimental results demonstrate the effectiveness of our proposed framework.

cs.LG↗

arXiv2022

Route to Time and Time to Route: Travel Time Estimation from Sparse Trajectories

Due to the rapid development of Internet of Things (IoT) technologies, many online web apps (e.g., Google Map and Uber) estimate the travel time of trajectory data collected by mobile devices. However, in reality, complex factors, such as network communication and energy constraints, make multiple trajectories collected at a low sampling rate. In this case, this paper aims to resolve the problem of travel time estimation (TTE) and route recovery in sparse scenarios, which often leads to the uncertain label of travel time and route between continuously sampled GPS points. We formulate this problem as an inexact supervision problem in which the training data has coarsely grained labels and jointly solve the tasks of TTE and route recovery. And we argue that both two tasks are complementary to each other in the model-learning procedure and hold such a relation: more precise travel time can lead to better inference for routes, in turn, resulting in a more accurate time estimation). Based on this assumption, we propose an EM algorithm to alternatively estimate the travel time of inferred route through weak supervision in E step and retrieve the route based on estimated travel time in M step for sparse trajectories. We conducted experiments on three real-world trajectory datasets and demonstrated the effectiveness of the proposed method.

cs.AI↗

arXiv2022

DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method

We introduce the so called DeepParticle method to learn and generate invariant measures of stochastic dynamical systems with physical parameters based on data computed from an interacting particle method (IPM). We utilize the expressiveness of deep neural networks (DNNs) to represent the transform of samples from a given input (source) distribution to an arbitrary target distribution, neither assuming distribution functions in closed form nor a finite state space for the samples. In training, we update the network weights to minimize a discrete Wasserstein distance between the input and target samples. To reduce computational cost, we propose an iterative divide-and-conquer (a mini-batch interior point) algorithm, to find the optimal transition matrix in the Wasserstein distance. We present numerical results to demonstrate the performance of our method for accelerating IPM computation of invariant measures of stochastic dynamical systems arising in computing reaction-diffusion front speeds through chaotic flows. The physical parameter is a large Peclét number reflecting the advection dominated regime of our interest.

cs.LG↗

arXiv2022

ST-ExpertNet: A Deep Expert Framework for Traffic Prediction

Recently, forecasting the crowd flows has become an important research topic, and plentiful technologies have achieved good performances. As we all know, the flow at a citywide level is in a mixed state with several basic patterns (e.g., commuting, working, and commercial) caused by the city area functional distributions (e.g., developed commercial areas, educational areas and parks). However, existing technologies have been criticized for their lack of considering the differences in the flow patterns among regions since they want to build only one comprehensive model to learn the mixed flow tensors. Recognizing this limitation, we present a new perspective on flow prediction and propose an explainable framework named ST-ExpertNet, which can adopt every spatial-temporal model and train a set of functional experts devoted to specific flow patterns. Technically, we train a bunch of experts based on the Mixture of Experts (MoE), which guides each expert to specialize in different kinds of flow patterns in sample spaces by using the gating network. We define several criteria, including comprehensiveness, sparsity, and preciseness, to construct the experts for better interpretability and performances. We conduct experiments on a wide range of real-world taxi and bike datasets in Beijing and NYC. The visualizations of the expert's intermediate results demonstrate that our ST-ExpertNet successfully disentangles the city's mixed flow tensors along with the city layout, e.g., the urban ring road structure. Different network architectures, such as ST-ResNet, ConvLSTM, and CNN, have been adopted into our ST-ExpertNet framework for experiments and the results demonstrates the superiority of our framework in both interpretability and performances.

cs.LG↗

arXiv2022

Pre-classification based stochastic reduced-order model for time-dependent complex system

We propose a novel stochastic reduced-order model (SROM) for complex systems by combining clustering and classification strategies. Specifically, the distance and centroid of centroidal Voronoi tessellation (CVT) are redefined according to the optimality of proper orthogonal decomposition (POD), thereby obtaining a time-dependent generalized CVT, and each class can generate a set of cluster-based POD (CPOD) basis functions. To learn the classification mechanism of random input, the naive Bayes pre-classifier and clustering results are applied. Then for a new input, the set of CPOD basis functions associated with the predicted label is used to reduce the corresponding model. Rigorous error analysis is shown, and a discussion in stochastic Navier-Stokes equation is given to provide a context for the application of this model. Numerical experiments verify that the accuracy of our SROM is improved compared with the standard POD method.

math.NA↗

arXiv2021

EpiMob: Interactive Visual Analytics of Citywide Human Mobility Restrictions for Epidemic Control

The outbreak of coronavirus disease (COVID-19) has swept across more than 180 countries and territories since late January 2020. As a worldwide emergency response, governments have implemented various measures and policies, such as self-quarantine, travel restrictions, work from home, and regional lockdown, to control the spread of the epidemic. These countermeasures seek to restrict human mobility because COVID-19 is a highly contagious disease that is spread by human-to-human transmission. Medical experts and policymakers have expressed the urgency to effectively evaluate the outcome of human restriction policies with the aid of big data and information technology. Thus, based on big human mobility data and city POI data, an interactive visual analytics system called Epidemic Mobility (EpiMob) was designed in this study. The system interactively simulates the changes in human mobility and infection status in response to the implementation of a certain restriction policy or a combination of policies (e.g., regional lockdown, telecommuting, screening). Users can conveniently designate the spatial and temporal ranges for different mobility restriction policies. Then, the results reflecting the infection situation under different policies are dynamically displayed and can be flexibly compared and analyzed in depth. Multiple case studies consisting of interviews with domain experts were conducted in the largest metropolitan area of Japan (i.e., Greater Tokyo Area) to demonstrate that the system can provide insight into the effects of different human mobility restriction policies for epidemic control, through measurements and comparisons.

cs.HC↗

arXiv2021

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the solution space of the parabolic partial differential equations (PDEs) in the forward problem with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problem extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in the parabolic inverse source problems, which is referred to as the POD algorithm in this paper. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD algorithm in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD algorithm for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the POD algorithm provides considerable computational savings over the finite element method.

math.NA↗

arXiv2021

Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials

In this paper, we analyze the convergence of the operator-compressed multiscale finite element method (OC MsFEM) for Schrödinger equations with general multiscale potentials in the semiclassical regime. In the OC MsFEM the multiscale basis functions are constructed by solving a constrained energy minimization. Under a mild assumption on the mesh size $H$, we prove the exponential decay of the multiscale basis functions so that localized multiscale basis functions can be constructed, which achieve the same accuracy as the global ones if the oversampling size $m = O(\log(1/H))$. We prove the first-order convergence in the energy norm and second-order convergence in the $L^2$ norm for the OC MsFEM and super convergence rates can be obtained if the solution possesses sufficiently high regularity. By analysing the regularity of the solution, we also derive the dependence of the error estimates on the small parameters of the Schrödinger equation. We find that the OC MsFEM outperforms the finite element method (FEM) due to the super convergence behavior for high-regularity solutions and weaker dependence on the small parameters for low-regularity solutions in the presence of the multiscale potential. Finally, we present numerical results to demonstrate the accuracy and robustness of the OC MsFEM.

math.NA↗