# 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 $c,C\in \left(0,\infty \right)$, such that:$$c{\u2225\mathbf{b}\u2225}_{2}\le {\u2225\sum _{\lambda \in \mathcal{J}}{b}_{\lambda}{\psi}_{\lambda}\u2225}_{H}\le C{\u2225\mathbf{b}\u2225}_{2},\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{1.em}{0ex}}\mathbf{b}:={\left\{{b}_{\lambda}\right\}}_{\lambda \in \mathcal{J}}\in {l}^{2}\left(\mathcal{J}\right).$$
- (ii)
- The functions are local in the sense that $\mathrm{diam}\left(\mathrm{supp}\phantom{\rule{0.166667em}{0ex}}{\psi}_{\lambda}\right)\le \tilde{C}\phantom{\rule{0.166667em}{0ex}}{2}^{-\left|\lambda \right|}$ 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 ${\u2225\mathbf{f}-{\mathbf{f}}_{j}\u2225}_{2}$ 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 ${k}_{0},{k}_{1},{k}_{2}\in \left(0,1\right)$, $K\in \mathbb{N}$. |

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 $\u03f5:={\u2225{\mathbf{r}}_{j}\u2225}_{2}$. |

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

## References

- Beylkin, G. Wavelets and fast numerical algorithms. In Different Perspectives on Wavelets 47; Daubechies, I., Ed.; Proceedings of Symposia in Applied Mathematics; American Mathematical Society: Providence, RI, USA, 1993; pp. 89–117. [Google Scholar]
- Shann, W.C.; Xu, J.C. Galerkin-wavelet methods for two-point boundary value problems. Numer. Math.
**1992**, 63, 123–142. [Google Scholar] - Hariharan, G.; Kannan, K. Haar wavelet method for solving some nonlinear parabolic equations. J. Math. Chem.
**2010**, 48, 1044–1061. [Google Scholar] [CrossRef] - Lepik, Ü. Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul.
**2005**, 68, 127–143. [Google Scholar] [CrossRef] - Lepik, Ü. Numerical solution of evolution equations by the Haar wavelet method. Appl. Math. Comput.
**2007**, 185, 695–704. [Google Scholar] [CrossRef] - Cattani, C. Haar wavelet spline. J. Interdiscip. Math.
**2001**, 4, 35–47. [Google Scholar] [CrossRef] - Cattani, C.; Bochicchio, I. Wavelet analysis of chaotic systems. J. Interdiscip. Math.
**2006**, 9, 445–458. [Google Scholar] - Ciancio, A.; Cattani, C. Analysis of Singularities by Short Haar Wavelet Transform. In Lecture Notes in Computer Science; Gervasi, O., Kumar, V., Tan, C.J.K., Taniar, D., Laganá, A., Mun, Y., Choo, H., Eds.; Springer: Berlin/Heidelberg, Germany, 2006; pp. 828–838. [Google Scholar]
- Kumar, V.; Mehra, M. Cubic spline adaptive wavelet scheme to solve singularly perturbed reaction diffusion problems. Int. J. Wavelets Multiresolut. Inf. Process.
**2007**, 15, 317–331. [Google Scholar] [CrossRef] - Jia, R.Q.; Zhao, W. Riesz bases of wavelets and applications to numerical solutions of elliptic equations. Math. Comput.
**2011**, 80, 1525–1556. [Google Scholar] [CrossRef] - Negarestani, H.; Pourakbari, F.; Tavakoli, A. Adaptive multiple knot B-spline wavelets for solving Saint-Venant equations. Int. J. Wavelets Multiresolut. Inf. Process.
**2013**, 11, 1–12. [Google Scholar] - Mehra, M. Wavelets and differential equations—A short review. In AIP Conference Proceedings 1146; Siddiqi, A.H., Gupta, A.K., Brokate, M., Eds.; American Institute of Physics: New York, NY, USA, 2009; pp. 241–252. [Google Scholar]
- Dahmen, W. Multiscale and wavelet methods for operator equations. Lect. Notes Math.
**2003**, 1825, 31–96. [Google Scholar] - Caflisch, R.E.; Gargano, F.; Sammartino, M.; Sciacca, V. Complex singularities and PDEs. Rivista Matematica Universita di Parma
**2015**, 6, 69–133. [Google Scholar] - Gargano, F.; Sammartino, M.; Sciacca, V.; Cassel, K.V. Analysis of complex singularities in high-Reynolds-number Navier-Stokes solutions. J. Fluid Mech.
**2014**, 747, 381–421. [Google Scholar] [CrossRef] - Weideman, J.A.C. Computing the dynamics of complex singularities of nonlinear PDEs. SIAM J. Appl. Dyn. Syst.
**2003**, 2, 171–186. [Google Scholar] [CrossRef] - Cohen, A.; Dahmen, W.; DeVore, R. Adaptive wavelet schemes for elliptic operator equations—Convergence rates. Math. Comput.
**2001**, 70, 27–75. [Google Scholar] [CrossRef] - Cohen, A.; Dahmen, W.; DeVore, R. Adaptive wavelet methods II - beyond the elliptic case. Found. Math.
**2002**, 2, 203–245. [Google Scholar] [CrossRef] - Dijkema, T.J.; Schwab, C.; Stevenson, R. An adaptive wavelet method for solving high-dimensional elliptic PDEs. Constr. Approx.
**2009**, 30, 423–455. [Google Scholar] [CrossRef] - Hilber, N.; Reichmann, O.; Schwab, C.; Winter, C. Computational Methods for Quantitative Finance; Springer: Berlin, Germany, 2013. [Google Scholar]
- Stevenson, R. Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal.
**2003**, 41, 1074–1100. [Google Scholar] [CrossRef] - Dahmen, W.; Kunoth, A. Multilevel preconditioning. Numer. Math.
**1992**, 63, 315–344. [Google Scholar] [CrossRef] - Urban, K. Wavelet Methods for Elliptic Partial Differential Equations; Oxford University Press: Oxford, UK, 2009. [Google Scholar]
- Černá, D.; Finěk, V. Approximate multiplication in adaptive wavelet methods. Cent. Eur. J. Math.
**2013**, 11, 972–983. [Google Scholar] [CrossRef] - Dijkema, T.J.; Stevenson, R. A sparse Laplacian in tensor product wavelet coordinates. Numer. Math.
**2010**, 115, 433–449. [Google Scholar] [CrossRef] - Cvejnová, D.; Černá, D.; Finěk, V. Hermite cubic spline multi-wavelets on the cube. In AIP Conference Proceedings 1690; Pasheva, V., Popivanov, N., Venkov, G., Eds.; American Institute of Physics: New York, NY, USA, 2015; No. 030006. [Google Scholar]
- Černá, D.; Finěk, V. On a sparse representation of an n-dimensional Laplacian in wavelet coordinates. Results Math.
**2016**, 69, 225–243. [Google Scholar] [CrossRef] - Černá, D. Numerical solution of the Black–Scholes equation using cubic spline wavelets. In AIP Conference Proceedings 1789; Pasheva, V., Popivanov, N., Venkov, G., Eds.; American Institute of Physics: New York, NY, USA, 2016; No. 030001. [Google Scholar]
- Cvejnová, D.; Šimůnková, M. Comparison of multidimensional wavelet bases. AIP Conf. Proc.
**2015**, 1690, 030008. [Google Scholar] - Schneider, A. Biorthogonal cubic Hermite spline multi-wavelets on the interval with complementary boundary conditions. Results Math.
**2009**, 53, 407–416. [Google Scholar] [CrossRef] - Dahmen, W.; Han, B.; Jia, R.Q.; Kunoth, A. Biorthogonal multi-wavelets on the interval: Cubic Hermite splines. Constr. Approx.
**2000**, 16, 221–259. [Google Scholar] [CrossRef] - Jia, R.Q.; Liu, S.T. Wavelet bases of Hermite cubic splines on the interval. Adv. Comput. Math.
**2006**, 25, 23–39. [Google Scholar] [CrossRef] - Shumilov, B.M. Multiwavelets of the third-degree Hermitian splines orthogonal to cubic polynomials. Math. Models Comput. Simul.
**2013**, 5, 511–519. [Google Scholar] [CrossRef] - Shumilov, B.M. Cubic multi-wavelets orthogonal to polynomials and a splitting algorithm. Numer. Anal. Appl.
**2013**, 6, 247–259. [Google Scholar] [CrossRef] - Xue, X.; Zhang, X.; Li, B.; Qiao, B.; Chen, X. Modified Hermitian cubic spline wavelet on interval finite element for wave propagation and load identification. Finite Elem. Anal. Des.
**2014**, 91, 48–58. [Google Scholar] [CrossRef] - Černá, D.; Finěk, V. Wavelet bases of cubic splines on the hypercube satisfying homogeneous boundary conditions. Int. J. Wavelets Multiresolut. Inf. Process.
**2015**, 13, 1550014. [Google Scholar] [CrossRef] - Zuhlsdorff, C. The pricing of derivatives on assets with quadratic volatility. Appl. Math. Financ.
**2001**, 8, 235–262. [Google Scholar] [CrossRef] - Dahmen, W. Stability of multiscale transformations. J. Fourier Anal. Appl.
**1996**, 4, 341–362. [Google Scholar] - Johnson, C.R. A Gershgorin-type lower bound for the smallest singular value. Linear Algebra Appl.
**1989**, 112, 1–7. [Google Scholar] [CrossRef] - Černá, D.; Finěk, V. Quadratic spline wavelets with short support for fourth-order problems. Results Math.
**2014**, 66, 525–540. [Google Scholar] [CrossRef] - Černá, D.; Finěk, V. A diagonal preconditioner for singularly pertubed problems. Bound. Value Probl.
**2017**, 2017, 22. [Google Scholar] [CrossRef]

**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.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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