The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and $\Hmat$ based methods, focus on structured low-rank approximations of subblocks in these systems. It is known that the ranks of these subblocks increase linearly with the wavenumber. We explore a data-sparse representation of BEM-matrices valid for a range of frequencies, based on extracting the known phase of the Green's function. Algebraically, this leads to a Hadamard product of a frequency matrix with an $\Hmat$. We show that the frequency dependency of this $\Hmat$ can be determined using a small number of frequency samples, even for geometrically complex three-dimensional scattering obstacles. We describe an efficient construction of the representation by combining adaptive cross approximation with adaptive rational approximation in the continuous frequency dimension. We show that our data-sparse representation allows to efficiently sample the full BEM-matrix at any given frequency, and as such it may be useful as part of an efficient sweeping routine.
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In many practices, scientists are particularly interested in detecting which of the predictors are truly associated with a multivariate response. It is more accurate to model multiple responses as one vector rather than separating each component one by one. This is particularly true for complex traits having multiple correlated components. A Bayesian multivariate variable selection (BMVS) approach is proposed to select important predictors influencing the multivariate response from a candidate pool with an ultrahigh dimension. By applying the sample-size-dependent spike and slab priors, the BMVS approach satisfies the strong selection consistency property under certain conditions, which represents the advantages of BMVS over other existing Bayesian multivariate regression-based approaches. The proposed approach considers the covariance structure of multiple responses without assuming independence and integrates the estimation of covariance-related parameters together with all regression parameters into one framework through a fast updating MCMC procedure. It is demonstrated through simulations that the BMVS approach outperforms some other relevant frequentist and Bayesian approaches. The proposed BMVS approach possesses the flexibility of wide applications, including genome-wide association studies with multiple correlated phenotypes and a large scale of genetic variants and/or environmental variables, as demonstrated in the real data analyses section. The computer code and test data of the proposed method are available as an R package.
A second order accurate, linear numerical method is analyzed for the Landau-Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non-convexity constraint of unit length of the magnetization. The numerical method is based on the second-order backward differentiation formula in time, combined with an implicit treatment of the linear diffusion term and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a point-wise level. This numerical scheme has shown extensive advantages in the practical computations for the physical model with large damping parameters, which comes from the fact that only a linear system with constant coefficients (independent of both time and the updated magnetization) needs to be solved at each time step, and has greatly improved the numerical efficiency. Meanwhile, a theoretical analysis for this linear numerical scheme has not been available. In this paper, we provide a rigorous error estimate of the numerical scheme, in the discrete $\ell^{\infty}(0,T; \ell^2) \cap \ell^2(0,T; H_h^1)$ norm, under suitable regularity assumptions and reasonable ratio between the time step-size and the spatial mesh-size. In particular, the projection operation is nonlinear, and a stability estimate for the projection step turns out to be highly challenging. Such a stability estimate is derived in details, which will play an essential role in the convergence analysis for the numerical scheme, if the damping parameter is greater than 3.
The multiple scattering theory (MST) is one of the most widely used methods in electronic structure calculations. It features a perfect separation between the atomic configurations and site potentials, and hence provides an efficient way to simulate defected and disordered systems. This work studies the MST methods from a numerical point of view and shows the convergence with respect to the truncation of the angular momentum summations, which is a fundamental approximation parameter for all MST methods. We provide both rigorous analysis and numerical experiments to illustrate the efficiency of the MST methods within the angular momentum representations.
We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called 'normex' approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named $d$-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles.
A fibration of graphs is an homomorphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. Recently, it has been shown that graph fibrations are useful tools to uncover symmetries and synchronization patterns in biological networks ranging from gene, protein,and metabolic networks to the brain. However, the inherent incompleteness and disordered nature of biological data precludes the application of the definition of fibration as it is; as a consequence, also the currently known algorithms to identify fibrations fail in these domains. In this paper, we introduce and develop systematically the theory of quasifibrations which attempts to capture more realistic patterns of almost-synchronization of units in biological networks. We provide an algorithmic solution to the problem of finding quasifibrations in networks where the existence of missing links and variability across samples preclude the identification of perfect symmetries in the connectivity structure. We test the algorithm against other strategies to repair missing links in incomplete networks using real connectome data and synthetic networks. Quasifibrations can be applied to reconstruct any incomplete network structure characterized by underlying symmetries and almost synchronized clusters.
We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nystr\"om discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. We consider two classes of CQ discretizations, one based on linear multistep methods and the other based on Runge-Kutta methods, in conjunction with Nystr\"om discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. We present a variety of accuracy tests that showcase the high-order in time convergence (up to and including fifth order) that the Nystr\"om CQ discretizations are capable of delivering for a variety of two dimensional scatterers and types of boundary conditions.
We study a class of bilevel integer programs with second-order cone constraints at the upper level and a convex quadratic objective and linear constraints at the lower level. We develop disjunctive cuts to separate bilevel infeasible points using a second-order-cone-based cut-generating procedure. To the best of our knowledge, this is the first time disjunctive cuts are studied in the context of discrete bilevel optimization. Using these disjunctive cuts, we establish a branch-and-cut algorithm for the problem class we study, and a cutting plane method for the problem variant with only binary variables. Our computational study demonstrates that both our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our test instances, where the non-linearities are linearized in a McCormick fashion.
We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Under certain assumptions about the distribution of the eigenvalues, we prove upper bounds on how the number of GMRES iterations grows with the frequency. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, we investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.
We study sparse linear regression over a network of agents, modeled as an undirected graph (with no centralized node). The estimation problem is formulated as the minimization of the sum of the local LASSO loss functions plus a quadratic penalty of the consensus constraint -- the latter being instrumental to obtain distributed solution methods. While penalty-based consensus methods have been extensively studied in the optimization literature, their statistical and computational guarantees in the high dimensional setting remain unclear. This work provides an answer to this open problem. Our contribution is two-fold. First, we establish statistical consistency of the estimator: under a suitable choice of the penalty parameter, the optimal solution of the penalized problem achieves near optimal minimax rate $\mathcal{O}(s \log d/N)$ in $\ell_2$-loss, where $s$ is the sparsity value, $d$ is the ambient dimension, and $N$ is the total sample size in the network -- this matches centralized sample rates. Second, we show that the proximal-gradient algorithm applied to the penalized problem, which naturally leads to distributed implementations, converges linearly up to a tolerance of the order of the centralized statistical error -- the rate scales as $\mathcal{O}(d)$, revealing an unavoidable speed-accuracy dilemma.Numerical results demonstrate the tightness of the derived sample rate and convergence rate scalings.
Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena. We train a Restricted Boltzmann Machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods. From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system. We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near criticality region of the Ising model. In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the Renormalization Group (RG) flow of the lattice model. Our results suggest an alternative explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points. Then from the reconstructed data we deduce the critical exponent associated to the magnetization to find satisfactory agreement with the actual physical value. We assume no prior knowledge about the criticality of the system and its Hamiltonian.
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