Null Space Basis Research Articles

Evanescent waves in a metabeam attached with lossy acoustic black hole pillars

In this paper, we suggest a metabeam having periodically distributed integrated acoustic black holes (IABHs). Compared to conventional embedded ABHs, the thickness of the base beam has not been modified, so it adapts well to application occasions with load-bearing requirements. The ABH pillars behave as dynamic vibration absorbers (DVAs) that not only provide abundant modes with high damping performance but also generate local resonance and Bragg scattering bandgaps to stop wave propagation. To characterize the evanescent waves, a wave and Rayleigh–Ritz method (WRRM) is proposed, based on the nullspace method (NSM) that assumes the response is constituted by the linear combination of the nullspace basis of the constraint matrix. The proposed WRRM is validated using numerical partial differential solutions. Analysis of the complex dispersion curves shows that multiple local resonance and Bragg scattering bandgaps take place in the whole frequency range of interest, and the damping effect can merge these bandgaps so as to form very wide attenuation bands. Not only that, remarkable evanescent waves are also observed in the passbands. The lattice constant and ABH length are studied parametrically. Results show that the larger the lattice constant, the more attenuation bands appear, and the longer ABH length can enhance the attenuation capability. Finally, the transmissibility and mean square velocity for a finite metabeam with 5 cells are investigated, indicating substantial wave damping in a broadband frequency range.

Downlink MIMO-RSMA With Successive Null-Space Precoding

In this paper, we consider the precoder design for an underloaded or critically loaded downlink multi-user multiple-input multiple-output (MIMO) communication system. We propose novel precoding and decoding schemes which enhance system performance based on rate splitting at the transmitter and single-stage successive interference cancellation at the receivers. The proposed successive null-space (SNS) precoding utilizes linear combinations of the null-space basis vectors of the successively augmented MIMO channel matrices of the users as precoding vectors to adjust the inter-user-interference experienced by the receivers. We formulate a non-convex weighted sum rate optimization problem for the precoding vectors and the associated power allocation for the proposed SNS-based MIMO-rate-splitting multiple access (RSMA) scheme. We obtain a suboptimal solution for this problem via successive convex approximation. Moreover, we study the robustness of the proposed precoding scheme to imperfect channel state information (CSI) at the base station via derivative-based sensitivity analysis. Our analysis and simulation results reveal the enhanced performance and robustness of the proposed SNS-based MIMO-RSMA scheme over several baseline multi-user MIMO schemes, especially for imperfect CSI.

Solving large linear least squares problems with linear equality constraints

We consider the problem of solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. While some classical approaches are theoretically well founded, they can face difficulties when the matrix of constraints contains dense rows or if an algorithmic transformation used in the solution process results in a modified problem that is much denser than the original one. We propose modifications with an emphasis on requiring that the constraints be satisfied with a small residual. We examine combining the null-space method with our recently developed algorithm for computing a null-space basis matrix for a “wide” matrix. We further show that a direct elimination approach enhanced by careful pivoting can be effective in transforming the problem to an unconstrained sparse-dense least squares problem that can be solved with existing direct or iterative methods. We also present a number of solution variants that employ an augmented system formulation, which can be attractive for solving a sequence of related problems. Numerical experiments on problems coming from practical applications are used throughout to demonstrate the effectiveness of the different approaches.

A null-space approach for large-scale symmetric saddle point systems with a small and non zero (2, 2) block

Null-space methods have long been used to solve large sparse n × n symmetric saddle point systems of equations in which the (2, 2) block is zero. This paper focuses on the case where the (1, 1) block is ill conditioned or rank deficient and the k × k (2, 2) block is non zero and small (k ≪ n). Additionally, the (2, 1) block may be rank deficient. Such systems arise in a range of practical applications. A novel null-space approach is proposed that transforms the system matrix into a nicer symmetric saddle point matrix of order n that has a non zero (2, 2) block of order at most 2k and, importantly, the (1, 1) block is symmetric positive definite. Success of any null-space approach depends on constructing a suitable null-space basis. We propose methods for wide matrices having far fewer rows than columns with the aim of balancing stability of the transformed saddle point matrix with preserving sparsity in the (1, 1) block. Linear least squares problems that contain a small number of dense rows are an important motivation and are used to illustrate our ideas and to explore their potential for solving large-scale systems.

A Bi-Level Consensus ADMM-Based Fully Distributed Inverter-Based Volt/Var Control Method for Active Distribution Networks

The distributed Volt/Var control solution is a promising one for active distribution networks (ADNs), but a critical issue is how to improve the distributed algorithm's convergence with less communication burden. In this paper, we propose a bi-level consensus alternation direction multiplier method-based fully distributed Volt/Var optimization algorithm (B-FADMM) for ADNs by exploiting inverter-based rapid control devices. In the first level, the Volt/Var optimal solution of subpartitions is obtained in parallel, and the second level completes the variable increment of each partition to correct the search direction of the first level. In the algorithm, firstly, an adaptive matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\omega $</tex-math></inline-formula> initialization strategy is proposed to adapt to the different magnitude of variables, which avoids non-convergence or slow convergence of the algorithm. Secondly, the null-space method is used in the second level of the algorithm to obtain the null-space basis matrix on the active constraint of the subpartition, which reduces the dimensions of the problem and the computational complexity. In addition, the conjugate gradient (CG) algorithm is used to update the dual multiplier of each partition only by exchanging coupling information among partitions, which protects information privacy. Experiments showed the proposed approach attains the same result as the centralized, the computational performance of the proposed method is far superior to some popular other distributed methods in terms of accuracy, computational efficiency, and scalability.

Determining failure modes devoid of redundant elements for pin-bar structures using null space of equilibrium matrix

The reliability of a pin-bar structure is dominated only by the failure modes devoid of redundant elements, that is, the structure will become geometrically variant only when all the failed elements in the failure mode are removed. A novel method is developed to search for all the combinations of such elements in a pin-bar structure, i.e., the minimal cut sets of system reliability. Based on the matrix consisting of the null space basis vectors of the structural equilibrium matrix (abbr. null space basis matrix), two lemmas indicating the characteristics of the minimal cut sets are proposed and proved. By investigating the row vectors of the null space basis matrix and their linear dependence, whether single or multiple elements can constitute a minimal cut set can be evaluated. A strategy to search for all the minimal cut sets is proposed based on the maximal linearly independent subsets of the row vectors of the null space basis matrix. A procedure is also provided to calculate the system reliability of pin-bar structures only using the obtained minimal cut sets. Two numerical examples are employed to verify the validity and computational efficiency of the proposed method.

Flatness-Based Reduced Hessian Method for Optimal Control of Aircraft

Numerical solutions of flatness-based reformulation of optimal control problems lead to an optimization problem with fewer variables and constraints. However, the expressions for the states and controls in terms of the flat output variables can be highly nonlinear and complicated. Hence, this may lead to expensive function evaluations within the optimization problem, degradation of convexity, and issues with convergence. Thus, a flatness-based reformulation of the optimal control problems may not be a viable alternative for computational guidance and control. Alternatively, a novel flatness-based reduced Hessian sequential quadratic programming algorithm is developed in this paper to solve optimization problems for a six-degree-of-freedom aircraft. An analytical null space basis is derived from the linearized differential constraints, which leads to a reduced-dimensional quadratic program with the discretized flat outputs as decision variables. Unlike flatness-based reformulation of the optimal control problem, the null space/reduced Hessian method does not introduce additional nonlinearity and preserves the convexity of the optimization problem. A nonlinear model predictive control problem is solved to demonstrate the reduced Hessian algorithm. The algorithm is validated and Monte Carlo trials are performed to assess the effectiveness of the approach. Computational studies show that the current approach is faster than methods that directly exploit sparsity up to a factor of five.

Beyond L∞ reaction wheel commanding: increased on-board autonomy through enhanced wheel failure handling

Latest large spacecraft and/or agility requirements have led to configurations of five or more reaction wheels. To avoid (1) handling a significant amount of pre-stored parameters on board and (2) intervening by ground (classical approach) to recover single- and multiple-wheel failures, the orthogonal null space basis must be robustly computed on board. Two deterministic methods are presented in this paper: (1) via virtual torque commands using L∞ and L2 commanding, (2) via generalized inverse. Furthermore, a cascading null space control approach that autonomously respects predefined wheel momentum operating bands is introduced. The presented work is dedicated to the demand of high on-board autonomy driven by more and more customers requiring just a working spacecraft with minimum interventions by ground. The presented solutions are already part of the Airbus high-power satellite avionics system Astrobus AS400 for current and future missions such as MetOp-SG. Furthermore, the presented functionalities are essential towards on-board real-time spacecraft reconfiguration which is; however, beyond the scope of this paper.

Solving separable nonlinear least squares problems using the QR factorization

We present a method for solving the separable nonlinear least squares problem min y , z ‖ F ( y , z ) ‖ , where F ( y , z ) ≡ A ( y ) z + b ( y ) with a full rank matrix A ( y ) ∈ R ( N + ℓ ) × N , y ∈ R n , z ∈ R N and the vector b ( y ) ∈ R N + ℓ , with small ℓ ≥ n . We show how this problem can be reduced to a smaller equivalent problem min y ‖ f ( y ) ‖ where the function f has only ℓ components. The reduction technique is based on the existence of a locally differentiable orthonormal basis for the nullspace of A T ( y ) . We use Newton’s method to solve the reduced problem. We show that successive iteration points are independent of the nullspace basis used at any particular iteration point; thus the QR factorization can be used to provide a local basis at each iteration. We show that the first and second derivative terms that arise are easily computed, so quadratic convergence is obtainable even for nonzero residual problems. For the class of problems with N much greater than n and ℓ the main cost per iteration of the method is one QR factorization of A ( y ) . We provide a detailed algorithm and some numerical examples to illustrate the technique.

An Improved Multi-parametric Programming Algorithm for Flux Balance Analysis of Metabolic Networks

Flux balance analysis has proven an effective tool for analyzing metabolic networks. In flux balance analysis, reaction rates and optimal pathways are ascertained by solving a linear program, in which the growth rate is maximized subject to mass-balance constraints. A variety of cell functions in response to environmental stimuli can be quantified using flux balance analysis by parameterizing the linear program with respect to extracellular conditions. However, for most large, genome-scale metabolic networks of practical interest, the resulting parametric problem has multiple and highly degenerate optimal solutions, which are computationally challenging to handle. An improved multi-parametric programming algorithm based on active-set methods is introduced in this paper to overcome these computational difficulties. Degeneracy and multiplicity are handled, respectively, by introducing generalized inverses and auxiliary objective functions into the formulation of the optimality conditions. These improvements are especially effective for metabolic networks because their stoichiometry matrices are generally sparse; thus, fast and efficient algorithms from sparse linear algebra can be leveraged to compute generalized inverses and null-space bases. We illustrate the application of our algorithm to flux balance analysis of metabolic networks by studying a reduced metabolic model of Corynebacterium glutamicum and a genome-scale model of Escherichia coli. We then demonstrate how the critical regions resulting from these studies can be associated with optimal metabolic modes and discuss the physical relevance of optimal pathways arising from various auxiliary objective functions. Achieving more than fivefold improvement in computational speed over existing multi-parametric programming tools, the proposed algorithm proves promising in handling genome-scale metabolic models.

Prestructuring sparse matrices with dense rows and columns via null space methods

SummarySeveral applied problems may produce large sparse matrices with a small number of dense rows and/or columns, which can adversely affect the performance of commonly used direct solvers. By posing the problem as a saddle point system, an unconventional application of a null space method can be employed to eliminate dense rows and columns. The choice of null space basis is critical in retaining the overall sparse structure of the matrix. A new one‐sided application of the null space method is also presented to eliminate either dense rows or columns. These methods can be considered techniques that modify the nonzero structure of the matrix before employing a direct solver and may result in improved direct solver performance.

Constraint-Preconditioned Inexact Newton Method for Hydraulic Simulation of Large-Scale Water Distribution Networks

Many sequential mathematical optimization methods and simulation-based heuristics for optimal control and design of water distribution networks rely on a large number of hydraulic simulations. In this paper, we propose an efficient inexact subspace Newton method for hydraulic analysis of water distribution networks. By using sparse and well-conditioned fundamental null space bases, we solve the nonlinear system of hydraulic equations in a lower-dimensional kernel space of the network incidence matrix. In the inexact framework, the Newton steps are determined by solving the Newton equations only approximately using an iterative linear solver. Since large water network models are inherently badly scaled, Jacobian regularization is employed to improve the condition number of these linear systems and guarantee positive definiteness. After presenting a convergence analysis of the regularized inexact Newton method, we use the conjugate-gradient (CG) method to solve the sparse reduced Newton linear systems. Since CG is not effective without good preconditioners, we propose tailored constraint preconditioners that are computationally cheap because they are based only on invariant properties of the null-space linear systems and do not change with flows and pressures. The preconditioners are shown to improve the distribution of eigenvalues of the linear systems and so enable a more efficient use of the CG solver. Since contiguous Newton iterates can have similar solutions, each CG call is warm-started with the solution for a previous Newton iterate to accelerate its convergence rate. Operational network models are used to show the efficacy of the proposed preconditioners and the warm-starting strategy in reducing computational effort.

Interpretation of numerically detected rank defects in GNSS data analysis problems in terms of deficiencies in reference system definition

New tools in the form of angular deficiency indices are developed for the physical interpretation of numerically detected rank deficiencies or weaknesses, in terms of defects or weaknesses in the definition of the reference system characteristics (origin, orientation and scale) to which coordinates and other unknown parameters refer. Within a least squares estimation procedure, the numerical detection is based on the subspace of the parameter space spanned by eigenvectors of the coefficient matrix of the normal equations, which correspond to zero or very small eigenvalues. The physical interpretation is based on the well-known theoretically derived basis of the null space of the design matrix, provided by the columns of the matrix appearing in the so-called inner constraints. This null space basis is elaborated for single-epoch (weekly) solutions of GNSS networks, for deformable networks with constant station velocities, as well as for the stacking problem of coordinate time series. The applicability of these new tools and their implementation strategy are demonstrated by simple numerical examples.

Sparsity and Nullity: Paradigms for Analysis Dictionary Learning

Sparse models in dictionary learning have been successfully applied in a wide variety of machine learning and computer vision problems, and as a result have recently attracted increased research interest. Another interesting related problem based on linear equality constraints, namely the sparse null space (SNS) problem, first appeared in 1986 and has since inspired results on sparse basis pursuit. In this paper, we investigate the relation between the SNS problem and the analysis dictionary learning (ADL) problem, and show that the SNS problem plays a central role, and may be utilized to solve dictionary learning problems. Moreover, we propose an efficient algorithm of sparse null space basis pursuit (SNS-BP) and extend it to a solution of ADL. Experimental results on numerical synthetic data and real-world data are further presented to validate the performance of our method.

Online moving horizon estimation of fluxes in metabolic reaction networks

Using online state and parameter estimation, concentrations and fluxes in bioprocesses can be estimated for use in monitoring, optimization and control applications. Existing methodologies, however, either ignore the dynamic nature of the problem, or focus on the extracellular concentration states and pay less attention to accurate flux estimates. These estimates are useful for online monitoring of the flux state of an organism, or for developing novel flux-based strategies for online control of bioreactors.In this contribution, the dynamic metabolic flux analysis model structure is combined with two kinetic flux models: a linear flux model and a nonlinear, more mechanistic flux model. The parameters of these models are estimated online through a moving horizon estimation strategy. The resulting algorithm is illustrated on two simulated case studies: a small-scale network, to assess the influence of important algorithm parameters on the final estimates, and a medium-scale network for Escherichia coli, to empirically test the performance of the methodology in a more realistic situation.An important parameter in this estimation strategy is the chosen noise level on the estimated parameters. This choice is not trivial, but is observed to have a significant influence on the resulting estimates. Furthermore, also the effect of the choice of the null space basis for the stoichiometric matrix of the metabolic reaction network was assessed. In the small-scale case study, it was found that a linear flux model with a specific parameter noise level was performing well for both state and flux estimation. The influence of the choice of the null space basis matrix on the estimation performance was much lower. The resulting scenario was evaluated in the medium-scale case study and found to be performing very well also in that case.

Minimal metabolic pathway structure is consistent with associated biomolecular interactions.

Pathways are a universal paradigm for functionally describing cellular processes. Even though advances in high-throughput data generation have transformed biology, the core of our biological understanding, and hence data interpretation, is still predicated on human-defined pathways. Here, we introduce an unbiased, pathway structure for genome-scale metabolic networks defined based on principles of parsimony that do not mimic canonical human-defined textbook pathways. Instead, these minimal pathways better describe multiple independent pathway-associated biomolecular interaction datasets suggesting a functional organization for metabolism based on parsimonious use of cellular components. We use the inherent predictive capability of these pathways to experimentally discover novel transcriptional regulatory interactions in Escherichia coli metabolism for three transcription factors, effectively doubling the known regulatory roles for Nac and MntR. This study suggests an underlying and fundamental principle in the evolutionary selection of pathway structures; namely, that pathways may be minimal, independent, and segregated.

Relaxing order basis computation

The computation of an order basis (also called sigma basis) is a fundamental tool for linear algebra with polynomial coefficients. Such a computation is one of the key ingredients to provide algorithms which reduce to polynomial matrices multiplication. This has been the case for column reduction or minimal nullspace basis of polynomial matrix over a field. In this poster, we are interested in the application of order basis to compute minimal matrix generators of a linear matrix sequence. In particular, we focus on the linear matrix sequence used in the Block Wiedemann algorithm.

On large-scale generalized inverses in solving two-by-two block linear systems

The goal is to analyze a role of generalized inverses in the solution of two-by-two block linear systems by means of an algorithm combining the Schur complement reduction with the null-space method. The efficient procedures for computing actions of generalized inversions to rectangular large-scale matrices are presented with a special attention to the Moore–Penrose inverse. A regularization technique for matrices with the known null-space basis is introduced that enables to get generalized inverses without necessity to recognize zero pivots. The theoretical results are confirmed by numerical examples.

An efficient algorithm for sparse null space basis problem using ABS methods

We propose a way to use the Markowitz pivot selection criterion for choosing the parameters of the extended ABS class of algorithms to present an effective algorithm for generating sparse null space bases. We explain in detail an efficient implementation of the algorithm, making use of the special MATLAB 7.0 functions for sparse matrix operations and the inherent efficiency of the ANSI C programming language. We then compare our proposed algorithm with an implementation of an efficient algorithm proposed by Coleman and Pothen with respect to the computing time and the accuracy and the sparsity of the generated null space bases. Our extensive numerical results, using coefficient matrices of linear programming problems from the NETLIB set of test problems show the competitiveness of our implemented algorithm.

Backlash elimination in parallel manipulators using actuation redundancy

SUMMARYIn this work, accuracy enhancement through backlash elimination is considered. When a nonredundantly actuated parallel manipulator is subjected to a wrench while following a trajectory, required actuator torque switching (going from positive to negative or vice versa) may occur. If backlash is present in the actuation hardware for a manipulator, torque switching compromises accuracy. When in-branch redundant actuation is added, a pseudoinverse torque solution requires smaller joint torques, but torque switching may still occur. A method is presented where concepts of exploiting a nullspace basis of the joint torques are used to ensure that single sense joint torques can be achieved for the actuated joints. The same sense torque solutions are obtained using nonlinear optimization. The methodology is applied to several examples simulating parallel manipulators in machining applications.