Verification in Scientific Computing: from Pristine to Practical to Perimeter-Extending Joseph M. Powers Department of Aerospace and Mechanical Engineering University of Notre Dame ASME V&V 2020 Virtual Verification and Validation Symposium 20 May 2020
<latexit sha1_base64="oMI0ewaJRa5wigy3zNytjNT9H+c=">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</latexit> Outline • Signal v. Noise: first resolve the physics, then verify! sZ • Pristine: convergence, asymptotic convergence ( y a − y e ) 2 dx L 2 = || y a − y e || 2 = rates, multi-scale physics. • Practical: scarce computational resources, error di ffi cult to define, what should referees expect. • Perimeter-Extending: nonlinear dynamics, transition to chaos. • Focus on continuum calculus-based models of reacting fluid dynamics. Henrick, 2008 J. M. Powers ASME V&V 2020 20 May 2020 2
Verification v. Validation • Verification: solving the equations right. • Validation: solving the right equations. • Pat Roache informed me in 1990 I was doing verification. (I was, but didn’t know it.) • Seemed unnecessary. • I was wrong. The need exists. • Widespread misunderstanding of V&V. • Getting it right is important! • Focus here is solution verification: ASME V&V20: “Estimates the numerical accuracy of a particular calculation.” Roache, 2009 J. M. Powers ASME V&V 2020 20 May 2020 3
<latexit sha1_base64="Frf4ipkqnK3AsdsCDN5DIeLvNU=">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</latexit> Verification and Calculus y = x 2 • Cathartic moment in 1978 dy dx = 2 x when I saw a finite � di ff erence estimation of the dy � = 2 � dx derivative approached the � x =1 y ( x + ∆ x ) − y ( x ) dy prediction given by dx = lim ∆ x ∆ x → 0 Newton’s calculus begun in 1665. • Getting the low order estimate “right” is important! • We can (and should) do high order corrections later! Powers and Sen, 2015 J. M. Powers ASME V&V 2020 20 May 2020 4
Contentions • Getting a prediction that resolves the modeled physics is ultimately the most important. • This is often not achieved. • Low order methods, with appropriate resolution, can get the “signal.” • Once this “signal” has been identified, one can and should verify it. (“ h -refinement”). • Once this “signal” has been identified, high order methods may be used for enhanced accuracy and e ffi ciency (“ p -refinement”). J. M. Powers ASME V&V 2020 20 May 2020 5
<latexit sha1_base64="1tTELsfo9Ow5Oobx/CQRMWlzfA=">AB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkoignrRoh48VrAf0ISy2W7apZtN2J0INfSXePGgiFd/ijd/ije3aQ/a+mDg8d4M/OCRHANjvNlFZaWV1bXiuljc2t7bK9s9vUcaoa9BYxKodEM0El6wBHARrJ4qRKBCsFQyvJ37rgSnNY3kPo4T5EelLHnJKwEhdu+zdMAEA7AHpC0a1ecqpMDLxJ3RiqX32GOetf+9HoxTSMmgQqidcd1EvAzoBTwcYlL9UsIXRI+qxjqCQR036WHz7Gh0bp4TBWpiTgXP09kZFI61EUmM6IwEDPexPxP6+TQnjmZ1wmKTBJp4vCVGCI8SQF3OKURAjQwhV3NyK6YAoQsFkVTIhuPMvL5LmcdU9qZ7fOZXaFZqiPbRATpCLjpFNXSL6qiBKErRE3pBr9aj9Wy9We/T1oI1m9lDf2B9/ABA1pXj</latexit> <latexit sha1_base64="Zwc5ZiM0RseNTe+ULQK60BlzdyA=">AB+3icbVDLSsNAFJ3UV42vWJduBovgqiQiqAuxqAuXFewDmlAm0k7dCYJMzdiKf0VN10o4tZvcO9G/Bunj4W2HrhwOde7r0nTAX4LrfVm5peWV1Lb9ub2xube84u4WaTjJFWZUmIlGNkGgmeMyqwEGwRqoYkaFg9bB3PfbrD0xpnsT30E9ZIEkn5hGnBIzUcgr+DRNAMGBfc4l9IFnLKboldwK8SLwZKV5+2Bfp6MutJxPv53QTLIYqCBaNz03hWBAFHAq2ND2M81SQnukw5qGxkQyHQwmtw/xoVHaOEqUqRjwRP09MSBS674MTack0NXz3lj8z2tmEJ0FAx6nGbCYThdFmcCQ4HEQuM0VoyD6hCquLkV0y5RhIKJyzYhePMvL5Lack7KZ3fucXyFZoij/bRATpCHjpFZXSLKqiKHpET+gZvVhDa2S9Wm/T1pw1m9lDf2C9/wBm/ZcN</latexit> <latexit sha1_base64="xcUPKjf/yfAe+8T3ORkAQdvSjU=">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</latexit> <latexit sha1_base64="fHMvt/tNHwU8nWhFZwdyuAHJpo=">AB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkoigqCRT14rGA/oAls920SzebsDsRaugv8eJBEa/+FG/+FG9u0x609cHA470ZuYFieAaHOfLKiwtr6yuFdLG5tb2V7Z7ep41R1qCxiFU7IJoJLlkDOAjWThQjUSBYKxheT/zWA1Oax/IeRgnzI9KXPOSUgJG6dtm7YQIBnyBPSBp164VScHXiTujFQuv8Mc9a796fVimkZMAhVE647rJOBnRAGngo1LXqpZQuiQ9FnHUEkipv0sP3yMD43Sw2GsTEnAufp7IiOR1qMoMJ0RgYGe9ybif14nhfDMz7hMUmCSTheFqcAQ40kKuMcVoyBGhCquLkV0wFRhILJqmRCcOdfXiTN46p7Uj2/cyq1KzRFEe2jA3SEXHSKaugW1VEDUZSiJ/SCXq1H69l6s96nrQVrNrOH/sD6+AE9xJXh</latexit> Convergence of the Forward Euler Method dy = y (0) = 1 − y, dt e − t = y noise relaxation time constant , τ ∼ 1 , = n = 1 , . . . , N y n − ∆ ty n , y n +1 || y N − e − N ∆ t || . = error • : Solution not captured. ∆ t > τ signal • : Solution captured. ∆ t ∼ τ noise • : Solution expensively captured. ∆ t < τ J. M. Powers ASME V&V 2020 20 May 2020 6
Signal v. Noise • First order of business: tune to the signal to steer clear of the noise. • Getting the low order estimate “right” is important! • A simple AM radio, tuned to the station, conveys the signal with some noise. • A sophisticated FM radio, still properly tuned, conveys the signal with less noise. Silver, 2012 J. M. Powers ASME V&V 2020 20 May 2020 7
Signal v. Noise in Computational Simulation Henrick, 2008 Signal or Noise? Signal Signal J. M. Powers ASME V&V 2020 20 May 2020 8
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