{"paper":{"title":"Inexact Adjoint Gradients and Directional Tolerances for Full-Potential Airfoil Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Inexact adjoint gradients satisfy exact descent inequalities under directional tolerances derived from residual bounds.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Humberto Gimenes Macedo, Lu\\'is Felipe Bueno","submitted_at":"2026-05-17T18:52:16Z","abstract_excerpt":"This paper develops a framework connecting discrete adjoint gradient-error analysis with an optimization method that uses directional error tolerances, and applies it to airfoil shape optimization governed by a conservative full-potential flow solver on body-fitted structured meshes. The theoretical part derives the reduced discrete adjoint formula for scalar objectives constrained by a state equation and analyzes how inexact state and adjoint residuals propagate into the reduced gradient. For residuals that are affine in the state variable, the gradient error is bounded by a linear combinatio"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"On compact sets of decision variables, a uniform version of this bound is obtained, leading to a directional tolerance condition under which the inexact gradient satisfies an exact descent inequality. The resulting inexact general directions method inherits convergence properties under uniformly bounded, diminishing, and Armijo-type step-size rules.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The gradient-error bound and subsequent directional tolerance hold when the state and adjoint residuals are affine in the state variable; this assumption is invoked to obtain the linear combination bound on gradient error (see theoretical part of abstract).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Connects inexact discrete adjoint gradient bounds to directional tolerances that guarantee descent and convergence in full-potential airfoil pressure-matching optimization.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Inexact adjoint gradients satisfy exact descent inequalities under directional tolerances derived from residual bounds.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5830b153fc7438e7ea29d16fb5f02d0af9fcab41753905f35b6e86066e00b948"},"source":{"id":"2605.17599","kind":"arxiv","version":1},"verdict":{"id":"af958ac3-3e81-4705-9cbe-58359c36920b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:17:35.607333Z","strongest_claim":"On compact sets of decision variables, a uniform version of this bound is obtained, leading to a directional tolerance condition under which the inexact gradient satisfies an exact descent inequality. The resulting inexact general directions method inherits convergence properties under uniformly bounded, diminishing, and Armijo-type step-size rules.","one_line_summary":"Connects inexact discrete adjoint gradient bounds to directional tolerances that guarantee descent and convergence in full-potential airfoil pressure-matching optimization.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The gradient-error bound and subsequent directional tolerance hold when the state and adjoint residuals are affine in the state variable; this assumption is invoked to obtain the linear combination bound on gradient error (see theoretical part of abstract).","pith_extraction_headline":"Inexact adjoint gradients satisfy exact descent inequalities under directional tolerances derived from residual bounds."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17599/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.540901Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:21:37.848991Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.576390Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:21:57.505975Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"7fb711ff36ef0a3388930b8078e5d8025c4bc44f8918dbb390ceba71507967a6"},"references":{"count":7,"sample":[{"doi":"","year":2016,"title":"D. 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