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Optimal Damping with a Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models
No full textEfficient pricing of multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods become competitive compared to alternative techniques because the integrand in the frequency space has often higher regularity than in the physical space. However, when designing a numerical quadrature method for most of these Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of the damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of the high dimensionality. To address these challenges, we propose an efficient numerical method for pricing European multi-asset options based on two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of damping parameters based on a proposed heuristic optimization rule. Second, we use sparsification and dimension-adaptivity techniques to accelerate the convergence of the quadrature in high dimensions. Our extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some L'evy models demonstrates the advantages of adaptivity and our damping rule on the numerical complexity of the quadrature methods. Moreover, our approach achieves substantial computational gains compared to the Monte Carlo method
Numerical solution of an optimal control problem with probabilistic and almost sure state constraints
Get PDFWe consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau-Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated
Ontologies for models and algorithms in applied mathematics and related disciplines
Get PDFIn applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies
Nutrient Control for a Viscous Cahn–Hilliard–Keller–Segel Model with Logistic Source Describing Tumor Growth
No full textIn this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn--Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller--Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established
Fine properties of geodesics and geodesic lambda-convexity for the Hellinger--Kantorovich distance
No full textWe study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger--Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton--Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic lambda-convexity with respect to the Hellinger--Kantorovich distance
Graph-to-local limit for the nonlocal interaction equation
Get PDFWe study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study
Finite Volumes for Complex Applications X—Volume 2, Hyperbolic and Related Problems: FVCA10, Strasbourg, France, October 30, 2023–November 03, 2023
No full textThis volume comprises the second part of the proceedings of the 10th International Conference on Finite Volumes for Complex Applications, FVCA, held in Strasbourg, France, during October 30 to November 3, 2023. The Finite Volume method, and several of its variants, is a spatial discretization technique for partial differential equations based on the fundamental physical principle of conservation. Recent decades have brought significant success in the theoretical understanding of the method. Many finite volume methods are also built to preserve some properties of the continuous equations, including maximum principles, dissipativity, monotone decay of the free energy, asymptotic stability, or stationary solutions. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differentialequations becomes particularly important for multiphysics and multiscale applications. In recent years, the efficient implementation of these methods in numerical software packages, more specifically to be used in supercomputers, has drawn some attention. The first volume contains all invited papers, as well as the contributed papers focusing on finite volume schemes for elliptic and parabolic problems. They include structure-preserving schemes, convergence proofs, and error estimates for problems governed by elliptic and parabolic partial differential equations. This volume is focused on finite volume methods for hyperbolic and related problems, such as methods compatible with the low Mach number limit or able to exactly preserve steady solutions, the development and analysis of high order methods, or the discretization of kinetic equations
On a two-scale phasefield model for topology optimization
Get PDFIn this article, we consider a gradient flow stemming from a problem in two-scale topology optimization. We use the phase-field method, where a Ginzburg--Landau term with obstacle potential is added to the cost functional, which contains the usual compliance but also an additional contribution including a local volume constraint in a penalty term. The minimization of such an energy by its gradient-flow is analyzed in this paper. We use an regularization and discretization of the associated state-variable to show the existence of weak solutions to the considered system. Download Document
Finite volumes for simulation of large molecules
Get PDFWe study a finite volume scheme for simulating the evolution of large molecules within their reduced state space. The finite volume scheme under consideration is the SQRA scheme developed by Lie, Weber and Fackeldey. We study convergence of a more general family of FV schemes in up to 3 dimensions and provide a convergence result for the SQRA-scheme in arbitrary space dimensions
Impact of noise on spinodal dewetting of liquid-liquid films
No full textSpinodal dewetting provides fundamental insights into the physics at interfaces, such as van der Waals forces driving dewetting, dissipation processes or thermal fluctuations. The dewetting process of liquid bilayer systems still raises open problems involving two coupled moving interfaces. Comparison of experimental results of spinodally dewetting liquid polystyrene films from liquid polymethylmethacrylate substrates, with predictions from linear stability analysis, we demonstrate that both the spinodal wavelength and the rupture times show significant differences. Key for this discrepancy is the altered mode selection process due to the initial surface roughness of the liquid-air and liquid-liquid interfaces, which is perturbed by partially correlated colored noise in the linearly unstable region. The strong effect of noise on mode selection and rupture time is confirmed by comparing experimental results with numerical solutions of the full dynamic nonlinear model and suggest new strategies to include thermal fluctuations into modeling these processes