Anthony P. Austin

Anthony P. Austin, Lloyd N. Trefethen

The trigonometric interpolants to a periodic function $f$ in equispaced points converge if $f$ is Dini-continuous, and the associated quadrature formula, the trapezoidal rule, converges if $f$ is continuous. What if the points are perturbed? With equispaced grid spacing $h$, let each point be perturbed by an arbitrary amount $\le αh$, where $α\in [\kern .5pt 0,1/2)$ is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be be trouble for $α\ge 1/4$. We show that convergence of both the interpolants and the quadrature estimates is guaranteed for all $α<1/2$ if $f$ is twice continuously differentiable, with the convergence rate depending on the smoothness of $f$. More precisely it is enough for $f$ to have $4α$ derivatives in a certain sense, and we conjecture that $2α$ derivatives is enough. Connections with the Fejér--Kalmár theorem are discussed.

Anthony P. Austin, Zichao Wendy Di, Sven Leyffer et al.

Tomography can be used to reveal internal properties of a 3D object using any penetrating wave. Advanced tomographic imaging techniques, however, are vulnerable to both systematic and random errors associated with the experimental conditions, which are often beyond the capabilities of the state-of-the-art reconstruction techniques such as regularizations. Because they can lead to reduced spatial resolution and even misinterpretation of the underlying sample structures, these errors present a fundamental obstacle to full realization of the capabilities of next-generation physical imaging. In this work, we develop efficient and explicit recovery schemes of the most common experimental error: movement of the center of rotation during the experiment. We formulate new physical models to capture the experimental setup, and we devise new mathematical optimization formulations for reliable inversion of complex samples. We demonstrate and validate the efficacy of our approach on synthetic data under known perturbations of the center of rotation.

Robert L. Bassett, Austin Van Dellen, Anthony P. Austin

We investigate the vulnerability of computer-vision-based signal classifiers to adversarial perturbations of their inputs, where the signals and perturbations are subject to physical constraints. We consider a scenario in which a source and interferer emit signals that propagate as waves to a detector, which attempts to classify the source by analyzing the spectrogram of the signal it receives using a pre-trained neural network. By solving PDE-constrained optimization problems, we construct interfering signals that cause the detector to misclassify the source even though the perturbations to the spectrogram of the received signal are nearly imperceptible. Though such problems can have millions of decision variables, we introduce methods to solve them efficiently. Our experiments demonstrate that one can compute effective and physically realizable adversarial perturbations for a variety of machine learning models under various physical conditions.