Alphabeta MathNumerical Optimization Inverse Problems and Uncertainty Quantification
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
11 pagesA computational graph here is a finite, topologically ordered real-arithmetic program with differentiable primitive maps and a scalar output.
1 definition, 5 algorithms, 4 lemmas, 8 theorems, 1 remarkExamples & counterexamples →Finite-basis electronic-structure models can be formulated as optimization of a real energy over rank-r Hermitian projectors.
3 definitions, 8 lemmas, 2 propositions, 6 theoremsExamples & counterexamples →For a finite independent product input and a square-integrable real output, Independent product inputs have an orthogonal Hoeffding decomposition constructs orthogonal Hoeffding ANOVA components with variance sum.
4 definitions, 3 algorithms, 2 lemmas, 7 theoremsExamples & counterexamples →We fix the inequality convention g i(x)≤ 0 and the Lagrangian L(x,ν,λ)=f(x)+ν^T h(x)+λ^Tg(x) with λ≥0.
3 definitions, 2 lemmas, 1 proposition, 11 theoremsExamples & counterexamples →- Nonlinear equations and Newton methods12 results
For an open U⊆ℝⁿ with n≥1 and a square residual F:U→ℝⁿ, Newton's correction solves DF(x)s=-F(x) and updates x by s.
6 algorithms, 4 lemmas, 2 theoremsExamples & counterexamples → This page defines forward and central difference stencils and proves their truncation and bounded-evaluation-error bounds.
5 definitions, 1 lemma, 6 theoremsExamples & counterexamples →This page separates the main error sources in regularized inverse problems.
4 lemmas, 1 theorem, 1 corollaryExamples & counterexamples →This page extends the canonical gradient and damped-Newton methods with global convex guarantees.
1 algorithm, 1 lemma, 4 theoremsExamples & counterexamples →The input is a finite family of independent real random variables with a specified product law and an explicit admissible moment cutoff.
7 definitions, 1 algorithm, 4 lemmas, 9 theorems, 3 corollaries, 1 remarkExamples & counterexamples →This page separates rare-event proposal quality from the validity of a particular estimator.
3 definitions, 3 algorithms, 5 lemmas, 7 theorems, 1 counterexample, 1 remarkExamples & counterexamples →Quadratic trust-region models and the actual-to-predicted reduction ratio fixes the Euclidean model, feasible ball, reductions, and ratio.
2 definitions, 5 algorithms, 8 lemmas, 3 theoremsExamples & counterexamples →
Level 1
11 pagesDevelop log barriers and central paths for smooth convex inequalities and conic programs; prove the exact central-point primal-dual barrier gaps m/t and ν/t together with the…
3 definitions, 2 algorithms, 6 lemmas, 5 theoremsExamples & counterexamples →This page separates derivatives of a finite computational graph from derivatives of the exact ODE flow.
3 definitions, 6 lemmas, 9 theorems, 1 corollaryExamples & counterexamples →For a C² residual map r:U⊆ℝⁿ→ℝᵐ, the objective φ=‖r‖ 2²/2 has gradient J^Tr and Hessian J^TJ+∑ i r i∇²r i.
2 algorithms, 3 lemmas, 4 theoremsExamples & counterexamples →This page develops certified global search on compact boxes. Its spatial branch-and-bound procedure records valid lower bounds, verified feasible upper bounds, empty-box…
4 definitions, 7 algorithms, 2 lemmas, 10 theoremsExamples & counterexamples →This page defines a regular minimum-energy path on a supplied smooth potential domain and finite projected-gradient string and improved-tangent NEB procedures with explicit…
2 definitions, 4 algorithms, 5 lemmas, 3 propositions, 4 theoremsExamples & counterexamples →This page defines weak and strong Wolfe conditions, proves existence on a bounded-below descent ray, and derives the angle-weighted Zoutendijk sum.
1 definition, 2 algorithms, 5 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →For the composite objective F=f+g, the page assumes a convex continuously differentiable L-smooth term f and a proper closed convex extended-real term g.
6 definitions, 2 algorithms, 10 lemmas, 7 theoremsExamples & counterexamples →This page develops the revised simplex method for finite linear programs in standard form.
2 definitions, 5 algorithms, 1 lemma, 1 proposition, 6 theoremsExamples & counterexamples →This page develops first-order geometry and optimization for real orthonormal frames and their Grassmann quotient, plus the complex Grassmann geometry needed for Hermitian projector models.
5 definitions, 2 algorithms, 1 lemma, 5 propositions, 7 theoremsExamples & counterexamples →This page defines the self-consistent occupied-projector map only where a strict occupied/unoccupied spectral gap makes the rank-r projector unique.
2 definitions, 4 algorithms, 7 lemmas, 6 theorems, 1 corollaryExamples & counterexamples →This page develops compact-operator Tikhonov regularization from its spectral filter through source-condition rates and Morozov discrepancy selection.
1 definition, 3 algorithms, 7 lemmas, 6 theoremsExamples & counterexamples →
Level 2
8 pagesThis page defines full-memory BFGS Hessian and inverse updates, their secant and positive-definiteness properties, and the Wolfe curvature condition that makes the update well-defined.
1 definition, 3 algorithms, 8 lemmas, 3 theoremsExamples & counterexamples →- Coordinate and block descent methods17 results
This page defines randomized coordinate-gradient and proximal updates, cyclic exact block minimization, and the Lasso coordinate formula.
3 algorithms, 7 lemmas, 4 theorems, 3 counterexamplesExamples & counterexamples → Landweber iteration is given with an explicit discrepancy and finite-budget status; its spectral-filter identities yield exact-data convergence, source bias and noise bounds, then a source-conditioned discrepancy-stopping rate.
4 algorithms, 5 lemmas, 3 theorems, 1 corollaryExamples & counterexamples →- Nonlinear conjugate gradient methods7 results
This page defines finite-budget Fletcher–Reeves and safeguarded PR+ methods, including zero-gradient exit, restart, and search-failure statuses.
1 algorithm, 2 lemmas, 4 theoremsExamples & counterexamples → This page defines finite nonlinear observation designs, local and global identifiability, stacked sensitivities, and weighted information.
1 definition, 1 algorithm, 3 lemmas, 6 theorems, 1 remarkExamples & counterexamples →This page develops three distinct finite-dimensional methods. Douglas–Rachford splitting is treated for two maximal-monotone relations; its resolvent theorem is proved…
4 definitions, 4 algorithms, 9 lemmas, 4 theoremsExamples & counterexamples →This page defines filtration-adapted Robbins–Monro, SGD, and projected SGD, with conditional unbiasedness, bias, variance, and moment hypotheses stated separately.
2 definitions, 2 lemmas, 6 theoremsExamples & counterexamples →This page develops projected subgradient, proximal bundle, direct-search, and Nelder–Mead procedures while keeping finite-budget statuses separate from convergence guarantees.
1 definition, 4 algorithms, 11 lemmas, 3 theorems, 1 corollaryExamples & counterexamples →
Level 3
3 pagesSimulation objectives are expectations of random payoffs. A pathwise estimator differentiates a fixed-noise representation; it is unbiased when a common integrable derivative…
3 definitions, 1 algorithm, 1 proposition, 6 theorems, 1 counterexampleExamples & counterexamples →This pair uses g i≤0, λ i≥0, and L=f+ν^Th+λ^Tg, matching Inequality constrained programs, tangent cones, constraint qualifications, and KKT.
3 definitions, 8 algorithms, 2 lemmas, 8 theorems, 1 corollaryExamples & counterexamples →For a finite sum F=n⁻¹∑ i f i, this page defines SVRG with an independent uniform component index and random inner-iterate output, and SAGA with an independently sampled index and stored component gradients.
5 algorithms, 7 lemmas, 5 theorems, 2 counterexamplesExamples & counterexamples →
Level 4
2 pagesThis page separates a measurable-control existence model from the pointwise smooth-data branch required to evaluate Runge–Kutta stages.
5 definitions, 3 algorithms, 10 lemmas, 5 theorems, 1 counterexampleExamples & counterexamples →For a bounded coercive linear elliptic state equation with distributed L² controls, this page defines a closed pointwise box and a quadratic tracking objective with regularization α‖q‖ L²²/2, α 0.
2 definitions, 3 algorithms, 8 lemmas, 9 theorems, 1 corollaryExamples & counterexamples →