High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number …
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48 results for “Fully Nonlinear PDEs”
Paper develops techniques to solve complex PDEs involving higher cohomology forms.
problem Develop PDE techniques to study real (p, p) forms on Hermitian manifolds.
method Parabolic approach to establish existence of classical solutions.
result Existence of classical solutions for a large class of fully nonlinear equations.
We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient…
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
New machine learning methods solve complex PDEs with improved accuracy.
problem Solving fully nonlinear PDEs with convex Hamiltonian.
method Rewriting PDE in dual stochastic control form, estimating optimal feedback control with neural network, approximating value function with neural networks.
result Improved estimation of PDE solution and its derivatives, especially the second derivative.
Anisotropic minimal graphs over half-spaces are flat.
problem Characterizing minimal graphs over half-spaces.
method Maximum principle and fully nonlinear PDE theory.
result Anisotropic minimal graphs over half-spaces are flat.
The paper solves curvature measure problem in hyperbolic space.
problem Prescribed curvature measure problem in hyperbolic space.
method Establishing C^2 regularity estimates for solutions to fully nonlinear PDE.
result Existence of star-shaped k-convex bodies with prescribed curvature measures.
Interdisciplinary study linking potential theory and elliptic PDEs.
problem Understanding solutions to nonlinear elliptic PDEs.
method Combining geometric and potential theory approaches.
result Validity of comparison principle and existence/uniqueness of solutions.
In this note, we extend our previous work on the inverse problem. Inverse problem is a fully nonlinear geometric PDE on compact Kähler manifolds. Given a proper geometric condition, we prove that a large family of nonlinear geometric flows converges to the desired solution of the given PDE.
D2SRM solves complex PDEs using deep learning.
problem High-dimensional, Hessian-dependent fully nonlinear parabolic PDEs.
method Single scalar space-time network generating derivative-consistent approximations trained through residuals and penalties.
result Well-posedness and convergence theory established for globally Lipschitz equations.
Uniform estimates for complex equations on compact manifolds found.
problem Uniform estimates for form fully nonlinear PDEs on compact Hermitian manifolds.
method Local comparison with Monge-Ampère equations and finding an appropriate elliptic operator.
result A priori estimate for the equations.
Estimates for complex equations on manifolds derived from a conjecture.
problem Estimating solutions to complex equations on Hermitian manifolds.
method Developed second order estimates for fully nonlinear elliptic equations with gradient terms.
result Derived global estimates for an equation related to Gauduchon's conjecture.
Uniform bounds for complex equations using Monge-Ampère method.
problem Bounding solutions to complex equations.
method Auxiliary Monge-Ampère equation method.
result Uniform bounds remain valid even as background metrics degenerate.
Paper uses neural nets for financial optimization problems.
problem Financial optimization and derivative pricing problems.
method Neural networks and deep reinforcement learning for solving PDEs and dynamic optimization.
result Efficient resolution of nonlinear PDEs and dynamic optimization in finance.
This paper classifies the symmetry groups of a specific Monge-Ampère equation.
problem Classifying the symmetry groups of a specific Monge-Ampère equation.
method Developed Lie theory for fully nonlinear PDEs in Convex Geometry.
result Completely classified the symmetric groups of the -Minkowski problem.
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
In this paper, we present an initial attempt to learn evolution PDEs from data. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to …
Physics-informed deep learning for PDEs solves forward and inverse problems efficiently.
problem Solving forward and inverse problems in parametric PDEs efficiently and accurately.
method Physics-informed deep latent variable model (PDDLVM) combining deep neural networks, probabilistic modelling, and variational inference.
result Achieves up to three orders of magnitude speed-up compared to traditional FEM while providing coherent uncertainty estimates.
The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.
problem Characterizing equilibrium strategies and value functions for time-inconsistent stochastic control problems.
method Method of continuity and Banach's fixed point arguments, with Schauder prior estimates.
result Global well-posedness of nonlocal fully nonlinear PDEs with sharp a-priori estimates.
We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqu…
The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often criti…
GrADE: A graph based data-driven solver for time-dependent nonlinear partial differential equationsstat.ML
GrADE uses graph neural networks and Neural ODE for solving time-dependent nonlinear PDEs efficiently.
problem Solving time-dependent nonlinear PDEs is computationally challenging and time-consuming.
method GrADE combines graph neural networks for spatial modeling and Neural ODE for temporal modeling, using attention mechanisms.
result GrADE efficiently solves PDEs, demonstrating scalability and better accuracy compared to existing methods.
We prove an existence result for a "generalised" Monge-Ampère equation introduced earlier under some assumptions on a flat complex 3-torus. As an application we prove the existence of Chern connections on certain kinds of holomorphic vector bundles on complex 3-tori whose top Chern character forms are given representat…
We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy…
Sharp estimates for non-Kähler manifolds' PDEs are derived.
problem Sharp estimates for fully nonlinear PDEs on non-Kähler manifolds.
method Comparison with an auxiliary Monge-Ampère equation on a ball with Dirichlet boundary conditions.
result The method yields unique solutions and improves on existing methods.
In this paper, we propose the uncertain volatility models with stochastic bounds. Like the regular uncertain volatility models, we know only that the true model lies in a family of progressively measurable and bounded processes, but instead of using two deterministic bounds, the uncertain volatility fluctuates between …
We study a fully nonlinear PDE involving a linear combination of symmetric polynomials of the Kähler form on a Kähler manifold. A \emph{a priori} estimate is proven in general and a gradient estimate is proven in certain cases. Independently, we also provide a method-of-continuity proof via a path of Kähler metri…
Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equationsmath.NA
A new method infers parameters from PDEs using Gaussian processes.
problem Estimating unknown parameters in PDEs from noisy data.
method PDE-Informed Gaussian Process (PIGP) method.
result The method bypasses numerical solvers for PDEs and provides uncertainty quantification.
Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
Unified framework solves nonlinear PDEs and IPs using Gaussian processes.
problem Solving and identifying parameters in nonlinear PDEs and inverse problems.
method Gaussian process framework approximating solutions as MAP estimators, reducing to finite-dimensional optimization problem.
result Unified method converges in a small number of iterations for various PDEs.
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we …
Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEsmath.NA
Paper analyzes and proves convergence of a new method for solving complex PDEs.
problem Solving high-dimensional nonlinear PDEs and PIDEs with random neural networks.
method Random deep splitting method using random neural networks.
result The method converges to the unique viscosity solution of nonlinear PDEs and PIDEs.
In this paper, we study the asymptotic behavior of Asian option prices in the worst case scenario under an uncertain volatility model. We give a procedure to approximate the Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and cutting the obtained Black-S…
Bayesian methods solve complex nonlinear PDEs efficiently.
problem Solving nonlinear PDEs with high computational cost.
method Bayesian inference with approximate likelihood based on discretization.
result Probabilistic uncertainty quantification for PDE solutions is feasible.
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the solvability of a fully nonlinear elliptic PDE. The required a priori estimates are …
In his 1954 paper about the initial value problem for 2D hyperbolic nonlinear PDEs, P. Lax declared that he had "a strong reason to believe" that there must exist a well-defined class of "not genuinely nonlinear" nonlinear PDEs. In 1978 G. Boillat coined the term "completely exceptional" to denote it. In the case of $2…
This paper develops a fast algorithm for solving nonlinear PDEs using sparse Cholesky factorization.
problem Efficiently solving nonlinear PDEs with Gaussian processes and kernel methods.
method Sparse Cholesky factorization for near-linear complexity.
result Near-linear complexity algorithm for working with kernel matrices of nonlinear PDEs.
Study of nonlinear PDEs using derived geometry and BV formalism.
problem Understanding non-linear PDEs via derived geometric methods.
method Derived enhancement of de Rham complex, algebro-geometric techniques, BV formalism.
result Natural derived enhancement of de Rham complex for nonlinear PDEs.
Error estimates for nonlinear PDEs using kernel/GP methods.
problem Error analysis of kernel/GP methods for nonlinear and parametric PDEs.
method Sobolev space error estimates based on minimizing norm property of the solution.
result Dimension-benign convergence rates for smooth solutions.
Fundamental solutions found for PDEs in Finsler geometry.
problem Solving nonlinear PDEs in Finsler geometry.
method Introduced a non-isotropic Minkowski gauge and computed fundamental solutions.
result Explicit fundamental solutions computed for the PDEs.
Develops a new approach to study nonlinear PDEs and their singularities.
problem Understanding the propagation domains of solutions to nonlinear PDEs.
method Derived geometric machinery and sheaf theory to study nonlinear PDEs and their singular supports.
result Estimates the domains of propagation for solutions of non-linear systems.
Neural-net-induced Gaussian process (NNGP) regression inherits both the high expressivity of deep neural networks (deep NNs) as well as the uncertainty quantification property of Gaussian processes (GPs). We generalize the current NNGP to first include a larger number of hyperparameters and subsequently train the model…
Nonlinear PDEs risen when solving some optimization problems in finance, and their solutionsq-fin.CP
We consider a specific type of nonlinear partial differential equations (PDE) that appear in mathematical finance as the result of solving some optimization problems. We review some existing in the literature examples of such problems, and discuss the properties of these PDEs. We also demonstrate how to solve them nume…
Study moduli spaces of elliptic PDEs using derived -geometry.
problem Representability of moduli spaces of solutions of elliptic PDEs.
method Derived -geometry, stacks of relative jets, nonlinear Fredholm analysis.
result Moduli stack of solutions is relatively representable by quasi-smooth derived -schemes.
Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.
problem Generalize Picard iteration for nonlinear parabolic PDEs.
method Formulate Picard iteration as abstract state-transition model, derive generalization error bounds.
result Picard depth reduction reduces Picard truncation error without increasing estimation error.
New formulation tackles arbitrage in volatile markets using eigenvalue bounds.
problem Arbitrage opportunities in volatile markets beyond a certain time horizon.
method Formulated as a stochastic optimal control problem, solved via PDE.
result Characterized arbitrage time horizon through PDE solution.
Multifidelity Gaussian process regression for solving nonlinear partial differential equationsstat.ML
A new method uses multifidelity Gaussian process regression to solve nonlinear PDEs.
problem Efficiently solving nonlinear PDEs using kernel methods.
method Proposes a kernel learning approach based on cokriging for multifidelity simulations.
result Demonstrates improved performance on the Burgers' equation.