Browsing by Author "Aquino, Wilkins"

Many physical systems are modeled using partial dierential equations (PDEs) with uncertain or random inputs. For such systems, naively propagating a xed number of samples of the input probability law (or an approximation thereof) through the PDE is often inadequate to accurately quantify the risk associated with critical system responses. In addition, to manage the risk associated with system response and devise risk-averse controls for such PDEs, one must obtain the numerical solution of a risk-averse PDE-constrained optimization problem, which requires substantial computational eorts resulting from the discretization of the underlying PDE in both the physical and stochastic dimensions.

Bayesian Inverse problem, where unknown system parameters need to be inferred from some noisy data of the system response, is another important class of problems that suffer from excessive computational cost due to the discretization of the underlying PDE. To accurately characterize the inverse solution and quantify its uncertainty, tremendous computational eorts are typically required to sample from the posterior distribution of the system parameters given the data. Surrogate approximation of the PDE model is an important technique to expedite the inference process and tractably solve such problems.

In this thesis, we develop a goal-oriented, adaptive sampling and local reduced basis approximation for PDEs with random inputs. The method, which we denote by local RB, determines a set of samples and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. The local basis in a Voronoi cell is composed of the solutions at a xed number of closest samples as well as the gradient information in that cell. Thanks to the local nature of the method, computational cost of the approximation does not increase as more samples are included in the local RB model. We select the local RB samples in an adaptive and greedy manner using an a posteriori error indicator based on the residual of the approximation.

Additionally, we modify our adaptive sampling process using an error indicator that is specifically targeted for the approximation of coherent risk measures evaluated at quantities of interest depending on PDE solutions. This allow us to tailor our method to efficiently quantify the risk associated with the system responses. We then combine our local RB method with an inexact trust region method to eciently solve risk-averse optimization problems with PDE constraints. We propose a numerical framework for systematically constructing surrogate models for the trust-region subproblem and the objective function using local RB approximations.

Finally, we extend our local RB method to eciently approximate the Gibbs posterior distribution for inverse problems under uncertainty. The local RB method is employed to construct a cheap surrogate model for the loss function in the Gibbs posterior formula. To improve the accuracy of the surrogate approximation, we adopt a Sequential Monte Carlo framework to guide the progressive and adaptive construction of the local RB surrogate. The resulted method provides subjective and ecient inference of unknown system parameters under general distribution and noise assumptions.

We provide theoretical error bounds for our proposed local RB method and its extensions, and numerically demonstrate the performance of our methods through various examples.

Arterial stiffness plays a fundamental role in cardiovascular health. To assess arterial health noninvasively, we use arterial dispersion ultrasound vibrometry (ADUV): an acoustic radiation force (ARF) causes propagating waves in the arterial wall that are measured with ultrasound at multiple stages during the cardiovascular cycle. A long-standing problem in this field is developing a fast, reliable map from propagating waves to arterial health. Therefore, we propose an end-to-end classification framework: from ADUV data to arterial health. Our main framework contributions are the treatment of high-dimensional signals and the selection of interpretable features. In particular, we highlight the role of systolic and diastolic cardiovascular stages in classifying arterial health using ADUV signals.

Our findings, limited by dataset size, focus on interpretability of features and finding separation between the Healthy Subjects (HS) and Unhealthy Subjects (US). Our feature selection strategies included ranking by the training weights in Linear Support Vector Machines (LSVM), mutual information, and global search. The main goal was to select the best nf features from the set of wave speeds obtained over the cardiac cycle at various set frequencies. We then accessed performance using boostrapping accuracy. The LSVM Weights strategy (Holdout Accuracy = 82.1% ± ̆7.7% for a Systolic Feature Set) performed better than mutual information feature selection (Holdout Accuracy = 72.5% ± 9.6% for a Systolic Feature Set). However, both methods were surpassed by global search (Holdout Accuracy = 87.3% ± 7.8% for a Systolic Feature Set). Additionally, the best systolic, high cardiovascular pressure, feature sets (Holdout Accuracy = 87.3% ± 7.8% for Global Search) had higher mean holdout accuracy than the diastolic, low pressure, feature sets (Holdout Accuracy = 80.1% ± 8.5% for Global Search) for both LSVM weight feature selection and the exhaustive and global searches. The best set of features found (Holdout Accuracy = 90.4% ± 5.8%) was a combined diastolic and systolic set, but the single diastolic feature is at a higher pressure compared to other diastolic features. Therefore, separation exists between the HS and US groups, demonstrating that an end-to-end framework is feasible for this new class of arterial health biomarkers. Lastly, the interpretable framework identified that separation is particularly evident for high-pressure features when geometric features are excluded. Adding geometric features, specifically diameter and thickness, to the classifier input resulted in similar performance for the best feature set.

Identification and monitoring of damage have a growing importance in the maintenance of structures. With advances in high performance computing, numerical modeling tools are becoming more common in anomaly detection and structural damage localization. However, many methods presuppose that sufficiently many experimental data points are obtained. As these monitoring schemes are applied to higher risk, more detailed, and aging structures, a key challenge is the gathering of sufficient information through sensors such that modern damage estimation mechanisms can identify and diagnose anomalies with certainty.

To that end, we formulate a simultaneous inversion and optimal experimental design (OED) framework for models described by a set of discretized partial differential equations (PDEs). Abstractly, we construct and solve the corresponding (potentially nonlinear) inverse problem to attain a parameter estimator. Then, to devise the next best sensor placement, we linearize the OED problem around the newly determined estimator to generate the Fisher Information Matrix (FIM) for the next best sensor placement problem. In this manner, we let information gleaned from our measurements guide our sensor placement process: given a current estimated damage state with up-to-date sensor information, find the next best sensor using the FIM and update the damage estimator.

We will also explore three different strategies to enhance the framework. First, we introduce a Modified Error in Constitutive Equations (MECE) functional as a damage estimator. Using MECE will quasi-convexify the damage estimation problem, making the problem more resilient from being trapped in local minima. Next, we formulate a decision-centric, utility maximization framework for the OED problem. Mutual information (or relative entropy) is chosen as the utility criteria so that sensor locations maximize the information about the structural parameters. We provide a discussion on how prior knowledge can be incorporated. Finally, we develop a novel PDE-constrained optimization approach for generalized stress inversion. Given digital image correlation measurements on a structure, this method aims to infer body’s original stress field.

We will demonstrate all developed capabilities using a combination of numerical and experimental models of varying complexities. We show that these approaches can recover accurate parameter estimators (using a set number of sensors) in the presence of noisy measurements.

Dynamic conditions and harmful vibration are prevalent in many engineered systems. Computational optimization methods can usefully guide the design of structures and solid systems to obtain layouts with desired dynamic behaviors, such as minimized or tailored vibration response, while accounting for additional constraints. Due to resonance phenomena in steady-state dynamics, however, objectives used to control dynamic response possess many local minima that may represent poor solutions to a design problem, causing challenges for gradient-based optimization methods. Additionally, designs are typically characterized by high-dimensional parameter spaces, which require costly regularization operations and can lead to slow convergence. This dissertation presents strategies to address these obstacles in dynamic response design problems, seeking inspiration from solution strategies for the related class of medium inversion problems.

We propose an extension of the modified error in constitutive equations (MECE) method, used previously in material identification problems, as a new approach for frequency-domain dynamics topology optimization to mitigate solution dependence upon initial guess and considered loading conditions. The main idea of the MECE strategy is to relax the constitutive relations between stresses and strains and between inertial forces and displacements in the elastodynamic PDE-constraint. An additional penalty-like term measuring the discrepancy in these relations is added to the objective, so that both model error and the original response objective are minimized. The resulting optimization problem features an objective more amenable to minimization, as resonant responses are suppressed in the relaxed elastodynamic model. We show that this approach yields superior designs to conventional gradient-based optimization approaches that solely use a functional of displacements as the objective, while strictly enforcing the constitutive equations. We highlight the merits of our approach in a variety of scenarios for direct frequency response design, considering multiple frequency load cases and different frequency-response control objectives.

We next present an adaptive eigenspace basis (AEB) strategy for effective design dimensionality reduction in topology optimization problems. We construct the density field as a linear combination of eigenfunctions, computed for an elliptic operator defined over the design domain, and solve for the associated eigenfunction coefficients. Restriction to this finite-dimensional eigenspace drastically reduces the design dimension and imposes implicit regularization upon the solution. We furthermore develop a basis adaptation scheme, in which we iteratively recompute the eigenfunction basis to conform to the evolving solution field, enabling further dimension reduction and acceleration of the optimization process. We demonstrate the AEB method on topology optimization problems for static linear elasticity, benchmarking its performance against conventional TO implementations. We further present the formulation of an MECE frequency-response control problem using an AEB design parameterization. We demonstrate the combined methods provide the benefits of both dimensionality reduction and robust performance in multifrequency design scenarios.

We finally present a case study in dynamic design: a PDE-constrained optimization formulation for the design of continuously graded elastic cylinders to achieve passive cloaking from harmonic acoustic excitation, both at single frequencies and over extended bandwidths. We show that elastic parameters in a multilayered, constant-density cylinder may be selected such that the residual between the pressure field from an unobstructed spreading wave in a fluid and the pressure field produced by the cylindrical inclusion is minimized.