ROBUSTNESS IN NUMERICAL COMPUTATION IIVALIDATED ODE SOLVER

KWANG HEE KOSCHOOL OF MECHATRONICS

GWANGJU INSTITUTE OF SCIENCE AND TECHNOLOGY

SOLVING ODES

Traditional ODE solvers: Runge-Kutta, Adams-Bashforth, etc.

• Work well in general.• When two solution features are close to each other

• the step size selection becomes complex

• Incorrect step size may lead to a critical problem

• Looping or straying

SOLVING ODES

Traditional ODE solvers: Runge-Kutta, Adams-Bashforth, etc.

• Such problems happen since they control the size of each step solely based on controlling just the error alone.

• Do not consider the existence and uniqueness of solution.

VALIDATED INTERVAL ODE SOLVER

Tracing intersection curves

• Use the Validated ODE Solver• Phase I: Step size selection and a priori enclosure

computation• Determination of a region where existence and uniqueness of

the solution is validated.

• Phase II: Tight enclosure computation• Given an a priori enclosure, a tight enclosure at the next step is

computed minimizing the wrapping effect.

• Using compute a tight enclosure

][)(

))(()('

00 yty

tyfty

],[],~[)( 1 jjj tttallforyty

]~[ jy ][ 1jy

VALIDATED INTERVAL ODE SOLVERConceptual Illustration of Validated ODE Solver.

si si+1 si+2

Parameter

True solution curveS

olu

tion

Error bounds

VALIDATED INTERVAL ODE SOLVER

Phase I: Step size selection and a priori enclosure computation

• To compute a step-size hj and an a priori enclosure such that,

]~[])~([])([][

],[],~[)(

1

0

][][

1

j

k

ij

kkjj

iijj

jjj

yyfhyfhy

tttallforyty

]~[ jy

Tight Enclosure

Ex. Constant Enclosure Method

VALIDATED INTERVAL ODE SOLVER

Phase II: Tight enclosure computation

• Avoid the wrapping effect.

• QR decomposition method

VALIDATED INTERVAL ODE SOLVER

Example

APPLICATION TO SURFACE-TO-SURFACE INTERSECTIONS

In most cases, Rational Parametric Polynomial surface intersections are common.

Solution Methods

• Lattice Method• Subdivision Method• Marching Method (Tracing method)

Marching Method is a popular choice.

DERIVATION!!!

EXAMPLE: BICUBIC–BEZIER INTERSECTION

• Two rational bicubic-bezier patches.

• Starting point found by interval projected polyhedron(IPP) algorithm.

"Computation of the Solutions of Nonlinear Polynomial Systems" by E. C. Sherbrooke and N. M. Patrikalakis, Computer Aided Geometric Design, 10, No. 5, (1993) 379-405

OUTPUT FROM THE VALIDATED ODE SOLVER

With respect to the arc length parameter the a priori enclosures of the pre-images of the surfaces are connected. u

t v

s

s s

s

3D MAPPING OF THE PARAMETER BOXES

– tu – v

INTERSECTION OF THE BOXES IN 3DCollections of boxes obtained from the mapping from each of the surfaces, contain the true solution.

• Take union of the set of boxes obtained from each surface.• Take intersection of the two previously constructed sets.

During the intersection, we can in general obtain a substantial reduction in model space error.

The above is related to, and nicely complements, our previous work on interval solids.

”Topological and Geometric Properties of Interval Solid Models" by T. Sakkalis, G. Shen and N. M. Patrikalakis, Graphical Models. Vol. 63, No. 3, pp. 163-175, May 2001

INTERSECTION APPROXIMATED BY AN INTERVAL B-SPLINE

The result can be expressed as an interval B-spline curve.

Substantial reduction of data storage.

Essentially expressed as two B-spline curves representing,

• Spine curve.• An error curve representing half-width.

Slight increase in the width of the model space bound.

Approximation of measured data with interval B-splines. S. T. Tuohy, T. Maekawa, G. Shen and N. M. Patrikalakis. Computer-Aided Design, Vol. 29, No. 11, pp. 791-799, 1997.

VARIATION WITH TOLERANCE

Rel. Tolerance = 1.4x10-2 Rel. Tolerance = 2.6x10-4