# Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients

^{*}

## Abstract

**:**

## 1. Introduction

- Optimality: For a large class of differential equations, both linear and nonlinear, it was shown that this method converges and is asymptotically optimal in the sense that the storage and number of floating point operations, needed to resolve the problem with desired accuracy, depend linearly on the number of parameters representing the solution, and the number of these parameters is small. Thus, the computational complexity for all steps of the algorithm is controlled.
- High order-approximation: The method enables high order approximation. The order of approximation depends on the order of the spline wavelet basis.
- Sparsity: The solutions and the right-hand side of the equation have sparse representation in a wavelet basis, i.e., they are represented by a small number of numerically significant parameters. In the beginning, iterations start for a small vector of parameters, and the size of the vector increases successively until the required tolerance is reached. The differential operator is represented by a sparse or quasi-sparse matrix, and a procedure for computing the product of this matrix with a finite-length vector with linear complexity is known.
- Preconditioning: For a large class of problems, the matrices arising from a discretization using wavelet bases can be simply preconditioned by a diagonal preconditioner, and the condition numbers of these preconditioned matrices are uniformly bounded. It is important that the preconditioner is simple, such as the diagonal preconditioner, because in some implementations, only nonzero elements in columns of matrices corresponding to significant coefficients of solutions are stored and used.

**Example**

**1.**

## 2. Wavelet Bases

**Definition**

**1.**

- (i)
- Ψ is a Riesz basis for H, i.e., the closure of the span of Ψ is H, and there exist constants $c,C\in \left(\right)open="("\; close=")">0,\infty $, such that:$$c{\u2225\mathbf{b}\u2225}_{2}\le {\left(\right)}_{\sum _{\lambda \in \mathcal{J}}}H\in {l}^{2}\left(\mathcal{J}\right).$$
- (ii)
- The functions are local in the sense that $\mathrm{diam}\left(\right)open="("\; close=")">\mathrm{supp}\phantom{\rule{0.166667em}{0ex}}{\psi}_{\lambda}$ for all $\lambda \in \mathcal{J}$, and at a given level j, the supports of only finitely many wavelets overlap at any point x.

## 3. Construction of Scaling Functions

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**2.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**4.**

## 4. Construction of Wavelets

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**5.**

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**6.**

## 5. Wavelets on the Hypercube

## 6. Numerical Examples

- One starts with a variational formulation for a suitable wavelet basis, but instead of turning to a finite dimensional approximation, the continuous problem is transformed into an infinite-dimensional ${l}^{2}$-problem.
- Then, one proposes a convergent iteration for the ${l}^{2}$-problem.
- Finally, one derives an implementable version of this idealized iteration, where all infinite-dimensional quantities are replaced by finitely supported ones.

- Compute sparse representation ${\mathbf{f}}_{j}$ of the right-hand side $\mathbf{f}$, such that ${\left(\right)}_{\mathbf{f}}$ is smaller than a given tolerance ${\u03f5}_{j}^{1}$. The computation of a sparse representation insists on thresholding the smallest coefficients and working only with the largest ones. We denote the routine as ${\mathbf{f}}_{j}:=\mathrm{RHS}[\mathbf{f},{\u03f5}_{j}^{1}]$.
- Compute K steps of GMRESfor solving the system $\mathbf{A}\mathbf{v}={\mathbf{f}}_{j}$ with the initial vector ${\mathbf{v}}_{j}$. Each iteration of GMRES requires multiplication of the infinite-dimensional matrix with a finitely-supported vector. Since for the wavelet basis constructed in this paper, the matrix is sparse, it can be computed exactly. Otherwise, it is computed approximately with the given tolerance ${\u03f5}_{j}^{2}$ by the method from [24]. We denote the routine $\mathbf{z}=\mathbf{GMRES}[\mathbf{A},{\mathbf{f}}_{j},{\mathbf{v}}_{j},K]$.
- Compute sparse representation ${\mathbf{v}}_{j+1}$ of $\mathbf{z}$ with the error smaller than ${\u03f5}_{j}^{2}$. We denote the routine ${\mathbf{v}}_{j+1}:=\mathbf{COARSE}[\mathbf{z},{\u03f5}_{j}^{2}]$. It insists on thresholding the coefficients.

Algorithm 1 $u:=$SOLVE [ $\mathbf{A}$, $\mathbf{f}$, $\tilde{\u03f5}$ ] |

1. Choose ${k}_{0},{k}_{1},{k}_{2}\in \left(\right)open="("\; close=")">0,1$, $K\in \mathbb{N}$. |

2. Set $j:=0$, ${\mathbf{v}}_{0}:=0$ and $\u03f5:={\u2225\mathbf{f}\u2225}_{2}$. |

3. While $\u03f5>\tilde{\u03f5}$ |

$j:=j+1$, |

$\u03f5:={k}_{0}\u03f5$, |

${\u03f5}_{j}^{1}:={k}_{1}\u03f5$, |

${\u03f5}_{j}^{2}:={k}_{2}\u03f5$, |

${\mathbf{f}}_{j}:=\mathbf{RHS}[\mathbf{f},{\u03f5}_{j}^{1}]$, |

$\mathbf{z}:=\mathbf{GMRES}[\mathbf{A},{\mathbf{f}}_{j},{\mathbf{v}}_{j-1},K]$ |

${\mathbf{v}}_{j}:=\mathbf{COARSE}[\mathbf{z},{\u03f5}_{j}^{2}]$, |

Estimate ${\mathbf{r}}_{j}=\mathbf{f}-\mathbf{A}{\mathbf{v}}_{j}$ and set $\u03f5:={\left(\right)}_{{\mathbf{r}}_{j}}2$. |

end while, |

4. $\mathbf{u}:={\mathbf{v}}_{j}$, |

5. Compute approximate solution $\tilde{u}={\sum}_{{u}_{\lambda}\in \mathbf{u}}{u}_{\lambda}{\psi}_{\lambda}$. |

**RHS**and

**COARSE**, we refer to [17,18,23].

**Example**

**2.**

**Example**

**3.**

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 4.**The sparsity pattern of the matrices arising from a discretization using a wavelet basis constructed in this paper (left) and a wavelet basis from [25] (right) for the Black–Scholes equation with quadratic volatilities.

**Figure 5.**The approximate solution (left) and the derivative of the approximate solution (right) for Example 1.

**Figure 6.**Convergence history for Example 1. The number of basis functions and the ${L}^{\infty}$-norm of the error are in logarithmic scaling.

**Figure 7.**Contour plot (left) and 3D plot (right) of the approximate solution ${V}_{1}$ for Example 2.

**Figure 8.**Convergence history for Example 2. The number of basis functions and the ${L}^{\infty}$-norm of the error are in logarithmic scaling.

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Černá, D.; Finĕk, V.
Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients. *Axioms* **2017**, *6*, 4.
https://doi.org/10.3390/axioms6010004

**AMA Style**

Černá D, Finĕk V.
Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients. *Axioms*. 2017; 6(1):4.
https://doi.org/10.3390/axioms6010004

**Chicago/Turabian Style**

Černá, Dana, and Václav Finĕk.
2017. "Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients" *Axioms* 6, no. 1: 4.
https://doi.org/10.3390/axioms6010004