Browsing by Author "Finěk, Václav"
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- ItemAdaptive Scheme for Black-Scholes Equation using Hermite Cubic Spline Wavelets(2018) Finěk, VáclavThis contribution is devoted to the wavelet based adaptive numerical solution of the Black-Scholes equation for pricing European options. We apply the Crank-Nicolson scheme with Richardson extrapolation for time discretization and Hermite cubic spline wavelets with four vanishing moments for space discretization. The proposed scheme enables higher order approximation and exploits compression properties of wavelets. This scheme is the fourth order accurate both in time and in space. A numerical example is presented for the Black-Scholes equation with real data.
- ItemAdaptive wavelet based scheme for option pricing(IEEE, 345 E 47TH ST, NEW YORK, NY 10017 USA, 2018) Finěk, VáclavThis contribution deals with the numerical solution of the Black-Scholes equation. The Crank-Nicolson scheme is applied for time discretization and wavelets are applied for space discretization. Hermite cubic spline wavelets with four vanishing moments are adaptively used because they enable higher order approximations, are well-conditioned, have short supports, have a high potential in adaptive methods due to the four vanishing wavelet moments and mainly because arising stiffness matrices are sparse in wavelet coordinates. Due to irregularities of the initial data in the Black-Scholes model, the use of the second-order Crank-Nicolson scheme usually requires a certain amount of damping to compensate for the known weak stability of this scheme. We numerically show here that optimal convergence rate for a proposed adaptive wavelet discretization in space can be obtained without any damping and without any restriction on the time step. A numerical example is given for the Black-Scholes equation with real data from the Frankfurt Stock Exchange. We also compare numerical results for adaptive and non-adaptive wavelet discretization in space.
- ItemAdaptive wavelet scheme for convection-diffusion equations(Technická univerzita v Liberci, Česká republika, 2012) Černá, Dana; Finěk, VáclavOne of the most important part of adaptive wavelet methods is an efficient approximate multi- plication of stiffness matrices with vectors in wavelet coordinates. Although there are known algorithms to perform it in linear complexity, the application of them is relatively time con- suming and its implementation is very difficult. Therefore, it is necessary to develop a well- conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in any column is bounded by a constant. Then, matrix-vector multiplication can be performed exactly with linear complexity. We present here a wavelet basis on the interval with respect to which both the mass and stiffness matrices corresponding to the one-dimensional Laplacian are sparse. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in tensor product wavelet basis is also sparse. Moreover, the constructed basis has an excellent condition number. In this contribution, we shortly review this construction and show several numerical tests.
- ItemGalerkin method with new quadratic spline wavelets for integral and integro-differential equations(ELSEVIER, RADARWEG 29, 1043 NX AMSTERDAM, NETHERLANDS, 2019-10-18) Černá, Dana; Finěk, VáclavThe paper is concerned with the wavelet-Galerkin method for the numerical solution of Fredholm linear integral equations and second-order integro-differential equations. We propose a construction of a quadratic spline-wavelet basis on the unit interval, such that the wavelets have three vanishing moments and the shortest support among such wavelets. We prove that this basis is a Riesz basis in the space L-2(0, 1). We adapt the basis to homogeneous Dirichlet boundary conditions, and using a tensor product we construct a wavelet basis on the hyperrectangle. We use the wavelet-Galerkin method with the constructed bases for solving integral and integro-differential equations, and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices. We present numerical examples and compare the results with the Galerkin method using other quadratic spline wavelet bases and other methods.
- ItemMultiwavelets based on hermite cubic splines(Technická univerzita v Liberci, Česká republika, 2012) Černá, Dana; Finěk, Václav; Plačková, Gertathe convection-diffusion equation. We use an implicit scheme for the time discretization and
- ItemOn a sparse representation of laplacian(Technická univerzita v Liberci, Česká republika, 2012) Černá, Dana; Finěk, Václav; Ondračková., ZdenaThe paper is concerned with theoretical and computational issues of a numerical resolution of
- ItemOn the exact values of coefficients of coiflets(Elsevier Science Bv, 2014) Adolph, Christoph H.; Akhunzyanov, R.; Alekseev, M. G.; Alexakhin, V. Yu.; Alexandrov, Yu A.; Alexeev, G.D.; Amoroso, A.; Andrieux, Vincent; Anosov, V.; Austregesilo, Alexander; Badełek, Barbara; Balestra, Ferruccio; Barth, Jcl.; Baum, Günter; Beck, Reinhard; Bedfer, Yann; Berlin, Asher; Bernhard, Johannes; Bertini, Raimondo; Bicker, K.; Bieling, Jakob; Birsa, Renato; Bisplinghoff, Jens; Bodlak, Martin; Böer, Marie; Bordalo, Paula; Bradamante, Franco; Braun, Christopher; Bravar, Alessandro; Bressan, Andrea; Büchele, M.; Burtin, Etienne; Capozza, Luigi; Chiosso, Michela; Chung, Suhurk; Cicuttin, Andres; Crespo, María Liz; Curiel, Q.; Dalla Torre, Stefano; Dasgupta, Subinay; Dasgupta, Shivaji; Denisov, Oleg Yu; Donskov, Sergey; Doshita, Norihiro; Duic, Venicio; Duenweber, W.; Dziewiecki, Michał; Efremov, Anatoli Vasilievich; Eliay, C.; Eversheim, Paul Dieter; Eyrich, Wolfgang; Faessler, Martin; Ferrero, Andrea; Filin, Alex P.; Finger, Michael; Finger, Miroslav, Jr.; Fischer, H.; Franco, Celso; von Hohenesche, N. du Fresne; Friedrich, Jan Michael; Frolov, Vladimir; Garfagnini, Raffaello; Gautheron, F.; Gavrichtchouk, O.P.; Gerassimov, S.; Geyer, Reiner; Giorgi, Marcello A.; Gnesi, I.; Gobbo, Benigno; Goertz, S.; Gorzellik, M.; Grabmüller, Stefanie; Grasso, Antonino; Grube, Boris; Gushterski, R.; Guskov, Alexey; Guthoerl, T.; Haas, Florian; von Harrach, Dietrich; Hahne, Devin; Hashimoto, Ryo; Heinsius, F.H.; Herrmann, Felix; Hess, C.; Hinterberger, Frank; Hoeppner, Ch.; Horikawa, Naoaki; d'Hose, Nicole; Huber, Stefan; Ishimoto, Shigeru; Ivanov, Andrey; Ivanshin, Yu I.; Iwata, Takashi; Jahn, R.; Jary, Vladimír; Jasinski, Prometeusz Kryspin; Joerg, P.; Joosten, R.; Kabuss, E.; Kang, D.; Ketzer, B.; Khaustov, G. V.; Khokhlov, Yu. A.; Kisselev, Yu F.; Klein, F.; Klimaszewski, K.; Koivuniemi, J. H.; Kolosov, V. N.; Kondo, K.; Koenigsmann, K.; Konorov, I.; Konstantinov, V. F.; Kotzinian, A. M.; Kouznetsov, O.; Kral, Z.; Kraemer, M.; Kroumchtein, Z. V.; Kuchinski, N.; Kunne, F.; Kurek, K.; Kurjata, R. P.; Lednev, A. A.; Lehmann, A.; Levorato, S.; Lichtenstadt, J.; Maggiora, A.; Magnon, A.; Makke, N.; Mallot, G. K.; Marchand, C.; Martin, A.; Marzec, J.; Matousek, Jan; Matsuda, H.; Matsuda, T.; Meshcheryakov, G.; Meyer, W.; Michigami, T.; Mikhailov, Yu. V.; Miyachi, Y.; Nagaytsev, A.; Nagel, T.; Nerling, F.; Neubert, S.; Neyret, D.; Nikolaenko, V. I.; Novy, J.; Nowak, W. -D.; Nunes, A. S.; Orlov, I.; Olshevsky, A. G.; Ostrick, M.; Panknin, R.; Panzieri, D.; Parsamyan, B.; Paul, S.; Pesek, Michael; Peshekhonov, D.; Piragino, G.; Platchkov, S.; Pochodzalla, J.; Polak, J.; Polyakov, V. A.; Pretz, J.; Quaresma, M.; Quintans, C.; Ramos, S.; Reicherz, G.; Rocco, E.; Rodionov, V.; Rondio, E.; Rossiyskaya, N. S.; Ryabchikov, D. I.; Samoylenko, V. D.; Sandacz, A.; Sapozhnikov, M. G.; Sarkar, S.; Savin, I. A.; Sbrizzai, G.; Schiavon, P.; Schill, C.; Schlueter, T.; Schmidt, A.; Schmidt, K.; Schmitt, L.; Schmiden, H.; Schoenning, K.; Schopferer, S.; Schott, M.; Shevchenko, O. Yu.; Silva, L.; Sinha, L.; Sirtl, S.; Slunecka, M.; Sosio, S.; Sozzi, F.; Srnka, A.; Steiger, L.; Stolarski, M.; Šulc, Miroslav; Sulej, R.; Suzuki, H.; Szabelski, A.; Szameitat, T.; Sznajder, P.; Takekawa, S.; TerWolbeek, J.; Tessaro, S.; Tessarotto, F.; Thibaud, F.; Uhl, S.; Uman, I.; Vandenbroucke, M.; Virius, M.; Vondra, J.; Wang, L.; Weisrock, T.; Wilfert, M.; Windmolders, R.; Wislicki, W.; Wollny, H.; Zaremba, K.; Zavertyaev, M.; Zemlyanichkina, E.; Zhuravlev, N.; Ziembicki, M.; Černá, Dana; Finěk, Václav; Najzar, KarelIn 1989, R. Coifman suggested the design of orthonormal wavelet systems with vanishing moments for both scaling and wavelet functions. They were first constructed by I. Daubechies [15, 16], and she named them coiflets. In this paper, we propose a system of necessary conditions which is redundant free and simpler than the known system due to the elimination of some quadratic conditions, thus the construction of coiflets is simplified and enables us to find the exact values of the scaling coefficients of coiflets up to length 8 and two further with length 12. Furthermore for scaling coefficients of coiflets up to length 14 we obtain two quadratic equations, which can be transformed into a polynomial of degree 4 for which there is an algebraic formula to solve them. © Versita Warsaw and Springer-Verlag Berlin Heidelberg 2008.
- ItemOn the problem of variability of interval data(Technická univerzita v Liberci, Česká republika, 2012) Finěk, Václav; Matonoha, Ctirada vector which we have recently proposed in [1, 2]. The theoretical advantages of our scheme
- ItemOptimized construction of biorthogonal spline-wavelets(Technical university of Liberec, Czech Republic, 2008) Černá, Dana; Finěk, VáclavThe paper is concerned with the construction of wavelet bases on the interval derived from B-splines. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, Feauveau [5] while the construction of boundary wavelets is along the lines of [6], The disadvantage of popular bases from [6] is their bad condition which cause problems in practical applications. Some modifications which lead to better conditioned bases were proposed in [1, 7, 8, 9, 10], In this contribution, we further improve the condition of spline-wavelet bases on the interval. Quantitative properties of these bases are presented. © 2008 American Institute of Physics.
- ItemQUADRATIC SPLINE WAVELETS WITH SHORT SUPPORT SATISFYING HOMOGENEOUS BOUNDARY CONDITIONS(2018) Černá, Dana; Finěk, VáclavIn this paper, we construct a new quadratic spline-wavelet basis on the interval and on the unit square satisfying homogeneous Dirichlet boundary conditions of the first order. The wavelets have one vanishing moment and the shortest support among quadratic spline wavelets with at least one vanishing moment adapted to the same type of boundary conditions. The stiffness matrices arising from the discretization of the second-order elliptic problems using the constructed wavelet basis have uniformly bounded condition numbers, and the condition numbers are small. We present some quantitative properties of the constructed basis. We provide numerical examples to show that the Galerkin method and the adaptive wavelet method using our wavelet basis require fewer iterations than methods with other quadratic spline wavelet bases. Moreover, due to the small support of the wavelets, when using these methods with the new wavelet basis, the system matrix is sparser, and thus one iteration requires a smaller number of floating point operations than for other quadratic spline wavelet bases.
- ItemSparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients(MDPI, 2019-02-22) Černá, Dana; Finěk, VáclavWe propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efficiency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black–Scholes equation with a quadratic volatility.
- ItemWaveletové metody pro přibližné řešení parciálních diferenciálních rovnic(Technická Univerzita v Liberci, 2015) Cvejnová, Daniela; Finěk, VáclavDiplomová práce se zabývá numerickým řešením parciálních diferenciálních rovnic na čtvecové oblasti, konkrétně na intervalu [0,1]^2, a to pomocí waveletových bází. V první části jsou nadefinovány základní pojmy, jako jsou Hilbertovy a Sobolevovy prostory, Rieszova báze a wavelet. Dále je uveden koncept multirozkladu, který se využívá ke konstrukci waveletových bází. Je také zavedena waveletová báze na intervalu a uvedeny některé důležité vlastnosti waveletů. Ve druhé části jsou definovány spliny, po částech polynomiální funkce, kterých se ke konstrukci waveletů často využívá. Podrobněji se zde zabýváme B-spliny a Hermitovými kubickými spliny. Řešená úloha je představená v kapitole třetí, společně s odvozením její slabé formulace a s podmínkami existence a jednoznačnosti řešení. V rámci této práce byly implementovány tři různé waveletové báze, které jsou představeny v další kapitole. Pomocí těchto bází pak byla numericky řešena zadaná úloha. K řešení byla použita Galerkinova metoda. V poslední kapitole jsou uvedeny obdržené výsledky.
- ItemWavelety(Technická Univerzita v Liberci, 2013) Cvejnová, Daniela; Finěk, VáclavBakalářská práce se zabývá konstrukcemi waveletových bází a jejich využitím k numerickému řešení diferenciálních rovnic. V teoretické části jsou uvedeny definice základních pojmů jako je waveletová funkce, Rieszova báze nebo projekce. V této části je také zaveden koncept multirozkladu, který je později užíván pro konstrukci waveletové báze. Jsou zde definovány i biortogonální wavelety a zavedena waveletová báze na intervalu. Kapitola ''B-spliny'' pojednává o důležitém typu funkcí, které jsou pak také využity ke konstrukci waveletových bází. Detailně je probrán nejjednodušší příklad waveletu tzv. Haarův wavelet, které je založen na B-splinu nultého řádu. Ve druhé části práce je specifikována obyčejná diferenciální rovnice druhého řádu, která je následně přibližně řešena pomocí waveletů. Integrály ze slabé formulace úlohy jsou počítány ze škálové báze a následně jsou waveletou transformací převedeny do waveletové báze. Nakonec jsou uvedeny tři konstrukce waveletových bází a ty jsou aplikovány na řešení uvedené rovnice.