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A Comparison of Methods Used Hence Far for the FPE

Mrinal Kumar1, Ph.D. Student

Department of Aerospace Engineering, Texas A&M University

Nov. 2005

Finite Differences

Systems considered:

1D Transient, 2D Transient

Pros and Cons:

1. Unstable method - requires analysis of discretization scheme to evaluate its stability.:CON

2. Error is typically large due to the Zero Order Hold, and requires very fine grid size foracceptable answer.: CON

3. Positivity not ensured even with a very fine grid. The tail regions are especially suscep-tible to negative solution, causing error percolation into regions with high probabilitydensity: CON

4. Growth of domain size leads to an explosion in the number of grid points because thesame grid fineness needs to be maintained.: CON

5. Method cannot be generalized - every system requires a different discretization schemethat is stable, i.e., requires reformulation for every new system.: CON

6. Matrix to be inverted is very large in dimensions, requires special iterative algorithms,e.g. Gauss Siedel, and convergence is not guaranteed.: CON

7. Almost impossible for systems with state-dimension 2, for one or more of above citedreasons.: CON

8. It is the simplest possible implementation of the problem, only requires extensive book-keeping.: PRO

9. It served as a useful starting point towards the solution to a formidable problem.: PRO

1 cMrinal Kumar, 2005

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Results Obtained:

1D Transient System:: Unstable Solution, singularity makes the 1D system a patho-logical example.

2D Duffing Oscillator (x + x3 = 0):: Solution obtained for only moderately nonlinearcases (max = 0.2), and for only 1 orbit of propagation. Errors grow large after that.

Solution attempted for a case with separatrix:: Divergence observed almost instantly.

Global Functional Approximation

(Galerkin Method)

Systems Studied:

1D Transient, 2D stationary, 2D Transient.

Pros and Cons:

1. Given a good reference pdf, it gives accurate results, infact the most accurate resultsobtained hence far.:PRO

2. Positivity enforced by log-pdf transformation. Orthogonality of the polynomial basisused results in uncoupled non-linear equations for the basis functions coefficients,which are easy to integrate. :PRO

3. Especially suited for systems with polynomial dynamics, the integrals can be doneanalytically. :PRO

4. N domain issues, solution obtained over (,)N. :PRO

5. It automatically satisfies the boundary conditions. :PRO

6. No domain issues, solution obtained over (,)N. :PRO

7. Time required is the least of all methods considered so far. :PRO

8. The method is cumbersome for systems with non-polynomial dynamics. :CON - This

was the first reason why we looked for alternate schemes

9. It is not suitable for systems with pdf spread over multiple and/or asymmetric regions,as the global weight functions cannot account for all these regions at the same time.:CON - This was the second reason why we looked for a local approximation method.

10. If we desire to use basis functions other than polynomials, or, special functions, all thebenefits of orthogonality are lost, and there are no significant advantages. :CON

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Results Obtained:

1D transient:: Captures the solution with high accuracy, and even tracks down theterminal singular behavior of a dirac delta function for considerable duration. Positivityenforced

2D stationary:: Highly accurate results for ring-pdf and bimodal-pdf. However, log-pdftransform in this case leads to a quadratic matrix equation, which is solved iteratively,and requires a good starting guess.

2D transient:: Very accurate results for the ring pdf. Goes from initial state to sta-tionary pdf with small error. Time required 20-30 minutes.

Overall Method Evaluation

1. Highest accuracy so far. REASON: The log-transform comes as a blessing in disguise,despite transforming the equation into a non-linear one. It entails that the solutionwe seek is a polynomial function, instead of the exponential of a polynomial function.Consequently, a polynomial basis does the job with very small error. However, this re-sult is dependent on the reference pdf. It is expected that systems with non-polynomialdynamics will have much greater error.

2. Not recommended for future work, because the global nature of approximation is un-suitable complex pdfs. Also, dependence on reference solution is a major drawback.Although orthogonality properties are an advantage, such features are lost if one de-sires better basis functions that are deduced from prior knowledge of local behavior of

solution.

Local Approximation: MLPG with Polynomial Basis

Systems Studied:

2D stationary, 4D stationary (current)

Pros and Cons:

1. Local Approximation. :PRO

2. Can handle systems with any nature of system dynamics. :PRO

3. Much less susceptible to tuning issues, like a ref. pdf. :PRO

4. Local approximations made so far were local enough to not require positivity enforce-ment. :PRO

5. Requires a high density of nodes for acceptable solution. The reason is that a simplemonomial basis set is not good enough to learn a exponential surface on a coarse nodaldistribution. :CON

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6. Requires numerical integration. :CON

7. MLPG requires a MLS solution for each quadrature point - i.e., if there are 50 50nodes and 10 10 quadrature points per local regime, then one needs to solve >50 50 10 10 = 2.5 105 MLS problems. :CON

8. Recent experience with the 4D problem shows that there is difficulty in solving theMLS approximation, if the nodes are not close together. It leads to an ill-conditionedmatrix inversion. :CON

9. Boundary conditions are difficult to impose, and constitute the heuristic part of themethod - done via a penalty method. The difficulty is mainly because our problemrequires enforcement of Dirichlet BC over the artificial boundaries, which is a toughask. :CON

10. The most time consuming of all methods used so far. Right now, we cannot even beginto start solving the transient problem, e.g., stationary solution ( 1 time step in atransient problem) requires 20-25 minutes. :CON

Overall Method Evaluation:

1. Although it reduces the book-keeping as compared with traditional FEM in terms ofelement connectivity, there is significant book-keeping required for the management ofboundary nodes.

2. Although theoretically, it allows random placement of nodes, such an endeavor did not

yield better results, infact worse (more nodes were used in regions of high probabilitydensity).

3. The most computationally intensive method.

4. NOT recommended for future work, in its current state. The current form of thismethod will collapse for a high dimensional system. Also, with its current formulation,one cannot use different basis functions in different local regimes. This is due to thelimitation of the blending functions used in MLPG. Thus, basis enrichment is notpossible. The reason this method fails (despite its local nature) to match theresults produced by the global approach, is that a polynomial basis set is beingused, albeit locally, to approximate the exponential of polynomial functions.

It fails to achieve this beyond a certain degree of accuracy. On the otherhand, the global method utilized the polynomial basis to track polynomials,by virtue of the log-pdf transform.

The Direction to Take: Towards the DESIRED METHOD::

1. Desire a Local Approximation, with a coarse mesh, tentatively 5 - 8 nodes per dimension(86 2.63 105 DOF).

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2. Desire a Functional Approximation inside each local domain - i.e. desire element basedmethod.

3. Desire special functions for basis enrichment. Polynomials cannot give desired accuracy,especially in a coarse mesh. Need to implement the idea of Handbook Functions usedin the Generalized Finite Element methods. (ref: Strouboulis, Babuska, Kopps, 2000-2002). These handbook functions require some prior knowledge of the system, and canbe evaluated computationally. We can use the linear systems knowledge to build them.

4. Require a blending scheme to merge the various local approximations seamlessly. Thiscan be achieved by the PUM.

5. Challenge: Enforcement of boundary conditions: most GFEM methods deal with prob-lems with Neumann, and not Dirichlet boundary conditions. However, recent GFEM

literature deals with this issue.

6. Challenge: The coarseness of mesh might require positivity enforcement. If done bylog-pdf transform, will lead to complicated and coupled non-linear equations. However,the first task is to learn the evaluation of handbook functions.

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