The singular value decomposition (SVD) of a matrix is a powerful tool for many matrix computation problems. In this paper, we consider generalizing the standard SVD to analyze and compute the regularized solution of linear ill-posed problems that arise from discretizing the first kind Fredholm integral equations. For the commonly used quadrature method for discretization, a regularizer of the form $\|x\|_{M}^2:=x^TMx$ should be exploited, where $M$ is symmetric positive definite. To handle this regularizer, we give the weighted SVD (WSVD) of a matrix under the $M$-inner product. Several important applications of WSVD, such as low-rank approximation and solving the least squares problems with minimum $\|\cdot\|_M$-norm, are studied. We propose the weighted Golub-Kahan bidiagonalization (WGKB) to compute several dominant WSVD components and a corresponding weighted LSQR algorithm to iteratively solve the least squares problem. All the above tools and methods are used to analyze and solve linear ill-posed problems with the regularizer $\|x\|_{M}^2$. A WGKB-based subspace projection regularization method is proposed to efficiently compute a good regularized solution, which can incorporate the prior information about $x$ encoded by the regularizer $\|x\|_{M}^2$. Several numerical experiments are performed to illustrate the fruitfulness of our methods.
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In this paper, we consider the numerical approximation of a time-fractional stochastic Cahn--Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order $\alpha\in(0,1)$ and a fractional time-integral noise of order $\gamma\in[0,1]$. The numerical scheme approximates the model by a piecewise linear finite element method in space and a convolution quadrature in time (for both time-fractional operators), along with the $L^2$-projection for the noise. We carefully investigate the spatially semidiscrete and fully discrete schemes, and obtain strong convergence rates by using clever energy arguments. The temporal H\"older continuity property of the solution played a key role in the error analysis. Unlike the stochastic Allen--Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity in the underlying model adds complexity and challenges to the error analysis. To overcome these difficulties, several new techniques and error estimates are developed. The study concludes with numerical examples that validate the theoretical findings.
In this paper, we provide a detailed theoretical analysis of the numerical scheme introduced in J. Comput. Phys. 436 (2021) 110253 for the reaction kinetics of a class of chemical reaction networks that satisfies detailed balance condition. In contrast to conventional numerical approximations, which are typically constructed based on ordinary differential equations (ODEs) for the concentrations of all involved species, the scheme is developed using the equations of reaction trajectories, which can be viewed as a generalized gradient flow of physically relevant free energy. The unique solvability, positivity-preserving, and energy-stable properties are proved for the general case involving multiple reactions, under a mild condition on the stoichiometric matrix.
In this paper, we propose a method for solving a PPAD-complete problem [Papadimitriou, 1994]. Given is the payoff matrix $C$ of a symmetric bimatrix game $(C, C^T)$ and our goal is to compute a Nash equilibrium of $(C, C^T)$. In this paper, we devise a nonlinear replicator dynamic (whose right-hand-side can be obtained by solving a pair of convex optimization problems) with the following property: Under any invertible $0 \leq C \leq 1$, every orbit of our dynamic starting at an interior strategy of the standard simplex approaches a set of strategies of $(C, C^T)$ such that, for each strategy in this set, a symmetric Nash equilibrium strategy can be computed by solving the aforementioned convex mathematical programs. We prove convergence using previous results in analysis (the analytic implicit function theorem), nonlinear optimization theory (duality theory, Berge's maximum principle, and a theorem of Robinson [1980] on the Lipschitz continuity of parametric nonlinear programs), and dynamical systems theory (a theorem of Losert and Akin [1983] related to the LaSalle invariance principle that is stronger under a stronger assumption).
In this paper, we consider the hull of an algebraic geometry code, meaning the intersection of the code and its dual. We demonstrate how codes whose hulls are algebraic geometry codes may be defined using only rational places of Kummer extensions (and Hermitian function fields in particular). Our primary tool is explicitly constructing non-special divisors of degrees $g$ and $g-1$ on certain families of function fields with many rational places, accomplished by appealing to Weierstrass semigroups. We provide explicit algebraic geometry codes with hulls of specified dimensions, producing along the way linearly complementary dual algebraic geometric codes from the Hermitian function field (among others) using only rational places and an answer to an open question posed by Ballet and Le Brigand for particular function fields. These results complement earlier work by Mesnager, Tang, and Qi that use lower-genus function fields as well as instances using places of a higher degree from Hermitian function fields to construct linearly complementary dual (LCD) codes and that of Carlet, Mesnager, Tang, Qi, and Pellikaan to provide explicit algebraic geometry codes with the LCD property rather than obtaining codes via monomial equivalences.
The variable-order fractional Laplacian plays an important role in the study of heterogeneous systems. In this paper, we propose the first numerical methods for the variable-order Laplacian $(-\Delta)^{\alpha({\bf x})/2}$ with $0 < \alpha({\bf x}) \le 2$, which will also be referred as the variable-order fractional Laplacian if $\alpha({\bf x})$ is strictly less than 2. We present a class of hypergeometric functions whose variable-order Laplacian can be analytically expressed. Building on these analytical results, we design the meshfree methods based on globally supported radial basis functions (RBFs), including Gaussian, generalized inverse multiquadric, and Bessel-type RBFs, to approximate the variable-order Laplacian $(-\Delta)^{\alpha({\bf x})/2}$. Our meshfree methods integrate the advantages of both pseudo-differential and hypersingular integral forms of the variable-order fractional Laplacian, and thus avoid numerically approximating the hypersingular integral. Moreover, our methods are simple and flexible of domain geometry, and their computer implementation remains the same for any dimension $d \ge 1$. Compared to finite difference methods, our methods can achieve a desired accuracy with much fewer points. This fact makes our method much attractive for problems involving variable-order fractional Laplacian where the number of points required is a critical cost. We then apply our method to study solution behaviors of variable-order fractional PDEs arising in different fields, including transition of waves between classical and fractional media, and coexistence of anomalous and normal diffusion in both diffusion equation and the Allen-Cahn equation. These results would provide insights for further understanding and applications of variable-order fractional derivatives.
When applying Hamiltonian operator splitting methods for the time integration of multi-species Vlasov-Maxwell-Landau systems, the reliable and efficient numerical approximation of the Landau equation represents a fundamental component of the entire algorithm. Substantial computational issues arise from the treatment of the physically most relevant three-dimensional case with Coulomb interaction. This work is concerned with the introduction and numerical comparison of novel approaches for the evaluation of the Landau collision operator. In the spirit of collocation, common tools are the identification of fundamental integrals, series expansions of the integral kernel and the density function on the main part of the velocity domain, and interpolation as well as quadrature approximation nearby the singularity of the kernel. Focusing on the favourable choice of the Fourier spectral method, their practical implementation uses the reduction to basic integrals, fast Fourier techniques, and summations along certain directions. Moreover, an important observation is that a significant percentage of the overall computational effort can be transferred to precomputations which are independent of the density function. For the purpose of exposition and numerical validation, the cases of constant, regular, and singular integral kernels are distinguished, and the procedure is adapted accordingly to the increasing complexity of the problem. With regard to the time integration of the Landau equation, the most expedient approach is applied in such a manner that the conservation of mass is ensured.
This work studies a parabolic-ODE PDE's system which describes the evolution of the physical capital "$k$" and technological progress "$A$", using a meshless in one and two dimensional bounded domain with regular boundary. The well-known Solow model is extended by considering the spatial diffusion of both capital anf technology. Moreover, we study the case in which no spatial diffusion of the technology progress occurs. For such models, we propound schemes based on the Generalized Finite Difference method and proof the convergence of the numerical solution to the continuous one. Several examples show the dynamics of the model for a wide range of parameters. These examples illustrate the accuary of the numerical method.
In this paper we propose and analyze a novel multilevel version of Stein variational gradient descent (SVGD). SVGD is a recent particle based variational inference method. For Bayesian inverse problems with computationally expensive likelihood evaluations, the method can become prohibitive as it requires to evolve a discrete dynamical system over many time steps, each of which requires likelihood evaluations at all particle locations. To address this, we introduce a multilevel variant that involves running several interacting particle dynamics in parallel corresponding to different approximation levels of the likelihood. By carefully tuning the number of particles at each level, we prove that a significant reduction in computational complexity can be achieved. As an application we provide a numerical experiment for a PDE driven inverse problem, which confirms the speed up suggested by our theoretical results.
In this paper, we consider feature screening for ultrahigh dimensional clustering analyses. Based on the observation that the marginal distribution of any given feature is a mixture of its conditional distributions in different clusters, we propose to screen clustering features by independently evaluating the homogeneity of each feature's mixture distribution. Important cluster-relevant features have heterogeneous components in their mixture distributions and unimportant features have homogeneous components. The well-known EM-test statistic is used to evaluate the homogeneity. Under general parametric settings, we establish the tail probability bounds of the EM-test statistic for the homogeneous and heterogeneous features, and further show that the proposed screening procedure can achieve the sure independent screening and even the consistency in selection properties. Limiting distribution of the EM-test statistic is also obtained for general parametric distributions. The proposed method is computationally efficient, can accurately screen for important cluster-relevant features and help to significantly improve clustering, as demonstrated in our extensive simulation and real data analyses.
We build on the theory of ontology logs (ologs) created by Spivak and Kent, and define a notion of wiring diagrams. In this article, a wiring diagram is a finite directed labelled graph. The labels correspond to types in an olog; they can also be interpreted as readings of sensors in an autonomous system. As such, wiring diagrams can be used as a framework for an autonomous system to form abstract concepts. We show that the graphs underlying skeleton wiring diagrams form a category. This allows skeleton wiring diagrams to be compared and manipulated using techniques from both graph theory and category theory. We also extend the usual definition of graph edit distance to the case of wiring diagrams by using operations only available to wiring diagrams, leading to a metric on the set of all skeleton wiring diagrams. In the end, we give an extended example on calculating the distance between two concepts represented by wiring diagrams, and explain how to apply our framework to any application domain.
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