  Addressing the statistical challenge of computing the multivariate normal
(MVN) probability in high dimensions holds significant potential for enhancing
various applications. One common way to compute high-dimensional MVN
probabilities is the Separation-of-Variables (SOV) algorithm. This algorithm is
known for its high computational complexity of O(n^3) and space complexity of
O(n^2), mainly due to a Cholesky factorization operation for an n X n
covariance matrix, where $n$ represents the dimensionality of the MVN problem.
This work proposes a high-performance computing framework that allows scaling
the SOV algorithm and, subsequently, the confidence region detection algorithm.
The framework leverages parallel linear algebra algorithms with a task-based
programming model to achieve performance scalability in computing process
probabilities, especially on large-scale systems. In addition, we enhance our
implementation by incorporating Tile Low-Rank (TLR) approximation techniques to
reduce algorithmic complexity without compromising the necessary accuracy. To
evaluate the performance and accuracy of our framework, we conduct assessments
using simulated data and a wind speed dataset. Our proposed implementation
effectively handles high-dimensional multivariate normal (MVN) probability
computations on shared and distributed-memory systems using finite precision
arithmetics and TLR approximation computation. Performance results show a
significant speedup of up to 20X in solving the MVN problem using TLR
approximation compared to the reference dense solution without sacrificing the
application's accuracy. The qualitative results on synthetic and real datasets
demonstrate how we maintain high accuracy in detecting confidence regions even
when relying on TLR approximation to perform the underlying linear algebra
operations.
