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U ovom radu proučavali smo pravu ortogonalnu dekompoziciju (POD) i diskretnu empirijsku interpolacijsku metodu (DEIM) te njihovu primjenu na dobivanje modela smanjenog reda. POD baza ranga \(\ell\) za zadani skup vektora u Hilbertovom prostoru \(\mathcal{H}\) je ortonormalan niz vektora u \(\mathcal{H}\) odabran tako da minimizira srednju kvadratnu pogrešku najbolje aproksimacije danih vektora na potprostor razapet nizom. Kada je \(\mathcal{H}\) konačnodimenzionalan, POD se svodi na dekompoziciju singularnih vrijednosti. Opisali smo diskretnu i neprekidnu POD dekompoziciju za rješenja dinamičkog sustava te metodu uzoraka za računanje POD baze. DEIM je metoda aproksimacije nelinearne funkcije \(f\) kosom projekcijom na prostor blizak njezinoj trajektoriji. Vidjeli smo da je dobro taj prostor fiksirati da bude linearna ljuska vektora POD baze za trajektoriju od \(f\) . Predstavili smo dva algoritma za odabir nterpolacijskih indeksa koji određuju smjer kose projekcije, DEIM i Q-DEIM. Numeričkim eksperimentima potvrdili smo kvalitetu ovih metoda. Finom semidiskretizacijom nestacionarnih parcijalnih diferencijalnih jednadžbi dobivaju se dinamički sustavi visoke dimenzije koje je skupo računati. POD-Galerkinova metoda primjenom Galerkinove projekcije na prostor razapet POD bazom konstruira model niže dimenzije koji aproksimira puni model. Redukciju dimenzije sustava ovom metodom nije moguće u potpunosti provesti ako je problem nelinearan. Tome smo doskočili tako što smo zamijenili nelinearni dio njegovom DEIM aproksimacijom. Numeričkim primjerom nelinearnog prodiranja viskoznog fluida u poroznom sredstvu demonstrirali smo metode u praksi te potvrdili teoretska razmatranja. In this paper, we studied proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) as well as their application to obtain a reduced order model (ROM). For a given set of vectors in a Hilbert space \(\mathcal{H}\), POD basis of rank \(\ell\) is orthonormal sequence in \(\mathcal{H}\) chosen so as to minimize the mean square error of the best approximation of given vectors in a subspace spanned by sequence. In case of finite dimensional \(\mathcal{H}\), POD comes down to singular value decomposition. We describe discrete and continuous variant of POD for dynamical systems and a method of snapshots for determining the POD basis. DEIM is a method of approximation for nonlinear functions. It uses an oblique projection to the space close to the function trajectory. We saw that POD generated space is a good choice for that purpose. We presented two algorithms for selecting interpolation indices which determine the direction of the projection, DEIM and Q-DEIM. Quality of these methods was confirmed by two numerical experiments. High dimensional dynamical systems obtained by fine semi-discretization of non-stationary partial differential equations are rather costly to solve numerically. POD-Galerkin method uses Galerkin projection onto a POD generated subspace to construct a reduced order model which approximates the full model. Order reduction cannot be fully achieved if the problem is nonlinear. This difficulty can be overcome by replacing the nonlinear part with its DEIM approximation. We demonstrated the method in practice and confirmed the theoretical arguments using the example of nonlinear miscible viscous fingering in porous media.