Related papers: Regularity and Pathwise bounds for probabilistic s…

We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be…

Regularization robust preconditioners for PDE-constrained optimization problems have been successfully developed. These methods, however, typically assume that observation data is available throughout the entire domain of the state…

We make some remarks on a variant of the classical Tikhonov regularization in optimal control under PDEs which allows for a certain flexibility in dealing with non-linearities and state restrictions, in the sense that differential…

This work is motivated by the need to study the impact of data uncertainties and material imperfections on the solution to optimal control problems constrained by partial differential equations. We consider a pathwise optimal control…

Solving nonlinear optimal control problems is a challenging task, particularly for high-dimensional problems. We propose algorithms for model-based policy iterations to solve nonlinear optimal control problems with convergence guarantees.…

We consider piecewise-deterministic optimal control problems in which the environment randomly switches among several deterministic modes, and the goal is to optimize the expected cost up to the termination while taking the likelihood of…

We develop an algorithm for computing bounded reachability probability for hybrid systems, i.e., the probability that the system reaches an unsafe region within a finite number of discrete transitions. In particular, we focus on hybrid…

Probabilistic programming systems enable users to encode model structure and naturally reason about uncertainties, which can be leveraged towards improved Bayesian optimization (BO) methods. Here we present a probabilistic program embedding…

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated PDEs numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs…

Sampling-based motion planners have proven to be efficient solutions to a variety of high-dimensional, geometrically complex motion planning problems with applications in several domains. The traditional view of these approaches is that…

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We find conditions…

In this survey paper, we study the optimal regularity of solutions to uniformly degenerate elliptic equations in bounded domains and establish the H\"older continuity of solutions and their derivatives up to the boundary.

We prove an existence and uniqueness result for Neumann boundary problem of a parabolic partial differential equation (PDE for short) with a singular nonlinear divergence term which can only be understood in a weak sense. A probabilistic…

We study rare events in systems of diffusive fields driven out of equilibrium by the boundaries. We present a numerical technique and use it to calculate the probabilities of rare events in one and two dimensions. Using this technique, we…

The flow equation approach is a robust framework applicable to a broad class of singular SPDEs, including those with fractional Laplacians, throughout the entire subcritical regime. Inspired by Wilson's renormalization group, this method…

Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed…

Accurate modeling of boundary conditions is crucial in computational physics. The ever increasing use of neural networks as surrogates for physics-related problems calls for an improved understanding of boundary condition treatment, and its…

Ensemble forecast post-processing is a necessary step in producing accurate probabilistic forecasts. Conventional post-processing methods operate by estimating the parameters of a parametric distribution, frequently on a per-location or…

An adaptive regularization strategy for stabilizing Newton-like iterations on a coarse mesh is developed in the context of adaptive finite element methods for nonlinear PDE. Existence, uniqueness and approximation properties are known for…

We consider constrained bilinear optimal control of second-order linear evolution partial differential equations (PDEs) with a reaction term on the half line, where control arises as a time-dependent reaction coefficient and constraints are…