Solution of the Wave Equation with Interval and Random Parameters Andrew Pownuk 1 Jazmin Quezada 1 Iwona Skalna 2 1 The University of Texas at El Paso, El Paso, Texas, USA 2 AGH University of Science and Technology, Krakow, Poland 20th Joint NMSU/UTEP Workshop on Mathematics, Computer Science, and Computational Sciences 1 / 32
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Solution of the Wave Equationwith Interval and Random Parameters
Andrew Pownuk1
Jazmin Quezada1
Iwona Skalna2
1 The University of Texas at El Paso, El Paso, Texas, USA2 AGH University of Science and Technology, Krakow, Poland
20th Joint NMSU/UTEP Workshop on Mathematics,Computer Science, and Computational Sciences
1 / 32
Outline
1 Wave Equation
2 Uncertain Parameters
3 Methods for the solution of the wave equationThe Finite Difference MethodThe Finite Element MethodFourier Series
4 Equations with interval parameters
5 Equations with random and interval parameters
6 Conclusions
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WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Wave Equation
Wave equation∂2u
∂t2= c2∂
2u
∂x2,
where c =
√E
ρ.
Initial-boundary value problem for the wave equation∂2u∂t2 = c2 ∂2u
∂x2 , (x , t) ∈ [0, L]× [0,T ]u(0, t) = 0, t ∈ [0,T ]u(L, t) = 0, t ∈ [0,T ]u(x , 0) = u0(x) x ∈ [0, L]v(x , 0) = v0(x) x ∈ [0, L]
3 / 32
WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Wave Equation
Initial-boundary value problem for the wave equation
4 / 32
WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Wave Equation
Typical solution of the wave equation.
5 / 32
WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Interval parameters (worst case analysis)
Solution of the equation with interval parameters for given(x , t) can be defined as the following set:
E ,ρ,n etc.) and A is the smallest interval that contains the setA.
Function u is a solution of a PDE with the interval parameters
∂2u
∂t2= c2∂
2u
∂x2
6 / 32
WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Random parameters
Solution of the equation with random parameters u(x , t, r(ω))for given (x , t) can be defined as the function of randomvariable r(ω) = (r1(ω), ..., rn(ω)).
Function u is a solution of a partial differential equation withrandom parameters (for example E ,ρ,n etc.)
∂2u
∂t2= c2∂
2u
∂x2
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WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Random and interval parameters
Solution of the equation with uncertain parametersu(x , t, r(ω), p) for given (x , t) can be defined as a function ofrandom variable r(ω) and interval parameter p.
Function u is a solution of a partial differential equation withrandom and interval parameters (for example E , ρ, n etc.)
∂2u
∂t2= c2∂
2u
∂x2
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WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Methods for the solution of the wave equation
The Finite Difference Method
discretization of the second order differential equationdiscretization of the first order differential equation
The Finite Element Method
weak formulationmodal analysis
Fourier Series
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WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
The Finite Difference Method
Differential equation
∂2u
∂t2= c2∂
2u
∂x2
Discretization
ui ,j−1 − 2ui ,j + ui ,j+1
∆t2= c2 ui−1,j − 2ui ,j + ui+1,j
∆x2
ui ,j+1 = 2ui ,j − ui ,j−1 +c2∆t2
∆x2(ui−1,j − 2ui ,j + ui+1,j)
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WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
The Finite Difference Method
Differential equation ∂u∂t = v∂v∂t = c2 ∂2u
∂x2
Discretizationui ,j+1 = ui ,j + vi ,j∆t
vi ,j+1 = vi ,j + c2∆t∆x2 (ui−1,j − 2ui ,j + ui+1,j)
In order to find maximum/minimum of the function u it ispossible to apply a modified version of the steepest descentalgorithm.
1 Given x0, set k = 0.
2 dk = −∇f (xk). If dk = 0 then stop.
3 Solve minαf (xk + αdk) for the step size αk . If we know
second derivative H then αk =dTk dk
dTk H(xk )dk
.
4 Set xk+1 = xk + αkdk , update k = k + 1. Go to step 1.
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WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Parameter dependent probabilistic solution
For every specific value of the interval parameters p1, ..., pm itis possible to calculate the probabilistic solutionu(x , t, ω, p1, ..., pm).Now it is possible to calculate extreme value of the probabilisticevents. For example probability of failure. Probability of failure
Pf = PΩω ∈ Ω : g(u(ω)) ≤ 0 ∈ [P f ,P f ]
Lower bound of the probability of failure
P f = minPΩω ∈ Ω : g(u(ω), p) ≤ 0 : pi ∈
[pi, pi
]Upper bound of the probability of failure
P f = maxPΩω ∈ Ω : g(u(ω), p) ≤ 0 : pi ∈
[pi, pi
]25 / 32
WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Upper and lower solution with random parameters
In the specific case when the upper and lower solution does notdepend on the combination of random parameters for everyspecific ω ∈ Ω it is possible to calculate upper and lowersolution u(x , t, ω), u(x , t, ω) and then calculate all kinds ofprobabilistic results by using for example Monte Carlosimulations. For example it is possible to calculate theprobability that umin < u < umax in the following way:
If c is a random parameter, then now it is possible to considerupper and lower probability of different events by using abovedescribed interval solution.
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WaveEquation
UncertainParameters
Methods forthe solution ofthe waveequation
The FiniteDifferenceMethod
The FiniteElement Method
Fourier Series
Equationswith intervalparameters
Equationswith randomand intervalparameters
Conclusions
Conclusions
Interval solution of the wave equation can be calculated byusing different numerical methods and appropriateoptimization algorithms. In this paper, 3 numericalmethods and one optimization method were presented.
If there are interval and random parameters it is possibleto calculate upper and lower bound of the probability ofdifferent events.
If the interval solution depend always on the samecombination of parameters, then it is possible to computeupper and lower probabilistic solution and computedifferent probabilistic events by using this interval solution.