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The spectral clustering algorithm is often used as a binary clustering method for unclassified data by applying the principal component analysis. To study theoretical properties of the algorithm, the assumption of homoscedasticity is often supposed in existing studies. However, this assumption is restrictive and often unrealistic in practice. Therefore, in this paper, we consider the allometric extension model, that is, the directions of the first eigenvectors of two covariance matrices and the direction of the difference of two mean vectors coincide, and we provide a non-asymptotic bound of the error probability of the spectral clustering algorithm for the allometric extension model. As a byproduct of the result, we obtain the consistency of the clustering method in high-dimensional settings.

Conventionally, piecewise polynomials have been used in the boundary elements method (BEM) to approximate unknown boundary values. Since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for high dimensional domains, the unknown values are approximated by the RBFs in this paper. Therefore, a new formulation of BEM, called radial BEM, is obtained. To calculate singular boundary integrals of the new method, we propose a new distribution for boundary source points that removes singularity from the integrals. Therefore, the boundary integrals are calculated precisely by the standard Gaussian quadrature rule (GQR) with n = 16 quadrature nodes. Several numerical examples are presented to check the efficiency of the radial BEM versus standard BEM and RBF collocation method for solving partial differential equations (PDEs). Analytical and numerical studies presented in this paper admit the radial BEM as a perfect version of BEM which will enrich the contribution of BEM and RBFs in solving PDEs, impressively.

We analyse a numerical scheme for a system arising from a novel description of the standard elastic--perfectly plastic response. The elastic--perfectly plastic response is described via rate-type equations that do not make use of the standard elastic-plastic decomposition, and the model does not require the use of variational inequalities. Furthermore, the model naturally includes the evolution equation for temperature. We present a low order discretisation based on the finite element method. Under certain restrictions on the mesh we subsequently prove the existence of discrete solutions, and we discuss the stability properties of the numerical scheme. The analysis is supplemented with computational examples.

In this paper, we are interested in constructing a scheme solving compressible Navier--Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy under a standard hyperbolic type CFL constraint on the time step size, e.g., $\Delta t=\mathcal O(\Delta x)$. Strang splitting is used to approximate convection and diffusion operators separately. For the convection part, i.e., the compressible Euler equation, the high order accurate postivity-preserving Runge--Kutta discontinuous Galerkin method can be used. For the diffusion part, the equation of internal energy instead of the total energy is considered, and a first order semi-implicit time discretization is used for the ease of achieving positivity. A suitable interior penalty discontinuous Galerkin method for the stress tensor can ensure the conservation of momentum and total energy for any high order polynomial basis. In particular, positivity can be proven with $\Delta t=\mathcal{O}(\Delta x)$ if the Laplacian operator of internal energy is approximated by the $\mathbb{Q}^k$ spectral element method with $k=1,2,3$. So the full scheme with $\mathbb{Q}^k$ ($k=1,2,3$) basis is conservative and positivity-preserving with $\Delta t=\mathcal{O}(\Delta x)$, which is robust for demanding problems such as solutions with low density and low pressure induced by high-speed shock diffraction. Even though the full scheme is only first order accurate in time, numerical tests indicate that higher order polynomial basis produces much better numerical solutions, e.g., better resolution for capturing the roll-ups during shock reflection.

We develop a hybrid scheme based on a finite difference scheme and a rescaling technique to approximate the solution of nonlinear wave equation. In order to numerically reproduce the blow-up phenomena, we propose a rule of scaling transformation, which is a variant of what was successfully used in the case of nonlinear parabolic equations. A careful study of the convergence of the proposed scheme is carried out and several numerical examples are performed in illustration.

Along with the proliferation of electric vehicles (EVs), optimizing the use of EV charging space can significantly alleviate the growing load on intelligent transportation systems. As the foundation to achieve such an optimization, a spatiotemporal method for EV charging demand prediction in urban areas is required. Although several solutions have been proposed by using data-driven deep learning methods, it can be found that these performance-oriented methods may suffer from misinterpretations to correctly handle the reverse relationship between charging demands and prices. To tackle the emerging challenges of training an accurate and interpretable prediction model, this paper proposes a novel approach that enables the integration of graph and temporal attention mechanisms for feature extraction and the usage of physic-informed meta-learning in the model pre-training step for knowledge transfer. Evaluation results on a dataset of 18,013 EV charging piles in Shenzhen, China, show that the proposed approach, named PAG, can achieve state-of-the-art forecasting performance and the ability in understanding the adaptive changes in charging demands caused by price fluctuations.

In this paper, we present a discontinuity and cusp capturing physics-informed neural network (PINN) to solve Stokes equations with a piecewise-constant viscosity and singular force along an interface. We first reformulate the governing equations in each fluid domain separately and replace the singular force effect with the traction balance equation between solutions in two sides along the interface. Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface, we hereby use a network consisting of two fully-connected sub-networks that approximate the pressure and velocity, respectively. The two sub-networks share the same primary coordinate input arguments but with different augmented feature inputs. These two augmented inputs provide the interface information, so we assume that a level set function is given and its zero level set indicates the position of the interface. The pressure sub-network uses an indicator function as an augmented input to capture the function discontinuity, while the velocity sub-network uses a cusp-enforced level set function to capture the derivative discontinuities via the traction balance equation. We perform a series of numerical experiments to solve two- and three-dimensional Stokes interface problems and perform an accuracy comparison with the augmented immersed interface methods in literature. Our results indicate that even a shallow network with a moderate number of neurons and sufficient training data points can achieve prediction accuracy comparable to that of immersed interface methods.

This paper introduces non-linear dimension reduction in factor-augmented vector autoregressions to analyze the effects of different economic shocks. I argue that controlling for non-linearities between a large-dimensional dataset and the latent factors is particularly useful during turbulent times of the business cycle. In simulations, I show that non-linear dimension reduction techniques yield good forecasting performance, especially when data is highly volatile. In an empirical application, I identify a monetary policy as well as an uncertainty shock excluding and including observations of the COVID-19 pandemic. Those two applications suggest that the non-linear FAVAR approaches are capable of dealing with the large outliers caused by the COVID-19 pandemic and yield reliable results in both scenarios.

We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any orders, by combining the use of neural networks with a Monte Carlo branching algorithm. In comparison with other deep learning PDE solvers, it also allows us to check the consistency of the learned neural network function. Numerical experiments presented show that this algorithm can outperform deep learning approaches based on backward stochastic differential equations or the Galerkin method, and provide solution estimates that are not obtained by those methods in fully nonlinear examples.

The HEat modulated Infinite DImensional Heston (HEIDIH) model and its numerical approximation are introduced and analyzed. This model falls into the general framework of infinite dimensional Heston stochastic volatility models of (F.E. Benth, I.C. Simonsen '18), introduced for the pricing of forward contracts. The HEIDIH model consists of a one-dimensional stochastic advection equation coupled with a stochastic volatility process, defined as a Cholesky-type decomposition of the tensor product of a Hilbert-space valued Ornstein-Uhlenbeck process, the mild solution to the stochastic heat equation on the real half-line. The advection and heat equations are driven by independent space-time Gaussian processes which are white in time and colored in space, with the latter covariance structure expressed by two different kernels. First, a class of weight-stationary kernels are given, under which regularity results for the HEIDIH model in fractional Sobolev spaces are formulated. In particular, the class includes weighted Mat\'ern kernels. Second, numerical approximation of the model is considered. An error decomposition formula, pointwise in space and time, for a finite-difference scheme is proven. For a special case, essentially sharp convergence rates are obtained when this is combined with a fully discrete finite element approximation of the stochastic heat equation. The analysis takes into account a localization error, a pointwise-in-space finite element discretization error and an error stemming from the noise being sampled pointwise in space. The rates obtained in the analysis are higher than what would be obtained using a standard Sobolev embedding technique. Numerical simulations illustrate the results.