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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.

We develop a novel method of virtual sources to formulate boundary integral equations for exterior wave propagation problems. However, by contrast to classical boundary integral formulations, we displace the singularity of the Green's function by a small distance $h>0$. As a result, the discretization can be performed on the actual physical boundary with continuous kernels so that any naive quadrature scheme can be used to approximate integral operators. Using on-surface radiation conditions, we combine single- and double-layer potential representations of the solution to arrive at a well-conditioned system upon discretization. The virtual displacement parameter $h$ controls the conditioning of the discrete system. We provide mathematical guidance to choose $h$, in terms of the wavelength and mesh refinements, in order to strike a balance between accuracy and stability. Proof-of-concept implementations are presented, including piecewise linear and isogeometric element formulations in two- and three-dimensional settings. We observe exceptionally well-behaved spectra, and solve the corresponding systems using matrix-free GMRES iterations. The results are compared to analytical solutions for canonical problems. We conclude that the proposed method leads to accurate solutions and good stability for a wide range of wavelengths and mesh refinements.

This paper presents a novel approach to construct regularizing operators for severely ill-posed Fredholm integral equations of the first kind by introducing parametrized discretization. The optimal values of discretization and regularization parameters are computed simultaneously by solving a minimization problem formulated based on a regularization parameter search criterion. The effectiveness of the proposed approach is demonstrated through examples of noisy Laplace transform inversions and the deconvolution of nuclear magnetic resonance relaxation data.

This paper presents the method for the localization and grabbing of the long straight conductor based only on the magnetic field generated by the alternating current flowing through the conductor. The method uses two magnetometers mounted on the robot arm end-effector for localization. This location is then used to determine needed robot movement in order to grab the conductor. The method was tested in the laboratory conditions using the Schunk LWA 4P 6-axis robot arm.

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.

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.

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.

Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms and software capabilities for quadratization of non-autonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semi-discretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both non-autonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.

The semi-empirical nature of best-estimate models closing the balance equations of thermal-hydraulic (TH) system codes is well-known as a significant source of uncertainty for accuracy of output predictions. This uncertainty, called model uncertainty, is usually represented by multiplicative (log-)Gaussian variables whose estimation requires solving an inverse problem based on a set of adequately chosen real experiments. One method from the TH field, called CIRCE, addresses it. We present in the paper a generalization of this method to several groups of experiments each having their own properties, including different ranges for input conditions and different geometries. An individual (log-)Gaussian distribution is therefore estimated for each group in order to investigate whether the model uncertainty is homogeneous between the groups, or should depend on the group. To this end, a multi-group CIRCE is proposed where a variance parameter is estimated for each group jointly to a mean parameter common to all the groups to preserve the uniqueness of the best-estimate model. The ECME algorithm for Maximum Likelihood Estimation is adapted to the latter context, then applied to relevant demonstration cases. Finally, it is tested on a practical case to assess the uncertainty of critical mass flow assuming two groups due to the difference of geometry between the experimental setups.

For a singular integral equation on an interval of the real line, we study the behavior of the error of a delta-delta discretization. We show that the convergence is non-uniform, between order $O(h^{2})$ in the interior of the interval and a boundary layer where the consistency error does not tend to zero.