Real-time computation of optimal control is a challenging problem and, to solve this difficulty, many frameworks proposed to use learning techniques to learn (possibly sub-optimal) controllers and enable their usage in an online fashion. Among these techniques, the optimal motion framework is a simple, yet powerful technique, that obtained success in many complex real-world applications. The main idea of this approach is to take advantage of dynamic motion primitives, a widely used tool in robotics to learn trajectories from demonstrations. While usually these demonstrations come from humans, the optimal motion framework is based on demonstrations coming from optimal solutions, such as the ones obtained by numeric solvers. As usual in many learning techniques, a drawback of this approach is that it is hard to estimate the suboptimality of learned solutions, since finding easily computable and non-trivial upper bounds to the error between an optimal solution and a learned solution is, in general, unfeasible. However, we show in this paper that it is possible to estimate this error for a broad class of problems. Furthermore, we apply this estimation technique to achieve a novel and more efficient sampling scheme to be used within the optimal motion framework, enabling the usage of this framework in some scenarios where the computational resources are limited.
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We introduce CartiMorph, a framework for automated knee articular cartilage morphometrics. It takes an image as input and generates quantitative metrics for cartilage subregions, including the percentage of full-thickness cartilage loss (FCL), mean thickness, surface area, and volume. CartiMorph leverages the power of deep learning models for hierarchical image feature representation. Deep learning models were trained and validated for tissue segmentation, template construction, and template-to-image registration. We established methods for surface-normal-based cartilage thickness mapping, FCL estimation, and rule-based cartilage parcellation. Our cartilage thickness map showed less error in thin and peripheral regions. We evaluated the effectiveness of the adopted segmentation model by comparing the quantitative metrics obtained from model segmentation and those from manual segmentation. The root-mean-squared deviation of the FCL measurements was less than 8%, and strong correlations were observed for the mean thickness (Pearson's correlation coefficient $\rho \in [0.82,0.97]$), surface area ($\rho \in [0.82,0.98]$) and volume ($\rho \in [0.89,0.98]$) measurements. We compared our FCL measurements with those from a previous study and found that our measurements deviated less from the ground truths. We observed superior performance of the proposed rule-based cartilage parcellation method compared with the atlas-based approach. CartiMorph has the potential to promote imaging biomarkers discovery for knee osteoarthritis.
Identification of nonlinear dynamical systems has been popularized by sparse identification of the nonlinear dynamics (SINDy) via the sequentially thresholded least squares (STLS) algorithm. Many extensions SINDy have emerged in the literature to deal with experimental data which are finite in length and noisy. Recently, the computationally intensive method of ensembling bootstrapped SINDy models (E-SINDy) was proposed for model identification, handling finite, highly noisy data. While the extensions of SINDy are numerous, their sparsity-promoting estimators occasionally provide sparse approximations of the dynamics as opposed to exact recovery. Furthermore, these estimators suffer under multicollinearity, e.g. the irrepresentable condition for the Lasso. In this paper, we demonstrate that the Trimmed Lasso for robust identification of models (TRIM) can provide exact recovery under more severe noise, finite data, and multicollinearity as opposed to E-SINDy. Additionally, the computational cost of TRIM is asymptotically equal to STLS since the sparsity parameter of the TRIM can be solved efficiently by convex solvers. We compare these methodologies on challenging nonlinear systems, specifically the Lorenz 63 system, the Bouc Wen oscillator from the nonlinear dynamics benchmark of No\"el and Schoukens, 2016, and a time delay system describing tool cutting dynamics. This study emphasizes the comparisons between STLS, reweighted $\ell_1$ minimization, and Trimmed Lasso in identification with respect to problems faced by practitioners: the problem of finite and noisy data, the performance of the sparse regression of when the library grows in dimension (multicollinearity), and automatic methods for choice of regularization parameters.
Knowing the actual precipitation in space and time is critical in hydrological modelling applications, yet the spatial coverage with rain gauge stations is limited due to economic constraints. Gridded satellite precipitation datasets offer an alternative option for estimating the actual precipitation by covering uniformly large areas, albeit related estimates are not accurate. To improve precipitation estimates, machine learning is applied to merge rain gauge-based measurements and gridded satellite precipitation products. In this context, observed precipitation plays the role of the dependent variable, while satellite data play the role of predictor variables. Random forests is the dominant machine learning algorithm in relevant applications. In those spatial predictions settings, point predictions (mostly the mean or the median of the conditional distribution) of the dependent variable are issued. The aim of the manuscript is to solve the problem of probabilistic prediction of precipitation with an emphasis on extreme quantiles in spatial interpolation settings. Here we propose, issuing probabilistic spatial predictions of precipitation using Light Gradient Boosting Machine (LightGBM). LightGBM is a boosting algorithm, highlighted by prize-winning entries in prediction and forecasting competitions. To assess LightGBM, we contribute a large-scale application that includes merging daily precipitation measurements in contiguous US with PERSIANN and GPM-IMERG satellite precipitation data. We focus on extreme quantiles of the probability distribution of the dependent variable, where LightGBM outperforms quantile regression forests (QRF, a variant of random forests) in terms of quantile score at extreme quantiles. Our study offers understanding of probabilistic predictions in spatial settings using machine learning.
We propose an automated nonlinear model reduction and mesh adaptation framework for rapid and reliable solution of parameterized advection-dominated problems, with emphasis on compressible flows. The key features of our approach are threefold: (i) a metric-based mesh adaptation technique to generate an accurate mesh for a range of parameters, (ii) a general (i.e., independent of the underlying equations) registration procedure for the computation of a mapping $\Phi$ that tracks moving features of the solution field, and (iii) an hyper-reduced least-square Petrov-Galerkin reduced-order model for the rapid and reliable estimation of the mapped solution. We discuss a general paradigm -- which mimics the refinement loop considered in mesh adaptation -- to simultaneously construct the high-fidelity and the reduced-order approximations, and we discuss actionable strategies to accelerate the offline phase. We present extensive numerical investigations for a quasi-1D nozzle problem and for a two-dimensional inviscid flow past a Gaussian bump to display the many features of the methodology and to assess the performance for problems with discontinuous solutions.
The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain-decomposition (DD) methods and reduced-order modelling (ROM). In particular, we consider an optimisation-based domain-decomposition algorithm for the parameter-dependent stationary incompressible Navier-Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal-control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward-facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain-decomposition algorithm.
Navigating automated driving systems (ADSs) through complex driving environments is difficult. Predicting the driving behavior of surrounding human-driven vehicles (HDVs) is a critical component of an ADS. This paper proposes an enhanced motion-planning approach for an ADS in a highway-merging scenario. The proposed enhanced approach utilizes the results of two aspects: the driving behavior and long-term trajectory of surrounding HDVs, which are coupled using a hierarchical model that is used for the motion planning of an ADS to improve driving safety.
We consider the problem of computing a sparse binary representation of an image. To be precise, given an image and an overcomplete, non-orthonormal basis, we aim to find a sparse binary vector indicating the minimal set of basis vectors that when added together best reconstruct the given input. We formulate this problem with an $L_2$ loss on the reconstruction error, and an $L_0$ (or, equivalently, an $L_1$) loss on the binary vector enforcing sparsity. This yields a so-called Quadratic Unconstrained Binary Optimization (QUBO) problem, whose solution is generally NP-hard to find. The contribution of this work is twofold. First, the method of unsupervised and unnormalized dictionary feature learning for a desired sparsity level to best match the data is presented. Second, the binary sparse coding problem is then solved on the Loihi 1 neuromorphic chip by the use of stochastic networks of neurons to traverse the non-convex energy landscape. The solutions are benchmarked against the classical heuristic simulated annealing. We demonstrate neuromorphic computing is suitable for sampling low energy solutions of binary sparse coding QUBO models, and although Loihi 1 is capable of sampling very sparse solutions of the QUBO models, there needs to be improvement in the implementation in order to be competitive with simulated annealing.
We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$ stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented.
Velocity limit (VL) has been widely adopted in many variants of particle swarm optimization (PSO) to prevent particles from searching outside the solution space. Several adaptive VL strategies have been introduced with which the performance of PSO can be improved. However, the existing adaptive VL strategies simply adjust their VL based on iterations, leading to unsatisfactory optimization results because of the incompatibility between VL and the current searching state of particles. To deal with this problem, a novel PSO variant with state-based adaptive velocity limit strategy (PSO-SAVL) is proposed. In the proposed PSO-SAVL, VL is adaptively adjusted based on the evolutionary state estimation (ESE) in which a high value of VL is set for global searching state and a low value of VL is set for local searching state. Besides that, limit handling strategies have been modified and adopted to improve the capability of avoiding local optima. The good performance of PSO-SAVL has been experimentally validated on a wide range of benchmark functions with 50 dimensions. The satisfactory scalability of PSO-SAVL in high-dimension and large-scale problems is also verified. Besides, the merits of the strategies in PSO-SAVL are verified in experiments. Sensitivity analysis for the relevant hyper-parameters in state-based adaptive VL strategy is conducted, and insights in how to select these hyper-parameters are also discussed.
Nonlinear extensions to the active subspaces method have brought remarkable results for dimension reduction in the parameter space and response surface design. We further develop a kernel-based nonlinear method. In particular we introduce it in a broader mathematical framework that contemplates also the reduction in parameter space of multivariate objective functions. The implementation is thoroughly discussed and tested on more challenging benchmarks than the ones already present in the literature, for which dimension reduction with active subspaces produces already good results. Finally, we show a whole pipeline for the design of response surfaces with the new methodology in the context of a parametric CFD application solved with the Discontinuous Galerkin method.
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