2D Black-Scholes-Barenblatt Equation

Mathematical description

The PDE is given by

$\partial_tu=-\frac{1}{2}\operatorname{Tr}(\sigma^2\operatorname{diag}(\bold{x}^2)\operatorname{Hess}_\bold{x}u)+r(u-\nabla u \cdot \bold{x}),$

where $\sigma=0.04$ and $r=0.05$ .

The domain is defined as

$(\bold{x},t) \in [0,1]^2 \times (0,1]$

The terminal condition is

$u(\bold{x},1)=c||\bold{x}||_2^2,$

where $c \sim \mathcal{U}(0,1).$

Dataset file information

File name: 2D_Black-Scholes-Barenblatt_r0.05_sigma0.4.hdf5

File hierarchy:

/
|-- tensor
|-- x-coordinate
|-- y-coordinate
|-- t-coordinate

Datasets information:

• tensor
• Shape: (1000, 101, 128, 128)
• Data Type: float
• Description: 1000 solution samples, each corresponding to a solution under one terminal condition.
• x-coordinate
• Shape: (128,)
• Data Type: float
• Description: x-axis spatial coordinates
• y-coordinate
• Shape: (128,)
• Data Type: float
• Description: y-axis spatial coordinates
• t-coordinate
• Shape: (101,)
• Data Type: float
• Description: temporal coordinates

Data generation method

According to the paper, the equation admits the explicit solution: $u(\bold{x},t)=\operatorname{exp}((r+\sigma^2)(1-t))g(x)$ . We generate data directly based on the explicit solution. The python script is below:

import torch
import h5py

# 2D
Nt = 101
Nx = 128
Ny = 128
spatial_dim = 2
num_samples = 1000

# create coordinates
T, X, Y = torch.meshgrid(torch.linspace(0, 1, Nt), torch.linspace(0, 1, Nx), torch.linspace(0, 1, Ny), indexing="ij")
coords = torch.stack([T, X, Y], dim=spatial_dim+1)
# print(coords[0,0,:,:], coords[0,:,0,:], coords[:,0,0,:])
coords = torch.reshape(coords, [-1, spatial_dim+1])

# random coefficient c
C = torch.rand(num_samples)

dataset_path = "2D_Black-Scholes-Barenblatt_r0.05_sigma0.4.hdf5"
with h5py.File(dataset_path, 'w') as f:
    f.create_dataset("x-coordinate", data=torch.linspace(0, 1, Nx))
    f.create_dataset("y-coordinate", data=torch.linspace(0, 1, Ny))
    f.create_dataset("t-coordinate", data=torch.linspace(0, 1, Nt))
    tensor = f.create_dataset("tensor", (num_samples, Nt, Nx, Ny), dtype='f')
    tensor.attrs["c"] = C
    for i in range(num_samples):
        U = C[i] * torch.exp((r + sigma**2) * (1 - coords[:,0])) * torch.linalg.vector_norm(coords[:,1:], ord=2, dim=1)**2
        U = torch.reshape(U, [Nt, Nx, Ny, -1])
        tensor[i] = U.squeeze(-1)