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Stochastic Wave Dynamics and Uncertainty Quantification · Method in space. I Multigrid Preconditioned Defect Correction (PDC) Method for efficient and scalable iterative solution

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  • General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

    Users may download and print one copy of any publication from the public portal for the purpose of private study or research.

    You may not further distribute the material or use it for any profit-making activity or commercial gain

    You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

    Downloaded from orbit.dtu.dk on: Mar 31, 2021

    Stochastic Wave Dynamics and Uncertainty Quantification

    Engsig-Karup, Allan Peter; Bigoni, Daniele; Glimberg, Stefan Lemvig

    Publication date:2013

    Document VersionPublisher's PDF, also known as Version of record

    Link back to DTU Orbit

    Citation (APA):Engsig-Karup, A. P., Bigoni, D., & Glimberg, S. L. (2013). Stochastic Wave Dynamics and UncertaintyQuantification. Poster session presented at 38th Woudschoten Conference, Zeist, Netherlands.

    https://orbit.dtu.dk/en/publications/bd73ddf9-bbd7-4109-8df7-446bc5fd8342

  • Stochastic Wave Dynamicsand Uncertainty Quantification

    Allan P. Engsig-Karup, Daniele Bigoni and Stefan L. GlimbergTechnical University of Denmark (DTU)

    MotivationTo address challenges in reliable prediction of extremeevents for the design and safe operation of marinesystems, simulation-based engineering tools can beused. Such tools are increasingly cost-efficient and canbe used to quantify uncertainties and evaluate impacthereof in critical engineering design problems wheremeasurements are infeasible, impractical or too costly.To characterise uncertainties in wave dynamics andwave-structure responses our objective is to considermodern spectral techniques for uncertainty quantifica-tion to describe stochastic properties as accurately aspossible in practical times. The spectral techniques pro-vides the basis for meeting the accuracy requirementsince these techniques may achieve much faster conver-gence rates than conventional techniques. To make ourapproaches practical we seek to combine knowledge inmodern algorithms and many-core hardware technolo-gies in a framework to enable efficient stochastic hydro-dynamics calculations. Our scope is relevant for com-putationally intensive (fx. large scale problems) wheremany simulations are intractable by conventional tech-niques and approaches.

    Case studiesIn a first preliminary step, we revisit some classicalbenchmarks for applications and investigate feasibilityof using spectral techniques for quantification of uncer-tainty in wave dynamics. Possible uncertainty sourcesare assumed to beI Boundary conditions (wave generation signal, e.g.

    wave period and wave amplitude).I Bathymetry function (sea bed), h(x, ω).

    ContributionsI Stochastic formulation of fully nonlinear and

    dispersive wave equations.I Investigation of spectral uncertainty quantification

    techniques.I Integration of our research in fast hydrodynamics

    simulations with spectral uncertainty quantificationtechniques.

    Figure 1 : Snap shot of deterministic wave fieldproduced by Berkhoff experiment.

    Deterministic formulationTo describe nonbreaking irrotational ocean waves, a fullynonlinear and dispersive water wave is used. Dynamicand kinematic free surface boundary conditions are

    ∂tζ = −∇ζ · ∇φ̃+ w̃(1 +∇ζ · ∇ζ),

    ∂tφ̃ = −gζ −1

    2

    (∇φ̃ · ∇φ̃− w̃2(1 +∇ζ · ∇ζ)

    ),

    where φ̃ = φ(x, ζ, t), ζ(x, t) and w̃ = ∂zφ|z=ζ arefree surface quantities and g gravitational acceleration.A Laplace problem needs to be solved

    φ = φ̃, z = ζ(x, t),

    ∇2φ+ ∂zzφ = 0, −h ≤ z < ζ(x, t),∂zφ+∇h · ∇φ = 0, z = −h.

    from which closure is obtained by (u, w) = (∇, ∂z)φ.

    Stochastic formulationTo enable quantification of uncertainties, a stochasticformulation is obtained by introducing ω ∈ Ω as ran-dom input of the system defined in the probability space(Ω,F ,P), where Ω is the sample space, F is a σ-fieldand P is a probability measure. This makes the solutiona random quantity ζ(x, t, ω) : D̄FS × [0, T ] × Ω → Rand φ(x, t, ω) : D̄ × [0, T ] × Ω → R. D̄ is the closedspatial domain volume with FS indicating the restrictionto the free surface, D̄ = {x|x ∈ ξ}. A parametrizationof the stochastic model is required in order to solve it nu-merically. A random vector Z : Ω→ Rd, is introduced tocharacterise random inputs, where d ≥ 1 the stochasticdimension. The stochastic formulation is then

    ∂tζ(x, t,Z) = −∇ζ · ∇φ̃+ w̃(1 +∇ζ · ∇ζ),

    ∂tφ̃(x, t,Z) = −gζ −1

    2

    (∇φ̃ · ∇φ̃− w̃2(1 +∇ζ · ∇ζ)

    ),

    where for any (random) sea state, the Laplace problemis fulfilled to obtain closure. This is a stochastic systemwhere unknown variables are random processes.

    Generalized Polynomial ChaosWe use generalized Polynomial Chaos (gPC) to createsurrogate functions of stochastic variables of the form

    f(z) ≈ f̃(z) = PNf(z) =N∑i=0

    f̂iΦi(z), f̂i =(f,Φi)ρz‖Φi‖ρz

    .

    From these, cheap and exponentially accurate statisticsfor uncertainty quantification can be obtained

    E[f(z)] ≈ E[f̃(z)] = f̂0,

    Var[f(z)] ≈ Var[f̃(z)] =N∑i=1

    f̂2i ‖Φi‖2ρz.

    The unknown gPC expansion coefficients are deter-mined from a solution ensemble by forward propagationof uncertainties via a stochastic collocation method.

    Massively Parallel ComputingTo enable fast analysis and resolution of large mar-itime areas, we take advantage of distributed massivelyparallel high-performance and heterogenous computingon modern many-core hardware. A massively paral-lel solver has been prototyped in our in-house GPU-Lab library and can be used for stochastic wave dy-namics calculations. The fast model enable accelerationof sampling-based (non-intrusive) UQ algorithms in thefield of study.

    Figure 2 : Absolute run timings in single precision forheterogenous multi-GPU configurations as a function ofnumber of grid points for iterative PDC method.I Efficient multigrid Preconditioned Defect Correction

    (PDC) method for arbitrary-order discretizations.I Minimal memory requirements via short recurrence

    iterative PDC method, matrix-free stencilsimplementations of sparse operators and single ormixed-precision calculations.

    I Fast massively parallel execution on hardwaresystems of arbitrary size ranging from desktops tosuper clusters via hybrid MPI-CUDA.

    I Fault tolerance and resilience via robust multileveliterative methods.

    I Predictable and scalable performance.

    Discretization MethodsTo develop a numerical model we useI Tuneable numerics provides tradeoffs between

    accuracy and efficiency.I A flexible-order boundary-fitted Finite Difference

    Method in space.I Multigrid Preconditioned Defect Correction (PDC)

    Method for efficient and scalable iterative solution ofLaplace problem every Runge-Kutta stage.

    I Data-parallel domain decomposition methodimplemented for distributed computations.

    For the time discretization we useI An explicit fourth-order Runge-Kutta method in time.

    Due to bounded operator eigenspectra conditionalCFL stability without strict step size penalisation dueto high-order numerics and/or refined grids.

    I A parallel in time (Parareal) discretization to introducealgorithmic concurrency.

    For the stochastic discretization we useI A spectral stochastic collocation method for forward

    propagation of parametric uncertainty in input data.

    Numerical results

    Figure 3 : Uncertainty quantification of harmonicscontributions to the steady one-wave period solution inWhalin experiment (T = 2s) with respect to waveheight and wave period. The shaded areas show onestandard deviation from the mean (full lines).

    PerspectivesI Promising practical aspects for spectral uncertainty

    quantification techniques in maritime engineering forlow-dimensional stochastic problem.

    I Speedup solutions via parallel computations tosignificantly improve analysis in practical times (butdoes not resolve curse of dimensionality).

    I Stochastic simulation and uncertainty quantificationbecoming increasingly important for reliable analysisof impact of uncertainties on engineering designs.

    I Next steps: better predictions of wave statistics inlarge (near-costal) finite depths areas and reliableestimations of extreme events.

    ReferencesI Engsig-Karup, A. P., Glimberg, L. S., Nielsen, A. S.

    and Lindberg, O. 2013. Fast hydrodynamics onheterogenous many-core hardware. Part of: RaphäelCouturier (Ed). Designing Scientific Applications onGPUs, 2013, CRC Press / Taylor & Francis Group.

    I Engsig-Karup, A. P., Madsen, M. G. and Glimberg, S.L. A massively parallel GPU-accelerated model foranalysis of fully nonlinear free surface waves.E-published in International Journal for NumericalMethods in Fluids, July, 2011.

    I Engsig-Karup, A.P., Bingham, H.B. and Lindberg, O.2009 An efficient flexible-order model for 3D nonlinearwater waves. Journal of Computational Physics, 288,pp. 2100–2118.

    Contacts: Allan: [email protected], Daniele: [email protected], Stefan: [email protected]