LSDO Lab
Methods

Graph-accelerated MDO under uncertainty

Context
Design under uncertainty requires repeated model evaluations to estimate how uncertain inputs affect performance and constraints.
Gap
Tensor-product quadrature grows exponentially with input dimension, and conventional evaluations repeat operations that depend on only a subset of inputs.
Research goal
Reduce uncertainty-propagation cost by combining computational graph structure with quadrature and sensitivity-based dimension reduction.

Selected studies

References
Tensor-grid graph accelerationGraph-informed quadratureGradient-enhanced dimension reductionActive-subspace graph acceleration

Reusing operations across uncertainty samples

AMTC partitions a model's computational graph by dependence on uncertain inputs and reuses repeated operations across tensor-grid samples. The piston, aircraft, and air-taxi benchmarks retain the same polynomial-chaos quadrature points and estimates while reducing model-evaluation cost through this reuse.

Primary paper

Bingran Wang, Mark Sperry, Victor E. Gandarillas, John T. Hwang. Accelerating Model Evaluations in Uncertainty Propagation on Tensor Grids Using Computational Graph Transformations. Aerospace Science and Technology 145 (2024), p. 108843, 2024.

PDF
Computational graphs before and after AMTC show the factorization of a full tensor-grid evaluation into reusable subgraphs.
AMTC factors a full tensor-grid evaluation into subgraphs that depend on subsets of the uncertain inputs, allowing repeated operations to be evaluated once and reused. Fig. 2, Wang et al., 2024 [1]
Piston-model error is plotted against equivalent model-evaluation cost for Monte Carlo, Kriging, designed quadrature, full-grid NIPC, and AMTC-accelerated full-grid NIPC.
On the piston benchmark, AMTC-accelerated full-grid NIPC reaches the same polynomial-chaos estimate with substantially fewer equivalent model evaluations than ordinary full-grid NIPC. Fig. 4, Wang et al., 2024 [1]

Matching quadrature structure to model dependencies

Graph dependencies determine which uncertain inputs to group in a partially tensor-structured quadrature rule. Coupled with AMTC, these rules reduce cost relative to full-grid and unstructured designed quadrature in the four- and six-dimensional aircraft tests; the advantage depends on exploitable graph sparsity and the required accuracy.

Primary paper

Bingran Wang, Nicholas C. Orndorff, John T. Hwang. Graph-accelerated non-intrusive polynomial chaos expansion using partially tensor-structured quadrature rules for uncertainty quantification. Aerospace Science and Technology, 2024.

DOIPDF
Three three-dimensional point sets illustrate fully tensor-structured, partially tensor-structured, and unstructured quadrature rules.
Partially tensor-structured quadrature preserves tensor-product structure within selected input groups while using fewer points than a full tensor grid. Fig. 2, Wang et al., 2024 [2]
Relative error is plotted against equivalent model-evaluation cost for designed quadrature, full-grid NIPC, and partially tensor-structured NIPC.
With AMTC, tailored quadrature reduces cost for most tested error levels in the air-taxi problem; without AMTC, designed quadrature performs better. Fig. 9, Wang et al., 2024 [2]

Improving uncertainty estimates with response gradients

GUDR adds univariate gradient terms to ordinary univariate dimension reduction to approximate interactions between uncertain inputs. The rotor and aircraft tests improve standard-deviation estimates by about an order of magnitude. Linear cost scaling requires efficient automatic differentiation, including second derivatives.

Primary paper

Bingran Wang, Nicholas C. Orndorff, Mark Sperry, John T. Hwang. A gradient-enhanced univariate dimension reduction method for uncertainty propagation. Aerospace Science and Technology, 2024.

DOIPDF
Computational graph of the gradient-enhanced univariate dimension-reduction approximation shows function and derivative evaluations for each uncertain input.
GUDR combines univariate function approximations with their derivatives before assembling the reduced uncertainty-propagation approximation. Fig. 2, Wang et al., 2024 [3]
Relative error is plotted against equivalent function-evaluation cost for Monte Carlo, NIPC, univariate dimension reduction, and gradient-enhanced univariate dimension reduction.
GUDR improves standard-deviation estimates over UDR in the aircraft-design test, with a modest increase in evaluation cost. Fig. 12, Wang et al., 2024 [3]

Extending graph acceleration to many uncertain inputs

AS-AMTC constructs polynomial-chaos approximations in active variables and arranges quadrature points so graph operations can still be reused. An air-taxi uncertainty problem with 81 inputs demonstrates improved accuracy over the compared active-subspace estimators, while retaining the approximation error introduced by dimension reduction.

Primary paper

Bingran Wang, Nicholas C. Orndorff, Mark Sperry, John T. Hwang. Extension of graph-accelerated non-intrusive polynomial chaos to high-dimensional uncertainty quantification through the active subspace method. Aerospace Science and Technology, 2025.

DOIPDF
Relative error is plotted against equivalent function-evaluation cost for Monte Carlo and active-subspace Kriging, NIPC, and AMTC methods.
In the high-dimensional air-taxi test, AS-AMTC combines active-subspace reduction with graph acceleration and attains lower error at comparable evaluation cost than the other active-subspace estimators. Fig. 7, Wang et al., 2025 [4]

References

  1. Bingran Wang, Mark Sperry, Victor E. Gandarillas, John T. Hwang. Accelerating Model Evaluations in Uncertainty Propagation on Tensor Grids Using Computational Graph Transformations. Aerospace Science and Technology 145 (2024), p. 108843, 2024.
    PDF
  2. Bingran Wang, Nicholas C. Orndorff, John T. Hwang. Graph-accelerated non-intrusive polynomial chaos expansion using partially tensor-structured quadrature rules for uncertainty quantification. Aerospace Science and Technology, 2024.
    DOIPDF
  3. Bingran Wang, Nicholas C. Orndorff, Mark Sperry, John T. Hwang. A gradient-enhanced univariate dimension reduction method for uncertainty propagation. Aerospace Science and Technology, 2024.
    DOIPDF
  4. Bingran Wang, Nicholas C. Orndorff, Mark Sperry, John T. Hwang. Extension of graph-accelerated non-intrusive polynomial chaos to high-dimensional uncertainty quantification through the active subspace method. Aerospace Science and Technology, 2025.
    DOIPDF
  5. Tae H. Ha, Keunseok Lee, John T. Hwang. Large-scale multidisciplinary optimization under uncertainty for electric vertical takeoff and landing aircraft. AIAA Scitech 2020 Forum, 2020.
    DOIPDF
  6. Luca Scotzniovsky, John T. Hwang. A Fast, Memory-Efficient Panel Method for Large-Scale Multidisciplinary Design Optimization Under Uncertainty Using Graph-Based Modeling. AIAA AVIATION 2025 Forum, 2025.
    PDF
  7. Bingran Wang, Nicholas C. Orndorff, Anugrah J. Joshy, John T. Hwang. Graph-accelerated large-scale multidisciplinary design optimization under uncertainty of a laser-beam-powered aircraft. AIAA SCITECH 2024 Forum, 2024.
    DOIPDF
  8. Bingran Wang, Marius L. Ruh, Aoran Tian, Luca Scotzniovsky, John T. Hwang. Large-scale MDO under uncertainty of an eVTOL aircraft using dimension reduction via global sensitivity analysis. AIAA AVIATION FORUM AND ASCEND 2025, 2025.
    DOIPDF

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