Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the l1-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm (LISTA).

Finding optimal feedback controllers for nonlinear dynamic systems from data is hard. Recently, Bayesian optimization (BO) has been proposed as a powerful framework for direct controller tuning from experimental trials. For selecting the next query point and finding the global optimum, BO relies on a probabilistic description of the latent objective function, typically a Gaussian process (GP). As is shown herein, GPs with a common kernel choice can, however, lead to poor learning outcomes on standard quadratic control problems. For a first-order system, we construct two kernels that specifically leverage the structure of the well-known Linear Quadratic Regulator (LQR), yet retain the flexibility of Bayesian nonparametric learning. Simulations of uncertain linear and nonlinear systems demonstrate that the LQR kernels yield superior learning performance.

This paper derives new formulations for designing dominant pole placement based proportional-integral-derivative (PID) controllers to handle second order processes with time delays (SOPTD). Previously, similar attempts have been made for pole placement in delay-free systems. The presence of the time delay term manifests itself as a higher order system with variable number of interlaced poles and zeros upon Pade approximation, which makes it difficult to achieve precise pole placement control. We here report the analytical expressions to constrain the closed loop dominant and non-dominant poles at the desired locations in the complex s-plane, using a third order Pade approximation for the delay term. However, invariance of the closed loop performance with different time delay approximation has also been verified using increasing order of Pade, representing a closed to reality higher order delay dynamics. The choice of the nature of non-dominant poles e.g. all being complex, real or a combination of them modifies the characteristic equation and influences the achievable stability regions. The effect of different types of non-dominant poles and the corresponding stability regions are obtained for nine test-bench processes indicating different levels of open-loop damping and lag to delay ratio. Next, we investigate which expression yields a wider stability region in the design parameter space by using Monte Carlo simulations while uniformly sampling a chosen design parameter space. Various time and frequency domain control performance parameters are investigated next, as well as their deviations with uncertain process parameters, using thousands of Monte Carlo simulations, around the robust stable solution for each of the nine test-bench processes.

The need to predict or fill-in missing data, often referred to as matrix completion, is a common challenge in today's data-driven world. Previous strategies typically assume that no structural difference between observed and missing entries exists. Unfortunately, this assumption is woefully unrealistic in many applications. For example, in the classic Netflix challenge, in which one hopes to predict user-movie ratings for unseen films, the fact that the viewer has not watched a given movie may indicate a lack of interest in that movie, thus suggesting a lower rating than otherwise expected. We propose adjusting the standard nuclear norm minimization strategy for matrix completion to account for such structural differences between observed and unobserved entries by regularizing the values of the unobserved entries. We show that the proposed method outperforms nuclear norm minimization in certain settings.

In this dissertation, we address a problem of safe and efficient intersection crossing traffic management of autonomous and connected ground traffic. Toward this objective, an algorithm that is called the Discrete-time occupancies trajectory based Intersection traffic Coordination Algorithm (DICA) is proposed. All vehicles in the system are Connected and Autonomous Vehicles (CAVs) and capable of wireless Vehicle-to-Intersection communication. In the proposed framework, an intersection coordinates the motions of CAVs based on their proposed DTOTs to let them cross the intersection efficiently while avoiding collisions. In case when there is a collision between vehicles' DTOTs, the intersection modifies conflicting DTOTs to avoid the collision and requests CAVs to approach and cross the intersection according to the modified DTOTs. We then prove that the basic DICA is deadlock free and also starvation free. We also show that the basic DICA is conservative in computational complexity and improve it by several computational approaches. Next, we addressed the problem of evacuating emergency vehicles as quickly as possible through autonomous and connected intersection traffic in this dissertation. The proposed intersection control algorithm Reactive DICA aims to determine an efficient vehicle-passing sequence which allows the emergency vehicle to cross an intersection as soon as possible while the travel times of other vehicles are minimally affected. When there are no emergency vehicles within the intersection area, the vehicles are controlled by DICA. When there are emergency vehicles entering communication range, we prioritize emergency vehicles through optimal ordering of vehicles. A genetic algorithm is proposed to solve the optimization problem which finds the optimal vehicle sequence that gives the emergency vehicles the highest priority.

Research evidence in Cyber-Physical Systems (CPS) shows that the introduced tight coupling of information technology with physical sensing and actuation leads to more vulnerability and security weaknesses. But, the traditional security protection mechanisms of CPS focus on data encryption while neglecting the sensors which are vulnerable to attacks in the physical domain. Accordingly, researchers attach utmost importance to the problem of state estimation in the presence of sensor attacks. In this work, we present SecSens, a novel approach for secure nonlinear state estimation in the presence of modeling and measurement noise. SecSens consists of two independent algorithms, namely, SecEKF and SecOPT, which are based on Extended Kalman Filter and Maximum Likelihood Estimation, respectively. We adopt a holistic approach to introduce security awareness among state estimation algorithms without requiring specialized hardware, or cryptographic techniques. We apply SecSens to securely localize and time synchronize networked mobile devices. SecSens provides good performance at run-time several order of magnitude faster than the state of art solutions under the presence of powerful attacks. Our algorithms are evaluated on a testbed with static nodes and a mobile quadrotor all equipped with commercial ultra-wide band wireless devices.

A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical models of the physical world. In the current era of abundance of data and advanced machine learning capabilities, the natural question arises: How can we automatically uncover the underlying laws of physics from high-dimensional data generated from experiments? In this work, we put forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time. Specifically, we approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks. The first network acts as a prior on the unknown solution and essentially enables us to avoid numerical differentiations which are inherently ill-conditioned and unstable. The second network represents the nonlinear dynamics and helps us distill the mechanisms that govern the evolution of a given spatiotemporal data-set. We test the effectiveness of our approach for several benchmark problems spanning a number of scientific domains and demonstrate how the proposed framework can help us accurately learn the underlying dynamics and forecast future states of the system. In particular, we study the Burgers', Korteweg-de Vries (KdV), Kuramoto-Sivashinsky, nonlinear Schrödinger, and Navier-Stokes equations.

Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus, approximation theory, optimization and linear algebra. This article provides a very brief introduction to the basic ideas that underlie deep learning from an applied mathematics perspective. Our target audience includes postgraduate and final year undergraduate students in mathematics who are keen to learn about the area. The article may also be useful for instructors in mathematics who wish to enliven their classes with references to the application of deep learning techniques. We focus on three fundamental questions: what is a deep neural network? how is a network trained? what is the stochastic gradient method? We illustrate the ideas with a short MATLAB code that sets up and trains a network. We also show the use of state-of-the art software on a large scale image classification problem. We finish with references to the current literature.

When nodes in a mobile network use relative noisy measurements with respect to their neighbors to estimate their positions, the overall connectivity and geometry of the measurement network has a critical influence on the achievable localization accuracy. This paper considers the problem of deploying a mobile robotic network implementing a cooperative localization scheme based on range measurements only, while attempting to maintain a network geometry that is favorable to estimating the robots' positions with high accuracy. The quality of the network geometry is measured by a "localizability" function serving as potential field for robot motion planning. This function is built from the Cramér-Rao bound, which provides for a given geometry a lower bound on the covariance matrix achievable by any unbiased position estimator that the robots might implement using their relative measurements. We describe gradient descent-based motion planners for the robots that attempt to optimize or constrain different variations of the network's localizability function, and discuss ways of implementing these controllers in a distributed manner. Finally, the paper also establishes formal connections between our statistical point of view and maintaining a form of weighted rigidity for the graph capturing the relative range measurements.

We consider the problem of phase retrieval, i.e. that of solving systems of quadratic equations. A simple variant of the randomized Kaczmarz method was recently proposed for phase retrieval, and it was shown numerically to have a computational edge over state-of-the-art Wirtinger flow methods. In this paper, we provide the first theoretical guarantee for the convergence of the randomized Kaczmarz method for phase retrieval. We show that it is sufficient to have as many Gaussian measurements as the dimension, up to a constant factor. Along the way, we introduce a sufficient condition on measurement sets for which the randomized Kaczmarz method is guaranteed to work. We show that Gaussian sampling vectors satisfy this property with high probability; this is proved using a chaining argument coupled with bounds on VC dimension and metric entropy.

Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the oscillatory properties of the solution. However, the high-frequency wave pattern becomes very complicated in the presence of multiple scattering obstacles. We consider a boundary integral equation formulation of the Helmholtz equation in two dimensions involving several obstacles, for which ray tracing schemes have been previously proposed. The existing analysis of ray tracing schemes focuses on periodic orbits between a subset of the obstacles. One observes that the densities on each of the obstacles converge to an equilibrium after a few iterations. In this paper we present an asymptotic approximation of the phases of those densities in equilibrium, in the form of a Taylor series. The densities represent a full cycle of reflections in a periodic orbit. We initially exploit symmetry in the case of two circular scatterers, but also provide an explicit algorithm for an arbitrary number of general 2D obstacles. The coefficients, as well as the time to compute them, are independent of the wavenumber and of the incident wave. The results may be used to accelerate ray tracing schemes after a small number of initial iterations.

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the ...

Hierarchies are of fundamental interest in both stochastic optimal control and biological control due to their facilitation of a range of desirable computational traits in a control algorithm and the possibility that they may form a core principle of sensorimotor and cognitive control systems. However, a theoretically justified construction of state-space hierarchies over all spatial resolutions and their evolution through a policy inference process remains elusive. Here, a formalism for deriving such normative representations of discrete Markov decision processes is introduced in the context of graphs. The resulting hierarchies correspond to a hierarchical policy inference algorithm approximating a discrete gradient flow between state-space trajectory densities generated by the prior and optimal policies.

In this paper, we present an initial attempt to learn evolution PDEs from data. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to uncover the underlying hidden PDE models. The basic idea of the proposed PDE-Net is to learn differential operators by learning convolution kernels (filters), and apply neural networks or other machine learning methods to approximate the unknown nonlinear responses. Comparing with existing approaches, which either assume the form of the nonlinear response is known or fix certain finite difference approximations of differential operators, our approach has the most flexibility by learning both differential operators and the nonlinear responses. A special feature of the proposed PDE-Net is that all filters are properly constrained, which enables us to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network. These constrains are carefully designed by fully exploiting the relation between the orders of differential operators and the orders of sum rules of filters (an important concept originated from wavelet theory). We also discuss relations of the PDE-Net with some existing networks in computer vision such as Network-In-Network (NIN) and Residual Neural Network (ResNet). Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

For recursive circular filtering based on circular statistics, we introduce a general framework for estimation of a circular state based on different circular distributions, specifically the wrapped normal distribution and the von Mises distribution. We propose an estimation method for circular systems with nonlinear system and measurement functions. This is achieved by relying on efficient deterministic sampling techniques. Furthermore, we show how the calculations can be simplified in a variety of important special cases, such as systems with additive noise as well as identity system or measurement functions. We introduce several novel key components, particularly a distribution-free prediction algorithm, a new and superior formula for the multiplication of wrapped normal densities, and the ability to deal with non-additive system noise. All proposed methods are thoroughly evaluated and compared to several state-of-the-art solutions.

Tensor robust principal component analysis (TRPCA) has received a substantial amount of attention in various fields. Most existing methods, normally relying on tensor nuclear norm minimization, need to pay an expensive computational cost due to multiple singular value decompositions (SVDs) at each iteration. To overcome the drawback, we propose a scalable and efficient method, named Parallel Active Subspace Decomposition (PASD), which divides the unfolding along each mode of the tensor into a columnwise orthonormal matrix (active subspace) and another small-size matrix in parallel. Such a transformation leads to a nonconvex optimization problem in which the scale of nulcear norm minimization is generally much smaller than that in the original problem. Furthermore, we introduce an alternating direction method of multipliers (ADMM) method to solve the reformulated problem and provide rigorous analyses for its convergence and suboptimality. Experimental results on synthetic and real-world data show that our algorithm is more accurate than the state-of-the-art approaches, and is orders of magnitude faster.

We adopt the perspective of an aggregator, which seeks to coordinate its purchase of demand reductions from a fixed group of residential electricity customers, with its sale of the aggregate demand reduction in a two-settlement wholesale energy market. The aggregator procures reductions in demand by offering its customers a uniform price for reductions in consumption relative to their predetermined baselines. Prior to its realization of the aggregate demand reduction, the aggregator must also determine how much energy to sell into the two-settlement energy market. In the day-ahead market, the aggregator commits to a forward contract, which calls for the delivery of energy in the real-time market. The underlying aggregate demand curve, which relates the aggregate demand reduction to the aggregator's offered price, is assumed to be affine and subject to unobservable, random shocks. Assuming that both the parameters of the demand curve and the distribution of the random shocks are initially unknown to the aggregator, we investigate the extent to which the aggregator might dynamically adapt its offered prices and forward contracts to maximize its expected profit over a time window of $T$ days. Specifically, we design a dynamic pricing and contract offering policy that resolves the aggregator's need to learn the unknown demand model with its desire to maximize its cumulative expected profit over time. In particular, the proposed pricing policy is proven to incur a regret over $T$ days that is no greater than $O(\log(T)\sqrt{T})$.

In this work we establish the first linear convergence result for the stochastic heavy ball method. The method performs SGD steps with a fixed stepsize, amended by a heavy ball momentum term. In the analysis, we focus on minimizing the expected loss and not on finite-sum minimization, which is typically a much harder problem. While in the analysis we constrain ourselves to quadratic loss, the overall objective is not necessarily strongly convex.

To run quantum algorithms on emerging gate-model quantum hardware, quantum circuits must be compiled to take into account constraints on the hardware. For near-term hardware, with only limited means to mitigate decoherence, it is critical to minimize the duration of the circuit. We investigate the application of temporal planners to the problem of compiling quantum circuits to newly emerging quantum hardware. While our approach is general, we focus on compiling to superconducting hardware architectures with nearest neighbor constraints. Our initial experiments focus on compiling Quantum Alternating Operator Ansatz (QAOA) circuits whose high number of commuting gates allow great flexibility in the order in which the gates can be applied. That freedom makes it more challenging to find optimal compilations but also means there is a greater potential win from more optimized compilation than for less flexible circuits. We map this quantum circuit compilation problem to a temporal planning problem, and generated a test suite of compilation problems for QAOA circuits of various sizes to a realistic hardware architecture. We report compilation results from several state-of-the-art temporal planners on this test set. This early empirical evaluation demonstrates that temporal planning is a viable approach to quantum circuit compilation.

In this paper, we define a general class of abstract aerial robotic systems named Laterally Bounded Force (LBF) vehicles, in which most of the control authority is expressed along a principal thrust direction, while in the lateral directions a (smaller and possibly null) force may be exploited to achieve full-pose tracking. This class approximates well platforms endowed with non-coplanar/non-collinear rotors that can use the tilted propellers to slightly change the orientation of the total thrust w.r.t. the body frame. For this broad class of systems, we introduce a new geometric control strategy in SE(3) to achieve, whenever made possible by the force constraints, the independent tracking of position-plus-orientation trajectories. The exponential tracking of a feasible full-pose reference trajectory is proven using a Lyapunov technique in SE(3). The method can deal seamlessly with both under- and fully-actuated LBF platforms. The controller guarantees the tracking of at least the positional part in the case that an unfeasible full-pose reference trajectory is provided. The paper provides several experimental tests clearly showing the practicability of the approach and the sharp improvement with respect to state of-the-art approaches.

Optimal path planning problems for rigid and deformable (bendable) cuboid robots are considered by providing an analytic safety constraint using generalized $L_p$ norms. For regular cuboid robots, level sets of weighted $L_p$ norms generate implicit approximations of their surfaces. For bendable cuboid robots a weighted $L_p$ norm in polar coordinates implicitly approximates the surface boundary through a specified level set. Obstacle volumes, in the environment to navigate within, are presumed to be approximately described as sub-level sets of weighted $L_p$ norms. Using these approximate surface models, the optimal safe path planning problem is reformulated as a two stage optimization problem, where the safety constraint depends on a point on the robot which is closest to the obstacle in the obstacle's distance metric. A set of equality and inequality constraints are derived to replace the closest point problem, which is then defines additional analytic constraints on the original path planning problem. Combining all the analytic constraints with logical AND operations leads to a general optimal safe path planning problem. Numerically solving the problem involve conversion to a nonlinear programing problem. Simulations for rigid and bendable cuboid robot verify the proposed method.

Learning customer preferences from an observed behaviour is an important topic in the marketing literature. Structural models typically model forward-looking customers or firms as utility-maximizing agents whose utility is estimated using methods of Stochastic Optimal Control. We suggest an alternative approach to study dynamic consumer demand, based on Inverse Reinforcement Learning (IRL). We develop a version of the Maximum Entropy IRL that leads to a highly tractable model formulation that amounts to low-dimensional convex optimization in the search for optimal model parameters. Using simulations of consumer demand, we show that observational noise for identical customers can be easily confused with an apparent consumer heterogeneity.

In this paper we consider autonomous driving of miniature race cars. The viability kernel is used to efficiently generate finite look-ahead trajectories that maximize progress while remaining recursively feasible with respect to static obstacles (e.g., stay inside the track). Together with a low-level model predictive controller, this method makes real-time autonomous racing possible. The viability kernel computation is based on space discretization. To make the calculation robust against discretization errors, we propose a novel numerical scheme based on game theoretical methods, in particular the discriminating kernel. We show that the resulting algorithm provides an inner approximation of the viability kernel and guarantees that, for all states in the cell surrounding a viable grid point, there exists a control that keeps the system within the kernel. The performance of the proposed control method is studied in simulation where we determine the effects of various design choices and parameters and in experiments on an autonomous racing set-up maintained at the Automatic Control Laboratory of ETH Zurich. Both simulation and experimental results suggest that the more conservative approximation using the discriminating kernel results in safer driving style at the cost of a small increase in lap time.

In this paper we propose a new Koopman operator approach to the decomposition of nonlinear dynamical systems using Koopman Gramians. We introduce the notion of an input-Koopman operator, and show how input-Koopman operators can be used to cast a nonlinear system into the classical state-space form, and identify conditions under which input and state observable functions are well separated. We then extend an existing method of dynamic mode decomposition for learning Koopman operators from data known as deep dynamic mode decomposition to systems with controls or disturbances. We illustrate the accuracy of the method in learning an input-state separable Koopman operator for an example system, even when the underlying system exhibits mixed state-input terms. We next introduce a nonlinear decomposition algorithm, based on Koopman Gramians, that maximizes internal subsystem observability and disturbance rejection from unwanted noise from other subsystems. We derive a relaxation based on Koopman Gramians and multi-way partitioning for the resulting NP-hard decomposition problem. We lastly illustrate the proposed algorithm with the swing dynamics for an IEEE 39-bus system.

This work proposes a solution for the longitudinal and lateral control problem of urban autonomous vehicles using a gain scheduling LPV control approach. Using the kinematic and dynamic vehicle models, a linear parameter varying (LPV) representation is adopted and a cascade control methodology is proposed for controlling both vehicle behaviours. In particular, for the control design, the use of both models separately lead to solve two LPV LMI-LQR problems. Furthermore, to achieve the desired levels of performance, an approach based on cascade design of the the kinematic and dynamic controllers has been proposed. This cascade control scheme is based on the idea that the dynamic closed loop behaviour is designed to be faster than the kinematic closed loop one. The obtained gain scheduling LPV control approach, jointly with a trajectory generation module, has presented suitable results in a simulated city driving scenario.

In this paper we present a fast and efficient method for the reconstruction of Magnetic Resonance Images (MRI) from severely under-sampled data. From the Compressed Sensing theory we have mathematically modeled the problem as a constrained minimization problem with a family of non-convex regularizing objective functions depending on a parameter and a least squares data fit constraint. We propose a fast and efficient algorithm, named Fast NonConvex Reweighting (FNCR) algorithm, based on an iterative scheme where the non-convex problem is approximated by its convex linearization and the penalization parameter is automatically updated. The convex problem is solved by a Forward-Backward procedure, where the Backward step is performed by a Split Bregman strategy. Moreover, we propose a new efficient iterative solver for the arising linear systems. We prove the convergence of the proposed FNCR method. The results on synthetic phantoms and real images show that the algorithm is very well performing and computationally efficient, even when compared to the best performing methods proposed in the literature.

This paper presents a novel partial differential equation (PDE)-based framework for controlling an ensemble of robots, which have limited sensing and actuation capabilities and exhibit stochastic behaviors, to perform mapping and coverage tasks. We model the ensemble population dynamics as an advection-diffusion-reaction PDE model and formulate the mapping and coverage tasks as identification and control problems for this model. In the mapping task, robots are deployed over a closed domain to gather data, which is unlocalized and independent of robot identities, for reconstructing the unknown spatial distribution of a region of interest. We frame this task as a convex optimization problem whose solution represents the region as a spatially-dependent coefficient in the PDE model. We then consider a coverage problem in which the robots must perform a desired activity at a programmable probability rate to achieve a target spatial distribution of activity over the reconstructed region of interest. We formulate this task as an optimal control problem in which the PDE model is expressed as a bilinear control system, with the robots' coverage activity rate and velocity field defined as the control inputs. We validate our approach with simulations of a combined mapping and coverage scenario in two environments with three target coverage distributions.

Following the seminal work of Nesterov, accelerated optimization methods have been used to powerfully boost the performance of first-order, gradient-based parameter estimation in scenarios where second-order optimization strategies are either inapplicable or impractical. Not only does accelerated gradient descent converge considerably faster than traditional gradient descent, but it also performs a more robust local search of the parameter space by initially overshooting and then oscillating back as it settles into a final configuration, thereby selecting only local minimizers with a basis of attraction large enough to contain the initial overshoot. This behavior has made accelerated and stochastic gradient search methods particularly popular within the machine learning community. In their recent PNAS 2016 paper, Wibisono, Wilson, and Jordan demonstrate how a broad class of accelerated schemes can be cast in a variational framework formulated around the Bregman divergence, leading to continuum limit ODE's. We show how their formulation may be further extended to infinite dimension manifolds (starting here with the geometric space of curves and surfaces) by substituting the Bregman divergence with inner products on the tangent space and explicitly introducing a distributed mass model which evolves in conjunction with the object of interest during the optimization process. The co-evolving mass model, which is introduced purely for the sake of endowing the optimization with helpful dynamics, also links the resulting class of accelerated PDE based optimization schemes to fluid dynamical formulations of optimal mass transport.

: In this paper a method to make inputting electrical model upon factors that affect melting process of high ultra power(UHP) electric furnace by using fuzzy rule and regression model is suggested and its effectiveness is verified with simulation experiment.

These are the proceedings of the workshop on Formal Verification of Autonomous Vehicles, held on September 19th, 2017 in Turin, Italy, as an affiliated workshop of the International Conference on integrated Formal Methods (iFM 2017). The workshop aim is to bring together researchers from the formal verification community that are developing formal methods for autonomous vehicles as well as researchers working, e.g., in the area of control theory or robotics, interested in applying verification techniques for designing and developing of autonomous vehicles.