Jonathan Feinberg

PhD Student
Mobile: +47 932 14 474
E-mail: jonathfe@simula.no
Affiliations: Simula School of Research and Innovation AS Computational Geoscience

Research Interests


The core of my interest is Probability Theory and in particular Uncertainty Quantification. The topic can also be subdivided into the following groups
 
  • Bayesian Probability Theory
  • Monte Carlo Simulation
  • Polynomial Chaos Theory
  • Probabilistic Approach to Inverse Problems
  • Reliability Analysis
  • Scientific Computing



Polynomial Chaos Expansions


Polynomial chaos expansion is a method for creating polynomial representations of the solution of differential equations using a Galerkin projection. By applying polynomials tailored to the equation's uncertain parameter's probability distribution, the representation converges in probability measure with exponential rate against the polynomial order.
 
My current work focuses on dealing with complex dependency structures in the random properties of the model parameters. In the classical theory, this is dealt with by performing a Rosenblatt transformation which can be both complex to implement and add extra noise to the chaos approximation. The new method mitigates this problem by not requiring a transformation.

Instead of solving the differential equations joint with the chaos representation, it is possible to solve the equations as a black box. This allows for collocation methods like Monte Carlo simulation. However, if the computational cost for solving the equations is high, it becomes unfeasible to use because of the methods large sample size requirement. An alternative is to use Probabilistic Collocation method which utilizes polynomial chaos expansion to reduced the number of collocation nodes required for acceptable accuracy.
 
 
 

Uncertainty Quantification Toolkit

 
The different tools for performing uncertainty analysis, can often become tedious to implement. I am therefore working on a creating a uncertainty quantification python library. I am making an emphasis on making it easy to use while at the same time making it modular enough to make it simple for others to extend the work.
 
The current library covers currently the following topics.
  • Vectorized multivariate polynomials with compatibility with the Python numpy library
  • Modular building blocks for constructing complex multivariate joint probability distributions
  • Polynomial chaos ODE solver
  • Collection of adaptive and static schemes for performing the probabilistic collocation method.


























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Monte Carlo simulation samples behave in the same way the underlying analytically probabilty distribution that often can not be obtained.
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Hermite Polynomials; Orthogonal on a functional inner product weigted on the Normal distribution.
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Solution of stochastic differential equations are non-Gaussian non-stationary random fields.

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