We propose and analyze a block coordinate descent proximal algorithm (BCD-prox) for simultaneous filtering and parameter estimation of ODE models. As we show on ODE systems with up to d=40 dimensions, as compared to state-of-the-art methods, BCD-prox exhibits increased robustness (to noise, parameter initialization, and hyperparameters), decreased training times, and improved accuracy of both filtered states and estimated parameters. We show how BCD-prox can be used with multistep numerical discretizations, and we establish convergence of BCD-prox under hypotheses that include real systems of interest.

We develop a nonparametric method to infer the drift and diffusion functions of a stochastic differential equation. With this method, we can build predictive models starting with repeated time series and/or high-dimensional longitudinal data. Typical use of the method includes forecasting the future density or distribution of the variable being measured. The key innovation in our method stems from efficient algorithms to evaluate a likelihood function and its gradient. These algorithms do not rely on sampling, instead, they use repeated quadrature and the adjoint method to enable the inference to scale well as the dimensionality of the parameter vector grows. In simulated data tests, when the number of sample paths is large, the method does an excellent job of inferring drift functions close to the ground truth. We show that even when the method does not infer the drift function correctly, it still yields models with good predictive power. Finally, we apply the method to real data on hourly measurements of ground level ozone, showing that it is capable of reasonable results.

We provide an algorithm to build quantile regression trees in O(N log N) time, where N is the number of instances in the training set. Quantile regression trees are regression trees that model conditional quantiles of the response variable, rather than the conditional expectation as in standard regression trees. We build quantile regression trees by using the quantile loss function in our node splitting criterion. The performance of our algorithm stems from new online update procedures for both the quantile function and the quantile loss function. We test the quantile tree algorithm in three ways, comparing its running time against implementations of standard regression trees, demonstrating its ability to recover a known set of nonlinear quantile functions, and showing that quantile trees yield smaller test set errors (computed using mean absolute deviation) than standard regression trees. The tests include training sets with up to 16 million instances. Overall, our results enable future use of quantile regression trees for large-scale data mining.

Given a power network consisting of nodes (generators/loads) and edges (lines), there exist a set of constraints that must be satisfied in order for the system to be operational. When one or more power lines are cut, the bus phases and load/generator power values may need to be altered in order to restore the system to operation. The load shedding problem is to find the smallest adjustment to the loads that achieves this restoration. In this work, we show how to solve this nonlinear optimization problem by solving a sequence of linear programs. On random graphs with 1500 edges, we find that our method is at least 40 times faster than competing nonlinear optimization methods. We show that our method is capable of solving load shedding problems for a real graph with 19840 edges, and that the method scales with an O(n2) running time where n is the number of edges. For real subgraphs with hundreds of edges, we are able to rapidly solve all possible load shedding problems in which at most two lines are deleted. This work takes a first step towards a scalable load shedding algorithm capable of handling large networks that will arise in the future.

We consider a general class of 2-D passive propagation media, represented as a planar graph where nodes are capacitors connected to a common ground and edges are inductors. Capacitances and inductances are fixed in time but vary in space. Kirchhoff's laws give the time dynamics of voltage and current in the system. By harmonically forcing input nodes and collecting the resulting steady-state signal at output nodes, we obtain a linear, analog device that transforms the inputs to outputs. We pose the lattice synthesis problem: given a linear transformation, find the inductances and capacitances for an inductor-capacitor circuit that can perform this transformation. Formulating this as an optimization problem, we numerically demonstrate its solvability using gradient-based methods. By solving the lattice synthesis problem for various desired transformations, we design several devices that can be used for signal processing and filtering.