Error Estimates of Stochastic Optimal Neumann Boundary Control Problems

Gunzburger, Max; Lee, Hyung-Chun; Lee, Jangwoon

doi:10.1137/100801731

Cited by 57 publications

(47 citation statements)

References 19 publications

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“…There are two main ways of measuring this spatial-stochastic quantity: the expected value of spatial mismatch(see e.g. [8,7] more ref's) and the spatial mismatch of averages of the statistical quantities of interest. More precisely, we consider the minimization cost functionals of the type…”

Section: A Generalized Methodology For the Solution Of Stochastic Idementioning

confidence: 99%

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Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations

Trenchea¹

2012

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Section: A Generalized Methodology For the Solution Of Stochastic Idementioning

confidence: 99%

“…[13,14,1,2,15,10,9,7]) one can prove that the problem (0.10)-(0.3) has a unique optimal pair that is characterized by a maximum principle type result.…”

Section: A Generalized Methodology For the Solution Of Stochastic Idementioning

confidence: 99%

Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations

Trenchea¹

2012

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“…In tables and figures for computational results, for simplicity, we use DP = ∑ N n=1 p n , where p n is the maximum degree of polynomials in a y n -direction and DOF as the number DP (DOF ) Relative Error for u Relative Error for ξ Relative Error for f 1 (2) 1.198032987982e-01 1.787953125199e-01 1.720684070324e-01 3 (4) 9.562740236029e-03 1.524223854302e-02 1.341233756062e-02 5 (6) 5.570898785792e-04 1.057333186124e-03 8.615989678717e-04 7 (8) 2.795461789913e-05 6.309221819675e-05 4.901139558725e-05 9 (10) 1.286833647831e-06 3.391536066446e-06 2.567358708683e-06 11 (12) 4.560494644988e-08 1.362462543327e-07 1.019093421438e-07 (2) 9.401289567930e-02 1.987307188342e-01 1.752373609462e-01 3 (6) 1.021973093216e-02 2.469886703133e-02 1.951724713049e-02 5 (12) 1.005212637175e-03 3.036986195881e-03 2.380145213483e-03 7 (20) 9.899411004937e-05 3.609536395869e-04 2.891208795498e-04 9 (30) 9.451679011477e-06 3.927211535465e-05 3.149626856263e-05 11 (42) 3.335005157516e-07 2.710535612892e-06 2.420352397715e-06 of degrees of freedom of the discretization with respect to the random parameter space. For instance, if we use p = (p 1 , p 2 ) = (5, 4), then DP = 9 and DOF = 30.…”

Section: Numerical Setting In Our Numerical Experiments We Use Thatmentioning

confidence: 99%

“…We would like to mention here that ∇ means differentiation with respect to x ∈ D only. To analyze our optimal control problem using the h × p version approach, we use the Karhunan-Loéve (KL) expansion [6,7,8,9,10,11,12,13,14,15], transform stochastic PDEs to deterministic high dimensional PDEs, and present a priori error estimates of the h × p version of the SGFEM to the transformed model equation. After that, by using the method of Lagrange multipliers, we derive the optimality system of equations.…”

Section: Introductionmentioning

confidence: 99%

THE h × p FINITE ELEMENT METHOD FOR OPTIMAL CONTROL PROBLEMS CONSTRAINED BY STOCHASTIC ELLIPTIC PDES

Lee

2015

Journal of the Korea Society for Industrial and Applied Mathema

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ABSTRACT. This paper analyzes the h × p version of the finite element method for optimal control problems constrained by elliptic partial differential equations with random inputs. The main result is that the h × p error bound for the control problems subject to stochastic partial differential equations leads to an exponential rate of convergence with respect to p as for the corresponding direct problems. Numerical examples are used to confirm the theoretical results.

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“…Error estimates of stochastic optimal Neumann boundary control problems [9] We study mathematically and computationally optimal control problems for stochastic partial differential equations with Neumann boundary conditions. The control objective is to minimize the expectation of a cost functional, and the control is of the deterministic, boundary-value type.…”

Section: Brief Descriptions Of Other Workmentioning

confidence: 99%

Advanced Numerical Methods for Computing Statistical Quantities of Interest from Solutions of SPDES

Gunzburger¹

2012

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The public reportmg burd en for th1s collection of mform ation is estimated to average 1 hour per response. includmg the tim e for rev>ewing instruct>ons. searclhing exi sting data so urces. gathering and mmnta1mng the data needed . and co mpleting and rev1ew 1 ng the collection of 1nforma t1 on. Send co mments regarding th1s burden est1m ate or any other aspect o fth1s collection of mformat1on . 1nclud1 ng suggesti ons for reduc1 ng the burden. to the Department of De fense. ExecutiV e Serv1ce D>rectorate (0704 -0 188) Respo ndents should be aware that notwithstanding any other prov1 s1 on of law. no person shall be sub;ect to any penalty for falling to comply w1th a collect> on of 1 nform at1on if 1 t does not display a currently v alid OMB co ntrol number PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Department of Scientific Computing REPORT NUMBERFlorida State University SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) AFOSR SPONSOR/MONITOR'S REPORT NUMBER(S)AFRL-OSR-VA-TR-2012-0282 DISTRIBUTION/AVAILABILITY STATEMENT A SUPPLEMENTARY NOTES ABSTRACTComputational simulation-based predictions are central to science and engineering and to risk assessment and decision making in economics, public policy, and military venues, including several of importance to Air Force missions. Unfortunately, predictions are often fraught with uncertainty so that effective means for quantifying that uncertainty are of pararuount importance. The research effort investigates and resolves important algorithmic, mathematical, and practical issues related to the efficient, accurate, and robust computational determination of the quantities of interest used by engineers and decision makers that are determined from solutions of partial differential equations. 1S. SUBJECT TERMS ADVANCED NUMERICAL METHODS FOR COMPUTING STATISTICAL QUANTITIES OF INTEREST FROM SOLUTIONS OF SPDESFINAL REPORT -FA9550-08-1-0415 Max Gunzburger Department of Scientific ComputingFlorida State University gunzburg@fsu.edu AbstractComputational simulation-based predictions are central to science and engineering and to risk assessment and decision making in economics, public policy, and military venues, including several of importance to Air Force missions. Unfortunately, predictions are often fraught with uncertainty so that effective means for quantifying that uncertainty are of paramount importance. The research effort investigates and resolves important algorithmic, mathematical, and practical issues related to the efficient, accurate, and robust computational determination of the quantities of interest used by engineers and decision makers that are determined from solutions of partial differential equations having random inputs. Notable accomplishments include the development of a novel approach to discretizing white noise and its application to the stochastic Navier-Stokes-Boussinesq system; development of efficient methods for solving high-dimensional backward stochastic differential equations;...

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Error Estimates of Stochastic Optimal Neumann Boundary Control Problems

Cited by 57 publications

References 19 publications

Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations

Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations

THE h × p FINITE ELEMENT METHOD FOR OPTIMAL CONTROL PROBLEMS CONSTRAINED BY STOCHASTIC ELLIPTIC PDES

Advanced Numerical Methods for Computing Statistical Quantities of Interest from Solutions of SPDES

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