Project 2

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

A non-intrusive method referred to as Stochastic Collocation (SC) addresses the limitation of GPCE in Project 1. SC methods utilize interpolation methods and project a set of deterministic simulations, evaluated using carefully chosen sampled points based on the Smolyak algorithm, onto a polynomial basis. This approach is very useful when endeavouring to quantify uncertainty in models implemented with complex deterministic code which cannot be easily modified.

In this project, we developed an adaptive hierarchical sparse grid collocation (ASGC) method. We utilize a piecewise multi-linear hierarchical basis sparse grid interpolation approach towards adaptivity that addresses the issues of locality and curse-of-dimensionality. The basic idea here is to use a piecewise linear hat function as a hierarchical basis function by dilation and translation on equidistant interpolation nodes. Then the stochastic function can be represented by a linear combination of these basis functions. The corresponding coefficients are just the hierarchical increments between two successive interpolation levels (hierarchical surpluses) and . The magnitude of the hierarchical surplus reflects the local regularity of the function. For a smooth function, this value decreases to zero quickly with increasing interpolation level. On the other hand, for a non-smooth function, a singularity is indicated by the magnitude of the hierarchical surplus. The larger this magnitude is, the stronger the singularity. Thus, the hierarchical surplus serves as a natural error indicator for the sparse grid interpolation. When this value is larger than a predefined threshold, we simply add the 2N neighboring points to the current point. A key motivation towards using this framework is its linear scaling with dimensionality, in contrast to the N-dimensional tree (2^N) scaling of the h-type adaptive framework. In addition, such a framework guarantees that a user-defined error threshold is met. We will also show that it is rather easier with this approach to extract realizations, higher-order statistics, and the probability density function (PDF) of the solution.

The following figures shows the sparse grid when ASGC is used to interpolate an irregular function with line singularity:

Fig 1: Exact function (left) and Interpolated function (right).

Fig 2. The corresponding adaptive sparse grid.

The following figure shows the solution of the well-known Kraichnan–Orszag (K–O) problem which has the input stochastic discontinuity and the failure of GPCE:

Fig 3. Evolution of the variance of the solution for 1D random input. Top left: y1, Top right: y2,

Bottom left: y3, Bottom right: Adaptive sparse grid with ε = 10−2.

Finally, we applied ASGC to detect the critical temperature of well-known stochastic Rayleigh–Bénard problem with random boundary temperature around the neighborhood of the critical point.

Fig 4. Steady-state δNu (left) versus hot wall temperature using ASGC and

the corresponding adaptive sparse grid with threshold ε = 0.01 (right).

For the conductive regime, Nu = 1 and Nu > 1 when heat convection occurs. Thus, the difference δNu ≡ Nu (θh) − 1 provides a measure of the heat transfer enhancement. This result is provided in the left figure.. The result is reconstructed from the hierarchical surplus of the solution. It is noted that, the critical value is about 0.541, since below this value δNu = 0.0 and an essentially linear increase of δNu with θh is observed beyond this value. This can be further verified from the corresponding adaptive sparse grid in the same figure shown on the right in Fig 4.

Fig 5. Prediction of the u velocity (left column), v velocity (middle column) and temperature (right column)

when θh = 0.667891 using ASGC (top row) and the solution of the deterministic problem using the same θh (bottom row).

To further verify the results, we sample a uniform random variable for hot wall temperature from the convection regime and reconstruct the solution from the hierarchical surpluses. At the same time, we run a deterministic problem using the same realization of the random variable. The results compares very well in the figure 5.

Fig 6. Comparison of the variance of the u velocity (left column), v velocity (middle column)

and temperature (right column) using ASGC (top row) and MC-SOBOL method with 10000 iteration (bottom row)

However, there are still some limitations to this problem. Although ASGC depends weekly on the dimensionality of the stochastic space, it is still difficult so solve extremely high dimensional problem (~100) due to the limited computational source. In addition, due to its logarithm term in the error estimate, it needs much more collocation point to achieve a satisfied accuracy. These two issues motivates the HDMR method.

A detailed introduction to conventional sparse grid collocation method can be found here. The paper and PowerPoint presentation can be downloaded from here.