Numerical Robustness Research Articles

Robust Dynamic Mode Decomposition

This paper develops a robust dynamic mode decomposition (RDMD) method endowed with statistical and numerical robustness. Statistical robustness ensures estimation efficiency at the Gaussian and non-Gaussian probability distributions, including heavy-tailed distributions. The proposed RDMD is statistically robust because the outliers in the data set are flagged via projection statistics and suppressed using a Schweppe-type Huber generalized maximum-likelihood estimator that minimizes a convex Huber cost function. The latter is solved using the iteratively reweighted least-squares algorithm that is known to exhibit an excellent convergence property and numerical stability than the Newton algorithms. Several numerical simulations using canonical models of dynamical systems demonstrate the excellent performance of the proposed RDMD method. The results reveal that it outperforms several other methods proposed in the literature.

A Constrained Block-Term Tensor Decomposition Framework for Spectrum Cartography

Joint spectrum cartography and disaggregation from sparse spatial observations has been proven to be theoretically feasible based on block-term tensor decomposition (BTD) model. However, the existing BTD framework suffers from inherent drawbacks in terms of numerical stability, complexity and noise robustness. To combat with these drawbacks, we propose a new Constrained-BTD (CBTD) framework in this letter by fully utilizing practical traits of geographical power density spectrum (PSD) and spatial loss field (SLF). The cornerstone of CBTD framework is formulating the joint PSD and SLF estimation as a constrained matrix factorization problem, instead of addressing the factors of BTD. Further, a projection gradient-based (PG) algorithm, which has sublinear convergence, is proposed to handle the restrictions of PSD and SLF by projection on manifolds. Compared with the baseline methods, simulations verify that the proposed approach obtains better performances in terms of stability, complexity and noise robustness.

Contingency Analysis Based on Partitioned and Parallel Holomorphic Embedding

In the steady-state contingency analysis, the traditional Newton-Raphson method suffers from non-convergence issues when solving post-outage power flow problems, which hinders the integrity and accuracy of security assessment. In this paper, we propose a novel robust contingency analysis approach based on holomorphic embedding (HE). The HE-based simulator provides theoretical convergence guarantee, which is desirable because it avoids the influence of numerical issues and provides a credible security assessment conclusion. In addition, based on the multi-area characteristics of real-world power systems, a partitioned HE (PHE) method is proposed with an interface-based partitioning of HE formulation. The PHE method does not undermine the numerical robustness of HE and significantly reduces the computation burden in large-scale contingency analysis. The PHE method is further enhanced by parallel or distributed computation to become parallel PHE (P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> HE). Tests on a 458-bus system, a synthetic 419-bus system and a large-scale 21 447-bus system demonstrate the advantages of the proposed methods in robustness and efficiency.

Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables

This paper considers fast and provably accurate algorithms for approximating smooth functions on the d-dimensional torus, \(f: \mathbb {T}^d \rightarrow \mathbb {C}\), that are sparse (or compressible) in the multidimensional Fourier basis. In particular, suppose that the Fourier series coefficients of f, \(\{c_\mathbf{k} (f) \}_{\mathbf{k} \in \mathbb {Z}^d}\), are concentrated in a given arbitrary finite set \(\mathscr {I} \subset \mathbb {Z}^d\) so that $$\begin{aligned} \min _{\text{\O}mega \subset \mathscr {I} ~s.t.~ \left| \text{\O}mega \right| =s }\left\| f - \sum _{\mathbf{k} \in \text{\O}mega } c_\mathbf{k} (f) \, \mathbb {e}^{ -2 \pi \mathbb {i} {{\mathbf {k}}}\cdot \circ } \right\| _2 < \epsilon \Vert f \Vert _2 \end{aligned}$$holds for \(s \ll \left| \mathscr {I} \right| \) and \(\epsilon \in (0,1)\) small. In such cases we aim to both identify a near-minimizing subset \(\text{\O}mega \subset \mathscr {I}\) and accurately approximate its associated Fourier coefficients \(\{ c_\mathbf{k} (f) \}_{\mathbf{k} \in \text{\O}mega }\) as rapidly as possible. In this paper we present both deterministic and explicit as well as randomized algorithms for solving this problem using \(\mathscr {O}(s^2 d \log ^c (|\mathscr {I}|))\)-time/memory and \(\mathscr {O}(s d \log ^c (|\mathscr {I}|))\)-time/memory, respectively. Most crucially, all of the methods proposed herein achieve these runtimes while simultaneously satisfying theoretical best s-term approximation guarantees which guarantee their numerical accuracy and robustness to noise for general functions. These results are achieved by modifying several different one-dimensional Sparse Fourier Transform (SFT) methods to subsample a function along a reconstructing rank-1 lattice for the given frequency set \(\mathscr {I} \subset \mathbb {Z}^d\) in order to rapidly identify a near-minimizing subset \(\text{\O}mega \subset \mathscr {I}\) as above without having to use anything about the lattice beyond its generating vector. This requires the development of new fast and low-memory frequency identification techniques capable of rapidly recovering vector-valued frequencies in \(\mathbb {Z}^d\) as opposed to recovering simple integer frequencies as required in the univariate setting. Two different multivariate frequency identification strategies are proposed, analyzed, and shown to lead to their own best s-term approximation methods herein, each with different accuracy versus computational speed and memory tradeoffs.

Efficient numerical treatment of a gradient damage model for materials undergoing large deformations

AbstractIn this contribution we present a gradient damage formulation (cf. [1]) for hyperelastic materials undergoing large deformations, which is based on the small strain formulation of [2]. Through discretizing the gradient extended damage field with the neighbored element method the damage update can be performed at the element level. Although local update iterations remain present, the global system of equations goes without any additional degrees of freedom compared to purely elastic computations. Furthermore, in the case of severely damaged material, an element erosion based stabilization technique is considered. Computing efficiency and numerical robustness is demonstrated in several numerical examples.

Reynolds-Stress Model Computations of NASA Juncture Flow Experiment

Numerical simulations of the NASA Juncture Flow Experiment are carried out, employing a Reynolds-stress model. A transformation of the length-scale determining variable is suggested for improving the numerical robustness of the model. Results obtained with different unstructured flow solvers on different sets of grids show very good agreement, except for velocity and Reynolds-stress profiles in the vicinity of the predicted separation region. A recently developed length-scale correction and, particularly, prescribing transition on the fuselage and wing increase the predicted size of the separation region, where the variation is on the order of the experimental uncertainty. Detailed comparison of velocity and Reynolds-stress profiles with experimental data reveal a general deficiency of the model in predicting the peak of the streamwise normal stress and the distribution of all Reynolds stresses in the vicinity of the separation region. However, all predicted sizes of the separation region are in fair agreement with the experimental data.

Theoretical and computational improvements to the algebraic method for discovering cooperative rigid-unit modes

A linear-algebraic algorithm for identifying rigid-unit modes in networks of interconnected rigid units has recently been demonstrated. This article presents a series of enhancements to the original algorithm, which greatly improve its conceptual simplicity, numerical robustness, computational efficiency and interpretability. The improvements include the efficient isolation of constraints, the observation of variable-block separability, the use of singular value decomposition and a quantitative measure of solution inexactness.

Further assessment of a time domain impedance boundary condition for the numerical simulation of noise-absorbing materials

In regard to the mitigation of environmental noise across major industry sectors, the present study focuses on the numerical prediction of passive noise reduction devices. Here, it is further explored how the noise attenuation induced by locally reacting noise absorbing materials (also called acoustic liners) can be simulated using a time domain highly accurate Computational AeroAcoustics (CAA) method. To this end, it is assessed how a classical Time Domain Impedance Boundary Condition (TDIBC) can effectively model acoustic liners of practical interest, including when the latter are exposed to realistic conditions (grazing flow and noise excitation). The investigation consists in numerically reproducing two experimental campaigns initially performed at NASA Langley Research Center. Two different materials are considered (honeycomb superimposed with perforate or wiremesh resistive face-sheet), each being characterized by a specific noise attenuation behaviour ( e.g. dependency on the flow conditions and/or noise excitation). Each material is tested under various flow conditions ( e.g. grazing flow of Mach up to 0.5) and/or noise source excitation ( e.g. multiple tones of level up to 140 dB each). The results demonstrate the ability of the underlying CAA/TDIBC approach to simulate realistic acoustic liners in non-trivial configurations, with enough physical accuracy ( e.g. correct capture of the noise attenuation characteristics) and numerical robustness ( e.g. absence of instabilities). The study also reveals that, independent from the CAA/TDIBC approach itself, some specific pre-processing tasks (e.g. impedance eduction and subsequent TDIBC calibration) may play a bigger role than expected, in practice.

Optimization Tool for the Strategic Outline and Sizing of District Heating Networks Using a Geographic Information System

The implementation of district heating networks into cities is a main topic in policy planning that looks for sustainable solutions to reduce CO2 emissions. However, their development into cities is generally limited by a high initial investment cost. The development of optimization methods intended to draft efficient systems using heating consumption profiles into a prescribed geographic area are useful in this purpose. Such tools are already referred to in the scientific literature, yet they are often restricted to limit the computational load. To bridge this gap, the present contribution proposes a multi-period mixed integer linear programming model for the optimal outline and sizing of a district heating network maximizing the net cash flow based on a geographic information system. This methodology targets a large range of problem sizes from small-scale to large-scale heating networks while guaranteeing numerical robustness. For sake of simplicity, the developed model is first applied to a scaled down case study with 3 available heating sources and a neighborhood of 16 streets. The full-scale model is presented afterwards to demonstrate the applicability of the tool for city-scale heating networks with around 2000 streets to potentially connect within a reasonable computational time of around only one hour.

Efficient and Robust Distributed Matrix Computations via Convolutional Coding

Distributed matrix computations - matrix-matrix or matrix-vector multiplications - are well-recognized to suffer from the problem of stragglers (slow or failed worker nodes). Much of prior work in this area is (i) either sub-optimal in terms of its straggler resilience, or (ii) suffers from numerical problems, i.e., there is a blow-up of round-off errors in the decoded result owing to the high condition numbers of the corresponding decoding matrices. Our work presents a convolutional coding approach to this problem that removes these limitations. It is optimal in terms of its straggler resilience, and has excellent numerical robustness as long as the workers' storage capacity is slightly higher than the fundamental lower bound. Moreover, it can be decoded using a fast peeling decoder that only involves add/subtract operations. Our second approach has marginally higher decoding complexity than the first one, but allows us to operate arbitrarily close to the storage capacity lower bound. Its numerical robustness can be theoretically quantified by deriving a computable upper bound on the worst case condition number over all possible decoding matrices by drawing connections with the properties of large block Toeplitz matrices. All above claims are backed up by extensive experiments done on the AWS cloud platform.

Cooperative Localization of 3D Mobile Networks via Relative Distance and Velocity Measurement

This letter mainly considers the cooperative localization problem of 3D mobile cluster network. With the development of Doppler frequency shift measurement technique, high-precision relative velocity measurement has become possible. In this letter, we first incorporate the relative velocity information into the 3D cooperative localization framework. Then the quasi-Newton algorithm BFGS is used to solve the framework, and the vBFGS algorithm suitable for solving the relative position of the mobile cluster network is constructed. Simulation experiments show that using relative velocity measurement to assist positioning can significantly improve positioning accuracy. Compared with the state-of-the-art algorithm, vBFGS has higher positioning accuracy and faster convergence rate for low-connectivity networks, and has better numerical stability and robustness for high-speed moving models.

Application of regularized Savitzky–Golay filters to identification of time-varying systems

Savitzky–Golay (SG) filtering is a classical signal smoothing technique based on the local least squares approximation of the analyzed signal by a linear combination of known functions of time (originally — powers of time, which corresponds to polynomial approximation). It is shown that the regularized version of the SG algorithm can be successfully applied to identification of time-varying finite impulse response (FIR) systems. Such a solution is possible owing to the recently proposed preestimation technique, which converts the problem of identification of a time-varying FIR system into the problem of smoothing of the appropriately generated preestimates of system parameters. The resulting fast regularized local basis function estimators, optimized using the empirical Bayes approach, have very good parameter tracking capabilities, favorably comparing with the state-of-the-art in terms of accuracy, computational complexity and numerical robustness.

Nonlinear cyclic behavior of 2D complex RC structures by a plastic-damage concrete model with smeared reinforcement approach

In this work the damage model developed by authors for RC membrane elements is presented and discussed. In particular, the consequences of simplifying assumptions on the accuracy of the numerical results are deeply investigated referring to an extensive set of numerical simulations. Numerical results are compared with experimental data of monotonic and cyclic inelastic tests under uniform and non-uniform stress distributions, including also the hysteretic response of shear critical members. The performed analyses demonstrate a proper numerical robustness associated with a significant computational efficiency, essential features to perform large scale simulations of shear critical RC structures.

A quasi-monolithic phase-field description for mixed-mode fracture using predictor–corrector mesh adaptivity

In this work, we develop a mixed-mode phase-field fracture model employing a parallel-adaptive quasi-monolithic framework. In nature, failure of rocks and rock-like materials is usually accompanied by the propagation of mixed-mode fractures. To address this aspect, some recent studies have incorporated mixed-mode fracture propagation criteria to classical phase-field fracture models, and new energy splitting methods were proposed to split the total crack driving energy into mode-I and mode-II parts. As extension in this work, a splitting method for masonry-like materials is modified and incorporated into the mixed-mode phase-field fracture model. A robust, accurate and efficient parallel-adaptive quasi-monolithic framework serves as basis for the implementation of our new model. Three numerical tests are carried out, and the results of the new model are compared to those of existing models, demonstrating the numerical robustness and physical soundness of the new model. In total, six models are computationally analyzed and compared.

High-order accurate kinetic-energy and entropy preserving (KEEP) schemes on curvilinear grids

High-order accurate kinetic energy and entropy preserving (KEEP) schemes in generalized curvilinear coordinates are proposed for stable and non-dissipative numerical simulations. The proposed schemes are developed on the basis of the physical relation that the fluxes in the Euler equations in generalized curvilinear coordinates can be derived by taking the inner product between the inviscid fluxes and the area vectors used for the coordinate transformation. To satisfy this physical relation discretely, this study proposes to interpret the area vector components as another individual variable and discretize the area vectors in the same way as other physical variables, such as the density and velocity. Consequently, the convective and pressure-related terms are discretized in a new split convective form, “quartic split form”, and quadratic split form, respectively. The high-order extension is straightforward, referring to the high-order formulations proposed for kinetic energy preserving schemes in a previous study. Numerical tests of vortex convection, inviscid Taylor-Green vortex, and turbulent boundary layer flow are conducted to assess the order of accuracy, the kinetic energy and entropy preservation property, and numerical robustness of the proposed KEEP schemes. The proposed high-order accurate KEEP schemes successfully perform long-time stable computations without numerical dissipation by preserving the total kinetic energy and total entropy well, even on a largely-distorted computational grid.

Directed percolation and numerical stability of simulations of digital memcomputing machines.

Digital memcomputing machines (DMMs) are a novel, non-Turing class of machines designed to solve combinatorial optimization problems. They can be physically realized with continuous-time, non-quantum dynamical systems with memory (time non-locality), whose ordinary differential equations (ODEs) can be numerically integrated on modern computers. Solutions of many hard problems have been reported by numerically integrating the ODEs of DMMs, showing substantial advantages over state-of-the-art solvers. To investigate the reasons behind the robustness and effectiveness of this method, we employ three explicit integration schemes (forward Euler, trapezoid, and Runge-Kutta fourth order) with a constant time step to solve 3-SAT instances with planted solutions. We show that (i) even if most of the trajectories in the phase space are destroyed by numerical noise, the solution can still be achieved; (ii) the forward Euler method, although having the largest numerical error, solves the instances in the least amount of function evaluations; and (iii) when increasing the integration time step, the system undergoes a "solvable-unsolvable transition" at a critical threshold, which needs to decay at most as a power law with the problem size, to control the numerical errors. To explain these results, we model the dynamical behavior of DMMs as directed percolation of the state trajectory in the phase space in the presence of noise. This viewpoint clarifies the reasons behind their numerical robustness and provides an analytical understanding of the solvable-unsolvable transition. These results land further support to the usefulness of DMMs in the solution of hard combinatorial optimization problems.

Insights from exact exchange-correlation kernels

The exact exchange-correlation (xc) kernel ${f}_{\text{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of linear response time-dependent density functional theory is computed over a wide range of frequencies for three canonical one-dimensional finite systems. Methods used to ensure the numerical robustness of ${f}_{\text{xc}}$ are set out. The frequency dependence of ${f}_{\text{xc}}$ is found to be largely due to its analytic structure, i.e., its singularities at certain frequencies, which are required in order to capture particular transitions, including those of double excitation character. However, within the frequency range of the first few interacting excitations, ${f}_{\text{xc}}$ is approximately $\ensuremath{\omega}$ independent, meaning the exact adiabatic approximation ${f}_{\text{xc}}(\ensuremath{\omega}=0)$ remedies the failings of the local density approximation and random phase approximation for these lowest transitions. The key differences between the exact ${f}_{\text{xc}}$ and its common approximations are analyzed, and cannot be eliminated by exploiting the limited gauge freedom in ${f}_{\text{xc}}$. The optical spectrum benefits from using as accurate as possible an ${f}_{\text{xc}}$ and ground-state xc potential, while maintaining exact compatibility between the two is of less importance.

Design and Implementation of Attitude and Heading Reference System with Extended Kalman Filter Based on MEMS Multi-Sensor Fusion

The accuracy of attitude and heading measurement, as well as the system real-time performance are basic indicators used to evaluate an attitude and heading reference system (AHRS). In order to improve the attitude and heading measurement accuracy under dynamic complex environment, the AHRS system should also have numerical stability and calculation robustness. The AHRS system based on MEMS multi-sensor fusion can realize fusion processing of data measured by multiple sensors, so as to calculate and obtain the optimal carrier attitude and heading information, conduct real-time output, and improve the accuracy and reliability of attitude and heading measurement. For the AHRS system consisting of MEMS gyroscope, accelerometer and triaxial magnetometer, attitude and heading detection principle and algorithm based on MEMS multi-sensor fusion were proposed in this study: The information of the system itself was firstly used to discriminate motion state of the carrier within the filtering cycle, and then Kalman filtering was conducted using different measured information according to motion state to correct the attitude error angle caused by gyroscopic drift. On this basis, an attitude fusion algorithm based on extended Kalman filtering technology was designed for time update process of Kalman filtering, output information of accelerometer was taken as observed quantity under certain conditions to realize measurement updating process of Kalman filtering, and then attitude angle was calculated. In an optical fiber attitude and heading system project in practical engineering, a vehicle field test analysis was carried out simultaneously with the system using ordinary attitude algorithm, and the results showed that the extended Kalman filtering algorithm designed according to the simulation results could realize multi-sensor information fusion, improve measurement accuracy and realize accurate attitude positioning, so as to provide simpler and more flexible criteria for carrier motion status. The results have verified the accuracy and reliability of the algorithm, so it is feasible in practical engineering.

Development of parametric reduced-order model for consequence estimation of rare events

Computational fluid dynamics (CFD) models have been widely used in the chemical process industry to analyze various aspects of high-consequence rare events. However, CFD models are computationally intensive in nature, and therefore, several developments have been made in building computationally efficient models. The existing computationally efficient models in the field of high-consequence rare events are temporally static in nature and do not represent system evolution with time, which is crucial for consequence modeling of rare events. Further, the consequences depend on various parameters in addition to inputs. Since it is not affordable to construct a new model for every parameter value, incorporation of inputs and parameters is an additional challenge in developing a consequence model. Hence, to address these challenges, this work proposes a k-nearest neighbor (kNN)-based parametric reduced-order model (PROM) for consequence estimation of rare events to enhance numerical robustness with respect to parameter change. Specifically, local (with respect to parameters) ROMs are constructed based on multivariable output-error state space (MOESP) algorithm for a range of parameters, and they are linearly interpolated to estimate consequences at a new parameter value. The effectiveness of the proposed model is demonstrated through a case study of supercritical carbon dioxide release rare event.

A Cell-Based Smoothed Finite Element Method for Arbitrary Polygonal Element to Solve Incompressible Laminar Flow

In this paper, a cell-based smoothed finite element method using the arbitrary [Formula: see text]-sided polygonal element (CS-FEM-Poly) is developed to solve fluid mechanics problems. A stabilization method, characteristic-based split coupled with stabilized pressure gradient projection (CBS/SPGP), is employed to deal with numerical oscillations for CS-FEM-Poly. We validate CS-FEM-Poly and test its numerical behaviors using triangular, quadrilateral and polygonal elements on four typical numerical examples. Numerical results show that the CS-FEM-Poly based on CBS/SPGP produces well-agreed solutions with the exact solutions of benchmarks, and gives desirable convergence rate as compared with FEM. In the backward-facing step flow example, the numerical robustness for concave elements is manifested for CS-FEM-Poly. The proposed CS-FEM-Poly exhibits the remarkable potential for polygonal mesh or glued polygonal elements in the hybrid mesh to solve incompressible flows.