Numerical methods for structural anomaly detection using model order reduction and data-driven techniques

The objective of this thesis is to provide a mathematical and computational framework for the proactive maintenance of complex systems with a particular application to structural health monitoring (SHM). SHM techniques rely primarily on sensor responses to assess the risk associated with a structure of interest and seek to provide a support for automated decision-making strategy. An efficient integration of experimental measurements and numerical models is needed to exhaustively describe environmental and operational scenarios that a structure undergoes during its life time.

We propose a simulation-based approach that combines the solution of a parametric time-dependent partial differential equation (PDE) for multiple input parameters with data-driven techniques to discriminate between healthy and damaged configurations. This process exploits an offline-online decomposition of tasks. The dataset of synthetic sensor measurements is generated offline by repeatedly solving a parametric PDE for a predefined configuration under several healthy variations. A reduced order model is employed to overcome the computational bottleneck associated with the many-query context by building a projection-based reduced basis method in combination with a Laplace-domain solver. Weeks method is adopted to numerically invert the Laplace transform and transform the signals to the time domain. The datasets are used to train various one-class machine learning algorithms in a semi-supervised setting, sensor by sensor. Finally, the outputs of the classifiers are used to assess the state of damage of the structure online. Using a decision-level fusion strategy, we provide insight on the existence, location, and severity of possible damages.

The performance of SHM depends critically on the quality of the sensor measurements although their availability is often limited due to budget constraints and installation difficulties. We therefore propose a strategy to systematically place a fixed number of sensors on a structure of interest to minimize uncertainty at unsensed locations. The so-called inducing points, an outcome of sparse Gaussian processes, originally introduced to overcome the computational burden associated with performing a regression task with standard Gaussian processes, are here used to guide the sensor placement. A clustering approach is employed to select the sensor locations among a set of inducing points, computed for different input parameters.

We apply this methodology to 2D and 3D problems to mimic the vibrational behavior of complex structures under the effect of an active source and show the effectiveness of the approach for (i) detecting damaged geometries and (ii) identifying the locations for a network of sensors. This framework considers the realistic case where damage types and locations are a priori unknown, thus overcoming the main limitation of existing simulation-based damage detection and sensor placement strategies for SHM.

Fourier Sampling in Signal Processing and Numerical Linear Algebra

Amir Zandieh

This thesis focuses on developing efficient algorithmic tools for processing large datasets. In many modern data analysis tasks, the sheer volume of available datasets far outstrips our abilities to process them. This scenario commonly arises in tasks including parameter tuning of machine learning models (e.g., Google Vizier) and training neural networks. These tasks often require solving numerical linear algebraic problems on large matrices, making the classical primitives prohibitively expensive. Hence, there is a crucial need to efficiently compress the available datasets. In other settings, even collecting the input dataset is extremely expensive, making it vital to design optimal data sampling strategies. This is common in applications such as MRI acquisition and spectrum sensing. The fundamental questions above are often dual to each other, and hence can be addressed using the same set of core techniques. Indeed, exploiting structured Fourier sparsity is a recurring source of efficiency in this thesis, leading to both fast numerical linear algebra methods and sample efficient data acquisition schemes. One of the main results that we present in this thesis is the first Sublinear-time Model-based Sparse FFT algorithm that achieves a nearly optimal sample complexity for recovery of every signal whose Fourier transform is well approximated by a small number of blocks (e.g., such structure is common in spectrum sensing). Our method matches in sublinear time the result of Baraniuk et. al. (2010), which started the field of model-based compressed sensing. Another highlight of this thesis includes the first Dimension-independent Sparse FFT algorithm that, computes the Fourier transform of a sparse signal in sublinear runtime in any dimension. This is the first algorithm that just like the FFT of Cooley and Tukey is dimension independent and avoids the curse of dimensionality inherent to all previously known techniques. Finally, we give a Universal Sampling Scheme for the reconstruction of structured Fourier signals from continuous measurements. Our approach matches the classical results of Slepian, Pollak, and Landau (1960s) on the reconstruction of bandlimited signals via Prolate Spheroidal Wave Functions and extends these results to a wide class of Fourier structure types. Besides having classical applications in signal processing and data analysis, Fourier techniques have been at the core of many machine learning tasks such as Kernel Matrix Approximation. The second half of this thesis is dedicated to finding compressed and low-rank representations of kernel matrices, which are the primary means of computation with large kernel matrices in machine learning. We build on Fourier techniques and achieve spectral approximation guarantees to the Gaussian kernel using an optimal number of samples, significantly improving upon the classical Random Fourier Features of Rahimi and Recht (2008). Finally, we present a nearly-optimal Oblivious Subspace Embedding for high-degree Polynomial kernels which leads to nearly-optimal embeddings of the high-dimensional Gaussian kernel. This is the first result that does not suffer from an exponential loss in the degree of the polynomial kernel or the dimension of the input point set, providing exponential improvements over the prior works, including the TensorSketch of Pagh (2013) and application of the celebrated Fast Multipole Method of Greengard and Rokhlin (1986) to the kernel approximation problem.

EPFL2020

Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

Jan Sickmann Hesthaven, Babak Maboudi Afkham

Reduced basis methods are popular for approximately solving large and complex systems of dierential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present an approach for reduced model construction, that preserves the symplectic symmetry of dissipative Hamiltonian systems. The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath. This allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, in the stability of the solution. Accuracy and stability of the method are illustrated through the numerical simulation of the dissipative wave equation and a port-Hamiltonian model of an electric circuit.

2017

Model order reduction for large-scale structures with local nonlinearities

Mengwu Guo, Jan Sickmann Hesthaven, Zhenying Zhang

In solid mechanics, linear structures often exhibit (local) nonlinear behavior when close to failure. For instance, the elastic deformation of a structure becomes plastic after being deformed beyond recovery. To properly assess such problems in a real-life application, we need fast and multi-query evaluations of coupled linear and nonlinear structural systems, whose approximations are not straight forward and often computationally expensive. In this work, we propose a linear-nonlinear domain decomposition, where the two systems are coupled through the solutions on the linear-nonlinear interface. After necessary sensitivity analysis, e.g. for structures with a high dimensional parameter space, we adopt a non-intrusive method, e.g. Gaussian processes regression (GPR), to solve for the solution on the interface. We then utilize different model order reduction techniques to address the linear and nonlinear problems individually. To accelerate the approximation, we employ again the non-intrusive GPR for the nonlinearity, while intrusive model order reduction methods, e.g. the conventional reduced basis (RB) method or the static-condensation reduced- basis-element (SCRBE) method, are employed for the solution in the linear subdomain. We provide several numerical examples to demonstrate the effectiveness of our method.

ELSEVIER SCIENCE SA2019

Fourier Sampling in Signal Processing and Numerical Linear Algebra

Amir Zandieh

EPFL2020