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Dimensional reduction techniques have long been used to visualize the structure and geometry of high dimensional data. However, most widely used techniques are difficult to interpret due to nonlinearities and opaque optimization processes.…
This paper presents a methodology for using varying sample sizes in sequential quadratic programming (SQP) methods for solving equality constrained stochastic optimization problems. The first part of the paper deals with the delicate issue…
The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of…
We propose a stochastic optimization method for minimizing loss functions, expressed as an expected value, that adaptively controls the batch size used in the computation of gradient approximations and the step size used to move along such…
A procedure based on a Mixture Density Model for correcting experimental data for distortions due to finite resolution and limited detector acceptance is presented. Addressing the case that the solution is known to be non-negative, in the…
Discretizing an analytic function on a uniform real-space grid is often done via a straightforward collocation method. This is ubiquitous in all areas of computational physics and quantum chemistry. An example in Density Functional Theory…
Density functional theory calculations use a significant fraction of current supercomputing time. The resources required scale with the problem size, internal workings of the code and the number of iterations to convergence, the latter…
We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Mathematically Parker transport equation (PTE) describing non-stationary transport of charged particles in the…
We introduce a distributed adaptive quadrature method that formulates multidimensional integration as a hierarchical domain decomposition problem on multi-GPU architectures. The integration domain is recursively partitioned into subdomains…
Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior…
This paper outlines an approach to the approximation of probability density functions by quadratic forms of weighted orthonormal basis functions with positive semi-definite Hermitian matrices of unit trace. Such matrices are called…
Dynamic treatment regimes (DTRs) have received an increasing interest in recent years. DTRs are sequences of treatment decision rules tailored to patient-level information. The main goal of the DTR study is to identify an optimal DTR, a…
We propose a higher-order dimensionality reduction framework based on the Trace Ratio (TR) optimization problem. We establish conditions for existence and uniqueness of solutions and clarify the theoretical connection between the Trace…
A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the…
The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed…
This paper introduces a differential dynamic programming (DDP) based framework for polynomial trajectory generation for differentially flat systems. In particular, instead of using a linear equation with increasing size to represent…
An optimization-based approach for the Tucker tensor approximation of parameter-dependent data tensors and solutions of tensor differential equations with low Tucker rank is presented. The problem of updating the tensor decomposition is…
As parallel computing trends towards the exascale, scientific data produced by high-fidelity simulations are growing increasingly massive. For instance, a simulation on a three-dimensional spatial grid with 512 points per dimension that…
The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination…
We introduce the Directional Gradient-Curvature (DGC) method, a novel approach for filling gaps in gridded environmental data. DGC is based on an objective function that measures the distance between the directionally segregated normalized…