The Ensemble Kalman inversion (EKI), proposed by Iglesias et al. for the solution of Bayesian inverse problems of type $y=A u^\dagger +\varepsilon$, with $u^\dagger$ being an unknown parameter and $y$ a given datum, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles $\{u^j(t)\}_{j=1}^J$, whose initial empirical measure is sampled from the prior, evolving over an artificial time $t$ towards an approximate solution of the inverse problem. Using spectral techniques, we provide a complete description of the \new{deterministic} dynamics of EKI and their asymptotic behavior in parameter space. In particular, we analyze dynamics of deterministic EKI, averaged quantities of stochastic EKI, and mean-field EKI. We show that in the linear Gaussian regime, the Bayesian posterior can only be recovered with the mean-field limit and not with finite sample sizes or deterministic EKI. Furthermore, we show that -- even in the deterministic case -- residuals in parameter space do not decrease monotonously in the Euclidean norm and suggest a problem-adapted norm, where monotonicity can be proved. Finally, we derive a system of ordinary differential equations governing the spectrum and eigenvectors of the covariance matrix.
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We propose a structure-preserving parametric finite element method (SP-PFEM) for discretizing the surface diffusion of a closed curve in two dimensions (2D) or surface in three dimensions (3D). Here the "structure-preserving" refers to preserving the two fundamental geometric structures of the surface diffusion flow: (i) the conservation of the area/volume enclosed by the closed curve/surface, and (ii) the decrease of the perimeter/total surface area of the curve/surface. For simplicity of notations, we begin with the surface diffusion of a closed curve in 2D and present a weak (variational) formulation of the governing equation. Then we discretize the variational formulation by using the backward Euler method in time and piecewise linear parametric finite elements in space, with a proper approximation of the unit normal vector by using the information of the curves at the current and next time step. The constructed numerical method is shown to preserve the two geometric structures and also enjoys the good property of asymptotic equal mesh distribution. The proposed SP-PFEM is "weakly" implicit (or almost semi-implicit) and the nonlinear system at each time step can be solved very efficiently and accurately by the Newton's iterative method. The SP-PFEM is then extended to discretize the surface diffusion of a closed surface in 3D. Extensive numerical results, including convergence tests, structure-preserving property and asymptotic equal mesh distribution, are reported to demonstrate the accuracy and efficiency of the proposed SP-PFEM for simulating surface diffusion in 2D and 3D.
We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy $\gamma(\theta)$ -- anisotropic surface diffusion -- in two dimensions, while $\theta$ is the angle between the outward unit normal vector and the vertical axis. By introducing a positive definite surface energy (density) matrix $G(\theta)$, we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized in space by the parametric finite element method and area/mass conservation and energy dissipation are established for the semi-discretization. Then the problem is further discretized in time by a (semi-implicit) backward Euler method so that only a linear system is to be solved at each time step for the full-discretization and thus it is efficient. We establish well-posedness of the full-discretization and identify some simple conditions on $\gamma(\theta)$ such that the full-discretization keeps energy dissipation and thus it is unconditionally energy-stable. Finally the ES-PFEM is applied to simulate solid-state dewetting of thin films with anisotropic surface energies, i.e. the motion of an open curve under anisotropic surface diffusion with proper boundary conditions at the two triple points moving along the horizontal substrate. Numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed ES-PFEM.
We consider multivariate centered Gaussian models for the random variable $Z=(Z_1,\ldots, Z_p)$, invariant under the action of a subgroup of the group of permutations on $\{1,\ldots, p\}$. Using the representation theory of the symmetric group on the field of reals, we derive the distribution of the maximum likelihood estimate of the covariance parameter $\Sigma$ and also the analytic expression of the normalizing constant of the Diaconis-Ylvisaker conjugate prior for the precision parameter $K=\Sigma^{-1}$. We can thus perform Bayesian model selection in the class of complete Gaussian models invariant by the action of a subgroup of the symmetric group, which we could also call complete RCOP models. We illustrate our results with a toy example of dimension $4$ and several examples for selection within cyclic groups, including a high dimensional example with $p=100$.
The proposed method re-frames traditional inverse problems of electrocardiography into regression problems, constraining the solution space by decomposing signals with multidimensional Gaussian impulse basis functions. Impulse HSPs were generated with single Gaussian basis functions at discrete heart surface locations and projected to corresponding BSPs using a volume conductor torso model. Both BSP (inputs) and HSP (outputs) were mapped to regular 2D surface meshes and used to train a neural network. Predictive capabilities of the network were tested with unseen synthetic and experimental data. A dense full connected single hidden layer neural network was trained to map body surface impulses to heart surface Gaussian basis functions for reconstructing HSP. Synthetic pulses moving across the heart surface were predicted from the neural network with root mean squared error of $9.1\pm1.4$%. Predicted signals were robust to noise up to 20 dB and errors due to displacement and rotation of the heart within the torso were bounded and predictable. A shift of the heart 40 mm toward the spine resulted in a 4\% increase in signal feature localization error. The set of training impulse function data could be reduced and prediction error remained bounded. Recorded HSPs from in-vitro pig hearts were reliably decomposed using space-time Gaussian basis functions. Predicted HSPs for left-ventricular pacing had a mean absolute error of $10.4\pm11.4$ ms. Other pacing scenarios were analyzed with similar success. Conclusion: Impulses from Gaussian basis functions are potentially an effective and robust way to train simple neural network data models for reconstructing HSPs from decomposed BSPs. The HSPs predicted by the neural network can be used to generate activation maps that non-invasively identify features of cardiac electrical dysfunction and can guide subsequent treatment options.
We consider energy minimization for data-intensive applications run on large number of servers, for given performance guarantees. We consider a system, where each incoming application is sent to a set of servers, and is considered to be completed if a subset of them finish serving it. We consider a simple case when each server core has two speed levels, where the higher speed can be achieved by higher power for each core independently. The core selects one of the two speeds probabilistically for each incoming application request. We model arrival of application requests by a Poisson process, and random service time at the server with independent exponential random variables. Our model and analysis generalizes to today's state-of-the-art in CPU energy management where each core can independently select a speed level from a set of supported speeds and corresponding voltages. The performance metrics under consideration are the mean number of applications in the system and the average energy expenditure. We first provide a tight approximation to study this previously intractable problem and derive closed form approximate expressions for the performance metrics when service times are exponentially distributed. Next, we study the trade-off between the approximate mean number of applications and energy expenditure in terms of the switching probability.
Given a graph whose nodes may be coloured red, the parity of the number of red nodes can easily be maintained with first-order update rules in the dynamic complexity framework DynFO of Patnaik and Immerman. Can this be generalised to other or even all queries that are definable in first-order logic extended by parity quantifiers? We consider the query that asks whether the number of nodes that have an edge to a red node is odd. Already this simple query of quantifier structure parity-exists is a major roadblock for dynamically capturing extensions of first-order logic. We show that this query cannot be maintained with quantifier-free first-order update rules, and that variants induce a hierarchy for such update rules with respect to the arity of the maintained auxiliary relations. Towards maintaining the query with full first-order update rules, it is shown that degree-restricted variants can be maintained.
Gaussian processes that can be decomposed into a smooth mean function and a stationary autocorrelated noise process are considered and a fully automatic nonparametric method to simultaneous estimation of mean and auto-covariance functions of such processes is developed. Our empirical Bayes approach is data-driven, numerically efficient and allows for the construction of confidence sets for the mean function. Performance is demonstrated in simulations and real data analysis. The method is implemented in the R package eBsc that accompanies the paper.
We continue the investigation on the spectrum of operators arising from the discretization of partial differential equations. In this paper we consider a three field formulation recently introduced for the finite element least-squares approximation of linear elasticity. We discuss in particular the distribution of the discrete eigenvalues in the complex plane and how they approximate the positive real eigenvalues of the continuous problem. The dependence of the spectrum on the Lam\'e parameters is considered as well and its behavior when approaching the incompressible limit.
We propose a novel dimensionality reduction method for maximum inner product search (MIPS), named CEOs, based on the theory of concomitants of extreme order statistics. Utilizing the asymptotic behavior of these concomitants, we show that a few dimensions associated with the extreme values of the query signature are enough to estimate inner products. Since CEOs only uses the sign of a small subset of the query signature for estimation, we can precompute all inner product estimators accurately before querying. These properties yield a sublinear MIPS algorithm with an exponential indexing space complexity. We show that our exponential space is optimal for the $(1 + \epsilon)$-approximate MIPS on a unit sphere. The search recall of CEOs can be theoretically guaranteed under a mild condition. To deal with the exponential space complexity, we propose two practical variants, including sCEOs-TA and coCEOs, that use linear space for solving MIPS. sCEOs-TA exploits the threshold algorithm (TA) and provides superior search recalls to competitive MIPS solvers. coCEOs is a data and dimension co-reduction technique and outperforms sCEOs-TA on high recall requirements. Empirically, they are very simple to implement and achieve at least 100x speedup compared to the bruteforce search while returning top-10 MIPS with accuracy at least 90% on many large-scale data sets.
Machine learning algorithms have found several applications in the field of robotics and control systems. The control systems community has started to show interest towards several machine learning algorithms from the sub-domains such as supervised learning, imitation learning and reinforcement learning to achieve autonomous control and intelligent decision making. Amongst many complex control problems, stable bipedal walking has been the most challenging problem. In this paper, we present an architecture to design and simulate a planar bipedal walking robot(BWR) using a realistic robotics simulator, Gazebo. The robot demonstrates successful walking behaviour by learning through several of its trial and errors, without any prior knowledge of itself or the world dynamics. The autonomous walking of the BWR is achieved using reinforcement learning algorithm called Deep Deterministic Policy Gradient(DDPG). DDPG is one of the algorithms for learning controls in continuous action spaces. After training the model in simulation, it was observed that, with a proper shaped reward function, the robot achieved faster walking or even rendered a running gait with an average speed of 0.83 m/s. The gait pattern of the bipedal walker was compared with the actual human walking pattern. The results show that the bipedal walking pattern had similar characteristics to that of a human walking pattern.
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