Peridynamics for Data Estimation, Image Compression/Recovery, and Model Reduction

Abstract

The existing interpolation and regression methods are highly data-specific, challenge-specific, or approach-specific. Peridynamic approach provides a single mathematical framework for diverse data-sets and multi-dimensional data manipulation and model order reduction. The mathematical framework based on the Peridynamic Differential Operator (PDDO) provides a unified approach to transfer information within a set of discrete data, and among data sets in multi-dimensional space. The robustness and capability of this approach have been demonstrated by considering various real or fabricated data concerning two- or three-dimensional applications. The numerical results concern interpolation of real data in two and three dimensions, interpolation to approximate a three-dimensional function, adaptive data recovery in three-dimensional space, recovery of missing pixels in an image, adaptive image compression and recovery, and free energy evaluation through model reduction.

Appendices

Appendix 1 Peridynamic differential operator

According to the 2-order TSE in a 2-dimensional space, the following expression holds

$$f\left(\mathbf{x}+\xi \right)=f\left(\mathbf{x}\right)+{\xi }_{1}\frac{\partial f\left(\mathbf{x}\right)}{\partial {x}_{1}}+{\xi }_{2}\frac{\partial f\left(\mathbf{x}\right)}{\partial {x}_{2}}+\frac{1}{2!}{\xi }_{1}^{2}\frac{{\partial }^{2}f\left(\mathbf{x}\right)}{\partial {x}_{1}^{2}}+\frac{1}{2!}{\xi }_{2}^{2}\frac{{\partial }^{2}f\left(\mathbf{x}\right)}{\partial {x}_{2}^{2}}+{\xi }_{1}{\xi }_{2}\frac{{\partial }^{2}f\left(\mathbf{x}\right)}{\partial {x}_{1}\partial {x}_{2}}+R$$

(97)

where R is the small remainder term. Multiplying each term with PD functions, ${g}_{2}^{{p}_{1}{p}_{2}}(\xi )$, and integrating over the domain of interaction (family), ${H}_{\mathbf{x}}$, results in

$$\begin{array}{c}\underset{{H}_{x}}{\int }f(\mathbf{x}+\xi ){g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )dV=f\left(\mathbf{x}\right)\underset{{H}_{\mathbf{x}}}{\int }{g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )d{H}_{{\mathbf{x}}^{{\prime}}}+\frac{\partial f\left(\mathbf{x}\right)}{\partial {x}_{1}}\underset{{H}_{\mathbf{x}}}{\int }{\xi }_{1}{g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )d{H}_{{\mathbf{x}}^{{\prime}}}\\ +\frac{\partial f\left(\mathbf{x}\right)}{\partial {x}_{2}}\underset{{H}_{\mathbf{x}}}{\int }{\xi }_{2}{g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )d{H}_{{\mathbf{x}}^{^{\prime}}}+\frac{{\partial }^{2}f\left(\mathbf{x}\right)}{\partial {x}_{1}^{2}}\underset{{H}_{\mathbf{x}}}{\int }\frac{1}{2!}{\xi }_{1}^{2}{g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )d{H}_{{\mathbf{x}}^{{\prime}}}\\ +\frac{{\partial }^{2}f\left(\mathbf{x}\right)}{\partial {x}_{2}^{2}}\underset{{H}_{\mathbf{x}}}{\int }\frac{1}{2!}{\xi }_{2}^{2}{g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )d{H}_{{\mathbf{x}}^{{\prime}}}+\frac{{\partial }^{2}f\left(\mathbf{x}\right)}{\partial {x}_{1}\partial {x}_{2}}\underset{{H}_{\mathbf{x}}}{\int }{\xi }_{1}{\xi }_{2}{g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )d{H}_{{\mathbf{x}}^{{\prime}}}\end{array}$$

(98)

in which the point x is not necessarily symmetrically located in the domain of interaction. The initial relative position, $\xi$, between the material points x and ${\mathbf{x}}^{{\prime}}$ can be expressed as $\xi =\mathbf{x}-{\mathbf{x}}^{{\prime}}$. This ability permits each point to have its own unique family with an arbitrary position. Therefore, the size and shape of each family can be different, and they significantly influence the degree of non-locality. The degree of the interaction between the material points in each family is specified by a non-dimensional weight function, $w(\left|\xi \right|)$, which can vary from point to point. The interactions become more local with a decreasing family size. Thus, the family size and shape are important parameters. In general, the family of a point can be non-symmetric due to non-uniform spatial discretization. Each point has its own family members in the domain of interaction (family), and occupies an infinitesimally small entity such as volume, area, or a distance.

The PD functions are constructed such that they are orthogonal to each term in the TS expansion as

$$\frac{1}{{n}_{1}!{n}_{2}!}\underset{{H}_{\mathbf{x}}}{\int }{\xi }_{1}^{{n}_{1}}{\xi }_{2}^{{n}_{2}}{g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )dV={\delta }_{{n}_{1}{p}_{1}}{\delta }_{{n}_{2}{p}_{2}}$$

(99)

with $({n}_{1},{n}_{2},p,q=\mathrm{0,1},2)$ and ${\delta }_{{n}_{i}{p}_{i}}$ is a Kronecker symbol.

Enforcing the orthogonality conditions in the TSE leads to the non-local PD representation of the function itself and its derivatives as

$$f(\mathbf{x})=\underset{{H}_{\mathbf{x}}}{\int }f(\mathbf{x}+\xi ){g}_{2}^{00}(\xi )d{H}_{{\mathbf{x}}^{{\prime}}}$$

(100)

$$\left\{\begin{array}{c}\frac{\partial f(\mathbf{x})}{\partial {x}_{1}}\\ \frac{\partial f(\mathbf{x})}{\partial {x}_{2}}\end{array}\right\}=\underset{{H}_{\mathbf{x}}}{\int }f(\mathbf{x}+\xi )\left\{\begin{array}{c}{g}_{2}^{10}(\xi )\\ {g}_{2}^{01}(\xi )\end{array}\right\}d{H}_{{\mathbf{x}}^{{\prime}}}$$

(101)

$$\left\{\begin{array}{c}\frac{{\partial }^{2}f(\mathbf{x})}{\partial {x}_{1}^{2}}\\ \frac{{\partial }^{2}f(\mathbf{x})}{\partial {x}_{2}^{2}}\\ \frac{{\partial }^{2}f(\mathbf{x})}{\partial {x}_{1}\partial {x}_{2}}\end{array}\right\}=\underset{{H}_{\mathbf{x}}}{\int }f(\mathbf{x}+\xi )\left\{\begin{array}{c}{g}_{2}^{20}(\xi )\\ {g}_{2}^{02}(\xi )\\ {g}_{2}^{11}(\xi )\end{array}\right\}d{H}_{{\mathbf{x}}^{{\prime}}}$$

(102)

The PD functions can be constructed as a linear combination of polynomial basis functions:

$$\begin{array}{c}{g}_{2}^{\,{p}_{1}{p}_{2}}(\xi )={a}_{00}^{\,{p}_{1}{p}_{2}}{w}_{00}(\left|\xi \right|)+{a}_{10}^{\,{p}_{1}{p}_{2}}{w}_{10}(\left|\xi \right|){\xi }_{1}+{a}_{01}^{\,{p}_{1}{p}_{2}}{w}_{01}(\left|\xi \right|){\xi }_{2}\\ +{a}_{20}^{\,{p}_{1}{p}_{2}}{w}_{20}(\left|\xi \right|){\xi }_{1}^{2}+{a}_{02}^{\,{p}_{1}{p}_{2}}{w}_{02}(\left|\xi \right|){\xi }_{2}^{2}+{a}_{11}^{\,{p}_{1}{p}_{2}}{w}_{11}(\left|\xi \right|){\xi }_{1}{\xi }_{2}\end{array}$$

(103)

where ${a}_{{q}_{1}{q}_{2}}^{\,{p}_{1}{p}_{2}}$ are the unknown coefficients, ${w}_{{q}_{1}{q}_{2}}(\left|\xi \right|)$ are the influence functions, and ${\xi }_{1}$ and ${\xi }_{2}$ are the components of the vector $\xi$. Assuming ${w}_{{p}_{1}{p}_{2}}(\left|\xi \right|)=w(\left|\xi \right|)$ and submitting the PD functions into the orthogonality equation lead to a system of algebraic equations for the determination of the coefficients as

$$\mathbf{A}\mathbf{a}=\mathbf{b}$$

(104)

in which

$$\mathbf{A}=\underset{{H}_{\mathbf{x}}}{\int }w\left(\left|\xi \right|\right)\left[\begin{array}{cccccc}1& {\xi }_{1}& {\xi }_{2}& {\xi }_{1}^{2}& {\xi }_{2}^{2}& {\xi }_{1}{\xi }_{2}\\ {\xi }_{1}& {\xi }_{1}^{2}& {\xi }_{1}{\xi }_{2}& {\xi }_{1}^{3}& {\xi }_{1}{\xi }_{2}^{2}& {\xi }_{1}^{2}{\xi }_{2}\\ {\xi }_{2}& {\xi }_{1}{\xi }_{2}& {\xi }_{2}^{2}& {\xi }_{1}^{2}{\xi }_{2}& {\xi }_{2}^{3}& {\xi }_{1}{\xi }_{2}^{2}\\ {\xi }_{1}^{2}& {\xi }_{1}^{3}& {\xi }_{1}^{2}{\xi }_{2}& {\xi }_{1}^{4}& {\xi }_{1}^{2}{\xi }_{2}^{2}& {\xi }_{1}^{3}{\xi }_{2}\\ {\xi }_{2}^{2}& {\xi }_{1}{\xi }_{2}^{2}& {\xi }_{2}^{3}& {\xi }_{1}^{2}{\xi }_{2}^{2}& {\xi }_{2}^{4}& {\xi }_{1}{\xi }_{2}^{3}\\ {\xi }_{1}{\xi }_{2}& {\xi }_{1}^{2}{\xi }_{2}& {\xi }_{1}{\xi }_{2}^{2}& {\xi }_{1}^{3}{\xi }_{2}& {\xi }_{1}{\xi }_{2}^{3}& {\xi }_{1}^{2}{\xi }_{2}^{2}\end{array}\right]d{H}_{{\mathbf{x}}^{{\prime}}}$$

(105)

$$\mathbf{a}=\left[\begin{array}{cccccc}{a}_{00}^{00}& {a}_{00}^{10}& {a}_{00}^{01}& {a}_{00}^{20}& {a}_{00}^{02}& {a}_{00}^{11}\\ {a}_{10}^{00}& {a}_{10}^{10}& {a}_{10}^{01}& {a}_{10}^{20}& {a}_{10}^{02}& {a}_{10}^{11}\\ {a}_{01}^{00}& {a}_{01}^{10}& {a}_{01}^{01}& {a}_{01}^{20}& {a}_{01}^{02}& {a}_{01}^{11}\\ {a}_{20}^{00}& {a}_{20}^{10}& {a}_{20}^{01}& {a}_{20}^{20}& {a}_{20}^{02}& {a}_{20}^{11}\\ {a}_{02}^{00}& {a}_{02}^{10}& {a}_{02}^{01}& {a}_{02}^{20}& {a}_{02}^{02}& {a}_{02}^{11}\\ {a}_{11}^{00}& {a}_{11}^{10}& {a}_{11}^{01}& {a}_{11}^{20}& {a}_{11}^{02}& {a}_{11}^{11}\end{array}\right]$$

(106)

and

$$\mathbf{b}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 2& 0& 0\\ 0& 0& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$$

(107)

After determining the coefficients ${a}_{{q}_{1}{q}_{2}}^{{p}_{1}{p}_{2}}$ via $\mathbf{a}={\mathbf{A}}^{-1}\mathbf{b}$, then the PD functions ${g}_{2}^{{p}_{1}{p}_{2}}(\xi )$ can be constructed. The detailed derivations and the associated computer programs can be found in [8].

Appendix 2 Ordinary Kriging

As described by Cressie [4], the observed (known) data at spatial locations, ${s}_{1},.....,{s}_{n}$, are modeled as a random process denoted by $Z(s)$ in ordinary Kriging. It is assumed that the random process satisfies the decomposition

$$Z(s)\equiv \mu +\delta (s)$$

(108)

where $\mu$ is the expected value or mean of $Z(s)$ and $\delta (s)$ is the correlated error process. The mean, $\mu$, is unknown; however, it is assumed to be a constant. The predictor is defined as

$$p\left(Z;B\right)= \sum\limits_{i=1}^{n}{\lambda }_{i}Z({{{s}}}_{i})$$

(109)

subject to the constraint of

$$\sum\limits_{i=1}^{n}{\lambda }_{i}=1$$

(110)

The parameter, B, is defined as a block of spatial area whose location and geometry are known. The optimal predictor is obtained by minimizing the mean-squared prediction error defined as

$${\sigma }_{e}^{2}\equiv E{(Z\left(B\right)-p\left(Z; B\right))}^{2}$$

(111)

with respect to the coefficients ${\lambda }_{i}$ with $i=1,...,n$_.

Appendix 3 Temperature Data from Weather Stations in Arizona

Weather station name	Elevation (m)	Latitude	Longitude	Temp. (°F)
Robson Ranch	455.7	32.8118	−111.6313	65
Tacna 3 NE	98.8	32.7225	−113.9191	68
Fry	2194.6	35.07	−111.84	52
Gunsight	1609.3	36.7044	−112.5833	50
Williams	2105.9	35.2413	−112.1929	56
Guthrie	1097.3	32.8819	−109.3092	53
Rincon	2511.6	32.2056	−110.5481	60
Frazier Wells	2063.5	35.8456	−113.055	52
Bisbee 1 WNW	1694.7	31.4475	−109.9288	61
Tempe ASU	355.7	33.4258	−111.9216	71
Wittmann 1 SE	513	33.7776	−112.523	71
Painted Desert National Park	1755.6	35.068	−109.7688	50
Union Pass	1072.9	35.2247	−114.3747	62
Humbug Creek	1600.2	34.1164	−112.3006	60
Hurricane	1659.6	36.6992	−113.2072	47
Saguaro National Park	938.8	32.1794	−110.7363	57
Willcox	1271	32.2553	−109.8369	58
Oak Creek	1500.8	34.9417	−111.7517	56
Greer	2499.4	34.06	−109.45	57
Safford Agricultural Center	900.4	32.815	−109.6808	63
Betatakin	2220.8	36.6778	−110.5411	44
Tohono Chul	770.2	32.3391	−110.9808	63
Empire	1417.3	31.7806	−110.6347	69
Four Springs	1999.5	36.7939	−112.0422	41
Teec Nos Pos	1612.4	36.9233	−109.09	42
Patagonia Paton Center	1232.6	31.53923	−110.76028	72
Payson	1516.4	34.2431	−111.3028	61
Chiricahua	1645.9	32	−109.35	62
Paria Point	2205.2	36.7278	−111.8219	49
Cherry	1554.5	34.5964	−112.0481	58
Sanders	1784	35.2239	−109.3222	52
San Carlos Reservoir	771.8	33.1819	−110.5261	46
Goodwin Mesa	1280.2	34.75	−113.3	63
Sunset Crater National Monument	2127.5	35.3694	−111.5436	42
Sunrise Mountain	2856	33.9733	−109.563	42
Flagstaff	2133.6	35.145	−111.675	56
Hopi	1885.5	35.8103	−110.2069	53
Selles	721.2	31.91	−111.8975	69
Iron Springs	1804.4	34.5853	−112.5019	61
Hopkins	2170.2	31.6753	−110.88	66
Tucson International Airport	776.9	32.1313	−110.9552	70
Mormon Mountain	2286	34.94	−111.52	53
Catalina State Park	825.1	32.4177	−110.9302	61
Sunset Point	902.2	34.1953	−112.1417	64
Tweeds Point	1585	36.5819	−113.7319	51
Douglas Bisbee Inter. Airport	1251.2	31.4583	−109.6061	66
St. Johns Industrial Air Park	1747.4	34.51833	−109.37917	52
Nixon Flats	1981.2	36.39	−113.1522	54
Black Rock	2158	36.7944	−113.7567	45
Tusayan	2042.2	35.99	−112.12	55
Stray Horse	2139.7	33.5406	−109.3169	56
Snowslide Canyon	2965.7	35.34	−111.65	47
Stanton	1097.3	34.1667	−112.7333	64
Olaf Knolls	883.9	36.5072	−113.8161	60
Kingman Airport	1042.4	35.2577	−113.933	63
Douglas	1231.4	31.345	−109.5394	66
Nogales 6 N	1054.9	31.4554	−110.968	73
Truxton Canyon	1630.7	35.7825	−113.7942	56
Lindbergh Hill	2682.2	36.2858	−112.0794	48
Bagdad	1199.4	34.5975	−113.1745	57
White Horse Lake	2188.5	35.14	−112.15	56
Alamo Dam	393.2	34.228	−113.5777	65
Walnut Creek	1551.4	34.9281	−112.8097	62
Walnut Canyon National Monument	2040.6	35.1721	−111.5097	43
Dry Lake	2264.1	33.3597	−109.8331	60
Casa Grande	426.7	32.8875	−111.7147	67
Duncan	1115.6	32.748	−109.1213	57
Dry Park	2653.6	36.45	−112.24	49
Heber Black Mesa Ranger Station	2008.6	34.3925	−110.558	56
Alpine	2447.8	33.8417	−109.1222	57
Prescott Love Field	1536.8	34.65167	−112.42083	62
Phoenix Airport	337.4	33.4277	−112.0038	68
Benson 6 SE	1124.7	31.8805	−110.2403	58
Yuma Proving Ground	98.8	32.8356	−114.3942	68
Winslow Airport	1489.3	35.0281	−110.7208	51
Fort Valley	2240.3	35.27	−111.74	56
Happy Jack Ranger Station	2279.9	34.7433	−111.4139	47
Bellemont Weather Forecast Office	2179.9	35.2302	−111.8221	50
Beaver Dam	588.6	36.9139	−113.9423	60
Sasabe	1094.2	31.483	−111.5436	75
Show Low Airport	1954.1	34.2639	−110.0075	50
Window Rock Airport	2054	35.6575	−109.06139	52
Moss Basin	1804.4	35.0336	−113.8925	58
Picacho 8 SE	603.8	32.6463	−111.4017	63
Jerome	1508.8	34.7522	−112.1114	49
Quartzsite	266.7	33.665	−114.2272	69
Page Municipal Airport	1313.7	36.92611	−111.44778	43
Warm Springs Canyon	2441.4	36.7	−112.23	51
Kitt Peak	2069.6	31.96018	−111.59787	60
Workman Creek	2103.1	33.81	−110.92	55
Yellow John Mountain	1877.6	36.1542	−113.5417	54
Carefree	771.1	33.8161	−111.9019	67
Grand Canyon National Park Airport	2013.5	35.94611	−112.15472	49
Montezuma Castle Nat.Monument	969.3	34.6105	−111.838	59
Scottsdale Municipal Airport	449	33.62278	−111.91056	68
Havasu	144.8	34.7872	−114.5617	69
Limestone Canyon	2072.6	34.1789	−110.2736	51
Kartchner Caverns	1429.5	31.8352	−110.3552	53
Buckskin Mountain	1950.7	36.9306	−112.1997	48
Mount Lemmon Willow Canyon	2141.8	32.3859	−110.69799	47
Apache Junction 5 NE	630.9	33.4625	−111.4813	66
Anvil Ranch	840.9	31.9793	−111.3837	63
Canyon de Chelly	1709.9	36.1533	−109.5394	47
Green Valley	883.9	31.893	−110.9977	59
Bright Angel Ranger Station	2438.4	36.2147	−112.0619	41
Organ Pipe Cactus Nat. Monument	511.5	31.9555	−112.8002	73
Fort Huachuca Pioneer Airfield	1453.3	31.60722	−110.42806	66
Globe	1112.5	33.3503	−110.6519	56
San Simon	1091.2	32.29329	−109.22691	61
Phantom Ranch	771.1	36.1066	−112.0947	58
Petrified Forest National Park	1659.9	34.7994	−109.885	49
Saint Johns	1764.8	34.5172	−109.4028	49
Meteor Crater	1687.1	35.0364	−111.0231	47
Music Mountain	1652	35.6147	−113.7939	56
Muleshoe Ranch	1272.5	32.4	−110.2708	67
Tombstone	1420.1	31.7119	−110.0686	56
Sierra Vista	1403.9	31.53699	−110.28073	53
Hilltop	1743.5	33.6183	−110.42	61
Elgin 5 S	1466.4	31.5907	−110.5087	67
Ajo	533.7	32.3698	−112.8599	66
Smith Peak	762	34.1158	−113.3472	65