Search

We propose and rigorously analyze a finite element method for the approximation of stationary Fokker--Planck--Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. In particular, the Cordes setting guarantees the existence and uniqueness of a square-integrable invariant measure. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings, based on the finite element scheme for stationary FPK problems developed in the first part. Finally, we demonstrate the performance of the methods through numerical experiments., Comment: 28 pages

In this paper, we provide an optimisation approach motivated by the Blind Source Separation, or also known as Independent Component Analysis, for cointegration between financial time series. Two methods for cointegration tests are introduced, namely decorrelation for the bivariate case and maximisation of nongaussianity for higher-dimensions. The advantages of our methods, especially the better performances in limited sample size, enable a wider range of application and accessibility for researchers and practitioners to identify cointegrating relationships.

The investigation of tumor invasion and metastasis dynamics is crucial for advancements in cancer biology and treatment. Many mathematical models have been developed to study the invasion of host tissue by tumor cells. In this paper, we develop a novel stochastic interacting particle-field (SIPF) algorithm that accurately simulates the cancer cell invasion process within the haptotaxis advection-diffusion (HAD) system. Our approach approximates solutions using empirical measures of particle interactions, combined with a smoother field variable - the extracellular matrix concentration (ECM) - computed by the spectral method. We derive a one-step time recursion for both the positions of stochastic particles and the field variable using the implicit Euler discretization, which is based on the explicit Green's function of an elliptic operator characterized by the Laplacian minus a positive constant. Our numerical experiments demonstrate the superior performance of the proposed algorithm, especially in computing cancer cell growth with thin free boundaries in three-dimensional (3D) space. Numerical results show that the SIPF algorithm is mesh-free, self-adaptive, and low-cost. Moreover, it is more accurate and efficient than traditional numerical techniques such as the finite difference method (FDM) and spectral methods.

In this note we adopt an approach by Grenier, Nguyen and Rodnianski in \cite{GNR} for studying the nonlinear Landau damping of the two-species Vlasov-Poisson system in the phase space $\mathbb{T}^d_x \times \mathbb{R}^d_v$ with the dimension $d\geq 1$. The main goal is twofold: one is to extend the one-species case to the two-species case where the electron mass is finite and the ion mass is sufficiently large, and the other is to modify the $G$-functional such that it involves the norm in $L^{d+1}$ instead of $L^2$ as well as derivatives up to only the first order., Comment: 29 pages

In this paper, we propose novel proper orthogonal decomposition (POD)--based model reduction methods that effectively address the issue of inverse crime in solving parabolic inverse problems. Both the inverse initial value problems and inverse source problems are studied. By leveraging the inherent low-dimensional structures present in the data, our approach enables a reduction in the forward model complexity without compromising the accuracy of the inverse problem solution. Besides, we prove the convergence analysis of the proposed methods for solving parabolic inverse problems. Through extensive experimentation and comparative analysis, we demonstrate the effectiveness of our method in overcoming inverse crime and achieving improved inverse problem solutions. The proposed POD model reduction method offers a promising direction for improving the reliability and applicability of inverse problem-solving techniques in various domains.

This paper aims to investigate the diffusion behavior of particles moving in stochastic flows under a structure-preserving scheme. We compute the effective diffusivity for normal diffusive random flows and establish the power law between spatial and temporal variables for cases with anomalous diffusion phenomena. From a Lagrangian approach, we separate the corresponding stochastic differential equations (SDEs) into sub-problems and construct a one-step structure-preserving method to solve them. Then by modified equation systems, the convergence analysis in calculating the effective diffusivity is provided and compared between the structure-preserving scheme and the Euler-Maruyama scheme. Also, we provide the error estimate for the structure-preserving scheme in calculating the power law for a series of super-diffusive random flows. Finally, we calculate the effective diffusivity and anomalous diffusion phenomena for a series of 2D and 3D random fields., Comment: 39pages, 10 figures, planning to submit for Journal of Scientific Computing or Numerische Mathematik

We study an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient to obtain the rate function is the computation of the principal eigenvalue $\lambda$ of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size, and we show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, suitable for the IPM. The IPM applies naturally to problems in unbounded domains, scales easily to high dimensions, and adapts to singular behaviors in the vanishing-noise limit. We show numerical examples in dimensions up to 16. The numerical results show that our numerical approximation of $\lambda$ converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. Our paper appears to be the first one to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.

In light of recent work in the global well-posedness of solutions for an ionic Vlasov-Poisson system, as demonstrated by Griffin-Pickering and Iacobelli, the current work focuses on the moment propagation of the corresponding system in quasi-neutral regime. Such moment propagation result relies on an estimate of $Q_*(t)=|V(t;0,x,v)-V(0;0,x,v)|$, where $V(s;t,x,v)$ represents the solution of the characteristic ordinary differential equation associated with the Vlasov-Poisson system.

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., Comment: on SIAM JUQ

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.

We introduce an efficient stochastic interacting particle-field (SIPF) algorithm with no history dependence for computing aggregation patterns and near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis system in three space dimensions (3D). The KS solutions are approximated as empirical measures of particles coupled with a smoother field (concentration of chemo-attractant) variable computed by the spectral method. Instead of using heat kernels causing history dependence and high memory cost, we leverage the implicit Euler discretization to derive a one-step recursion in time for stochastic particle positions and the field variable based on the explicit Green's function of an elliptic operator of the form Laplacian minus a positive constant. In numerical experiments, we observe that the resulting SIPF algorithm is convergent and self-adaptive to the high gradient part of solutions. Despite the lack of analytical knowledge (e.g. a self-similar ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study the emergence of finite time blowup in 3D by only dozens of Fourier modes and through varying the amount of initial mass and tracking the evolution of the field variable. Notably, the algorithm can handle at ease multi-modal initial data and the subsequent complex evolution involving the merging of particle clusters and formation of a finite time singularity.

We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed., Comment: 36 pages, 17 figures, planning to submit for SIAM/ASA Journal on Uncertainty Quantification

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.

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.

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.

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.

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.

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.

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.

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., Comment: 16 pages

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.

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.

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.

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., Comment: 25 pages, 16 figures and 6 tables