Block-diagonal Research Articles

In-Memory Transposable Multibit Multiplication Based on Diagonal Symmetry Weight Block

A possible approach to overcome the von Neumann bottleneck and meet the increasing demand for better computing performance is to computing in-memory (CIM). The results of the in-memory calculations are primarily reflected in the vertical bitline (BL) analog voltage. However, the nonlinearity of the BL discharge deteriorates with the increase in discharge voltage. In this study, we propose a diagonal symmetry weight block (DSWB) based on an eight-transistor (8T) static random access memory (SRAM) that can achieve multibit transposable operations. In addition, to guarantee linearity and complete multibit multiplication operations, we propose a cascode current mirror (CCM)-based multiplier. To achieve low-overhead and more efficient quantification, our proposed CIM macro uses a counter-type quantization circuit to read out the analog calculation results. We simulated the performance of the proposed 8T SRAM in a 28-nm complementary metal–oxide–semiconductor process. The integral nonlinearity (INL) of the proposed CCM-based CIM decreased by approximately 54.4% compared with the traditional CIM. Furthermore, the proposed in-memory multibit multiplication throughput density was 6.74 GOPS/kb; this throughput density improvement is approximately 3.3–10.5 times higher than the existing CIM works.

Matrix splitting preconditioning based on sine transform for solving two-dimensional space-fractional diffusion equations

Finite difference discretization of the two-dimensional space-fractional diffusion equations derives a complicated linear system consisting of identity matrix and four scaled block-Toeplitz with Toeplitz block (BTTB) matrices resulted from the left and right Riemann–Liouville fractional derivatives in different directions. Incorporating with the diffusion coefficients and the symmetric parts of the BTTB matrices, we construct a diagonal and symmetric splitting (DSS) iteration method, which is demonstrated to be convergent conditionally when the considered space-fractional diffusion equations have sufficiently close diffusion coefficients. By further replacing the symmetric Toeplitz matrices involved in the BTTB matrices with τ matrices, an approximated DSS (ADSS) preconditioner based on two-dimensional fast sine transform is designed to accelerate the convergence rates of the Krylov subspace iteration methods. In this way, the total computational complexity of the ADSS-preconditioned GMRES method will be of O(n2logn), where n2 represents the dimension of the corresponding discrete linear system. In addition, theoretical analysis demonstrates that the eigenvalues of the ADSS-preconditioned matrix are weakly clustered around a complex disk centered at 1 with the radius less than 1. Numerical experiments show that the ADSS-preconditioned GMRES method is much more efficient than the other existing methods, and can show h-independent convergence behavior.

Privacy-Preserving Parallel Computation of Matrix Determinant With Edge Computing

With the widespread deployment of secure outsourcing computation, the resource-constrained client can delegate intensive computation tasks to powerful servers. Matrix determinant computation is a fundamental mathematical operation that has been widely used in IoT applications. This operation is computationally expensive. Nevertheless, the existing secure outsourcing protocols for matrix determinant are all designed based on one cloud server, which cannot well meet the low-latency and real-time computing requirements for the client. To address this issue, we explore accelerating the computation of matrix determinant by parallel outsourcing based on two nearby edge servers and propose the first practical protocol. We use the matrix blocking technique to split the computation task into multiple subtasks, which are parallel outsourced to edge servers for accelerating the computation. Moreover, we propose a privacy-preserving matrix transformation technique for data privacy protection. This technique only involves the operations of matrix-vector multiplication and matrix-matrix addition. It achieves the lightweight computation for the client and supports computational indistinguishability for the blinded input and a uniform distribution. The correctness, privacy and verifiability of the proposed protocol are analyzed. Finally, the performance advantage of the proposed protocol is demonstrated through simulation experiments.

Multiview Regularized Discriminant Canonical Correlation Analysis: Sequential Extraction of Relevant Features From Multiblock Data.

One of the important issues associated with real-life high-dimensional data analysis is how to extract significant and relevant features from multiview data. The multiset canonical correlation analysis (MCCA) is a well-known statistical method for multiview data integration. It finds a linear subspace that maximizes the correlations among different views. However, the existing methods to find the multiset canonical variables are computationally very expensive, which restricts the application of the MCCA in real-life big data analysis. The covariance matrix of each high-dimensional view may also suffer from the singularity problem due to the limited number of samples. Moreover, the MCCA-based existing feature extraction algorithms are, in general, unsupervised in nature. In this regard, a new supervised feature extraction algorithm is proposed, which integrates multimodal multidimensional data sets by solving maximal correlation problem of the MCCA. A new block matrix representation is introduced to reduce the computational complexity for computing the canonical variables of the MCCA. The analytical formulation enables efficient computation of the multiset canonical variables under supervised ridge regression optimization technique. It deals with the "curse of dimensionality" problem associated with high-dimensional data and facilitates the sequential generation of relevant features with significantly lower computational cost. The effectiveness of the proposed multiblock data integration algorithm, along with a comparison with other existing methods, is demonstrated on several benchmark and real-life cancer data.

NOT ENTIRELY ICHTHYOSAUR: A MYSTERIOUS LAMNIFORM AND ICHTHYOPTERYGIAN-FALL ASSOCIATION FROM THE ABYSSAL UPPER CRETACEOUS OF THE NORTHERN APENNINES (ITALY)

ABSTRACT Axial remains of a large ichthyosaur and a medium-sized anacoracid shark from the deep-water sediments of the Cenomanian of the Northern Apennines (Northern Italy) are described in detail. The specimens were found closely associated (less than 0.3 m apart), and offer an invaluable window into the taphonomy and dead-fall stages of pelagic vertebrates in a Mesozoic abyssal plain. The anacoracid shark remains, initially misinterpreted as an ichthyosaur, consist of eight articulated vertebrae embedded in a block of dark arenaceous matrix, and represent the first occurrence of an articulated shark from the Northern Apennines. The ichthyosaur remains consist of seven discoidal vertebrae and several unidentified fragments. Due to the absence of diagnostic skeletal elements, both specimens are assigned only at higher taxonomic levels. The two fossils, which come from the same outcrop and possibly from the same stratigraphic horizon, share a common taphonomic history, in terms of both their preservation and diagenesis. Mineralogy of the matrices of both is dominated by manganese micro-nodules, consistent with the deposition of polymetallic nodules in bathyal-abyssal settings. Biostratinomic processes that impacted the two specimens also provide insight on the development of Mesozoic marine vertebrate-falls in the deep-sea. Localized pyrite framboids inside the bone spongiosa are possible evidence of the sulfophilic stage (microbially mediated sulfur mobilization during lipid decay) in the ichthyosaur fall. Burrows assigned to Taenidium on the surface of the shark block, interpreted as worm-like feeding burrows or arthropod locomotion and feeding trails, might represent evidence of the enrichment opportunistic stage. Although intriguing, we have no evidence to support the hypothesis that this peculiar association of two pelagic predators is due to ecological interaction between the two animals.

Generalized efficient robust predictive control for networked interval type-2 T–S fuzzy system with adaptive event-triggered scheme

This article studies the generalized efficient robust predictive control (GERPC) for the nonlinear systems denoted by interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy model over networks with consideration of packet dropout, bounded disturbance and adaptive event-triggered scheme (AETS). First, the packet dropout is described by a Bernoulli process, and an AETS with thresholds that can be self-regulated according to different situations is employed to decide whether to release data into the communication network, so as to reduce the data transmission frequency and the network burden; Second, taking into consideration of packet dropout and AETS, a unified IT2 T–S fuzzy control model based on GERPC is established, and the stability of closed-loop system with bounded disturbance is guaranteed by quadratic boundedness (QB) technique; Third, generalized efficient robust predictive controller is designed to obtain a larger initial feasible region, smaller online computation and acceptable control performance. The unconstrained control law is designed offline, and a new variable will be optimized online to achieve good control performance, which is introduced to adjust the control quantity. Then, the convex LMI reformulation method is utilized to handle non-convex terms in optimization problem, and the effectiveness of the GERPC algorithm is proved by the matrix partition technique. At last, the effectiveness of the designed controller based on the GERPC strategy is verified by a given simulation experiment.

Covariance-Matrix-Based Criteria for Network Entanglement.

Quantum networks offer a realistic and practical scheme for generating multiparticle entanglement and implementing multiparticle quantum communication protocols. However, the correlations that can be generated in networks with quantum sources and local operations are not yet well understood. Covariance matrices, which are powerful tools in entanglement theory, have been also applied to the network scenario. We present simple proofs for the decomposition of such matrices into the sum of positive semi-definite block matrices and, based on that, develop analytical and computable necessary criteria for preparing states in quantum networks. These criteria can be applied to networks where nodes share at most one source, such as all bipartite networks.

A Representation of the Drazin Inverse for the Sum of Two Matrices and the Anti-Triangular Block Matrices

In this paper, a new formula for the Drazin inverse for the Sum of Two Matrices is given under conditions weaker than those used in some current literature. Further, we apply our results to obtain new representations for the Drazin inverse of an anti-triangular block matrix under some conditions, which also extend some existing results.

Rosenbrock-W methods for stochastic Galerkin systems

The stochastic Galerkin approach is a widely used method for quantifying uncertainty in dynamical systems. A polynomial chaos expansion of physical variables is used to represent uncertainty. Thus the evolution equations for the dynamics of physical quantities are replaced by evolution equations for the polynomial chaos coefficients. The dimension of the stochastic Galerkin system is m times larger than the dimension of the original physical system, where m is the number of polynomial chaos coefficients. The cost of implicit time integration methods, required for stiff dynamical systems, increases considerably, for example, by a factor m3 when direct linear solvers are used.This work studies efficient implicit methods for the time integration of stiff stochastic Galerkin systems consisting of ordinary differential equations. Linearly-implicit Rosenbrock–Wanner schemes are considered. A block-diagonal approximation of the Jacobian of the stochastic Galerkin system is employed, where the diagonal blocks are the Jacobian of the physical system, evaluated at the mean solution value. We perform a theoretical study of numerical stability for this approximation, using a stochastic extension of the Dahlquist test equation, and propose a concept of linear stochastic Galerkin stability. A rigorous analysis is done for methods of order one and two. A third-order scheme is designed to investigate the stability properties by numerical experiments. Numerical results confirm the theoretical findings.

Cross-sectional evaluation of the association between greenness and cognitive performance in Mexican pre-pubertal boys

BackgroundEvidence shows that greenspace exposure benefits children's health and cognitive development. However, evidence assessing this association in young children in low- and middle-income economies is scarce. ObjectiveTo assess the association between exposure to greenness and cognitive performance in pre-pubertal boys living in Mexico City. MethodsCross-sectional study using data from 144 boys aged 6–11 years living in Mexico City in 2017 and enrolled in the “MetCog” study. Cognitive performance was evaluated through selected Wechsler Scale for Intelligence in Children Fourth Edition (WISC-IV) and Neuropsychological Assessment of Children (Evaluación Neuropsicológica Infantil, ENI) tests. Exposure to greenness was assessed through Normalised Difference Vegetation Index (NDVI) at 300, 500, 1500, 2000, and 3000 m buffer zones from children's residences. Multiple linear regression analysis was undertaken to assess associations between cognitive performance and greenness (aβ) with 95% confidence intervals (CI) and adjusted for potential confounding variables. Significance was set at q < 0.05 after False Discovery Rate (FDR) correction. ResultsA positive association was found between the NDVI Interquartile Range (IQR) at 2000 m and the WISC-IV block design test score (aβ 2000 = 1.18, 95% CI = 0.31, 2.06; q < 0.05), which assesses perceptual reasoning. Positive associations were found with NDVI IQR at 1500 m and WISC-IV block design (aβ1500 = 1.00, 95% CI = 0.14, 1.86) and matrix reasoning (aβ1500 = 0.83, 95% CI = 0.06, 1.61) scores, but neither survived FDR correction. No significant associations were found between NDVI IQR at any buffer size with other WISC-IV and ENI task scores. ConclusionsGreater exposure to greenness was associated with higher perceptual reasoning skills in 144 pre-pubertal boys living in Mexico City. Thus, urban planning should consider increasing vegetation in megacities, especially in neighbourhoods with high percentages of young children.

Structural brain correlates of non-verbal cognitive ability in 5-year-old children: Findings from the FinnBrain birth cohort study.

Non-verbal cognitive ability predicts multiple important life outcomes, for example, school and job performance. It has been associated with parieto-frontal cortical anatomy in prior studies in adult and adolescent populations, while young children have received relatively little attention. We explored the associations between cortical anatomy and non-verbal cognitive ability in 165 5-year-old participants (mean scan age 5.40 years, SD 0.13; 90 males) from the FinnBrain Birth Cohort study. T1-weighted brain magnetic resonance images were processed using FreeSurfer. Non-verbal cognitive ability was measured using the Performance Intelligence Quotient (PIQ) estimated from the Block Design and Matrix Reasoning subtests from the Wechsler Preschool and Primary Scale of Intelligence (WPPSI-III). In vertex-wise general linear models, PIQ scores associated positively with volumes in the left caudal middle frontal and right pericalcarine regions, as well as surface area in left the caudal middle frontal, left inferior temporal, and right lingual regions. There were no associations between PIQ and cortical thickness. To the best of our knowledge, this is the first study to examine structural correlates of non-verbal cognitive ability in a large sample of typically developing 5-year-olds. The findings are generally in line with prior findings from older age groups, with the important addition of the positive association between volume / surface area in the right medial occipital region and non-verbal cognitive ability. This finding adds to the literature by discovering a new brain region that should be considered in future studies exploring the role of cortical structure for cognitive development in young children.

The Complexity of the Super Subdivision of Cycle-Related Graphs Using Block Matrices

The complexity (number of spanning trees) in a finite graph Γ (network) is crucial. The quantity of spanning trees is a fundamental indicator for assessing the dependability of a network. The best and most dependable network is the one with the most spanning trees. In graph theory, one constantly strives to create novel structures from existing ones. The super subdivision operation produces more complicated networks, and the matrices of these networks can be divided into block matrices. Using methods from linear algebra and the characteristics of block matrices, we derive explicit formulas for determining the complexity of the super subdivision of a certain family of graphs, including the cycle Cn, where n=3,4,5,6; the dumbbell graph Dbm,n; the dragon graph Pm(Cn); the prism graph Πn, where n=3,4; the cycle Cn with a Pn2-chord, where n=4,6; and the complete graph K4. Additionally, 3D plots that were created using our results serve as illustrations.

THERMAL CONDUCTIVITY OF PARTITION BOARD BY POLYMER COMPOSITE WITH FILLER EMPTY FRIUT BUNCHES FIBRES

Utilization and management of Empty Fruit Bunches (EFB) fibre continue to develop as the main ingredient and additional material used in various industrial products. The technological breakthrough targeted in this study is the developed EFB fibre as a filler in polymer composite partition boards which are used as heat retainers in the interior construction of buildings. The partition board is a heat insulator, and its thermal conductivity is affected by mass density and porosity. The purpose of this study was to determine the heat resistance of partition board products to reduce the heat entering the room from outside by propagating through walls exposed to direct sunlight. The test method used is adopted from ASTM C177-13, namely the measurement of heat propagation with a modification of the heat source of 40 watts. In addition, mass density tests (referring to SNI 03-2105-2006) and water absorption (referring to ASTM D5229M-12) were also carried out on the product. The specimens were based on the formation of Singapore Highpolymer Chemical Product (SHCP) 2667 WNC polyester resin matrix partition board with weight fractions of 25%, 30%, and 35% chop strand mat (CSM). The test results show that the highest thermal conductivity value is found on the board with a weight fraction of 25%, namely 0.153 W/m.°C with a mass density of 1.16 g/cm3 and a water absorption capacity of 3.38%. However, the lowest thermal conductivity value was found in the fibre with weight fraction of 35%, namely 0.147 W/m. °C at a mass density of 1.24 g/cm3 and a water absorption capacity of 3.75%.

Inequalities for partial determinants of accretive block matrices

Let A=[A_{i,j}]^{m}_{i,j=1}in mathbf{M}_{m}(mathbf{M}_{n}) be an accretive block matrix. We write det1 and det2 for the first and second partial determinants, respectively. In this paper, we show that ∥det1(ReA)∥≤∥(tr(|A|)m)mIn∥\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \\bigl\\Vert \\operatorname{det}_{1} (\\operatorname{Re}A) \\bigr\\Vert \\leq \\biggl\\Vert \\biggl(\\frac{\\operatorname{tr}( \\vert A \\vert )}{m} \\biggr)^{m}I_{n} \\biggr\\Vert \\end{aligned}$$ \\end{document} and ∥det2(ReA)∥≤∥(tr(|A|)n)nIm∥\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \\bigl\\Vert \\operatorname{det}_{2} (\\operatorname{Re}A) \\bigr\\Vert \\leq \\biggl\\Vert \\biggl( \\frac {\\operatorname{tr}( \\vert A \\vert )}{n} \\biggr)^{n}I_{m} \\biggr\\Vert \\end{aligned}$$ \\end{document} hold for any unitarily invariant norm |cdot |. The two inequalities generalize some known results related to partial determinants of positive-semidefinite block matrices.

Mutual structure learning for multiple kernel clustering

Multiple kernel clustering (MKC) has garnered considerable attention in recent years, aiming to obtain an optimal partition from multiple base kernels. Existing MKC methods typically focus on either learning the pairwise structure by constructing an optimal kernel from the base kernels or learning the cluster structure by integrating multiple base partitions into a consensus one for clustering. However, these previous approaches overlook the potential for mutual structure learning between the two aspects, i.e., pairwise and cluster structure, of clustering. To address this limitation, we propose a novel multiple kernel clustering method, referred to as MSL-MKC, which simultaneously learns pairwise and cluster structure to achieve an improved consensus partition matrix for clustering. Specifically, MSL-MKC extends the adaptive neighbor graph learning approach into kernel space to construct a discriminative similarity graph from multiple base kernels for pairwise structure learning. The consensus partition is formulated by aligning it with multiple base partitions for cluster structure learning. We utilize a Laplacian regularization term to preserve the pairwise structure in the consensus partition and inject the cluster structure into the discriminative similarity graph. The proposed method integrates similarity graph learning, partition fusion, and Laplacian regularization into a unified framework, optimized using an iterative algorithm. Experimental results on various benchmark datasets demonstrate the efficacy of MSL-MKC. The demo code of this work is publicly available at https://github.com/guanyuezhen/MSL-MKC.

A semi-implicit semi-Lagrangian method for simulating immersed boundary motion under high inertia and elasticity

In this paper, we present an efficient and stable fractional-step 2D immersed boundary (IB) method for solving the interaction problems between bulk fluid and elastic interface, in particular, when the fluid inertia and the interfacial elasticity are the significant factors affecting its dynamics. In myriads of real-world applications, the effects of high inertia and elasticity are dominant. So the complex fluid dynamics under such harsh conditions is an important topic in computational physics and is inherently challenging due to high computational complexity. In turn, it requires to solve the governing equations of elastic interfacial motion in an implicit manner so that more stable simulations can be performed by relaxing the Courant-Friedrichs-Lewy (CFL) condition. The contributions of the proposed approach are three folds. First, an iteration-free semi-Lagrangian method is employed in Navier-Stokes (NS) equations. Second, the elastic force acting along the interface is treated semi-implicitly in IB formulations. Both approaches improve the numerical stability associated with the high fluidic inertia and interfacial elasticity. Finally, to solve the resulting linear system, our novel idea is to transform the original 3-by-3 block matrix system into a reduced 2-by-2 block matrix system using a discrete projection operator in a staggered grid, and then explicitly represent the exact solution via the Schur complement of the Helmholtz operator. Owing to this feature, we refer to this proposed approach as reduced immersed boundary method (rIBM). We show that the two systems are equivalent in theory, whereas the conventional immersed boundary projection method (IBPM) modifies the discrete momentum equation in the original system. A series of numerical tests is conducted to confirm the stability of the rIBM using relatively larger time-step sizes, specifically with Reynolds number and inverse capillary number equal to or larger than approximately 1000. By estimating the computational time, the numerical efficiency of the proposed method is further verified in comparison with the conventional IBPM and the Crank-Nicolson scheme-based IB method. In conclusion, the proposed approach not only improves the numerical stability, but also increases the computational speed, suitable for solving more realistic problems.

Block Aligner: an adaptive SIMD-accelerated aligner for sequences and position-specific scoring matrices.

Efficiently aligning sequences is a fundamental problem in bioinformatics. Many recent algorithms for computing alignments through Smith-Waterman-Gotoh dynamic programming (DP) exploit Single Instruction Multiple Data (SIMD) operations on modern CPUs for speed. However, these advances have largely ignored difficulties associated with efficiently handling complex scoring matrices or large gaps (insertions or deletions). We propose a new SIMD-accelerated algorithm called Block Aligner for aligning nucleotide and protein sequences against other sequences or position-specific scoring matrices. We introduce a new paradigm that uses blocks in the DP matrix that greedily shift, grow, and shrink. This approach allows regions of the DP matrix to be adaptively computed. Our algorithm reaches over 5-10 times faster than some previous methods while incurring an error rate of less than 3% on protein and long read datasets, despite large gaps and low sequence identities. Our algorithm is implemented for global, local, and X-drop alignments. It is available as a Rust library (with C bindings) at https://github.com/Daniel-Liu-c0deb0t/block-aligner.

CDRKD: An improved density peak algorithm based on kernel fuzzy measure in the overlapping community detection

Compared with other traditional community discovery algorithms, density peak clustering algorithm is more efficient in getting network structures through clustering. However, DPC needs to contain the distance information between all nodes as sources, so it cannot directly processing the complex network represented by the adjacency matrix. DPC introduces truncation distance when calculating the local density of nodes, which is usually set as a fixed value according to experience, and lacks self-adaptability for different network structures. A feasible solution to those problems is to combined rough set theory and kernel fuzzy similarity measures. In this work, we present overlapping community detection algorithm based on improved rough entropy fusion density peak. The algorithm applied rough set theory to attribute reduction of massive high-dimensional data. Another algorithm defines the similarity of sample points by the inner product between two vectors on the basis of fuzzy partition matrix. Finally, a community detection algorithm based on rough entropy and kernel fuzzy density peaks clustering (CDRKD) has proposed by combining the two algorithms above, we perform an extensive set of experiments to verify the effectiveness and feasibility of the algorithm.

Immiscible imbibition in fractured media: A dual-porosity microfluidics study

We use dual porosity microfluidics and fluorescence microscopy to investigate immiscible imbibition in the pore networks formed in fractured rocks, and to identify emergent pore-scale events that arise as a result of the interplay between advection-dominant flow in fractures (F) and capillary-driven matrix imbibition (M). The dimensionless ratio between the two time scales T = tF/tM defines the various displacement patterns: fracture-dominant advective invasion at low Τ-values leaves a higher residual non-wetting phase saturation; compact invasion is observed at intermediate T-values, and fractures act as capillary barriers during matrix-dominant capillary imbibition at high Τ-values. Experiments and analyses show effective capillary-driven corner flow during immiscible imbibition; in particular, corner flow imbibition displaces non-wetting fluids that were initially trapped in the matrix during fast advective invasion. In contrast to wetting fluid invasion and imbibition, injected non-wetting fluids invade and flow along fractures as soon as the capillary pressure reaches the fracture entry pressure, and there is no matrix invasion and drainage. The capillary pressure versus saturation curve for the fractured rock mass assumes that fractures and matrix blocks share the same capillary pressure at equilibrium; then, the combined pressure-saturation response is a function of their relative contributions to the total porosity. In the absence of gouge or precipitates, fractures determine the entry pressure while the matrix controls storativity.

Symbol based convergence analysis in block multigrid methods with applications for Stokes problems

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied.However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated symbols. In this paper, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f, which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies.