Continuous Differentiability of the Value Function of Semilinear Parabolic Infinite Time Horizon Optimal Control Problems on L 2 ( Ω ) Under Control Constraints.

An abstract framework guaranteeing the local continuous differentiability of the value function associated with optimal stabilization problems subject to abstract semilinear parabolic equations subject to a norm constraint on the controls is established. It guarantees that the value function satisfies the associated Hamilton-Jacobi-Bellman equation in the classical sense. The applicability of the developed framework is demonstrated for specific semilinear parabolic equations.

An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology.

In this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg-Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033-2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

Total Deep Variation: A Stable Regularization Method for Inverse Problems.

Various problems in computer vision and medical imaging can be cast as inverse problems. A frequent method for solving inverse problems is the variational approach, which amounts to minimizing an energy composed of a data fidelity term and a regularizer. Classically, handcrafted regularizers are used, which are commonly outperformed by state-of-the-art deep learning approaches. In this work, we combine the variational formulation of inverse problems with deep learning by introducing the data-driven general-purpose total deep variation regularizer. In its core, a convolutional neural network extracts local features on multiple scales and in successive blocks. This combination allows for a rigorous mathematical analysis including an optimal control formulation of the training problem in a mean-field setting and a stability analysis with respect to the initial values and the parameters of the regularizer. In addition, we experimentally verify the robustness against adversarial attacks and numerically derive upper bounds for the generalization error. Finally, we achieve state-of-the-art results for several imaging tasks.

Variational Networks: An Optimal Control Approach to Early Stopping Variational Methods for Image Restoration.

We investigate a well-known phenomenon of variational approaches in image processing, where typically the best image quality is achieved when the gradient flow process is stopped before converging to a stationary point. This paradox originates from a tradeoff between optimization and modeling errors of the underlying variational model and holds true even if deep learning methods are used to learn highly expressive regularizers from data. In this paper, we take advantage of this paradox and introduce an optimal stopping time into the gradient flow process, which in turn is learned from data by means of an optimal control approach. After a time discretization, we obtain variational networks, which can be interpreted as a particular type of recurrent neural networks. The learned variational networks achieve competitive results for image denoising and image deblurring on a standard benchmark data set. One of the key theoretical results is the development of first- and second-order conditions to verify optimal stopping time. A nonlinear spectral analysis of the gradient of the learned regularizer gives enlightening insights into the different regularization properties.

Time optimal control-based RF pulse design under gradient imperfections.

PURPOSE: This study incorporates a gradient system imperfection model into an optimal control framework for radio frequency (RF) pulse design. THEORY AND METHODS: The joint design of minimum-time RF and slice selective gradient shapes is posed as an optimal control problem. Hardware limitations such as maximal amplitudes for RF and slice selective gradient or its slew rate are included as hard constraints to assure practical applicability of the optimized waveforms. In order to guarantee the performance of the optimized waveform with possible gradient system disturbances such as limited system bandwidth and eddy currents, a measured gradient impulse response function (GIRF) for a specific system is integrated into the optimization. RESULTS: The method generates optimized RF and pre-distorted slice selective gradient shapes for refocusing that are able to fully compensate the modeled imperfections of the gradient system under investigation. The results nearly regenerate the optimal results of an idealized gradient system. The numerical Bloch simulations are validated by phantom and in-vivo experiments on 2 3T scanners. CONCLUSIONS: The presented design approach demonstrates the successful correction of gradient system imperfections within an optimal control framework for RF pulse design.

Inverse localization of earliest cardiac activation sites from activation maps based on the viscous Eikonal equation.

In this study we propose a novel method for identifying the locations of earliest activation in the human left ventricle from activation maps measured at the epicardial surface. Electrical activation is modeled based on the viscous Eikonal equation. The sites of earliest activation are identified by solving a minimization problem. Arbitrary initial locations are assumed, which are then modified based on a shape derivative based perturbation field until a minimal mismatch between the computed and the given activation maps on the epicardial surface is achieved. The proposed method is tested in two numerical benchmarks, a generic 2D unit-square benchmark, and an anatomically accurate MRI-derived 3D human left ventricle benchmark to demonstrate potential utility in a clinical context. For unperturbed input data, our localization method is able to accurately reconstruct the earliest activation sites in both benchmarks with deviations of only a fraction of the used spatial discretization size. Further, with the quality of the input data reduced by spatial undersampling and addition of noise, we demonstrate that an accurate identification of the sites of earliest activation is still feasible.

On a Monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics.

A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is also compared to two alternative algorithms for nonsmooth nonconvex optimization arising in optimal control and mathematical imaging.

Simultaneous multislice refocusing via time optimal control.

PURPOSE: Joint design of minimum duration RF pulses and slice-selective gradient shapes for MRI via time optimal control with strict physical constraints, and its application to simultaneous multislice imaging. THEORY AND METHODS: The minimization of the pulse duration is cast as a time optimal control problem with inequality constraints describing the refocusing quality and physical constraints. It is solved with a bilevel method, where the pulse length is minimized in the upper level, and the constraints are satisfied in the lower level. To address the inherent nonconvexity of the optimization problem, the upper level is enhanced with new heuristics for finding a near global optimizer based on a second optimization problem. RESULTS: A large set of optimized examples shows an average temporal reduction of 87.1% for double diffusion and 74% for turbo spin echo pulses compared to power independent number of slices pulses. The optimized results are validated on a 3T scanner with phantom measurements. CONCLUSION: The presented design method computes minimum duration RF pulse and slice-selective gradient shapes subject to physical constraints. The shorter pulse duration can be used to decrease the effective echo time in existing echo-planar imaging or echo spacing in turbo spin echo sequences.

Magnetic Resonance RF Pulse Design by Optimal Control With Physical Constraints.

Optimal control approaches have proved useful in designing RF pulses for large tip-angle applications. A typical challenge for optimal control design is the inclusion of constraints resulting from physiological or technical limitations that assure the realizability of the optimized pulses. In this paper, we show how to treat such inequality constraints, in particular, amplitude constraints on the B1 field, the slice-selective gradient, and its slew rate, as well as constraints on the slice profile accuracy. For the latter, a pointwise profile error and additional phase constraints are prescribed. Here, a penalization method is introduced that corresponds to a higher order tracking instead of the common quadratic tracking. The order is driven to infinity in the course of the optimization. We jointly optimize for the RF and slice-selective gradient waveform. The amplitude constraints on these control variables are treated efficiently by semismooth Newton or quasi-Newton methods. The method is flexible, adapting to many optimization goals. As an application, we reduce the power of refocusing pulses, which is important for spin echo-based applications with a short echo spacing. Here, the optimization method is tested in numerical experiments for reducing the pulse power of simultaneous multislice refocusing pulses. The results are validated by phantom and in-vivo experiments.

Robust [Formula: see text] Approaches to Computing the Geometric Median and Principal and Independent Components.

Robust measures are introduced for methods to determine statistically uncorrelated or also statistically independent components spanning data measured in a way that does not permit direct separation of these underlying components. Because of the nonlinear nature of the proposed methods, iterative methods are presented for the optimization of merit functions, and local convergence of these methods is proved. Illustrative examples are presented to demonstrate the benefits of the robust approaches, including an application to the processing of dynamic medical imaging.

PDE constrained optimization of electrical defibrillation in a 3D ventricular slice geometry.

A computational study of an optimal control approach for cardiac defibrillation in a 3D geometry is presented. The cardiac bioelectric activity at the tissue and bath volumes is modeled by the bidomain model equations. The model includes intramural fiber rotation, axially symmetric around the fiber direction, and anisotropic conductivity coefficients, which are extracted from a histological image. The dynamics of the ionic currents are based on the regularized Mitchell-Schaeffer model. The controls enter in the form of electrodes, which are placed at the boundary of the bath volume with the goal of dampening undesired arrhythmias. The numerical optimization is based on Newton techniques. We demonstrated the parallel architecture environment for the computation of potentials on multidomains and for the higher order optimization techniques.

On boundary stimulation and optimal boundary control of the bidomain equations.

The bidomain equations with Neumann boundary stimulation and optimal control of these stimuli are investigated. First an analytical framework for boundary control is provided. Then a parallel finite element based algorithm is devised and its efficiency is demonstrated not only for the direct problem but also for the optimal control problem. The computations realize a model configuration corresponding to optimal boundary defibrillation of a reentry phenomenon by applying current density stimuli.

Optimal control approach to termination of re-entry waves in cardiac electrophysiology.

This work proposes an optimal control approach for the termination of re-entry waves in cardiac electrophysiology. The control enters as an extracellular current density into the bidomain equations which are well established model equations in the literature to describe the electrical behavior of the cardiac tissue. The optimal control formulation is inspired, in part, by the dynamical systems behavior of the underlying system of differential equations. Existence of optimal controls is established and the optimality system is derived formally. The numerical realization is described in detail and numerical experiments, which demonstrate the capability of influencing and terminating reentry phenomena, are presented.

Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints.

Time optimal control governed by the internally controlled linear Fitzhugh-Nagumo equation with pointwise control constraint is considered. Making use of Ekeland's variational principle, we obtain Pontryagin's maximum principle for a time optimal control problem. Using the maximum principle, the bang-bang property of the optimal controls is established under appropriate assumptions.