Abstract. We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix T into a Cauchy-like matrix C, which is known to have low-numerical-rank offdiagonal blocks. Thus, we design a fast scheme for constructing a hierarchically semiseparable (HSS) matrix approximation to C, where the HSS generators have internal structures. Unlike classical HSS methods, our solver employs randomized sampling techniques together with fast Toeplitz matrixvector multiplications, and thus converts the direct compression of the off-diagonal blocks of C into the compression of much smaller blocks. A strong rank-revealing QR factorization method is used to generate/preserve certain special structures, and also to ensure stability. A fast ULV HSS factorization scheme is provided to take advantage of the special structures. We also propose a precomputation procedure for the HSS construction so as to further improve the efficiency. The complexity of these methods is significantly lower than some similar Toeplitz solvers for large matrix size n. Detailed flop counts are given, with the aid of a rank relaxation technique. The total cost of our methods includes O(n) flops for HSS operations and O(n log 2 n) flops for matrix multiplications via FFTs, where n is the order of T . Various numerical tests on classical examples, including ill-conditioned ones, demonstrate the efficiency, and also indicate that the methods are stable in practice. This work shows a practical way of using randomized sampling in the development of fast rank structured methods.

Esophageal squamous-cell carcinoma (ESCC), one of the most prevalent and lethal malignant disease, has a complex but unknown tumor ecosystem. Here, we investigate the composition of ESCC tumors based on 208,659 single-cell transcriptomes derived from 60 individuals. We identify 8 common expression programs from malignant epithelial cells and discover 42 cell types, including 26 immune cell and 16 nonimmune stromal cell subtypes in the tumor microenvironment (TME), and analyse the interactions between cancer cells and other cells and the interactions among different cell types in the TME. Moreover, we link the cancer cell transcriptomes to the somatic mutations and identify several markers significantly associated with patients’ survival, which may be relevant to precision care of ESCC patients. These results reveal the immunosuppressive status in the ESCC TME and further our understanding of ESCC.

Esophageal squamous cell carcinoma (ESCC) is prevalent in some geographical regions of the world. ESCC development presents a multistep pathogenic process from inflammation to invasive cancer; however, what is critical in these processes and how they evolve is largely unknown, obstructing early diagnosis and effective treatment. Here, we create a mouse model mimicking human ESCC development and construct a single-cell ESCC developmental atlas. We identify a set of key transitional signatures associated with oncogenic evolution of epithelial cells and depict the landmark dynamic tumorigenic trajectories. An early downregulation of CD8 + response against the initial tissue damage accompanied by the transition of immune response from type 1 to type 3 results in accumulation and activation of macrophages and neutrophils, which may create a chronic inflammatory environment that promotes carcinogen-transformed epithelial cell survival and proliferation. These findings shed light on how ESCC is initiated and developed.

We present some superfast (O((m + n) log 2 (m + n)) complexity) and stable structured direct solvers for m × n Toeplitz least squares problems. Based on the displacement equation, a Toeplitz matrix T is first transformed into a Cauchy-like matrix C, which can be shown to have small off-diagonal numerical ranks when the diagonal blocks are rectangular. We generalize standard hierarchically semiseparable (HSS) matrix representations to rectangular ones, and construct a rectangular HSS approximation to C in nearly linear complexity with randomized sampling and fast multiplications of C with vectors. A new URV HSS factorization and a URV HSS solution are designed for the least squares solution. We also present two structured normal equation methods. Systematic error and stability analysis for our HSS methods is given, which is also useful for studying other HSS and rank structured methods. We derive the growth factors and the backward error bounds in the HSS factorizations, and show that the stability results are generally much better than those in dense LU factorizations with partial pivoting. Such analysis has not been done before for HSS matrices. The solvers are tested on various classical Toeplitz examples ranging from well-conditioned to highly ill-conditioned ones. Comparisons with some recent fast and superfast solvers are given. Our new methods are generally much faster, and give better (or at least comparable) accuracies, especially for ill-conditioned problems.

Abstract. Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a ThickRestart version of the Lanczos algorithm with deflation ('locking') and a new type of polynomial filters obtained from a least-squares technique. The resulting algorithm can be utilized in a 'spectrumslicing' approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different sub-intervals independently from one another.

Abstract. In this paper, we propose a structured bisection method with adaptive randomized sampling for finding selected or all of the eigenvalues of certain real symmetric matrices A. For A with a low-rank property, we construct a hierarchically semiseparable (HSS) approximation and show how to quickly evaluate and update its inertia in the bisection method. Unlike some existing randomized HSS constructions, the methods here do not require the knowledge of the off-diagonal (numerical) ranks in advance. Moreover, for A with a weak rank property or slowly decaying offdiagonal singular values, we show an idea of aggressive low-rank inertia evaluation, which means that a compact HSS approximation can preserve the inertia for certain shifts. This is analytically justified for a special case, and numerically shown for more general ones. A generalized LDL factorization of the HSS approximation is then designed for the fast evaluation of the inertia. A significant advantage over standard LDL factorizations is that the HSS LDL factorization (and thus the inertia) of A − sI can be quickly updated with multiple shifts s in bisection. The factorization with each new shift can reuse about 60% of the work. As an important application, the structured eigensolver can be applied to symmetric Toeplitz matrices, and the cost to find one eigenvalue is nearly linear in the order of the matrix. The numerical examples demonstrate the efficiency and the accuracy of our methods, especially the benefit of low-rank inertia evaluations. The ideas and methods can be potentially adapted to other HSS computations where shifts are involved and to more problems without a significant low-rank property.

Summary This paper presents an efficient method to perform structured matrix approximation by separation and hierarchy (SMASH), when the original dense matrix is associated with a kernel function. Given the points in a domain, a tree structure is first constructed based on an adaptive partition of the computational domain to facilitate subsequent approximation procedures. In contrast to existing schemes based on either analytic or purely algebraic approximations, SMASH takes advantage of both approaches and greatly improves efficiency. The algorithm follows a bottom‐up traversal of the tree and is able to perform the operations associated with each node on the same level in parallel. A strong rank‐revealing factorization is applied to the initial analytic approximation in the separation regime so that a special structure is incorporated into the final nested bases. As a consequence, the storage is significantly reduced on one hand and a hierarchy of the original grid is constructed on the other hand. Due to this hierarchy, nested bases at upper levels can be computed in a similar way as the leaf level operations but on coarser grids. The main advantages of SMASH include its simplicity of implementation, its flexibility to construct various hierarchical rank structures, and a low storage cost. The efficiency and robustness of SMASH are demonstrated through various test problems arising from integral equations, structured matrices, etc.

Abstract. This paper introduces a robust preconditioner for general sparse symmetric matrices, that is based on low-rank approximations of the Schur complement in a Domain Decomposition (DD) framework. In this "Schur Low Rank" (SLR) preconditioning approach, the coefficient matrix is first decoupled by DD, and then a low-rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface points. The method avoids explicit formation of the Schur complement matrix. We show the feasibility of this strategy for a model problem, and conduct a detailed spectral analysis for the relationship between the low-rank correction and the quality of the preconditioning. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach.